%% v20.0 (August 21, 2026): PROMOTION TO THE v20 STAMP.
%%   Version and cover date only.  Promoted from v15.11 with no other change:
%%   not one character of body text, mathematics, float, caption or
%%   bibliography differs.  The promotion is safe to do mechanically because
%%   the cohort cites its companions BY TITLE, never by versioned filename --
%%   a scan of the printed text of all twelve deposit papers found zero
%%   version strings, so no cross-reference can be left pointing at a
%%   superseded file.  This is the version posted to arisaka-gg.org and
%%   deposited to Zenodo, so homepage, repository and DOI all carry one
%%   stamp.
%% ===================================================================
%% v15.8 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 7 of this paper's 21 cohort bibitems named a companion by a title
%% that companion does not carry.  Measured, not recalled: the truth is
%% each paper's printed cover, read from its deployed PDF; the markup is
%% each paper's own \GGTitlePage arguments, so the title sets in the same
%% faces the cover does.  Cohort-wide the pass touched 233 sites in 23
%% papers, and afterwards all 477 companion bibitems agree with their
%% covers.
%%
%% TWO CLASSES, and the difference is worth recording.
%%   RENAME PROPAGATION -- a paper was retitled and its citers were never
%%   updated.  The signature is the same wrong string in many papers:
%%   A6-GRIP's retired subtitle "Gi-4 Pillars (TGG, CCI, GFF, and GRIP) as
%%   the Geometric Backbone of GDOS" stood in seventeen papers, and
%%   C2-Hubble's retired "Two Hubble Constants as the Two Ends of One
%%   Six-Dimensional Geometric Corridor" in seventeen more.  W1-Symmetry,
%%   W2-Holonomy, P5-CPT and P6-Proton were all renamed and none of their
%%   citers followed.
%%   HOUSE SHORTHAND -- P5-CPT, W1-Symmetry and W2-Holonomy each wrote
%%   their own descriptive short form for every companion ("GDOS: the
%%   Boundary Law", "The Kaxion", "The Cosmological Inventory of the
%%   Descent").  Readable in prose and useless in a reference list,
%%   because none of those strings finds the paper.
%%
%% ONLY TITLE TEXT CHANGED.  Author blocks, years, availability lines,
%% Relevance notes, keys, labels and \emph/\textit wrappers are asserted
%% unchanged in count; the changelog comments are asserted BYTE-IDENTICAL.
%% ====================================================================
%% ===================================================================
%% v15.7 (August 18, 2026) -- FP-4: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%%
%% (1) The A11 bibliography entry said "Preprint, available at
%%     https://arisaka-gg.org/"; it now reads "In preparation; to be
%%     posted ... on completion.  No result of the present paper depends
%%     on it."
%% (2) THREE SITES DESCRIBED THE CAPSTONE'S CASCADE WRONGLY: a figure
%%     caption ("the seventeen-layer deployment"), the bibliography
%%     Relevance note ("its seventeen-layer cascade"), and a runtime
%%     paragraph ("its full sixteen-event theory").  Measured from A11
%%     v15.14 and not recalled -- the GRIPout longtable runs 1->2 through
%%     17->18 -- the canonical counts are EIGHTEEN LAYERS and SEVENTEEN
%%     GRIPout TRANSITIONS.  All three corrected.
%%     FROZEN BY EXACT TEXT in the same sentence as (2): "the seven-layer
%%     survey given in that section" is A9's OWN survey, a different
%%     quantity, and is left alone.
%% (3) catalogue -> catalog in printed text (Report 443's pair-list hole).
%% (4) A9 HAS SEVENTEEN PARTS AND SAID SIXTEEN.  The roadmap Section opens
%%     "the roadmap of the seventeen Parts" and "a panorama of the
%%     seventeen Parts of the present paper", then closes the same
%%     paragraph with "stands in the sixteen-Part arc".  Counted from
%%     this paper's own \GGPart / \GGAppendixPart headers and confirmed
%%     against the printed Overview: seventeen.  Corrected.
%% (5) The cohort-table caption promised "where all preprints are posted".
%%     Twenty-one will be; A11-Origin is held.  Reworded to match the
%%     A2-GG6 caption repaired in FP-2.
%%
%% A9 carries NO simultaneity claim, NO reading-set enumeration, NO roster
%% row for A11 and NO pointer into A11's internal Part structure; the
%% claim-derived probe checked all four before this rider was written.
%%
%% 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:
%%
%% * no showcase-headline macro: its showcase pages predate
%%   the pass and set their own headlines inline.
%% * A COHERENCE REPAIR, NOT NEW PROSE.  T-SM-60.1 said the
%%   electroweak-anomaly suppression is
%%   "e^{-4pi/alpha_W} ~ e^{-170}".  The SAME paper writes
%%   "e^{-4pi/alpha_W} approx 10^{-162}" in two other
%%   places, and 4pi/alpha_W is about 372, so e^{-372} is
%%   10^{-162} and e^{-170} is 10^{-74} -- the theorem
%%   carried the right formula against the wrong number.
%%   Repaired to the paper's own wording, per section 1.2.
%%
%% 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.
%%
%% polariser -> polarizer (1); recogniser -> recognizer (1).  Charter 1.1's pair list names these forms; spelling_audit.py's
%% PAIRS list did not carry them, so this paper has reported MIXED
%% PAIRS 0 and ALWAYS-BRITISH 0 at every gate while printing them.  The
%% cohort sweep of August 17, 2026 found sixty-one across twelve of the
%% twenty-four live papers; Report 406 section 6 has the inventory and
%% spelling_audit.py now carries the family as a RULE rather than a list.
%%
%% Substitution is on PRINTED TEXT ONLY.  Changelog comments are the
%% historical record, \label/\ref/\cite arguments are keys, and published
%% titles are quoted rather than written -- all three are masked, and every
%% frozen title is asserted present and unchanged after the edit.
%% NOTHING ELSE IN THIS PAPER CHANGES: no claim, no number, no float, no
%% label, and no page.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.2 -> v15.3, August 16, 2026.  THE SHOWCASE, COMPLETED WITH
%% ITS EXPOSURE PAGE.  Ninth of the twenty-four and the one that
%% needed the least: A9 ALREADY HAD THE SHOWCASE, and pages 5-7 are
%% where the whole pattern was read from.  Not one of them is
%% changed here.
%%   WHAT THEY LACKED IS THE EXPOSURE.  All three answer 'what it
%%   delivers' -- the thirteen best-measured parameters, the full
%%   28-parameter audit, the three-line gauge-coupling meeting.
%%   Every other showcase in this pass ends on what would end the
%%   paper, and A9, claiming the most, is where that page matters
%%   most.  So ONE page is added: Table 70, the consolidated
%%   falsification map -- seven channels, each with the number it
%%   turns on and the measurement that would end it.
%%   FOUR PAGES, NOT THREE.  The rule is one page one idea; three
%%   was the count that fell out of it, never the rule.  Nothing is
%%   removed to make room.
%%   The new page is UNNUMBERED while A9's showcase figures are
%%   numbered 1-3: numbering it would take Table 1 and shift all
%%   ninety of A9's tables, the defect caught in A1 v15.23.
%%   NOTHING IS COPIED: Table 70's tabular is filled into a savebox
%%   once in the front matter and used twice.
%% STAGE 0 FOUND A FALSE BASELINE FIRST.  Built without its ten
%% \includegraphics assets, A9 substitutes draft boxes -- and
%% because every include carries an explicit height with
%% keepaspectratio, the draft boxes take the declared height and the
%% page count comes out at 618, exactly matching the deployed PDF.
%% The assets were staged and the baseline rebuilt: 0 of 618 pages
%% differ from the deployed artifact.
%% NEW PROSE IN THIS SHIP: one headline and one showcase caption,
%% both on the new page.  Nothing else in the paper is rewritten.
%% ------------------------------------------------------------
%% -- cohort constant sweep, August 14, 2026.  The baryon bridge is
%%    quoted cohort-wide as eta10 = 273.9 Omega_b h^2 (A9's own value,
%%    the mean-baryon-mass form including helium).  The pure-proton
%%    3.6524e-3 = 1/273.79 is superseded.  Three printed quantities
%%    move with it: +0.53 -> +0.57 per cent, 16.5 -> 16.4 per cent,
%%    and -0.42 -> -0.38 per cent; and the prose claim 'within half a
%%    per cent' is retired, being false at +0.57 per cent.
%% ===================================================================
%% v20.71 (August 14, 2026)
%%
%% 1. HALF-ANGLE COHERENCE REPAIR.  The kinematics are retained; the
%%    inference that the factor of i is itself the CP violation is
%%    withdrawn.  A Majorana phase of pi is opposite intrinsic CP
%%    parity, a CP-CONSERVING configuration; what breaks CP here is
%%    that the flavoured asymmetries are individually nonzero while
%%    their sum vanishes exactly, and that differential washout does
%%    not treat the three flavours alike.  This is what this paper's
%%    own washout section already computes.
%%
%% 2. OBJECTIONS 41, 44 AND 49 RE-SWEPT.  All three graded the
%%    Majorana phases "Asserted, not Derived" against a codebook, an
%%    MSMF/OGN promotion audit and a benchmark table that all say
%%    Derived.  The Objections Part had not been re-swept after the
%%    promotion pass.
%%
%% 3. THE delta_PMNS pi COLLISION SETTLED.  Three sites claimed the
%%    base pi of delta_PMNS IS the Majorana parity alpha_31, and that
%%    the long-baseline programme therefore measures the origin of the
%%    matter asymmetry.  Appendix K.1.5 of this paper proves that no
%%    oscillation experiment can ever measure alpha_31 -- "a theorem,
%%    not a sensitivity statement" -- so the claim is withdrawn.  The
%%    convention reading of sec:PMNS stands.  Appendix K's middle
%%    reading is reframed from identity to common geometric origin.
%%    F-SM-3d remains a sharp falsifier of the ALPHABET; it is no
%%    longer sold as a falsifier of the matter account.  The two are
%%    now independently falsifiable, which is a gain and is said so.
%%
%% 4. E RELABELLED.  E = 7.1 [4.4, 11.4] is Level B's deficit, not the
%%    central-parameter integration's: 6.12/0.86 = 7.1, and the band
%%    is 6.12/1.38 and 6.12/0.54.  Both numbers are kept; the label is
%%    corrected and the central-parameter deficit (6.1, no band) is
%%    stated separately.  The ratio 0.17 becomes 0.16 (1.0/6.12).
%%
%% 5. THE BARYON CHAIN CARRIED TO ITS END.  New subsection: the bridge
%%    stated once, rho_b given, Omega_b connected to eta_B, the two
%%    roads named and the ledger led with -- identical to A10 and S2.
%%
%% 6. CONVERSION CONSTANT.  eta10 = 273.9 Omega_b h^2, this paper's own
%%    value, confirmed from first principles (T0 = 2.7255 K, mean mass
%%    per baryon with helium) and adopted cohort-wide.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.70  (August 13, 2026)
%%  Found while re-deriving A4-Euclid's imports, not by opening A9.
%%  THE "A LADDER" PARAGRAPH PUT BOTH ANCHORS IN THE WRONG PLACE.  It
%%  read: the ultraviolet rung r = 0 is "the Planck scale E_Pl ~ 1.22e19
%%  GeV" and the infrared rung r = K is "the lock scale M_lock ~ 3.85
%%  TeV".  Under A9's OWN convention r = ln(E_GUT/mu), so:
%%    * r = 0 is E_GUT = 2.28e16 GeV, not E_Pl -- the two differ by
%%      e^{2pi}.  A9's two running-coupling captions already say "with
%%      r = 0 at the unification point", and A9's string comparison says
%%      flatly "the present program sets the ultraviolet endpoint at
%%      E_GUT, not at the Planck scale".
%%    * r = K = 35 is E_GUT e^{-35} = 14.4 GeV, not 3.85 TeV.  M_lock is
%%      a Kaluza-Klein MASS anchored at the bulk scale M_6 = M_Pl/sqrt(pi),
%%      not a boundary scale on the interval.
%%  So A9 was right everywhere except in its own plain-English summary
%%  -- which is exactly where a general reader looks.  A sweep of every
%%  "r = 0", "ultraviolet rung", "UV end" and "ultraviolet endpoint"
%%  site finds this paragraph and no other; the script asserts that.
%%  The replacement carries A2-GG6's "two anchors" distinction, the one
%%  A4-Euclid imported at its Stage 1 to fix the identical confusion,
%%  in A9's own register, keeping the Lemma 3 pointer and the closing
%%  image.  Arithmetic evaluated in the script, not quoted:
%%    E_GUT = E_Pl e^{-2pi}        = 2.2799e16 GeV
%%    E_GUT e^{-35}                = 14.4 GeV
%%    M_6/E_GUT = e^{2pi}/sqrt(pi) = 302.1
%%    M_lock = (1/2) E_Pl e^{-35}  = 3.849 TeV
%% ===================================================================
% ===================================================================
%  A9-SM v20.69 (August 12, 2026) -- cohort-wide sweep, two defects.
%
%  (1) GU-5 IS A THREE-PLUS-TWO PARTITION, exactly as OGN-5 is
%      (M,N,L;J,K).  The separator before k_B is a SEMICOLON:
%      {c, G_N, hbar; k_B, Q_B}.  9 site(s) corrected here.
%      NOT touched, by design: the four-constant Planck unit system
%      {c, G_N, hbar, k_B}, which is a different object and takes a
%      comma; a semicolon there would be wrong.
%
%  (2) Internal affairs -- companion versions, dates, page counts and
%      build status -- do not belong in a preprint.
%%  ALSO: the subsection "Note on version and Phase numbers" is DELETED
%%  in full.  It listed eleven companion papers with their dates and page
%%  counts, declared them "current as of May 12, 2026", and reported the
%%  build state -- 0 errors / 0 undefined references / 0 undefined
%%  citations on pdflatex x 3 -- inside a preprint.  None of that is
%%  reader-facing content.  One bibliography entry used the deleted
%%  subsection as its first-citation target and is repointed.
%
%  No physics changes; no parameter, theorem or falsifier is touched.
% ===================================================================
%% ===================================================================
%%  A9-SM v20.68 (August 12, 2026) -- Stages 6, 7 and 8 of the neutrino
%%  expansion plan: the figures, the bibliography sweep, the final gate.
%%
%%  STAGE 6.  Part VIII had eight tables and NO figures.  It now carries
%%  six plates, every datum computed rather than sketched:
%%    fig:lg-descent      the r-axis, with ln(EPl/EGUT) = 2pi and
%%                        ln(EPl/Mlock) = K + ln2 = 35.693, both exact;
%%    fig:lg-regulator-landscape   the five candidates on a log axis with
%%                        the required decade shaded -- a figure that is
%%                        also the argument against itself;
%%    fig:lg-boltzmann    the system of eq:lg-boltz-N/D integrated afresh
%%                        with the paper's own parameters;
%%    fig:lg-three-levels the A/B/C bands against observation;
%%    fig:lg-chain        K = 35 to eta_B, with C4 marked in red;
%%    fig:lg-triangles    the two unitarity triangles at TRUE relative
%%                        area, ratio 307.9 -- the number the paper prints.
%%  Each measured by standalone compile on a 1500 mm sheet and placed by
%%  f = (7.79/9.0) x W / 159.2.  Measured W: 167.1, 174.2, 142.2, 183.9,
%%  121.9, 89.9 mm -> f = 0.908, 0.947, 0.773, 0.950, 0.663, 0.489.
%%
%%  INDEPENDENT VALIDATION recorded in the Boltzmann caption: integrating
%%  the paper's own system with no further freedom gives eta_B =
%%  (0.23-0.45)e-10 across c_tau in [5,10]e-3, against Level B's
%%  0.86 +0.52 -0.32.  They agree once the declared pattern-acceptance
%%  and washout-kernel factors are applied, and NOT BEFORE.  Said that way.
%%
%%  FINDING exposed by drawing the regulator landscape: Stage 5's
%%  Appendix L claimed the table "compares ELEVEN candidate regulators"
%%  and listed contents it does not have.  The table has FIVE rows.  I
%%  invented the list instead of reading the table -- the Report-320 error
%%  class, caught this time by a figure.  Three sites corrected, and the
%%  per-decade argument corrected with them: five candidates over sixteen
%%  orders sit ~three decades apart, so the coincidence is about the odds
%%  of a fair die -- weaker evidence than "one per decade" implied, and
%%  the text now says so.
%%
%%  STAGE 7 FINDING: the flavoured pattern (+3.4, +3.1, -6.6)e-3 is
%%  printed beside "note sum = 0".  The printed values sum to -1e-4.  The
%%  lemma is right; the rounding is the culprit; the text now says so
%%  instead of inviting the reader to add three numbers and find the
%%  lemma failing.
%%
%%  No parameter of the codebook changes; no falsifier criterion changes.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.67 (August 12, 2026) -- the accelerator corollaries
%%  (GG-Report-321's four follow-ups, author-approved).
%%
%%  NEW SECTION in Part VIII: "Accelerator corollaries: what a collider
%%  can and cannot prove."  The negative half: three locks on the direct
%%  channel (mixing 2.2e-15 vs best reach 1e-6; the cluster is
%%  leptophobic per P1-KK, so no gauge portal; the quasi-degeneracy
%%  closes on-shell portals) plus the blind-spot lifetime ctau = 2.4 um
%%  and Y^4 = 2.8e-25 on all indirect probes.  The positive half: the
%%  four-premise test matrix -- HL-LHC tests the address (F-SM-1) and
%%  the door (leptophobic Z'_B); DUNE/Hyper-K test the SIGN (the pi base
%%  of delta_PMNS IS the Majorana parity that decides matter vs
%%  antimatter); FCC-hh could see the Delta B = 3 door swing; the
%%  magnitude needs a calculation, not a beam (F-SM-7).
%%
%%  THE MIRROR-PORTAL QUESTION registered as OPEN with owners named
%%  (P5-CPT, P1-KK): does the mirror sector couple at gauge strength to
%%  anything at the lock?  A9 deliberately does not answer it -- the
%%  Report-320 lesson about reasoning from priors, applied prospectively.
%%
%%  F-SM-4 gains the collider reading: Delta B = 1, 2 events at any
%%  machine are fatal; a Delta B = 3 sphaleron event at 100 TeV would be
%%  the conversion vertex observed directly.  F-SM-3d gains its reading:
%%  the long-baseline programme measures the sign-setter of the baryon
%%  asymmetry.  ITS KILL WINDOW IS UNCHANGED.
%%
%%  Three new references: Keung-Senjanovic 1983, Tye-Wong 2015,
%%  Ellis-Sakurai 2016 (the last two cited as a contested pair, no side
%%  taken).  No parameter of the codebook changes; no falsifier
%%  criterion changes.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.66 (August 12, 2026) -- four author-raised corrections.
%%
%%  ROOT CAUSE of three of the four: the leptogenesis Part was added in
%%  v20.61 and that work was scoped to the NEW PART.  Every claim the
%%  paper already made about the origin of the matter asymmetry was left
%%  standing, and two of them contradicted the new Part outright.  The
%%  sweep here is scoped BY SUBJECT, not by text.
%%
%%  (1) THE LOCK-SCALE SEESAW, written out.  It was one subordinate
%%      clause.  New section: the mass matrix, and the closure
%%      m_nu = Y^2 v^2/M_N = pi v^2/E_GUT = m_star, in which M_lock
%%      CANCELS IDENTICALLY.  Stated WITHOUT triumph: because Y_nu is
%%      fixed by matching, the closure is automatic for any heavy scale
%%      and the neutrino spectrum carries NO information about M_N.  The
%%      content is that M_lock is fixed independently by the anomaly
%%      ledger, so Y_nu = 7.28e-7 is a prediction that could have been
%%      absurd and is not -- within a factor of four of the electron's.
%%      And the cancellation is also a LIMITATION, said in those words:
%%      no neutrino measurement can confirm or refute this seesaw.
%%      Priority framed as the author directed: a DERIVED TeV-scale
%%      seesaw, not a first one; Pilaftsis is cited on the same page.
%%      Also: |theta|^2 = 2.2e-15, so the TeV Majorana triplet is
%%      invisible -- and the two tenants at 3.85 TeV are separated.
%%
%%  (2) CPT.  Two sites grounded the two signs of electric charge in the
%%      standard CPT theorem, inverting the cohort's own order.  Regrounded
%%      on T-P4-1/T-P4-3/T-P4-5 (P4-Real) and T-P5-A/C/C2/1 (P5-CPT),
%%      read this session.  MY FIRST DRAFT OVERCLAIMED: it said the
%%      standard theorem is "recovered".  P5 explicitly disclaims that --
%%      the relation is COMPLEMENTARITY, Jost's scope exceeding the
%%      geometric account's.  Written that way instead.  "Pauli-Lueders-
%%      Schwinger" named three physicists A9 never cited; dropped, and
%%      Jost (1957) added properly.
%%
%%  (3) BARYON NUMBER IS CONSERVED.  "Baryon-number violation via the
%%      bulk anomaly inflow" was backwards: per P6-Proton the inflow
%%      ABSORBS the anomaly so that B survives.  The sole door is the
%%      topological SU(2)_L vertex at Delta B = N_gen n = 3n, welded cold
%%      at 1e-162 and open hot.  The proton cannot decay and baryons
%%      could be made FOR THE SAME REASON.  Leptogenesis is not preferred
%%      here; it is forced.  Appendix L gains the rule too.
%%
%%  (4) delta_CKM DOES NOT PAY FOR BARYOGENESIS.  Two sites said it
%%      supplies a Sakharov condition; Part VIII says the quark sector
%%      falls ten orders short.  Corrected, with the shortfall priced.
%%      A THIRD ERROR sat in the same sentence: quark-antiquark
%%      annihilation is at t ~ 1e-5 s, not "soon after recombination" --
%%      wrong by twelve orders of magnitude in time.
%%
%%  ALSO: the ten-open-problems entry still said the mechanism "lies
%%  outside the scope of the present paper" -- it has been in it since
%%  v20.61.  The Dirac-mass convention is pinned to m_D = Y_nu v and
%%  Appendix J corrected to match, with both conventions now stated for
%%  the student.  And F-SM-4 was UNDERSTATING itself: P6-Proton v1.25
%%  has discharged hypothesis H2.
%%
%%  No parameter of the 28-parameter codebook changes.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.65 (August 12, 2026) -- Stage 5: the appendix rebuilt as a
%%  graduate course.
%%
%%  THE PART OPENER now carries an in-depth review of nineteen standard
%%  textbooks in five tiers, each reviewed for what it does supremely
%%  well AND where it stops -- the stopping point stated as a specific
%%  unanswered question, not as a compliment -- a reading map built on
%%  \ref rather than on hand-written letters, and a list of TWENTY-ONE
%%  open items in four groups, of which two are marked OPEN.
%%
%%  THE NEUTRINO CHAPTER is rewritten end to end, 11k -> 60k characters.
%%  Oscillations are derived rather than quoted (including the 1.2669
%%  conversion constant and the wave-packet caveat); the MSW effect is
%%  derived from forward scattering with the solar resonance density
%%  computed; the seesaw is given in all three types with Casas-Ibarra;
%%  Step 2 goes from one paragraph to ten numbered limitations.
%%
%%  TWO NEW CHAPTERS.  Appendix K: Majorana neutrinos, leptonic CP
%%  violation and 0nubb.  Appendix L: leptogenesis and the origin of the
%%  baryon asymmetry.  Both in the Step 1 / Step 2 / Step 3 shape.
%%
%%  PROBLEMS: 60 -> 132, every numerical answer recomputed before being
%%  typeset and re-derived in the edit scripts' closures.
%%
%%  THREE FINDINGS, all recorded in the report:
%%
%%  (1) THE ALPHABET.  Two new chapters take the lettered-section count
%%      to 27.  LaTeX's \@Alph throws "Counter too large" past 26 -- the
%%      paper would NOT HAVE COMPILED.  \@Alph is extended (27 -> AA),
%%      tested standalone first.
%%
%%  (2) THE STAGE-1 REFRESH MISSED THIS CHAPTER.  It still carried
%%      Dm21 = 7.4e-5 and Dm31 = 2.5e-3 at five sites and used them to
%%      claim a ratio agreement of "33.8, within 7%".  Against NuFIT 6.1
%%      the measured ratio is 33.35 and the ledger's 36 is +7.96%.  More
%%      importantly the ratio HIDES THE STRUCTURE: the ledger nails
%%      Dm31 to +0.23% and misses Dm21 by -7.16%.  All of the
%%      discrepancy is in the SOLAR splitting, it is 4.2 sigma against
%%      JUNO alone, and the chapter now says so instead of averaging.
%%
%%  (3) ONE HAND-WRITTEN CHAPTER-LETTER REFERENCE ("M-chapter
%%      cosmology") survived every earlier guard because it does not
%%      match "Appendix~X".  Replaced by a \ref form.
%%
%%  No parameter of the 28-parameter codebook changes.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.64 (August 12, 2026) -- Stage 4: the falsification rebuild.
%%
%%  F-SM-3 is split into five separately-killable sub-channels, each with
%%  a pre-declared criterion, a facility and a date: 3a ordering, 3b the
%%  0nubb/Majorana nature (existing tag, meaning unchanged), 3c the mass
%%  sum, 3d the Dirac phase, 3e the atmospheric octant (new).  The
%%  leptogenesis chain becomes F-SM-7 -- NOT F-SM-6, which is already the
%%  dark-energy channel at eight sites.
%%
%%  F-SM-7 is a different kind of channel and the text says so: every
%%  other channel waits on a facility, this one waits on a calculation
%%  that could be done today, and it is the only one settleable without
%%  new data.  On the timeline plate it sits in the NOW band.
%%
%%  Also: the F-SM-1 collider statement now says why the CMS 4.55 TeV
%%  Randall-Sundrum KK-gluon exclusion does not transfer to a narrow
%%  state with suppressed ttbar branching.  Without that sentence a
%%  reader who knows the RS result concludes the cluster is dead.
%%
%%  No prediction changes; the channels are re-partitioned, not moved.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.63 (August 12, 2026) -- Stage 3 of the neutrino expansion:
%%  the existing lepton sections expanded in place.  Each mixing angle now
%%  carries its geometry rather than arriving from the master ansatz in
%%  three lines; the Dirac-phase reference-channel argument is written
%%  out; the Majorana rank-two argument is given; m_bb is worked rather
%%  than quoted; and the two unitarity triangles are compared.
%%
%%  ONE FINDING, and it is uncomfortable: scanning the two Majorana
%%  phases freely over the derived spectrum gives m_bb in [1.44, 3.74]
%%  meV, and the other admissible parity alpha_31 = 0 gives 3.62 meV.
%%  The derived (0, pi) lands at 1.80 -- the LOWEST quarter of the
%%  accessible range, close to maximal destructive interference.  This
%%  framework's Majorana prediction is therefore harder to confirm than a
%%  generic normal-ordering one, and the paper now says so with the
%%  numbers rather than only noting that 1.8 meV is small.
%%
%%  No parameter of the 28-parameter codebook changes; every printed
%%  value was recomputed independently before being written.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.62 (August 12, 2026) -- three riders closing the items left
%%  open by Reports 313 and 314.
%%
%%  (1) THE CMB-S4 RIDER.  The cohort instrument sweep of Report 307
%%      covered A10, W2 and C2 and missed A9, which carried seven live
%%      CMB-S4 sites -- two of them venue cards inside the falsification
%%      timeline plate.  All seven are plain forthcoming-instrument
%%      namings (unlike C2's, none is a dated funding conditional) and
%%      take the cohort replacement; one historical mention is added by
%%      design recording the July 2025 termination.
%%
%%  (2) THE T-SM CONCORDANCE.  The T-SM-NN.M identifiers are
%%      section-number-coupled, and ELEVEN OF TWENTY have drifted -- not
%%      the three found by spot-check in Report 313.  The naming section
%%      claimed the ids carry the section number and gave T-SM-60.1 as
%%      its example; that theorem prints as 75.1.  The tags are frozen
%%      (companions cite them), the claim is corrected, and a concordance
%%      is printed whose "prints as" column is a live \ref, so it cannot
%%      go stale again on a future insertion.
%%
%%  (3) PART VIII DEEPENED.  The one-loop self-energy computation behind
%%      the resonant Breit-Wigner form, and the Boltzmann collision terms
%%      derived rather than stated.
%%
%%  No parameter of the 28-parameter codebook changes.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.61 (August 12, 2026) -- Stage 2 of the neutrino expansion:
%%  the new Part "The Majorana door and leptogenesis at the lock scale",
%%  placed after Part X and before Part XI, label part:lepto.
%%
%%  A9 previously carried NO leptogenesis: one Discussions subsection, a
%%  bookkeeping paragraph, two provenance notes and three pointers, with
%%  every substantive statement registered as a consumer of A10 and P5.
%%  There was no eta_B formula, no CP-asymmetry expression, no Boltzmann
%%  equation, and exactly one occurrence of the word "sphaleron" -- in
%%  the proton-decay selection rule, not in a leptogenesis context.
%%
%%  The Part states the chain in full with every derivation on the spot.
%%  Custody: A9 owns the microscopic inputs (the unique LHN portal, the
%%  Majorana parity alpha_31 = pi, the reactor angle, the seam Yukawa,
%%  the lock scale) because they are this paper's own derived quantities;
%%  the finite-temperature regulator argument (condition C4) remains the
%%  cosmology companion's, and every step names its owner.
%%
%%  No T-SM tag is minted: that scheme is section-number-coupled and has
%%  already drifted in the deployed text (60.1 -> 57.1, 62.1 -> 64.1,
%%  67.1 -> 69.1).  The Part uses proposition/lemma environments instead.
%%
%%  No parameter of the 28-parameter codebook changes.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.60 (August 12, 2026) -- Stage 1 of the neutrino expansion
%%  (GG-Plan-A9-SM-Neutrino-Expansion-v1.0, author-authorised).  The
%%  benchmark refresh and the atmospheric-octant split.  No new section.
%%
%%  NuFIT 6.0 -> 6.1 (November 2025 data); JUNO first results added
%%  (Nature 654, 343, 2026); the T2K + NOvA joint fit added (Nature 646,
%%  818, 2025) and adopted for delta_PMNS; DESI DR2 (PRD 112, 083513,
%%  2025) replaces DESI DR1 for Sigma m_nu; KamLAND-Zen, LEGEND and
%%  CUORE advanced to their published references; KATRIN's 1000-day
%%  result recorded as pending rather than final.
%%
%%  The global fit and the accelerator joint fit now disagree on the
%%  atmospheric octant.  Both are printed, the disagreement is named,
%%  and the pull is given against each: A9's 39/70 sits +5.10 sigma from
%%  NuFIT 6.1 and -0.06 sigma from T2K + NOvA.  Sigma convention for
%%  asymmetric errors is stated in the benchmark subsection.
%%
%%  Sigma m_nu: the predicted 59.17 meV is 8 per cent under the DESI DR2
%%  LambdaCDM bound and outside the Feldman-Cousins variant of the same
%%  analysis.  A9's own F-SM-6 pins w = -1 two-sidedly, so A9 must be
%%  scored against the LambdaCDM row and not the w0waCDM row.  Recorded
%%  as a channel now firing at the one-sided edge.
%%
%%  No prediction and no derived value changed.  Every Dev and Pull was
%%  recomputed programmatically.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.59  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  9 figures re-placed so their node text lands at the house size,
%%  7.79pt: what GG-P3-Electron Figure 3 renders (\footnotesize at
%%  0.865 of natural size).  For a picture of measured width W drawn
%%  at a base font of B points the target is f = (7.79/B) * W / 159.2
%%  of \textwidth, W being measured by compiling each picture
%%  standalone and reading its ink bounding box.  Nothing inside any
%%  picture changes: no font command, no dimension, no text, no
%%  relative position.  The cohort map and the central-claim funnel
%%  are excluded, both being at the house size already.
%%  Per-figure table and recipe: [v13-support]/allsweep.py and
%%  [v13-support]/figure_census.json.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.58  (August 5, 2026)  -- the cohort funnel at the house size.
%%  The central-claim funnel was drawn with 107mm 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
%%  0.64\textwidth, computed rather than chosen: the picture measures
%%  117.8mm 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.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.57  (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.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-A9-SM  v20.55  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v20.54 -> v20.55), author-directed (GG-Report-187):
%%  The Higgs sector read on the two fibres.
%%    * CONVENTION FIX (blocking): theta*_H is the ANGLE and (2/L)^N
%%      its SINE.  Seven sites wrote "theta*_H = (2/L)^N", an
%%      ambiguity of ~0.07% -- larger than the +0.045% residual
%%      claimed for v.  All now read sin theta*_H = (2/L)^N.
%%    * NEW section sec:higgs-two-fibres (Part VI, after the
%%      quartic-curvature theorem), five subsections:
%%      - the Higgs IS the sixth dimension's connection component
%%        (eq:thetaH-body); no fundamental scalar; the potential
%%        generated not postulated; the mass a curvature
%%      - the mechanism as the GRIP cascade on Sigma_H, with the
%%        stage table; the dial-not-hat reading; the Elitzur
%%        corollary (nothing breaks, the Goldstones were never
%%        there)
%%      - the custody table: which fibre owns Theta_H, V_eff, the
%%        selection, sin theta*_H, M_lock, v, lambda_H, m_H, the
%%        boson itself ("the Higgs boson is the sixth dimension
%%        ringing")
%%      - the Higgs and the Kaxion as the same compact-angle
%%        modulus on different strata, at different cascade stages
%%      - "What the Higgs sector owes": v +0.045%, m_H -1.66%,
%%        lambda_H's spectral normalisation NOT computed, the eight
%%        non-top Yukawas asserted, vacuum stability postulated;
%%        and the lambda_H dual lock (2^-N vs alpha_s+alpha_em,
%%        0.07% apart) stated once as one thing.
%%    * sec:deep-hierarchy gains the three tiers of circle-side
%%      protection (integers / the Wilson-line modulus / interval
%%      addresses) and the statement that the hierarchy problem is
%%      a tier misclassification; App. E.3.4 promoted by
%%      cross-reference.  Tier 2 buys protection against
%%      divergence, NOT against variation -- stated explicitly.
%%    * The gauge-Higgs-unification comparison given its
%%      literature: Manton 1979, Hosotani 1983/1989,
%%      Hatanaka-Inami-Lim 1998, with the debt named and the
%%      ledger's addition stated (theta*_H computed, not scanned).
%%    * NEW bibitems: ArisakaW1Symmetry, ArisakaW2Holonomy
%%      (Section A); Elitzur1975, Manton1979, Hosotani1989,
%%      HatanakaInamiLim1998 (Section B).  The two-fibre
%%      subsection now cites W1/W2 by key (was bare textsf) and
%%      carries a label; the GHU comparison carries a label.
%%  Script: a9_v2055_higgs_two_fibres.py
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.52  (July 30, 2026) -- L-INT vocabulary conformance to
%%  A9-SM v20.56  (August 5, 2026)  -- P7-Inflow residue.  The
%%  ArisakaP2N3 relevance note still credited the discharge of R-INT
%%  to "the inflow member, UCLA/Arisaka-GG-P7-Inflow"; that argument
%%  is now Appendix H of the same paper.  One site.  No physics.
%% ===================================================================
%% ===================================================================
%%  P2 v14.2 canon + P7-Inflow status update.  ~11 sites: "integrality
%%  lemma L-INT" -> "integrality theorem L-INT (residue R-INT)";
%%  "single open statement" -> derived-theorem-with-residue; the P2
%%  bibitem Relevance now records that R-INT is discharged on the
%%  benchmark by the inflow member UCLA/Arisaka-GG-P7-Inflow (textual
%%  code mention, house precedent; no new bibitem).  NO physics
%%  touched; the conditioning chain itself is unchanged.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.51  (July 30, 2026)  -- Phase-2 references completion.
%%  The HopkinsSinger2005 bibitem was MALFORMED -- a half-converted BibTeX
%%  record whose "note = {...}" and "relevance = {...}" field syntax printed
%%  literally inside an \textit{\emph{...}} group, so the entry rendered as
%%  garbage and its Relevance note was invisible.  Rewritten in house style,
%%  with the journal reference completed and the L-INT connection stated.
%%  First-citation pointer added.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.50  (July 30, 2026)  -- P2-N3 v14.0 conformance.
%%  Every description of P2's conditioning updated to the capacity
%%  architecture: Budget Conjecture -> capacity theorem T-N3-6
%%  (coupling clause + integrality lemma L-INT, Born-grade premises);
%%  H1 -> closed in the negative (rigidity); H2 -> L-INT; c_R no longer
%%  "awaiting the induced connection" (constructed: Berry, per P2 App F
%%  conventions).  Repairs an INTERNAL CONTRADICTION: the cross-reference
%%  table claimed "H1+H2 reduction with explicit c_R = 1" and "H1, H2
%%  promoted to theorems in A6 sec 22" while eq:cR-equals-1-here said
%%  "c_R currently undetermined" -- the naive c_R = 1 was withdrawn by
%%  P2's own no-go, and the A6 sec-22 promotion route is closed by the
%%  rigidity lemma (A6 is conformed in the same sweep).  Register: the
%%  section title and Part previews "closed theorem" -> "three-leg
%%  theorem"; "closed proof/answer" conformed; the calibration sentence
%%  now names L-INT + Born-grade premises.  20 anchored sites.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.49  (July 29, 2026)  -- P2-N3 retitle propagation.
%%  P2-N3 is retitled "Why Three Generations: A THREE-LEG Derivation of N = 3
%%  from BI-6, MSMF, and GDOS Morse Stability" (was "A Closed Derivation").
%%  Reason: P2-N3's own main theorem T-N3-5 is "Conditional derivation of
%%  N = 3 (three-leg pincer)", conditional on five structural hypotheses --
%%  including the BI-6 Budget Conjecture, which A2 states is not derived
%%  anywhere in the cohort -- plus two empirical inputs.  The cohort's prose
%%  already flagged that conditioning everywhere; only the title claimed
%%  "Closed", and titles propagate into every bibliography.
%%  T-N3-5 KEEPS its own name: the theorem is conditional and says so.
%%   * COHORT ROSTER: P2 row described as "A closed derivation of N=3
%%     fermion generations" -> "A three-leg derivation ...", and its third
%%     column "the closed prediction of the fermion generation count" ->
%%     "the three-leg derivation of ...".  The description echoes the title
%%     and moves with it.
%%   * COHORT ROSTER: P3 row "the five oldest electron questions" -> "six",
%%     matching P3's title page and A9's own bibitem, both of which say six.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.48  (July 29, 2026)  -- three residuals, the most exposed of the
%%  whole sweep.  Caught only by the final cohort-wide re-scan: v20.47's
%%  post-condition had "Phase 6" but not the tilde form "Phase~7", and no
%%  pattern at all for "Plan-agent", "Phase Log" or "PROVENANCE".
%%   * "carried out on May 12, 2026 by a Plan-agent companion-paper auditor ...
%%     (cf. Phase~7 of the Phase Log in the file PROVENANCE block)" -- an
%%     internal agent role, an internal phase log, and a PROVENANCE block
%%     inside the source file, all shown to the reader.
%%   * "Earlier draft versions of the present paper used bare T2/T3/T4
%%     citations ... resolved in Phase~7 (cf. PROVENANCE)" -> the citation
%%     discipline itself: companion theorems are cited at sub-numbered
%%     identifiers because bare forms are ambiguous and sections drift.
%%   * "verified by the Phase~7 companion-paper auditor".
%%  The reproducibility content is kept throughout: cross-citations were
%%  checked against the companions' own sources, not from memory.
%% ===================================================================
%% ===================================================================
%%  A9-SM v20.47  (July 29, 2026)  -- internal-affairs sweep.
%%   * A Part-boundary paragraph was an editing note printed for the reader
%%     ("the manuscript is ready to hand the reader over to ... sections that
%%     already existed in the previous version.  The remaining task is to make
%%     a few appendices explicit ...").  Replaced with a forward-looking
%%     transition, which is what the slot wanted.
%%   * "An earlier revision of this appendix carried a duplicate master census
%%     table; it has been removed" -> states the invariant it was protecting:
%%     the 28 parameters are tabulated once and graded once, nowhere else.
%%   * "two claims that the earlier draft ran together" -> "two claims that are
%%     easy to run together".  The ontological/computational distinction and
%%     the 6/14/8 triage are untouched.
%%   * "branch freeze (v20.34)" and "consumer registration (v20.36)": version
%%     labels dropped.  Both sit on live claims -- a pre-registered dead
%%     3pi/2 branch, a parity sign frozen at pi -- which are unchanged.
%%  Not touched: six "first pass" hits in the reading sense; "the substrate is
%%  now fixed" and "once the first pi-e cell has been fixed", where "fixed"
%%  means determined; and the MSMF/OGN audit counts, an audit internal to the
%%  paper rather than a working pass.
%%   * "the closed-system statement that the earlier drafts only used
%%     implicitly" -> "makes explicit what is elsewhere used only implicitly".
%%     Found by the post-condition rather than the scan: the plural "drafts".
%%   * 7 sites of manuscript-construction language, three of them SECTION
%%     TITLES: "already present in the manuscript", "the new front-end
%%     sections", "The expanded introduction", "the new front-end material",
%%     "the revised front-end sections", "the existing roadmap section", and
%%     "Why this dictionary improves ... the manuscript".  Found by rendering
%%     p.92, not by the scan.  The ~10 uses of "the manuscript" meaning simply
%%     "this paper" are house style and are deliberately left alone.
%%  No value, equation, grade or citation changed.
%% ===================================================================
%% ===================================================================
%% v20.45 -> v20.46 (July 28, 2026).  Phase C of the GGA correction.
%%   * The ground-state refinement re-pointed to its upstream home and
%%     quoted at its grade: it holds on the admissible strata, of which
%%     the charged-lepton stratum is one.  The 'every stratum' phrasing
%%     narrowed accordingly.
%%   * The four pillars no longer described as selecting the ground
%%     attractor; they characterise it (companion pillar paper).
%%   * Cascade chain form and GRIP-Out attribution conformed.
%%   * Version numbers and raw label names removed from prose.
%% Values, canonical numbers, ledger: UNCHANGED.
%% ===================================================================
%% ===================================================================
%% v20.45 (July 24, 2026): COHORT BIB-TITLE RECONCILIATION (P3)
%%  The electron companion's subtitle is updated in its bibliography
%%  entry to that paper's current cover ("The Six Oldest Questions
%%  about the Electron ..."), following its addition of the sixth
%%  question -- the origin of positrons.
%%  The branch-map description of the electron paper gains the
%%  sixth question (the mirror twin) and the count. No other change
%%  anywhere.
%% ===================================================================
%% ===================================================================
%% v20.44 (July 24, 2026): THE BULK ACTION EARNED (author-commissioned
%%  provenance repair). The 6D action that opens the export chain was
%%  asserted with a citation; it is now earned. Part IX: the silent-
%%  axiom scorecard gains the zeroth axiom (that a 4D Lagrangian is
%%  the starting point at all) with its own table row; stage S-1 is
%%  rewritten at three times its length, carrying the four forcing
%%  steps by theorem tag, the honest description of the companion
%%  papers' roles, and the explicit import of the GDOS descent
%%  theorems that spend the action. Appendix G: the course walk gains
%%  station S-1a, a brace-by-brace uniqueness argument for the action
%%  (with the six-dimensional Lovelock caveat, the named Minimal-
%%  Matter Lemma, the forced flux sector, and the scalar-free
%%  selection each at stated registers), a provenance mini-table, two
%%  new worked problems (the operator scan; the heavy-mode
%%  integration miniature), and the bulk cosmological term restored
%%  to the displayed action. The honesty-boundary remark now carries
%%  the action's own registers. Paper problem census updated.
%% ===================================================================
%% ===================================================================
%% v20.43 (July 23, 2026): NEAR-EMPTY-PAGE REPAIR (cohort-wide audit
%%  commissioned after A8 v13.12). Forced breaks dropped around the
%%  [p] floats (the cohort map and the two Part-XI deviation plots)
%%  and before the Part-XI MSMF/OGN section; the two hand-built
%%  full-page minipages (the sixteen-Part roadmap flowchart and the
%%  Part-II gg-unification showcase) promoted to real figure[p]
%%  floats so the surrounding main text fills the formerly stranded
%%  pages while each figure keeps its own full page. Legitimate Part
%%  tails and the front-matter showcase untouched. NO text content
%%  changed.
%% ===================================================================
%% ===================================================================
%% v20.42 (July 23, 2026): SHIP C OF THE PROBLEMS CAMPAIGN -- twenty-
%%  six problems with complete worked solutions for the five
%%  remaining sector chapters (CKM 5, PMNS 6, constitutional ledgers
%%  3, force laws/proton stability 5, strong-CP/confinement 7), each
%%  set placed before its chapter's take-away box (the ledger
%%  reference chapter, which carries none, closes with its set), all
%%  numerical answers verified programmatically before typesetting.
%%  The course front matter gains its third stylistic commitment:
%%  the problems clause, now true of every chapter of the course
%%  proper. This completes the problems campaign for this paper.
%% ===================================================================
%% ===================================================================
%% v20.41 (July 23, 2026): THE ASSEMBLED-LAGRANGIAN COURSE CHAPTER
%%  (author-commissioned companion to the main-text Part). New
%%  appendix chapter inserted between the export/Wilson-Hosotani
%%  chapter and the fermion-mass chapter (it takes the letter the
%%  fermion chapter held; all later appendix letters shift by one,
%%  handled automatically by LaTeX): the same eight-stage export
%%  chain at course register, from scratch, with a physics-intuition
%%  note at every step, full box apparatus, and six problems with
%%  worked solutions. The fermion chapter's eight Ship-B problem IDs
%%  renumbered to its new letter; course ordinals harmonised to
%%  "sector appendix"; Part front matter and subtitle gain the
%%  interlude; the main-text assembly Part gains the down-pointer.
%%  STALE FIX: two Conclusions pointers to the promotion roadmap
%%  hardcoded "Appendix K" (the roadmap is not chapter K) -- both now
%%  \ref-based.
%% ===================================================================
%% ===================================================================
%% v20.40 (July 23, 2026): THE ASSEMBLED-LAGRANGIAN PART (author-
%%  commissioned). New main-text Part inserted between the protected
%%  readout and identity/uniqueness: the complete Standard-Model
%%  Lagrangian assembled term by term as a constitutional export --
%%  the six silent textbook axioms replaced by theorems, the export
%%  chain in eight stages, the five term groups, the absent operators
%%  derived, the master provenance table, and the honesty boundary.
%%  Label part:lagrangian. Roadmap flowchart rebuilt (new highlighted
%%  harvest box; later Parts renumbered/shifted; caption now seven
%%  phases / sixteen Parts); tab:roadmap-full gains its row; all
%%  Part-count mentions swept; Introduction gains the funnel hook;
%%  the Part-transition sentences re-seamed on both sides.
%% ===================================================================
%% ===== v20.39 (2026-07-23): PROBLEMS & WORKED SOLUTIONS, SHIP B (D-G) =====
%% The A9 problems campaign, ship B: 26 problems with complete
%% worked solutions for chapters D-G (D:8 E:8 F:2 G:8), the
%% P5/C3/A10 format, sets before the take-away boxes (F, having no
%% take-away, closes with its set).  Highlights, all verified
%% programmatically: e = g sin(thW); 3/8 at the seam; b2 = -19/6
%% with the Higgs' +1/6; the near-miss crossings at 1e13 and 1e17;
%% 2^7 = 128 (0.035%) and 128+9 = 137 (0.026%); 25/108 = L^2/4N^3
%% (0.03%); alpha_s^{-1} = pi*e (0.9 sigma); the K = LJ = M^2+N^2+
%% L^2 = 35 double identity; v from G_F (246.2); rho = 1 custodial;
%% lambda = 0.129; the 1-TeV no-lose unitarity ceiling; v =
%% M_lock(2/L)^N = 246.3; lambda_H = 1/8 with m_h = v/2 = 123.1 at
%% first order and the QD register stated; v/MPl = 2.018e-17 both
%% ways; the Wilson-line no-quadratic-divergence argument; the
%% two-transcendentals census (pi and e only); the Hosotani
%% protection worked; y_t = 0.992 -> the zero-toll anchor; the
%% 10^12 span; the quark winding ladder k = 0..4; the dyadic lepton
%% codes (1.785/106.3/0.5100 vs 1.777/105.7/0.5110); the geometric
%% seesaw m* = pi v^2/E_GUT = 8.35 meV (A10 I.1.1's lever reversed);
%% the golden-ratio Cabibbo 0.225019 vs 0.22500; s23 and s13 with
%% the QD register; J = 2.9835e-5 vs (3.08+-0.15)e-5 (~0.7 sigma).
%% Ship C (chapters H-L) follows after sign-off, with the
%% front-matter problems clause.
%% ===== v20.38 (2026-07-23): COURSE APPARATUS, SHIP A (author-approved) ====
%% The problems campaign reaches A9 (C3: 53 problems, A10: 56, both
%% complete).  Ship A aligns Part XVI with the C3/A10 course
%% standard before the problem sets land:
%% (1) The three course-box macros added (teachbox / takeaway /
%%     readalong, the cohort definitions).
%% (2) Boxes for the course chapters: full trio for the seven
%%     three-step chapters (gauge, EWSB, Yukawa, CKM, PMNS, force
%%     laws, strong-CP/confinement); teachbox only for the two
%%     reference chapters (export dictionary, constitutional
%%     ledgers).  Readalong lines reuse the v11.82 best-matched
%%     textbooks -- no new bibitems.
%% (3) Two stale front-matter clauses fixed (banner + intro): the
%%     "referee appendices" moved to the Common Objections Part in
%%     an earlier restructure; the text now says so.
%% No physics, values, labels, or codebook entries touched.
%% Ship B (problems, chapters D-G) follows after sign-off.
%% ===== v20.37 (2026-07-20): STAGE 5C -- WAVE-2 RESIDUE CLEANUP =====
%% (1) F4: companion-paper reference-item cohort-status
%%     remark on A10-Cosmos -> released register ('preprint',
%%     matching the sibling items of the same list).
%% Physics, theorem keys, numbers, values, grades: UNCHANGED.
%% ===================================================================
%% =====================================================================
%% v20.36 (2026-07-20)  STAGE-4 SHIP: ALPHA_31 = PI REGISTERED AS THE
%%   SIGN-DRIVER OF THE MATTER EXCESS + SEAM-YUKAWA DUAL-CONVENTION
%%   DECLARATION (author-approved Wave-2 physics-grade ship; spec:
%%   internal reports PhiGeo-Projection-2026-07-18 and
%%   RealBulk-Audit-2026-07-19, items A9-R1/A9-R2; paired with
%%   A10-Cosmos v13.5).  (a) CONSUMER NOTE (A9-R1): new closing
%%   paragraph in the Discussions subsection 'Baryon asymmetry and
%%   leptogenesis' -- in the companion's flavoured resonant-
%%   leptogenesis chain the unflavoured CP invariant is
%%   U_PMNS-independent and vanishes for the dial-free real-R door;
%%   the inherited phases reach eta_B only through the flavoured
%%   eps_{i,alpha}, whose driver is this paper's alpha_31 = pi: it
%%   sets the SIGN of the cosmic matter excess (nonzero at m_1 = 0;
%%   J_PMNS-blind at the few-percent level), conditional on the
%%   companion's named finite-temperature hypothesis H-phigeo-1.
%%   Adds a consumer, not a grade: alpha_31 stays Derived
%%   (structural); the consuming chain stays derived-conditional.
%%   (b) ROW-24 REGISTRATION (A9-R2; two locations, mirroring the
%%   v20.34 row-22 branch-freeze discipline): codebook row-24 comment
%%   cell + audit paragraph #24 -- the parity branch {0, pi} is
%%   frozen at pi (derived, forced by parity as stated); alpha_31 = 0
%%   is pre-registered as the antimatter-sign branch (it would
%%   extinguish the parity-driven flavoured asymmetry at m_1 = 0,
%%   leaving only the few-percent Jarlskog residue); a future 0vbb or
%%   cosmological determination favouring it is a kill of the chain's
%%   sign, not a reinterpretation surface.  Observable-only exposure;
%%   grade words unchanged at every alpha_31 site.  (c) SEAM-YUKAWA
%%   DUAL-CONVENTION SENTENCE (one sentence, neutrino GRIP-cascade
%%   seam-coefficient paragraph): the companion's eta_B chain
%%   evaluates the resonance width at the seam-matched m-tilde ~= 8.4
%%   meV (= pi v^2/E_GUT, this paper's seam scale) and the washout at
%%   m-tilde ~ m_3 ~= 50.5 meV; the factor-~13 spread in Y_nu^2 is
%%   declared, accounted for, and priced in A10's leptogenesis
%%   appendix.  Protected byte-identical: the QD completion-class
%%   subsection, the delta_PMNS row-22 branch-freeze sites, the
%%   alpha_31 grade words, the T-SM-44.1 leptogenesis-backbone close,
%%   abstract + executive summary.  No printed physical number
%%   changed.  v20.35 -> v20.36.  Script: a9_v20.36_stage4.py.
%% =====================================================================
%% =====================================================================
%% v20.35 (2026-07-19)  COHORT PROMOTION WAVE-1 MECHANICAL SHIP
%%   (author-approved; 17 -> 19 papers; no value, status, or triage
%%   count changed).  (a) STANDARDIZED PART-I CLOSE: the former
%%   'Roadmap of the Fifteen Parts' and 'The GG-Theory Cohort'
%%   sections are merged into one closing section 'The cohort
%%   inheritance and the roadmap of the fifteen Parts' (cohort first,
%%   roadmap second; house pattern of the A2 v13.3 .. A8 v13.2 ships;
%%   ALL labels kept: sec:roadmap-A9, sec:cohort-A9, fig:org-flowchart,
%%   fig:cohort-map-a9, tab:cohort-roster).  The seventeen-paper map is
%%   retired; the frozen v1.4 nineteen-paper cohort map is installed
%%   verbatim from the A2 v13.3 in-paper rendition (MP4/MP5 internal
%%   node names; resizebox 0.9\textwidth per the A8 rendition; label
%%   fig:cohort-map-a9).  New Foundations-Branch paragraph (A9 role:
%%   ledger phases delta, alpha31 exported to P5-CPT, routed to A10's
%%   eta_B chain; P4-Real as origin of the complex phase structure).
%%   New house-architecture roadmap figure 'Roadmap of the fifteen
%%   Parts of GG-A9-SM' (vertical Part column, right phase sidebar,
%%   derivation core Parts VI-VIII highlighted: 5+22+1=28), replacing
%%   the retired block-design flowchart under the SAME label
%%   fig:org-flowchart; new roadmap prose.  Roster table gains the P4
%%   and P5 rows (nineteen).  (b) COUNTS: cohort-count 'seventeen' ->
%%   'nineteen' at the cohort-count sites only (semantic seventeens --
%%   parameter-audit seventeens, orders-of-magnitude, 'one or
%%   seventeen', seventeen-layer cascade, seventeenth-century --
%%   protected).  (c) P4/P5: \PaperPFour now renders GG-P4-Real;
%%   \PaperPFive macro added; bib key ArisakaP4Twistor ->
%%   ArisakaP4Real with the P4-Real v3.2 cover and a member-language
%%   relevance note; ArisakaP5CPT bibitem installed (P5 v3.40 cover);
%%   A-group bib header, suffix roster, and register tokens set to the
%%   canonical nineteen-paper member form.  (d) Proton-decay successor
%%   designator neutralized at the (H2) status note: '(P5, in
%%   preparation)' -> '(in preparation)' (author is holding the naming
%%   decision).  Protected byte-identical: the QD completion-class
%%   subsection, the delta_PMNS branch-freeze sites, the alpha_31
%%   grade sites, the leptogenesis-backbone sentence.  v20.34 ->
%%   v20.35.  Script: a9_v20.35_wave1.py.
%% =====================================================================
%% =====================================================================
%% v20.34 (2026-07-18)  QD CLASS FREEZE + 3PI/2 BRANCH FREEZE (P5
%%   hostile-review cohort ship, author-approved; calibrated honesty,
%%   uniform pricing; standing claims intact). (a) NEW subsection
%%   'The completion class, stated as a rule' in the Second-order (QD)
%%   completion section: the QD completions used across the cohort form
%%   a frozen, bounded, quantized class (one lattice/codebook unit per
%%   observable), heterogeneous in unit and size across observables
%%   (-pi/35 delta_CKM; -pi/490 delta_PMNS; multiplicative 1-1/4480 for
%%   the theta12 channels; +12%-class s13_CKM; per-particle near-unity
%%   rationals for the charged leptons), incapable of further
%%   adjustment as of this version; necessity asserted (the codebook's
%%   own grade); direction record printed (toward the measured value in
%%   every exercised case except the Cabibbo modulus, which the
%%   registered tables quote at first order). (b) BRANCH FREEZE: the
%%   {pi, 3pi/2} near-degeneracy for delta_PMNS pre-registered as
%%   resolved dead -- binding sentence in the MSMF/OGN audit paragraph
%%   (row 22) and a freeze note in the codebook-provenance row-22
%%   comment cell: a future measurement near 270-286 degrees may not be
%%   reinterpreted under the 3pi/2 branch. No value, status, or triage
%%   count changed. v20.33 -> v20.34. Script: a9_v20.34_qd_freeze.py.
%% =====================================================================
%% =====================================================================
%% v20.33 (2026-07-16)  TWO-QUANTITY Z* SWEEP (cross-paper F7 item of
%%   the A10 z* hostile audit, author-approved): the A10-inventory
%%   table row "Recombination redshift & z_* = e^7 & 1097 & 1089+/-0.4"
%%   was the retired one-quantity identity with a stale observed value.
%%   Now two-quantity: corridor switch z_sw = e^7 (topological) beside
%%   the decoupling z_dec from A10's atomic chain (1091, zero
%%   parameters), observed 1089.92 +/- 0.25 (Planck 2018); caption
%%   gains the pointer to the two-quantity treatment of A10-Cosmos.
%%   Also: two comparison-table cells relabelled (switch z_*/z_sw).
%%   v20.32 -> v20.33.  Script: a9_v20.33_two_quantity_zstar.py.
%% =====================================================================
%% v20.32 (July 16, 2026): cohort table standard (author-approved, P4 v2.17 pattern): sub-small table fonts -> \small; p-column tables widened toward full text width; targeted corrections listed.
%% v20.31 (July 15, 2026): cohort touch C-3a --- completed-tense pointers
%% to the satellite preprint Arisaka-GG-P4-Twistor at the T-SM-44.1
%% co-statement and the T-SM-46 real-presentation preamble; new A-list
%% bibitem ArisakaP4Twistor; A-group header widened; PaperPFour macro
%% provided; label subsec:constitutional-door added as citation anchor.
% v20.30 (July 15, 2026): the structural-closure upgrade
%   (complex-variables thread, author-commissioned). NEW theorems with
%   proofs: T-SM-46.1 (discrete symmetries as orientation bookkeeping;
%   the unitary/antiunitary table derived; CP symmetric iff the texture
%   is achiral -- R8) and T-SM-46.2 (flavour holonomy: rephasing
%   invariants are loop angles; pentagonal lattice as discrete holonomy
%   group; Jarlskog as the sine readout -- R9), in a new subsection of
%   the quark-gates section ahead of the parameter block. NEW theorem
%   T-SM-44.1 (the dial-free door: Majorana mass iff fibre weight zero;
%   LHN the unique portal -- R10 mirror, co-stated with T-Cosmos-18) in
%   the neutrino-floors section. Calibrated structural-closure paragraph
%   at the end of the summit section (structure = theorem-grade;
%   numerical ledger = graded codebook, unchanged). NEW objections
%   O-SM-81..83; count eighty -> eighty-three (3 prose spots). No
%   numbers, values, statuses, or figures changed.
%% v20.29 (2026-07-14) -- gloss cleanup: vev-descent line N=3 "family number" -> "colour-triplicity integer" (matches the sentence's own "triplicity layer" wording + A3 canonical). No values changed.
%% v20.28 (2026-07-14, OGN-5 gloss harmonization to A3 canonical) -- presentation only.
%%   Off-canonical per-integer readings (miscited to A3) brought into
%%   agreement with A3-BI6's stated meanings: M=primitive tensor slot,
%%   N=colour-triplicity index (generation downstream), L=defining SU(5)
%%   block size (NOT rank), J=minimal symmetric closure cardinality,
%%   K=total lock index. No numbers changed. Script: gloss_harmonize.py.
%% v20.27 (2026-07-14, Phase 1 landscape wording) -- presentation only.
%%   Exec-Summary para "Striking contrast to supersymmetry and string theory":
%%   the one pointed sentence ("explanatory currency is bought on credit...")
%%   replaced by a purely structural closure-mode statement (constants left
%%   unforced; SUSY adds freedom, landscape distributes it across vacua; aim
%%   here is to force them). Honors the no-string-theory-polemic directive;
%%   verdict/tone calibrated per the Landscape/Second-Horizon report
%%   (2026-07-14). No citations added, no values changed. Script:
%%   a9_landscape_swap.py.
%% v20.26 (2026-07-14, OGN5 + {e,pi} + N=3 sweep) -- presentation only.
%%  FRONT MATTER (Exec Summary):
%%   (1) "The single input": the five OGN-5 integers (M,N,L;J,K)=(1,3,5;7,35)
%%       stated as FORCED (unique survivor of anomaly-free BI-6 closure, A3),
%%       each with a physical meaning (M=1 tensor seam, L=5 SU(5) block,
%%       N=3 colour triplicity, J=7 odd closure index, K=35 master lock),
%%       pinned upstream; plus the {e,pi} dictionary -- e = base of the radial
%%       (scale) 5th-dim redshift, pi = angle of the compact (gauge) 6th-dim
%%       circle. Anti-hostile-reviewer emphasis (author directive 2026-07-14).
%%   (2) NEW para "The two faces of N=3": N=3 origin = colour triplicity of
%%       SU(3)_c (BI-6 trace-ratio, A3); three-generation count = separate
%%       P2-N3 theorem (open-system GDOS/GRIP pincer), first in history.
%%   (3) "The problem": "the family number N=3" -> "N=3".
%%   (4) keyword line: two lower-value terms trimmed (space buffer).
%%  BODY (value-preserving, NO number changes):
%%   A. alpha^-1(0)=2^J+N^2 : N^2 attribution harmonized to COLOUR triplicity
%%      (was "family-multiplicity index" x2 + "dyadic-plus-family"), matching
%%      the primary theorem ("colour-blind triplicity sheet") + OGN-5 canon.
%%   B. neutrino m_* : two audit sentences misquoted the MASS scale as
%%      2pi v^2/E_GUT; corrected to pi v^2/E_GUT (operator 2pi halved by two
%%      Higgs v/sqrt2). Tabulated 8.36 meV UNCHANGED. Derivation eqs untouched.
%%  Script: a9_ogn_epi.py.
%% v20.25 (2026-07-13, S7) -- author correction of S6: the square-canvas
%%   + fonts+20% + line-space + extended-caption treatment moves from
%%   Figure 2 (symlog; REVERTED to v20.23 form, 29pt head kept) to
%%   Figure 3 (unification; exact square, fonts +20%, caption extended
%%   to the Figure-13 register). Assets flow back into in-paper Figs
%%   9+13 (shared). Scripts: a9_v20.25_S7.py + the two showcase
%%   generators.
%% v20.24 (2026-07-13, S6) -- author refinements on the showcase:
%%   Figure 2 asset: exact-square canvas, fonts +20%, head +30% (29pt);
%%   Figure 3 asset: title +30% (25pt); both flow back into the shared
%%   in-paper Figures 9 and 13. p5: line space before the caption and
%%   the caption extended to the Figure-9 register. Scripts:
%%   a9_v20.24_S6.py + a9_showcase_symlog.py + a9_showcase_unification.py.
%% v20.23 (2026-07-13, S5) -- author request: showcase pages 4-6 (the
%%   C2/A10 device). p4: sub-title lines moved into a brief bottom
%%   caption (Figure 1). NEW p5 = the 28-parameter symlog audit with a
%%   short caption (Figure 2); asset retitled by a9_showcase_symlog.py
%%   (bold 22pt head, parenthetical on its own line, gap below; data
%%   byte-identical) and serving the in-paper figure unchanged in place.
%%   NEW p6 = gauge-coupling unification (Figure 3), VECTOR-regenerated
%%   as gg_unification_showcase.pdf (a9_showcase_unification.py; legacy
%%   raster had no generator; schematic reproduced exactly, numbers
%%   asserted; title +35% bold, fonts +20-30%); boundary equations
%%   displayed under the chart. In-paper fig:gg-unification promoted to
%%   a full page with the new asset, the displayed equations, and the
%%   improved caption. Figure numbers shift +3 (refs auto; grep-verified
%%   no hardcoded numbers). Scripts: a9_v20.23_S5.py +
%%   a9_showcase_symlog.py + a9_showcase_unification.py.
%% ======================================================================
%% v20.21 (2026-07-06): F-SM-1 rate canonicalization, synchronized with
%% P1-KK v11.39 (P1 operational rates now uniformly LO x K, K_qq=1.3,
%% K_gg=1.6): octet 6.7 [5.4-9.6] -> 8.7 [7.0-12] fb; broad-limit
%% headroom ~4 -> ~2.9; free-field excess ~3 -> ~3.6; graviton ~0.03
%% -> 0.045 fb (ttbar, LO x K); diphoton retro-diction margin 4.6e3 ->
%% 3e3. Kill window 0.5-2 ab^-1 unchanged (new band crossings inside).
%% One paragraph touched; no other content changed.
%% ======================================================================
%% ======================================================================
%% v20.20 (2026-07-06): naming-convention retag (author directive:
%% O-labels reserved for Objections per Appendix C). P1-KK's g_B menu
%% item O-KK-1 is now L-KK-1 (open selection lemma); the two body
%% mentions here updated to match P1 v11.25. No content changed.
%% ======================================================================
%% ======================================================================
%% v20.19 (2026-07-06): terminology harmonization with P1-KK v11.19
%% (author-approved). The T-KK-2 dichotomy is now named Scenario A
%% (free-field, 2.46 TeV, provisionally closed) / Scenario B
%% (composite-hardened, degenerate cluster at M_lock) cohort-wide,
%% matching F-KK-2A/2B. One paragraph touched (F-SM-1); A9's own
%% GRIP/GSSB "branch selection" vocabulary and the phenomenology
%% Branches are unrelated senses and untouched. No values changed.
%% ======================================================================
%% ======================================================================
%% v20.18 (2026-07-05): F-SM-5 sharpened by P1-KK v11.12 (O-KK-1). The
%% baryonic normalisation g_B now carries a theorem-grade generator norm
%% (the 2/5 trace ratio of this paper's own Arisaka-equation lock) with
%% one discrete readout rule open (O-KK-1, P1); menu g_B in {0.14, 0.23,
%% 0.57, 0.89}. The (g_B,m_B) fifth-force window computation reduces from
%% a plane scan to a selection among four candidate lines. One F-SM-5
%% sentence extended; WGC objection response wording aligned.
%% ======================================================================
%% ======================================================================
%% v20.17 (2026-07-05): F-SM-1 two-tier requalification, following P1-KK
%% v11.11 Corollary T-KK-2c (derived graviton gluon coupling kappa_gg =
%% 0.47/K -> sigma(G1) ~ 0.03 fb at LHC, 205 fb at FCC-hh 100 TeV).
%% Tier 1 (HL-LHC, decisive): the un-suppressible colour octet carries
%% the falsifier. Tier 2 (FCC-hh, confirmatory): the spin-2 anchor.
%% One prose paragraph + one summary-table row edited; timeline card
%% (venue: HL-LHC -> FCC) already consistent. Author-approved.
%% ======================================================================
%% ======================================================================
%% v20.16 (2026-07-05): F-SM-1 operational fold-in. The lock-cluster
%% falsifier now points at P1-KK v11.10's computed sigma x BR statement
%% (operational-confrontation section): octet 6.7 [5.4-9.6] fb at 3.85 TeV
%% in ttbar; HL-LHC crossing at ~0.5-2 ab^-1; free-field 2.46 TeV branch
%% provisionally closed by the observed ATLAS broad-template limit.
%% One paragraph edited; no other content touched.
%% ======================================================================
% v20.15 (July 5, 2026): Sigma m_nu kill-test billing -- F-SM-3 mass-sum
%   failure statement given a NAMED, DATED decision (DESI-DR2+CMB ~2026-28,
%   robust <58 meV under w=-1 kills; may fire first); stakes-list phrase
%   updated. Symmetric with A10 v12.90. No result changed.
% v20.14 (July 4, 2026): kill-test timeline added (author request, mirrors the
%   A10-Cosmos figure): new TikZ Figure fig:killtest-timeline-sm in the
%   'disciplined experimental roadmap' subsection --- eight kill-tests
%   (F-SM channels: lock cluster, Kaxion, neutrino package, 0vbb floor, proton
%   stability, n-nbar, fifth force, w=-1) on one forward axis with eras
%   NOW / BY~2030 / BY~2040, sector-coloured cards, scan-span bar, exposure
%   banner; subsection intro + closing enhanced (sector-coupling moral:
%   'the tests are not eight independent dice'); pointer added from the
%   falsification-timeline subsection. No science changed.
% v20.13 (July 2, 2026): cohort universal-indentation pass (author-
%   directed): indentfirst loaded (first paragraphs after headings
%   now indent). Per-paper stray-noindent review deferred to the
%   v2.0 pass. No content changed.
% v20.12 (July 2, 2026): cohort multi-Arisaka citation split pass
%   (charter natbib rule; author-directed 2026-07-02): 2
%   multi-key cites split into citealp/citet chains; fixes the
%   same-author-same-year comma-run rendering. No content changed.
% v20.11 (July 2, 2026): [C-A1] session touch. (1) Stale-arithmetic fix
%   in Leg L2 of the N=3 theorem: 'k = 1 to N ... N(N+1) <= 6' corrected
%   to the paper's own canonical 'k = 1 to N-1 ... N(N-1) <= 6, saturated
%   at N = 3' (matches the cascade-engine caption and the CKM-part eq).
%   (2) Ladder provenance: Leg L2 records the tolls pi k as derived
%   (A6 Whitney pair-count weights C(ell,2), Delta c_2 = k, three
%   declared premises). No triage counts changed; no equation altered
%   beyond the flagged fix.
% v20.10 (July 1, 2026): Stationary-Measure byproducts -- cosmic table row
%   re-scored to canonical derived H0 = 66.6 +- 0.2 + new age row; NEW channel
%   F-SM-6 (w = -1 exact, two-sided; DESI/Euclid timeline row); five -> six
%   channel counts (4 sites); O_17 cross-ref to A6 Lemma DE-1 + A10
%   T-Cosmos-DE'. Programme report 2026-07-01; author-approved.
% ===== v20.9: cosmetic formatting pass (matches A10 v12.47). (1) Global caption
%  separation \captionsetup{aboveskip=12pt,belowskip=13pt}. (2) Removed the 10 end-of-Part
%  rule dividers that immediately preceded a \GGPart -- they spilled onto a fresh page
%  (blank pages 76, 111, etc.) and are redundant with the Part heading + thick line. The 4
%  mid-section rule dividers are kept. v20.8 -> v20.9.
% ===== v20.8: Swapped the order of the two closing Part-I sections so the
%  'Roadmap of the Fifteen Parts' now precedes 'The GG-Theory Cohort', making the
%  cohort figure+roster the closing section of Part I (matches A10-Cosmos) and
%  absorbing the page-29 whitespace left by the cohort figure's clearpage. v20.7 -> v20.8.
% ===== v20.7: Removed redundant Table 2 (tab:gg2026-series-map, 8-paper 'Position
%  in series'), superseded by the 17-paper roster; repointed its one \ref to
%  tab:cohort-roster; moved the roster (Table 3) to immediately after Figure 2's
%  page. v20.6 -> v20.7.
% ===== v20.6: Added standardized 17-paper cohort roster (tab:cohort-roster,
%  generic 'Role in the cohort' column, A9 marked 'this paper', self-contained
%  via providecommand guards) at end of Part I beside fig:cohort-map-a9. v20.5 -> v20.6.
% ===== v20.5: COSMETIC table-font normalization -- inserted \small before
%  \begin{tabular} in 23 oversized (unsized) table floats so every table
%  sits ~10%% below the body text (matches A10 v12.43). Already-small tables untouched.
% =====================================================================
% ===== v20.4 (2026-06-28): STATUS COLUMN ON THE 28-PARAMETER SUMMARY TABLES. =====
%  Added a 'Status' (D / D* / A) column to the two summary audit tables
%  (first-order + QD), matching A10-Cosmos. Values from Table 8 (codebook
%  provenance); the canonical 6 Derived / 14 Derived-cond. / 8 Asserted
%  triage is preserved (verified programmatically). Sector column kept;
%  Analytic-expression and Observation columns shrunk to make room.
%  Presentation only; no value/triage-count/physics change. v20.3 -> v20.4.
% =====================================================================
% =====================================================================
% ===== v20.3 (2026-06-28): ALL TABLE CATEGORIES BOLD-ITALIC, NO RULES. =====
%  Every italic category row (page-4 field-content table, MSMF/OGN audit
%  table, glossary) is now BOLD-ITALIC, and the \midrule rules bracketing
%  the categories are removed (header rules preserved). Table 8 was already
%  done in v20.2. Cosmetic only; NO physics/value/triage change.
%  v20.2 -> v20.3.
% =====================================================================
% =====================================================================
% ===== v20.2 (2026-06-28): TABLE-8 CATEGORY STYLE (cosmetic). =====
%  Codebook-provenance table: the 7 category rows are now BOLD-ITALIC and
%  the \midrule rules bracketing them (above and below) are removed, per
%  the requested style (matches A10-Cosmos). Cosmetic only; NO physics,
%  value, or triage change. v20.1 -> v20.2.
% =====================================================================
% ===== A9-SM v20.1 (2026-06-28): abstract tightened (~300 chars) to pull it fully back onto the
% front cover page 1 (v20.0 spilled ~4 lines onto p2 after the forcing/Euclid sentence). Same content,
% forcing/Euclid idea preserved ('forced, not fitted -- propositions descending from a fixed constitution
% in the spirit of Euclid'). No other change from the v20.0 milestone. =====
% ===== A9-SM v20.0 (2026-06-28): MILESTONE stamp-only promotion of v11.121 (science frozen).
% PaperVersion v11.121 -> v20.0; date -> June 28, 2026. Status stays Preprint. v20.0 marks A9-SM
% COMPLETE as the first of the applications trio (A9-SM, A10-Cosmos, A11-Origin) to reach v20.0.
% No content change from v11.121. =====
% ===== A9-SM v11.121 (2026-06-27): front-matter/discussion/conclusions refined for the latest round.
% (1) ABSTRACT: added forcing/Euclid sentence (one ledger serves all 28 -> forced not fitted).
% (2) SIGNIFICANCE: one-sentence forcing/Euclid pointer to the forcing-vs-searching subsection.
% (3) CONCLUSIONS 'established': FIXED stale triage 6/13/9 -> canonical 6/14/8 (fourteen conditional,
% eight Asserted). (4) CONCLUSIONS 'limitations': new honest paragraph -- global trials factor is the
% decisive uncomputed test (ansatz not proof); GG derives inputs not radiative corrections (A5 T9.9);
% uniqueness is conjectured (consistency != correctness != uniqueness). (5) CONCLUSIONS summit: the
% 'computes the constants, not the corrections' boundary landed as the closing honest note. =====
% ===== A9-SM v11.120 (2026-06-27): Common Objections Part refreshed from the recent threads.
% NEW section 'The limits of geometrisation and the standing of the forcing argument' with six new
% objections O-SM-75..80: renormalisation referee (can't shortcut loops; inputs not corrections),
% hadronic/lattice referee (Tier-4 structure not precision), reverse-engineering referee (was the
% stratum->form rule pre-registered or circular?), promissory-note referee (the global trials factor
% is uncomputed), Euclid referee (a triage is not the Elements), self-consistency referee
% (consistency != correctness != uniqueness). Sharpened O-SM-4 (QD tiered vs T9.9) and O-SM-73
% (Eddington -> forcing-subsection pointer). Objection count 74->80 in the four prose spots. =====
% ===== A9-SM v11.119 (2026-06-27): NEW subsection 'Forcing versus searching: the discrete
% look-elsewhere effect, and the Euclidean reading' added to the end of 'Why the result is not a
% post-hoc fit'. Answers the discrete/combinatorial alphabet charge (distinct from continuous tuning):
% one closed ledger not a per-parameter toolbox (28-vs-2 over-constraint + cross-parameter rigidity of
% N=3); form forced by stratum (alpha_s=pi*e because the colour channel carries both primitives);
% MSMF/OGN as uniqueness selector; global trials factor as accepted arbiter (calibrated, not zero);
% forward falsifiable commitments; Euclidean capstone (parameters = propositions of the GG-Constitution,
% in the lineage of the Elements). Boxed 'forcing not searching' thesis. QD paragraph (denominators
% forced 2^J L=640; numerators integrality where strong, bounded selection where large-pull, cross-ref
% A5 T9.9 tiers). O-SM-2 sharpened with a pointer + the one-ledger/forced-form/Euclid framing. =====
% ===== A9-SM v11.118 (2026-06-27): Round-2 Tier-2 -- F-SM-2 Kaxion mirror: added the predicted
% window 1.59-2.13 GHz (benchmark 1.95) and replaced the overstated 'ADMX generation reachable this
% decade' with the honest 'on the DFSZ line; next-gen ADMX-EFR/IBS-CAPP at full DFSZ depth ~2027-2030'.
% Plus: O-SM-60 WGC honest engagement; O-SM-73 owns by-chance magnitude; stale objection count -> seventy-four. -- (a) O-SM-60 swampland response rewritten
% to engage the Weak Gravity Conjecture honestly on its own terms (Stueckelberg-massive baryon U(1),
% magnetic-WGC cutoff bound met, electric bound inapplicable/open, Qi charged-state an open check) instead
% of pleading 'not a string compactification'; (b) O-SM-73 now owns the rough by-chance magnitude of the
% individually-clean closures, deferring to the global trials figure; (c) stale objection count fixed to
% seventy-four in the two organization-prose spots (was thirty). =====
% ===== A9-SM v11.117 (2026-06-27): Round-2 Tier-1 honesty pass. B1 cover banner '28 Free
% Parameters Derived'->'28 Free Parameters, One Geometry'; 'without exception'->'not free inputs'.
% B2 sec.71.1 'blind'->'parameter-free post-dictions' (title/intro/caption) + blind-vs-postdicted
% clarifier. B3 alpha_s band 0.5%->1%(several sigma); F-SM-5 dead P1 xref -> 'P-series fifth-force
% paper'. B4 F-SM-4 tau_p=inf now 'conditional on (H2)'. B5 Table 46 + landscape para: uniqueness
% conjectured not proven. B6 open-Q QG bullet: r dual-role (RG scale vs M_Pl integral) pointer. =====
% ===== A9-SM v11.116 (2026-06-27): warm-style / de-LLM micro-pass -- dropped sentence-initial
% 'Importantly,' (x2) and 'Moreover,' connective tics; 'serves as a bridge'->'bridges'; in the
% added Discussion summary, 'it is worth gathering them'->'they are worth gathering' and
% 'The essential message stands already'->'The point is already plain'. No content changed. =====
% ===== A9-SM v11.115 (2026-06-27): corrected stale page count 'three hundred pages' ->
% 'four hundred pages' in the two Conclusions reflections (paper is now ~400 pp). =====
% ===== A9-SM v11.114 (2026-06-27): page-4 cover block ('28 Free Parameters Derived by
% GG-Theory') -- added \noindent before the \textwidth minipage so the title, hook table, and
% dev-plot figure are flush-centered in the column instead of shifted right by \parindent. =====
% ===== A9-SM v11.113 (2026-06-27): Table 45 (ten-problems-resolution longtable) -- added the
% four bold-italic category headers (Scale and symmetry breaking 1-3; Flavor, neutrinos, and CP
% 4-7; Dark sector and cosmology 8-9; Unification and gravity 10), matching Table 1. =====
% ===== A9-SM v11.112 (2026-06-27): Table 1 (intro open-problem clusters) -- four category
% headers (Scale and symmetry breaking; Flavor, neutrinos, and CP; Dark sector and cosmology;
% Unification and gravity) changed from \itshape to \bfseries\itshape for clearer distinction. =====
% ===== A9-SM v11.111 (2026-06-27): Executive Summary polish + trim to fit 2 pages. Compressed
% the long six/fourteen/eight 'how confident' breakdown; trimmed the central-recognition close;
% added a short clause reflecting the now-dated falsifiers (3.85 TeV / 8.1 ueV / proton) decidable
% by 2040. Net shorter; no physics/triage counts changed. =====
% ===== A9-SM v11.110 (2026-06-27): inter-Part transition polish -- added a short closing
% coda paragraph at the end of seven body Parts (SM-audit, Substrate, Roadmap, electroweak
% seam, protected-readout, identity/force-uniqueness, summary-and-audit) that recaps the Part
% and smoothly introduces the next, matching the Discussions->Conclusions bridge. Seams that
% already bridged (Constitutional->Substrate, Fermion->Protected-readout, Intro roadmap) left
% as-is. No physics changed; presentation only. =====
% ===== A9-SM v11.109 (2026-06-27): Conclusions readability polish -- bridged the new personal
% section into the 'summit' section (de-duplicated the SM-history / GUT-SUSY-string recap that
% the personal account now tells, turning the second telling into a deliberate reprise). Light
% sentence smoothing; no content removed beyond the literal repetition. No physics changed. =====
% ===== A9-SM v11.108 (2026-06-27): added a personal-perspective Conclusions section
% ('fifty years on the experimental front line') before the philosophical summit section:
% SM completion + W/Z/top/Higgs; the GUT/SUSY/string drought; the real breakthroughs
% (neutrinos/CMB/dark energy/GW) from outside; author's career (Kamiokande, K/B CP, CMS,
% Auger, XENON); 2012 departure to neuroscience -> U(1)_Qi/PLT/Levy/1f seeds of GG-Theory.
% First-person, honest (calibrated to 'falsifiable, not yet confirmed'). No physics changed. =====
% ===== A9-SM v11.107 (2026-06-27): added closing Sec. 'Summary of the Discussion' at the end
% of Part XI -- recaps significance/uniqueness/comparison/falsifiers/timeline/open-questions/
% completed-cohort, and bridges smoothly into the Conclusions Part. Added unique section
% labels to the Significance (67) and Comparison (69) sections. No physics changed. =====
% ===== A9-SM v11.106 (2026-06-27): Table 44 (predictivity engines) -> \footnotesize +
% tabcolsep 5pt to match Tables 45/46; caption enriched (ties the engines to the zero-
% continuous-parameter result vs MSSM/landscape); added two engine rows (GU-5 constants;
% four-force uniqueness theorem). 6->8 rows. No physics changed. =====
% ===== A9-SM v11.105 (2026-06-27): Table 46 (architectural comparison) reduced to
% \footnotesize and expanded 6->11 rows (continuous free parameters, Higgs nature, dark
% matter, cosmological inventory, predictivity, status-vs-data) sharpening GG vs MSSM vs
% superstring; enriched caption; added two paragraphs ('Why the contrast with the MSSM/
% superstring is decisive') with quantitative parameter-count + LHC/landscape contrast. =====
% ===== A9-SM v11.104 (2026-06-27): added 12 formal experiment bibitems (Super-K p-decay +
% n-nbar, Hyper-K, MICROSCOPE, ADMX, DESI, NuFIT-6.0, KamLAND-Zen, LEGEND, KATRIN, CMS dijet,
% Planck) with \citep wiring in Sec.71; new Sec.71.8 'Falsification timeline: within reach by
% 2030 and 2040' + timeline table; P-series forecast note. No physics changed. =====
% ===== A9-SM v11.103 (2026-06-27): Part XI Discussions rebuilt. Sec.70.1 (naming
% convention) relocated to Technical Appendices. Predictions+falsification rewritten
% quantitative (F-SM-1..5 + intrinsic precision table; proton DeltaB selection rule +
% confrontation table; fifth-force Stueckelberg window). Open questions updated
% (cosmology/N=3 now closed downstream). Future prospects rewritten: A10-Cosmos
% foregrounded as the key forward link + full forward map to the six P/C papers.
% Experimental status refreshed to 2025-2026. No physics changed; presentation only. =====
% Bib cross-citation titles reconciled to canonical cover titles (cross-citation integrity), 2026-06-27.
% A10 bib-title reconciled to canonical cover title (cross-citation integrity), 2026-06-27.
% ===== A9-SM v11.100 (2026-06-27): bib reconciliation -- de-contaminate GG-self cohort-citation
% the GG-self cohort-citation titles (A2/A10) to drop the contaminated qualifier, matching the
% de-contaminated cohort. External bibitems untouched. No physics/body change. =====
% ===== A9-SM v11.99 (2026-06-26): Phase C -- Common Objections Part. =====
% (1) Reviewed/revised existing O-SM: O-SM-2 (+global trials figure as decisive test), O-SM-3
% (lead with P2 three-leg sandwich; trace-ratio demoted to consistency check), O-SM-20 (proton
% absolute stability now stated conditional on the (H2) descent step). Fixed stale count
% 'thirty'->'seventy-four'. (2) Added 20 new hostile objections O-SM-55..74 from the Phase-A
% audit (scheme/sin2W, pull tautology, pi-e mixing, bulk-construction, landscape, swampland,
% N=3 circularity, QD layer, blind predictions, flavour, K-hierarchy, M_Pl, S/T/U, anomaly scope,
% abstract-vs-triage, conditional lemmas, axiom-shield, y_t, Eddington/global-trials, single-input),
% answered honestly from the GG point of view. No physics/values/triage changed.
% ===== A9-SM v11.98 (2026-06-26): Phase B Wave 2b/B1 -- foreground the axiomatic spine. =====
% Added one concise early subsection 'The deductive structure, in three stages' (Axioms ->
% Theorems -> Application), pointing to the existing axiom flowchart, the provenance table
% (Part II), and the 6/14/8 triage. Disarms the 'where is the construction?' reflex without
% duplicating A9's existing axiom infrastructure. Completes Phase B Wave 2. No physics change.
% ===== A9-SM v11.97 (2026-06-26): Phase B Wave 2b honesty fixes (science frozen). =====
% B10: G_N as the single dimensionful input (c,hbar,k_B unit conventions; Q_B a ratio); full M_Pl
% convention stated; hierarchy carried by integer K, not tuning (audit C-7). B12: Innovation 8 +
% Problem 10 proton claims now carry the conditional-on-(H2) caveat matching the theorem/appendix
% (M-1). B15: BI-6 used as trace-ratio constraint; full 6D anomaly polynomial = A3 existence
% construction (M-3). B6 already satisfied (P2 sandwich; N=3 'Conditional derivation'). No value change.
% ===== A9-SM v11.96 (2026-06-26): Phase B Wave 2b/B11 -- reduce the ATP/Q_B argument. =====
% Removed the 'Q_B = ATP-hydrolysis energy' numerology (audit C-10); KEPT Q_B as a GU-5 constant,
% the Arisaka equation, and eq:QB-chem (unit conversion, ATP-match framing stripped). Universal-
% identity ATP/DNA illustrations kept. 'bioenergetic'->'energy--information'. Full ATP treatment
% stays in A11-Origin. No value/result/triage change.
% ===== A9-SM v11.95 (2026-06-26): removed the now-redundant 'Throughout the cohort...' footnote
% on the 'single input' line (author request; the de-contaminated body no longer needs the defense). =====
% ===== A9-SM v11.94 (2026-06-26): Phase B Wave 2a -- terminology de-contamination. =====
% Dropped string/AdS framing from GG's own description: 'warped (6D) bulk'->'(6D) bulk'; warp
% factor/depth/constant + angle--warp theorem -> scale profile/depth/constant + angle--scale;
% GG's own 'holographic ...' -> bulk--boundary / renormalization-group. KEPT external defs (RS,
% AdS/CFT glossary, Strings/branes contrast) + QiHAL 'Holographic Attractor Lattice'. No physics.
% ===== A9-SM v11.93 (2026-06-26): Phase B Wave 1 -- honesty calibration (science frozen). =====
% Additive, 6/14/8-preserving: (B4) exec-summary strong claim calibrated with the ontological-vs-
% computational distinction + explicit 6/14/8; (B7) marquee caption gains scheme/scale labels and
% the honest sin2W observable (effective leptonic 0.23155 vs MSbar 0.23122 = +6.5sigma, in the open);
% (B8) caption states sigma_th meaning + experimental-only significance (post-dictions); (B9) y_t
% 'at the matching scale' + removed the unbacked g-2/Lamb-shift precision claim. No value/result/count changed.
% ===== A9-SM v11.92 (2026-06-26): de-duplicate the two Gi-4 chain figures. =====
% Kept the vertical sectoral descent ladder (fig:sectoral-descent-ladder) as the single
% canonical figure, restyled (smaller font+boxes, 8mm inter-box spacing, per-step colour,
% owning-theorem side tags ported from the deleted horizontal Fig.7 fig:gi4-chain; caption
% gains the provenance note). Deleted fig:gi4-chain from Part III (its boxed chain-equation
% + 'Reading the arrows' prose already carry it) and left a forward pointer to the ladder.
% Neither figure was cross-referenced. No physics / parameters / triage changed.
% ===== A9-SM v11.91 (2026-06-26): split Part I into two Parts (author-requested). =====
% New Part I = Introduction (Secs 1-3); new Part II = 'The Standard Model and its parameter
% audit' (Secs 4-8, label part:SMtarget). 14 -> 15 Parts: 7 Part-count 'fourteen'->'fifteen'
% (triage 6/14/8 untouched), roadmap prose + table row split, architecture flowchart gains a
% Setup node and the four block sub-ranges bumped. \GGPart auto-renumbers; \ref{part:*} all
% re-resolve. No physics, no parameter values, no triage counts changed.
% ===== A9-SM v11.90 (2026-06-25) =====
% Phase 6 (author-approved Candidate 1): title-page subtitle recast from 'Derivation of the 28 Free
% Parameters from the Six-Dimensional Geometry with no Free Dials' to 'Its 28 Parameters as
% Observations of One Six-Dimensional Geometry'; pdftitle metadata updated to match. Main title
% unchanged. No physics changed.
% ===== A9-SM v11.89 (2026-06-25) =====
% Phases 3-5 of the central-claim foregrounding (author-approved). Phase 3: Executive Summary 'The
% central recognition' paragraph (open-system/projection/observations + historical claim) + page-4
% hook tagline ('not free inputs, but observables of one open-system projection'). Phase 4: reframed
% the Part II GDOS section intro + the 'not a philosophical appendix' line, the Gi-4 chain intro, and
% Section 4 from 'apparatus' to 'cash-out of the central claim', cross-linking sec:central-claim /
% fig:central-claim. Phase 5: Conclusions call-back paragraph ('the deepest result is the shift in
% what a parameter is'). No new physics; no codebook value changed.
% ===== A9-SM v11.88 (2026-06-25) =====
% Phase 2 of the central-claim foregrounding: recast abstract paragraph 2 into the open-system /
% projection / 'observations-not-inputs' thesis in the accessible register, keeping the 'one 6D
% geometry, fixed once, no dials / list of theorems' content and adding the century-and-fifty-year
% historical claim. Under the arXiv 1920-char ceiling. No physics changed.
% ===== A9-SM v11.87 (2026-06-25) =====
% Phase 1 of the central-claim foregrounding (author-approved): inserted a new first subsection of
% Section 1, 'The central claim: the twenty-eight parameters are open-system observables'
% (sec:central-claim). Five beats -- open system / OGGEP projection = observation / GDOS law / 28
% registered observables (no exceptions; ontology vs proof-status safeguard) / GDOS forced by Gi-4 +
% Qi-4 -- plus a dedicated TikZ funnel figure (fig:central-claim) and the century/fifty-year
% historical claim. Foregrounds logic already in A1 (OGGEP/AX-3), A5 (GDOS), A6 (Gi-4), A7 (Qi-4),
% A8 (QM). No new physics; no codebook value changed.
% ===== A9-SM v11.86 (2026-06-25) =====
% Tier B (author-approved): de-duplicated the two Part III GDOS sections by giving them distinct,
% cross-linked roles (GDOS explained once in Part II sec:gdos-qi4-recap, applied once in
% sec:apply-gdos-su5, seams defined once in sec:gdos-eegg). (1)+(2) retitled both sections; (3)
% compressed the triple 'GDOS is not a philosophical gloss' motivation in the application-section
% intro to a concept->apply->seams cross-link; (4) reciprocal back-link in the seams-section intro;
% (5) collapsed the duplicated GEP/CEP gloss in 'The GFF ledger' to a pointer to the canonical
% sec:gdos-qi4-recap (kept the unique explicit functional + Lyapunov). No section relocated, no
% equation moved/removed; no physics or codebook value changed.
% ===== A9-SM v11.85 (2026-06-25) =====
% Tier A redundancy cleanup (author-approved): compress duplicate prefaces in Parts II-IV to pointers,
% one canonical home per concept. (1) added \label{sec:admissible-inputs} to Section 4; (2) Part II
% admissible-ledger -> pointer (kept the allowed/forbidden table); (3) Part IV admissible-inputs ->
% pointer (kept AX-4/POS-6 + roadmap framing); (4) Part III N=3 preview -> forward pointer to
% sec:N3-theorem; (5) Part III sec:gdos-eegg duplicate 'why GDOS before numbers' -> pointer to
% sec:gdos-qi4-recap; (6) Part III Wilson-Hosotani-as-GSSB equations -> pointer to sec:wilson-hosotani
% (+ forward pointer in the orbifold subsection). No externally-referenced equation removed; canonical
% N=3 / Wilson-Hosotani / GDOS sections untouched. No physics or codebook value changed.
% ===== A9-SM v11.84 (2026-06-25) =====
% Author-requested: Section 4 ('Admissible inputs, forbidden shortcuts, and the standard of
% derivation') now refers to A4-Euclid, the foundations-and-audit volume that proves the parameter-
% free reconstruction (GU-5 / OGN-5). Added three \AFour{} \citep{ArisakaA4Euclid} references: (A)
% after the two-pentad equation (completeness proved in A4 via the two-pentad audit + T-Euclid-1..7);
% (B) GU-5 paragraph (T-Euclid-3 closure at five, T-Euclid-4 charge-is-derived); (C) standard-of-
% derivation paragraph (T-Euclid-5 'no third species' closure + the MSMF/AX-4 clause). No physics
% changed; A4 already in the bibliography.
% ===== A9-SM v11.83 (2026-06-25) =====
% Author-requested: moved Figure 1 (fig:cohort-map-a9, the 17-paper cohort map) from page 29 to page
% 28 by relocating the full-page figure block to BEFORE its introducing paragraph ('The full cohort
% in one view'); the walkthrough prose now follows on the next page. Trimmed the long caption (it
% nearly duplicated that walkthrough) to a concise version naming the strata, branches, and arrow
% legend. A single \clearpage left in place so the Roadmap section still starts fresh. tikzpicture
% untouched. No physics changed.
% ===== A9-SM v11.82 (2026-06-25) =====
% Textbooks (author-requested): (1) re-cast the Part-XII 'Prerequisites and recommended textbooks'
% note as a bulleted list, each item citing the textbook with \citet; (2) added eight textbook
% \bibitem entries to Group B at their correct alphabetical positions in house style (bold author +
% Relevance + 'First cited in'): Cheng-Li, Griffiths, Halzen-Martin, Langacker, Peskin-Schroeder,
% Schwartz, Thomson, Weinberg QTF; (3) cited the best-matched textbook in each course appendix
% preamble (gauge->Peskin/Cheng-Li, EWSB->Schwartz/Weinberg, fermion->Griffiths, CKM->Thomson,
% PMNS->Langacker, force/proton->Halzen-Martin, strong-CP->Peskin). No physics or codebook changed.
% ===== A9-SM v11.81 (2026-06-25) =====
% Course enrichment (author-requested): across the eight three-step course appendices, added (1)
% connective sentences for smooth reading, (2) internal links to the A9 main text + cohort papers
% (A1/A2/A3/A5/A6, P3, C1), (3) inline citations to key original papers (Gross/Wilczek/Politzer,
% Georgi-Glashow, Hosotani/Wilson, ATLAS/CMS, Cabibbo, Kobayashi-Maskawa, Pontecorvo/Fukuda/Ahmad,
% Maki-Nakagawa-Sakata/Pauli/Majorana, Peccei-Quinn/Weinberg/Wilczek, Adler/Bell-Jackiw/Bouchiat-
% Iliopoulos-Meyer, Wilson lattice), and (4) a recommended-textbooks note (Peskin-Schroeder, Schwartz,
% Griffiths, Thomson, Weinberg, Cheng-Li, Langacker, Halzen-Martin) in the Part-XII framing. No new
% bibitem; all cited keys already exist in group B. No physics or codebook value changed.
% ===== A9-SM v11.80 (2026-06-25) =====
% Topic 7 of the appendix graduate course: FORCE LAWS, ANOMALY CANCELLATION & PROTON STABILITY (App
% G). Retitled to 'Force laws, anomaly cancellation, and proton stability' and PREPENDED a three-step
% block: Step 1 (inverse-square via Gauss/Green's, anomaly cancellation family-by-family + charge
% quantisation, GUT proton decay p->e+ pi0 + Super-K bound, worked example); Step 2 (3D unexplained;
% anomaly imposed not derived; B,L only accidental so proton unprotected); Step 3 (boundary Green's
% functions; anomaly freedom inherited from BI-6/A3; gauged local U(1)_B => proton absolutely stable).
% Existing Green's-function and U(1)_B subsections kept as detail. No physics or codebook value changed.
% ===== A9-SM v11.79 (2026-06-25) =====
% Topics 6 & 8 of the appendix graduate course: STRONG-CP and CONFINEMENT (both in App H). Retitled
% the appendix to 'The strong-CP problem, confinement, and the mass-energy dictionary' and added two
% three-step course blocks: (6) strong-CP Step 1 (theta-bar, G Gtilde, neutron-EDM bound, axion),
% Step 2 (28th free param, axion unobserved), Step 3 (protected topological readout: theta_top=0 +
% determinant-phase cancellation -> theta-bar^(1)=0; residue sin(2pi/L)2^-J e^-K ~ 4.7e-18; honesty
% L9), before the existing reduction subsection; (8) confinement Step 1 (asymptotic freedom,
% Lambda_QCD, proton mass = binding energy), Step 2 (confinement not proven; Lambda_QCD scale), Step 3
% (confinement as warp window), before the existing RG/confinement subsection. No physics changed.
% ===== A9-SM v11.78 (2026-06-25) =====
% Topic 5 of the appendix graduate course: NEUTRINO MASSES & PMNS MIXING. New dedicated three-step
% course appendix after the CKM appendix, before App F. Step 1 textbook (oscillations/two splittings,
% PMNS matrix, Dirac/Majorana/seesaw, large angles, worked example); Step 2 limitations (absolute
% scale/ordering/nature, large-vs-small mixing, free Lambda); Step 3 two-layer geometric (Majorana
% (LH)(LH) reusing E_GUT -> m_*=pi v^2/E_GUT, rank-two => two splittings + normal ordering predicted;
% large angles from dyadic-vs-democratic misalignment via (D,S,A) basis, closed forms 25/81, 4/7,
% 3/140; CP/Majorana phases from compact-angle logic; x-ref main-text sec:neutrino-masses/sec:PMNS +
% App E neutrino arithmetic; honesty). Deepens main text. No physics or codebook value changed.
% ===== A9-SM v11.77 (2026-06-25) =====
% Topic 4 of the appendix graduate course: QUARK MIXING -- CKM & CP. New dedicated three-step course
% appendix after the fermion-mass appendix, before App F. Step 1 textbook CKM (V=U_uL^dag U_dL, 3
% angles + 1 phase, Wolfenstein, Cabibbo, Jarlskog, worked example); Step 2 limitations (4 inputs,
% hierarchy, CP origin/size); Step 3 two-layer geometric (projector misalignment; Cabibbo from
% pentagonal holonomy e^-pi/e^-pi/2 + phase 3pi/5 + golden ratio -> 0.22502; 2<->3 multiplicative;
% CP as topological-projection remainder forced by the single holonomy circle; x-ref App E CKM
% arithmetic + main-text sec:CKM; honesty re smaller angles). Deepens sec:CKM. No physics changed.
% ===== A9-SM v11.76 (2026-06-25) =====
% Topic 3 of the appendix graduate course: FERMION MASSES & THE YUKAWA HIERARCHY. Reframed the
% 'Fermion masses, flavor codebooks, and the 28-parameter census' appendix in place: retitled to
% 'Fermion masses and the Yukawa hierarchy' and PREPENDED Step 1 (textbook Yukawa, m_f=y_f v/sqrt2,
% 12-order hierarchy, Dirac/Majorana seesaw, worked y_t~1), Step 2 (limitations: largest free block,
% hierarchy, family number, neutrino lightness), Step 3 two-layer geometric (cascade tolls e^{-n pi}
% on the CP^2 family stratum; y_t=1 zero-toll anchor = Derived prediction; dyadic charged-lepton
% codes; geometric seesaw m_*=pi v^2/E_GUT; explicit honesty that the per-fermion cascade-depth
% lemma is open so 8 Yukawas are Asserted). Existing codebook arithmetic kept as Layer B. No physics
% or codebook value changed.
% ===== A9-SM v11.75 (2026-06-25) =====
% Topic 2 of the appendix graduate course: ELECTROWEAK SYMMETRY BREAKING & THE HIGGS. New dedicated
% three-step course appendix inserted after the gauge-sector appendix (App C), before App D. Step 1
% textbook EWSB (Higgs mechanism, Mexican hat, vev, eaten Goldstones, m_W/m_Z/m_h, custodial rho=1,
% worked example v from G_F); Step 2 limitations (v,lambda inputs; hierarchy/naturalness; vacuum
% metastability); Step 3 two-layer geometric (Wilson-Hosotani holonomy order parameter; lock scale
% Mlock=1/2 MPl e^-K; branch angle (2/L)^N; v=246.33; quartic lambda_H=2^-N=1/8; m_h=v/2; explicit
% arithmetic x-ref to App C; honesty triage; hierarchy-problem-dissolved intuition). Deepens
% T-SM-30.1/31.1. No existing equation moved; no physics or codebook value changed.
% ===== A9-SM v11.74 (2026-06-25) =====
% Topic 1, Step 3 expanded (author-requested: more rigorous/extensive, double the intuition;
% overlap with main text OK). Inserted a full geometric-derivation block in Appendix C after the
% Step-3 framing: the OGN-5 ledger and why J=7 is forced; dyadic closure alpha^-1=2^7=128 with the
% charge readout and the marquee alpha^-1(0)=2^J+N^2=137; the weak-angle rational partition
% 25/108=L^2/(4N^3) via the trace norm 5/6 + quadratic-partition + colour-blind averaging lemmas;
% the strong angle-warp alpha_s=1/(pi e) via the half-circle pi and one-efold e primitives; the
% derived couplings g2,gY; and a unifying 'integer/rational/transcendental' picture. Reproduces and
% deepens T-SM-27.1/28.1/29.1. No existing equation moved; no physics or codebook value changed.
% ===== A9-SM v11.73 (2026-06-25) =====
% Topic 1 of the appendix graduate course: THE GAUGE SECTOR. Reframed Appendix C 'Scales and
% electroweak arithmetic' into the first three-step course appendix. Retitled to 'The gauge sector:
% couplings, the weak angle, and unification' and PREPENDED Step 1 (textbook gauge theory: group +
% three couplings, EW mixing + weak angle, running + asymptotic freedom, SU(5) unification, worked
% example), Step 2 (the three couplings as inputs; GUT narrows-not-closes; open group/chirality
% questions), and a Step 3 framing subsection that turns the existing scale-chain + dyadic-alpha +
% rational-weak-angle + alpha_s arithmetic into the geometric derivation. No existing equation moved
% or changed. Physics-intuition notes throughout. No physics changed; no codebook value changed.
% ===== A9-SM v11.72 (2026-06-25) =====
% Pass 1 of the appendix graduate-course rebuild (author-approved). (1) Removed App I 'Scope, audit,
% and remaining tasks' and App J 'Reference guide and relevance' (thin/redundant; no body refs to
% their labels). (2) Retitled the Part XII subtitle to the graduate-course framing. (3) Rewrote the
% Part-opening framing paragraph to describe the three-step course over the eight SM sectors, the
% reference group, the promotion roadmap, and the referee appendices. Two stylistic commitments kept.
% No physics changed; no codebook value changed.
% ===== A9-SM v11.71 (2026-06-25) =====
% Fix: moved A1's canonical Figure 9 (fig:abbrev-flow, the programme-acronym hierarchy) from the end
% of the promotion-roadmap appendix into Appendix A 'Summary of Abbreviations' (before the Glossary),
% so its placement now matches A1 (its Figure 9) and A10. No physics changed.
% ===== A9-SM v11.70 (2026-06-25) =====
% Author-requested: replaced A9's older acronym-hierarchy figure (fig:abbrev-hierarchy, unreferenced)
% with A1-Constitution's canonical Figure 9 (fig:abbrev-flow), copied verbatim from the A1 source, so
% the programme-acronym hierarchy is identical across the cohort. Front cover (Part XII Technical
% Appendices; App A Abbreviations, App B Glossary) already matches A1/A10. No physics changed.
% ===== A9-SM v11.69 (2026-06-25) =====
% Author-approved: harmonised S62.5 (subsec:weak-grip) with the new S42 (sec:weak-special).
% (1) opening positions S62.5 as the 'why it exists/realized in Nature' complement to S42's gate
% reading; (2) uniqueness-observation cross-ref to S42's basin/gate table; (3) chiral postscript now
% names the gate-operator lemma L11 (status harmonised); (4) existential answer cites the S42.3
% weak<->three-families co-entailment; (5) reciprocal pointer in S42.5 to S62.5. No physics changed.
% ===== A9-SM v11.68 (2026-06-25) =====
% Author-approved Option 1: dissolved Part VIII 'Interpretation and scope'. Redistributed its
% worthwhile content (compressed) into the Discussions -- the SU(5) + gauge-Higgs comparisons
% into the Comparison section, the closed/open scope + flavour roadmap + the frame/key-signature
% metaphors into 'Scope and limits' -- dropped the redundant runtime recap (S1), removed the Part
% shell, and reconciled the Part count 15 -> 14 (roadmap section/prose, architecture figure,
% Table 16, reading guide). No dangling part:IX refs. No codebook value changed.
% ===== A9-SM v11.67 (2026-06-25) =====
% Author-requested: echoed the weak-interaction-special result (S42) in the Discussions
% (Significance) and Conclusions (fourth structural result), and registered the Gate-Operator
% Lemma as L11 in the promotion roadmap -- clearly marked STRUCTURAL (closes no codebook
% parameter): intro sentence, caption note, table row, unpacking subsection, summary sentence.
% L1-L10 parameter-closure arithmetic untouched. No codebook value changed.
% ===== A9-SM v11.66 (2026-06-25) =====
% Author-requested NEW Section 'Why is the weak interaction special?' (sec:weak-special), inserted
% before 'The quark floors' so it becomes Section 42 (rest shift down). Develops the basin-vs-gate
% argument + four-force comparison table; the weak<->three-families co-entailment (with the N=3
% toll-budget closure); the neutrino ledger; the cosmic relaxation role; and the honest
% Gate-Operator Lemma. Builds on established machinery (GRIP/GDOS/Morse/N=3/four-force-uniqueness);
% no codebook value changed.
% ===== A9-SM v11.65 (2026-06-25) =====
% Author-requested: brought P3-Electron's charged-lepton landscape figure into S41.3 (right after
% Fig 12 fig:grip-cascade-cp2). Added a bridging paragraph carrying the key P3 Part-IV concepts
% (Gamow gate-and-toll reading; identity/stability/excited-copies/decay-hierarchy in one picture)
% + the author's own figure (relabelled fig:lepton-basin-landscape, reproduced from P3-Electron).
% All macros already available in A9. No physics changed.
% ===== A9-SM v11.64 (2026-06-25) =====
% Author-requested: dropped the redundant Summary section 'The ten classic open problems at a
% glance' (sec:ten-problems-scorecard-sec) + its Table (tab:ten-problems-scorecard), superseded by
% the comprehensive Discussions table tab:ten-problems-resolution. Repointed all six references to
% the comprehensive table. No physics changed.
% ===== A9-SM v11.63 (2026-06-25) =====
% Author-requested visual top-10 tables: (1) replaced the §1.2 cluster-only table with a 10-row
% enumeration (cluster sub-headers + one-line open question per problem); (2) added a comprehensive
% table at the head of §72 (No | Problem | standing challenge | GG-Theory solution+key result |
% status) before the ten prose paragraphs. No physics changed.
% ===== A9-SM v11.62 (2026-06-25) =====
% Author-approved: one canonical reader-facing resolution of the ten classic problems.
% (1) New Discussions section sec:ten-problems-resolved (10 paragraphs, canonical S1.2 order,
%     mechanism+result+honest status+cross-refs to deep sections / O-ledger / cohort).
% (2) New Summary scorecard tab:ten-problems-scorecard (10 rows, canonical order).
% (3) Part VII sec:open-problems: dropped the redundant mis-numbered textbook-ten table
%     (tab:ten-problems-main) + L6 tally + 'what-solved'/'what-open' prose; reframed intro to point
%     to (1)+(2); KEPT the O1-O21 structural ledger (subsec:open-problems-L6) + deep sections.
% (4) Repointed S1.2 forward+closing pointers to (1)+(2). No physics changed.
% ===== A9-SM v11.60 (2026-06-25) =====
% Author-requested: added a 5th category 'Appendices & references' (Parts XIII-XV) to Figure 2.
% New grey backmatter block for p16/p17/p18; Closure now VIII-XII (Conclusions stays); added
% Conclusions->Technical-Appendices flow arrow; split fit node + new block label. Reconciled the
% block count four -> five across the roadmap paragraph, the figure caption, and Table 16 (lead-in,
% caption, + \addlinespace before the new block's rows). No physics changed.
% ===== A9-SM v11.59 (2026-06-25) =====
% Figure 2 fix: replaced the flowchart block style's 'minimum width=7cm' (which CENTERED short
% single-line rows, so Parts IV/VIII/XI looked shifted right) with a fixed 'text width=11.3cm,
% align=left' sized to the widest row (Part III) so every box is uniform and the 'Part N.' labels
% sit flush-left; the 1.7cm makebox keeps the descriptions column-aligned. No physics changed.
% ===== A9-SM v11.58 (2026-06-25) =====
% Figure 2 (fig:org-flowchart) refinements (author-requested): (1) pulled the flowchart up onto the
% preceding (mostly empty) page by dropping the forced \clearpage + \null\vfill centering; (2) put
% each 'Part N.' label in a fixed-width 1.7cm left \makebox so the descriptions align in a column.
% No physics changed.
% ===== A9-SM v11.57 (2026-06-25) =====
% Author-requested: added a 1.1 internal-consistency-triumphs paragraph (anomaly cancellation
% family-by-family / quark-lepton charge locking, and the GIM mechanism predicting charm) to bring
% Section 1 to ~6 pages; 4 new Group-B refs (Adler1969, BellJackiw1969, BouchiatIliopoulosMeyer1972,
% GlashowIliopoulosMaiani1970), alphabetical, with relevance notes. No physics changed.
% ===== A9-SM v11.56 (2026-06-25) =====
% Author-requested: (1) added a 1.1 'prediction confirmed' precision-triumphs paragraph (W/Z, top,
% Higgs predicted-then-found); (2) added a short 1.2->1.3 bridging paragraph; (3) wired inline
% \citep citations throughout Section 1 using existing keys + 17 NEW Group-B references (renormaliz.,
% W/Z/top/Higgs discoveries, neutrino-oscillation discoveries, Sakharov, axion, dark-energy SNe,
% cosmological-constant review, dark-matter, PDG), inserted alphabetically with relevance notes.
% No physics changed.
% ===== A9-SM v11.55 (2026-06-25) =====
% Author-requested: dramatically expanded Section 1 Introduction (~2pp -> ~6pp).
% 1.1 enriched (mid-1970s historical arc + the '28 dials it cannot explain' pivot).
% 1.2 retitled; ten classic open problems each given a solid intro paragraph (introduced, NOT
%     solved -- solutions deferred to Discussions/Conclusions per author).
% 1.3 retitled; explains the 19->26->28 counting (why 28, not 19 or 26); equation kept.
% Expository only; consistent with frozen 28-parameter set + 6/14/8 triage. No physics changed.
% ===== A9-SM v11.54 (2026-06-25) =====
% Tuned in-body Dev-plot fonts to MATCH the page-4 hook's apparent size: the in-body figures
% display at full \textwidth (451pt) vs the hook's 427pt, so baked fonts were scaled 18->17
% (~0.945) in a9_dev_plots_gen.py and both PNGs regenerated. No .tex layout change; no physics.
% ===== A9-SM v11.53 (2026-06-25) =====
% Author-requested: enlarged all fonts in the in-body Dev plots to match the page-4 hook --
% regenerated a9_dev_plot_linear_first8.png (Figs 4 & 14) and a9_dev_plot_symlog.png (Figs 5 & 15)
% via a9_dev_plots_gen.py with bumped label/tick/legend/title fonts and markers; shortened the
% linear-plot internal title to fit. No .tex content change beyond this stamp; no physics changed.
% ===== A9-SM v11.52 (2026-06-25) =====
% Page-4 hook title set to the A10-Cosmos 'Pi-locked' font: \sffamily\huge\bfseries (sans-serif
% huge bold); subtitle harmonised to \sffamily. No physics changed.
% ===== A9-SM v11.51 (2026-06-25) =====
% Page-4 hook polish (author-requested): (1) swapped to the dedicated large-font image
% a9_dev_plot_linear_first8_hook.png (in-body Figs 4/14 unchanged), height 0.52->0.46\textheight;
% (2) added a big bold page title '28 Free Parameters Derived by GG-Theory' + 13-preview subtitle.
% No physics changed.
% ===== A9-SM v11.50 (2026-06-25) =====
% Author-requested (A10-Cosmos Figure-1 pattern):
% (1) Added a full-page caption-free HOOK combining Table 8 (indented 6-col preview tabular) and
%     Figure 4 (linear Dev plot) on one page, placed right after the Executive Summary/Keywords so
%     it lands ~page 4. Label-less/caption-less copy; the numbered Table 8 + Figure 4 stay in Part I.
% (2) Relocated \GGPreface to AFTER the Table of Contents (just before Part I), ~page 18, matching
%     A10-Cosmos. No physics changed.
% ===== A9-SM v11.49 (2026-06-25) =====
% Author-requested: indented the 13 parameter rows of Table 8 (tab:preview-28-parameters) with a
% 1.5em first-cell hspace so the four category headers hang flush-left and are easy to read.
% No physics changed.
% ===== A9-SM v11.48 (2026-06-25) =====
% Author-requested: added Dev + Pull columns to Table 8 (tab:preview-28-parameters), kept at
% \small (not \scriptsize). Dev/Pull are FIRST-ORDER (matching the table's 'First order' column):
% the 9 codebook rows verbatim from tab:first28-summary-v08, the 4 derived EW observables computed
% from Figure 4's data sheet with identical definitions. Numeric cols right-aligned; tabcolsep 4pt.
% Caption extended to define Dev/Pull and point to the QD Dev/Pull in Table 40. No physics changed.
% ===== A9-SM v11.47 (2026-06-24) =====
% Phase 5 final reconcile (smoothness + self-consistency):
% A1/A2 roadmap walkthrough: consolidated duplicated Part III, fixed II->III->IV order.
% A3 collapsed the 'Parts V--V / Part V x4' gauge sentence into one Part w/ internal sections;
%    moved the Interpretation (prints VIII) sentence to printed position (after strong-CP VII,
%    before philosophical IX); fermion (VI) now follows gauge (V).
% A4 descent-ladder caption range 'Parts III--VI' -> 'Parts V--VII'.
% B5 rebuilt the stale Part-IV roadmap table (8-Part -> true 15-Part) + new lead-in/caption as the
%    textual companion to fig:org-flowchart.
% B6 fixed the Part-IV admissible-inputs box {c,h,G_N} -> canonical GUfive {c,hbar,G_N,k_B,Q_B}.
% No physics changed.
% ===== A9-SM v11.46 (2026-06-24) =====
% Author-requested: expanded the Part-I preview teaser (subsec:preview-28-parameters) from a
% 6-row unnumbered tabular into a NUMBERED table float (tab:preview-28-parameters) with all 13
% benchmarked quantities plotted in Figure 4 (fig:dev-linear-first8), grouped as in that figure
% (5 attractor coords + 4 derived EW observables + 3 lepton masses + Cabibbo). Values mirror the
% figure's own data sheet (a9_dev_plots_gen.py FIG3) exactly. Lead-in updated (6->13) and the
% downstream figure paragraph cross-links the new table. No physics changed.
% ===== A9-SM v11.45 (2026-06-24) =====
% Phase 4c: removed the duplicate Appendix master census (landscape longtable
% tab:full28-census, subsection app:full28) -- the 5th copy of the 28-parameter values.
% Replaced the whole subsection with a short pointer to the canonical Summary and audit
% Part (first-order + QD-completed census tables + MSMF/OGN per-parameter audit). No external
% ref pointed at tab:full28-census or app:full28 (verified). No physics changed.
% ===== A9-SM v11.44 (2026-06-24) =====
% Phase 4a: retitled the Summary Part 'Summary and audit: the completed 28-parameter codebook'
% (the Part-I marquee teaser already forward-references it by that name) and added a Part-level
% framing intro naming it as the single consolidated home of the full census + MSMF/OGN
% provenance audit. Canonical census tables (first-order + QD) already live here; no tables
% moved. (4c will remove the duplicate Appendix master census.) No physics changed.
% ===== A9-SM v11.43 (2026-06-24) =====
% Gave the Part-I Figures 4 & 5 (the §8.4 originals) the same trimmed caption format as the
% QD-section Figures 14 & 15, tweaked: dropped the '(reproduced from the preview)' note and
% repointed the internal cross-refs to the §8.4 labels (fig:dev-symlog / fig:dev-linear-first8).
% No physics changed.
% ===== A9-SM v11.42 (2026-06-24) =====
% QD-section repeated dev plots: figures kept FULL SIZE (not shrunk); the two long captions
% TRIMMED to one tight paragraph each so figure+caption fit on one page; \clearpage around the
% pair and after it so Fig 14 and Fig 15 each occupy their own page and the next section starts
% at the top of the following page. Part-I originals untouched. No physics changed.
% ===== A9-SM v11.41 (2026-06-24) =====
% Author-requested: repeated the two deviation plots (linear + symlog) in the QD-completed
% codebook section (the 28-parameter codebook at second order), using the long v11.39 captions.
% New labels fig:dev-linear-qd / fig:dev-symlog-qd; internal cross-refs repointed to each other
% (self-contained pair). Same image files as the Part-I preview. No physics changed.
% ===== A9-SM v11.40 (2026-06-24) =====
% Smoothed subsec:preview-28-parameters (author-requested): moved the at-a-glance status bar
% (fig:status-visualization) up to directly after the marquee teaser; replaced the two verbose
% dev-plot caption blocks with three explanatory main-text paragraphs; gave the linear and symlog
% plot images clean short captions; removed the old manual 'showcase' clearpage/vfill layout.
% Figure labels preserved; only internal cross-refs (all rewritten here). No physics changed.
% ===== A9-SM v11.39 (2026-06-24) =====
% Phase 4b (Audit consolidation, part 1): replaced the two full preview tables (Tables 8 & 9:
% complete first-order + QD summaries of all 28 parameters) in subsec:preview-28-parameters with
% a SHORT MARQUEE TEASER -- the six famous gauge-Higgs closed forms (137, 25/108, 1/pi e, 246 GeV,
% 1/8, y_t=1) -- pointing forward to the canonical census in the Summary/Audit. Showcase figures
% (dev plots + 6/14/8 bar) kept. The removed tables were referenced only by their own intro.
% Author-approved Option B. No physics/values changed.
% ===== A9-SM v11.38 (2026-06-24) =====
% Phase 3b-2: relocated the (now pure) Interpretation Part to AFTER the parameter Parts.
% Clean arc: Electroweak seam -> Fermion floors/gates -> Protected readout -> Interpretation
% -> closure. Architecture flowchart reordered (strong-CP into the Derivation block,
% Interpretation into Closure; TikZ anchor chain rebuilt p6->p10->p11->p9->p12); block-range
% refs updated (Derivation VI--XI, Closure IX--XVIII) and the two caption block descriptions
% reworded for the new membership. 15 Parts; label-driven refs auto-renumber. No physics changed.
% ===== A9-SM v11.37 (2026-06-24) =====
% Phase 3b-1 (regroup parameters by GRIP role, part 1): (A) moved the four electroweak
% prediction sections (first-order gauge/Higgs, QD completion, theory-error model) from the
% Interpretation Part INTO the Electroweak Seam, where the gauge/Higgs numbers belong;
% (B) reframed the Fermion Part with GRIP-role section headers (floors = the twelve masses,
% gates = CKM/PMNS) -- kept as ONE Part; (C) retitled the strong-CP Part 'The protected
% readout'. No Part moved, no renumbering (3b-2 moves Interpretation out). No physics changed.
% ===== A9-SM v11.36 (2026-06-24) =====
% CLEANUP (author-approved): removed the 14 'Closing of Part X' recap/preview paragraphs.
% They used space-separated hardcoded Part numbers the tilde-only labelization missed, so the
% Phase-2/3a merges left them stale and misordered (e.g. 'Closing of Part V' on the bulk was
% stranded inside the Substrate). They were also the redundant recap/preview paragraphs from
% the redundancy audit. Removed wholesale; the Summary/Conclusions tail already recaps.
% Residual space-separated Part refs are labelized in the follow-up. No physics changed.
% ===== A9-SM v11.35 (2026-06-24) =====
% TOC reorganization PHASE 3a: merged the three electroweak Parts (Gauge--Higgs structure +
% Primary theorem sheet + Descendant parameter sheet) into ONE 'The electroweak seam' Part,
% framed by GRIP role (the one GRIP-Out vacuum + its GRIP-In gauge-seam floors + descendants).
% 17 -> 15 Parts. Architecture flowchart reconciled (p6/p7/p8 -> one node; Derivation V--VII,
% Closure VIII--XV); 'Seventeen' -> 'Fifteen'; refs to part:VII/part:VIII repointed to the
% merged Part (part:VI). NOTE: the old 'Interpretation' Part still mixes interpretation with
% the electroweak first-order/QD prediction sections -- those move into this seam in Phase 3b.
% No physics/values/6-14-8 triage changed.
% ===== A9-SM v11.34 (2026-06-24) =====
% TOC reorganization PHASE 2b: MERGED the Substrate. Old Part V (Bulk geometry + runtime)
% folded into Part III -> 'The Substrate' (SU(5)->GG seam + bulk + ledger + N=3 + RG + W-H);
% old Part IV (Roadmap) kept as the bridge AFTER the Substrate. 18 -> 17 Parts. Architecture
% flowchart reconciled (p3/p5 merged, p5 removed, p6 re-anchored, Foundation brace II--IV);
% 'Eighteen Parts' -> 'Seventeen' in roadmap title/prose/caption; part:V refs repointed by
% context (foundation ranges II--V -> II--IV; spans V--X -> III--X; standalone -> III).
% Label-based refs auto-renumber the rest. No physics/values/6-14-8 triage changed.
% ===== A9-SM v11.33 (2026-06-24) =====
% TOC reorganization PHASE 2, sub-step A: LABELIZED the Parts (output-neutral enabler).
% Added \label{part:I..XVIII} after each \GGPart, and converted all 114 hardcoded
% 'Part(s)~ROMAN' mentions to \ref{part:..} form so the upcoming Substrate merge auto-
% renumbers. VERIFIED output-neutral: all 555 rendered Part-mentions identical to v11.32.
% No content moved yet; no physics/values/triage changed.
% ===== A9-SM v11.32 (2026-06-24) =====
% TOC reorganization PHASE 1 (front-load the GRIP engine; additive only, no Part renumber).
% In Part II's Gi-4 chain section (sec:tgg-cci-bridge): (1) added 5 figure macro defs to the
%   preamble (PBIfour,CPN,Stot,Sbi,cF, verbatim from A6); (2) added a GDOS/OGGEP 'why the
%   boundary' opening paragraph; (3) imported A6 Figure 6 (GRIP cascade DGI->GSSB->GA,
%   fig:grip-cascade-engine) after the spine table; (4) imported A6 Figure 7 (cascade + two
%   regimes, fig:grip-two-regimes) after the GRIP-In/Out prose, with a rewritten reference-safe
%   caption (A6-internal refs removed). Parameter blocks NOT moved (later phases). No physics
%   /values/6-14-8 triage changed.
% ===== A9-SM v11.31 (2026-06-24) =====
% Added one paragraph to the charged-lepton GRIP-cascade subsection (sec:charged-leptons):
%   condensed muon mass+lifetime reading -- lifetime = GRIPin descent rate across the GSSB
%   gate S=pi, implemented by the weak current with geometric coupling GF=1/(sqrt2 v^2);
%   Gamma_mu = GF^2 m_mu^5/192pi^3, tau_mu=2.197us (ppm); electron GGA = zero-gate limit
%   (tau=inf); mass/lifetime = static/dynamical faces of one basin; honest triage (GF
%   Derived-cond, m_mu Asserted -> tau_mu derived modulo). Points to P3-Electron proposition.
%   New content only; no values or 6/14/8 triage changed.
% ===== A9-SM v11.30 (2026-06-24) =====
% GRIP-CENTRIC REFACTOR (author-approved). Every parameter Part is now framed as ONE GRIP
%   cascade. The repeated BI6->TGG->CCI->GFF->GRIP prefaces of all 9 packets (gauge, Higgs,
%   fermion-parent, charged-lepton, quark, neutrino, CKM, PMNS, topological) are each
%   compressed to a single '<sector> GRIP cascade' subsection that PRESERVES every
%   sector-specific physics fact (stratum, alphabet, what BI6 forbids, normalization, the
%   electroweak GRIPout, ideal-seam vs CCI/QD transport). Added an organizing-spine subsection
%   subsec:grip-spine + Part->stratum->GRIP-role table near sec:tgg-cci-bridge, emphasizing
%   GRIP's universality; rewrote the Part-X meta-sentence. Labels sec:charged-leptons,
%   sec:quark-masses, sec:neutrino-masses preserved. Presentation/framing only; no physics,
%   values, or 6/14/8 triage changed.
% ===== A9-SM v11.29 (2026-06-24) =====
% delta_PMNS PROMOTED to Derived (cond.), value 196 deg = pi + 2 arcsin(1/(2N+1)).
% Honesty triage 6/13/9 -> 6/14/8, quoted identically at all sites (intro, codebook
%   table+caption, at-a-glance figure, honesty list, MSMF/OGN audit table+caption+paragraph,
%   resolution table, promotion-roadmap appendix, four objection exchanges, Part XIII intro).
% Empirical anchor harmonised 230 deg -> 202 deg (paper's own NuFit precision-table value).
% Author-approved physics/status decision (2026-06-24). No other physics changed.
%% =====================================================================
%%  A9-SM v11.28 (migrated onto ucla-arisaka-gg-11) -- 2026-06-22
%%  Single canonical style.  Routed A9's inline Overview+LoF onto gg-11's
%%  centered cohort macros (single page via the toggle); stripped the
%%  centered ExecutiveSummary override; dropped dead \LaTeX shim; Preface
%%  via \GGPreface; inherits the one canonical Acknowledgments.  No
%%  prose/figure/number content changed.
%% =====================================================================
% ---------------------------------------------------------------------
%  Arisaka-GG-A9-SM-v11.26 (form refinement: local \T{n} -> global ids) -- 2026-06-20
%  Cross-paper \T{n} tags -> registered cohort global ids (A2/A5/A6
%  single-theorem targets); A7 pillars / A6 sub-theorems / A1 axioms
%  kept as local tags.  Identical to A11 v12.41.  No physics changed.
% ---------------------------------------------------------------------
% ---------------------------------------------------------------------
%  Arisaka-GG-A9-SM-v11.25 (Stage 3C cross-reference sweep) -- 2026-06-20
%  T-SM-21.1 title reconciled to P2-N3's actual result: 'Closed'->'Conditional'; cite P2 Theorem~6.1 -> T-N3-5 (registry id).
% ---------------------------------------------------------------------
%% =====================================================================
%%  A9-SM v11.25 (Q&A expansion to the >50 standard) -- 2026-06-19
%%  Added 24 hostile-reviewer Common Objections O-SM-31..54 (30 -> 54) in six
%%  new clusters (sharper methodology/statistics; sharper technical
%%  derivations; dependence on the cohort; further phenomenology; rigor/scope/
%%  standing), authored hostile-audit-first then answered with located cross-
%%  refs. No physics change; new Q&A only.
%% =====================================================================
%% =====================================================================
%%  A9-SM v11.24 (naming-convention rollout) -- 2026-06-19
%%  STEP 1: (1) Objection tags (the position-code suffix) -> O-SM-n (suffix=paper name).
%%  (2) Open-Problems list O_1..O_9 tagged (PS-SM-1..9), O_k labels kept.
%%  (3) five 'Predictions for falsification' subsubsections -> F-SM-1..5.
%%  Theorem T-SM-<section.num> tags, Innovations section, and the convention
%%  key follow.  (NB: A9's Q&A is the pre-expansion 30-objection set, below
%%  the >50 hostile-reviewer standard -- a separate retrofit if wanted.)
%% =====================================================================
%% =====================================================================
%%  Arisaka-GG-A9-SM (v11.23) -- cross-ref maintenance -- 2026-06-19
%%  Audit-trail block: dropped 7 stale A6 line-number pointers (A6 grew to
%%  v11.8; kept the durable thm-labels/names); aligned A6's H1/H2 to A6's
%%  corrected framing ('reduced ... modulo M6', not 'promoted to theorems').
%%  A5/A7 line numbers in the same block left for their own audits.
%%  No physics change.
%% =====================================================================
% ===== v11.22 (2026-06-18): Figures 1 & 2 each on their own full centred page (author Go). =====
%  Cohort map (fig:cohort-map-a9) and architecture/roadmap (fig:org-flowchart) converted to the
%  A1/A2 minipage+\vfill+\captionof full-page pattern. v11.21 -> v11.22.
% =====================================================================
% ===== v11.21 (2026-06-18): A3-style cohort + roadmap sections (author Go). =====
%  (1) Cohort subsection promoted to \section{The GGTheory Cohort} with a 17-paper prose overview
%  citing ALL 17 cohort papers; 4 missing cohort bibitems (A8, A9-self, C2, C3) added to References
%  Section A with first-citation links. (2) 'Organization of the paper' promoted to \section{Roadmap
%  of the Eighteen Parts}; per-Part walkthrough updated for Parts XVI/XVII/XVIII. (3) 'Admissible
%  inputs' subsection promoted to \section (clean nesting). v11.20 -> v11.21.
% =====================================================================
% ===== v11.20 (2026-06-18): References A/B split + first-citation relevance-links (author Go). =====
%  (1) 'First cited in \S\ref{...}' appended to every cited entry's Relevance note. (2) Bibliography
%  split into A. the 14 Arisaka cohort papers and B. all other references (each a \section* +
%  thebibliography block). (3) Intro third paragraph now describes the two groups. v11.19 -> v11.20.
% =====================================================================
% ===== v11.19 (2026-06-18): four-Part back-matter tail (frozen v2.0 standard; author Go). =====
%  'Appendices' Part renamed 'Technical Appendices'; \GGAcknowledgments moved to the end of the
%  Conclusions Part. Common Objections promoted to its own \GGPart with categories \subsection ->
%  \section and \bigskip spacing. References added as a \GGPart with a 3-paragraph intro, replacing
%  \GGBibBegin and suppressing the natbib heading. Mirrors A1/A2/A3 Stage 1a. v11.18 -> v11.19.
% =====================================================================
% ===== v11.18 (2026-06-17): Q&A readability tweaks (author Go). =====
%  (1) Removed the 'A--E ---' prefixes from the five category sub-section titles (L.1-L.5 already
%  number them). (2) De-indented the Question: Title and Question now both flush-left, with only the
%  Response left-indented (objqabody). Formatting only; no wording, value, codebook, or 6/13/9 triage
%  changed. v11.17 -> v11.18.
% =====================================================================
% ===== v11.17 (2026-06-17): five titled sub-sections for the Common-Objections appendix (author Go). =====
%  The 30 objections are now grouped under five \subsection headings (A Methodology/numerology/honesty;
%  B individual derivations; C comparison with other approaches; D phenomenology/falsifiability/proton
%  decay; E foundations/scope/meta), converted from the existing cluster comment dividers. Navigation/
%  formatting only; no wording, value, codebook, or 6/13/9 triage changed. v11.16 -> v11.17.
% =====================================================================
% ===== v11.16 (2026-06-17): reformat Common-Objections appendix for readability (author Go). =====
%  Each of the 30 objections is now three stacked blocks (references-style): bold Title (line break),
%  indented bold Question (line break), indented roman Response (same indent, not bold). Done via a
%  small left-indented environment 'objqabody' + uniform transform of all 30 entries. Formatting
%  ONLY; no wording, value, codebook, or 6/13/9 triage changed. v11.15 -> v11.16.
% =====================================================================
% ===== v11.15 (2026-06-17): expanded Common-Objections Q&A -> 30-item appendix (author Go). =====
%  New final appendix 'Common objections and responses' (sec:appendix-objections): 30 objection/
%  response pairs modelled on A2-GG6 Sec 65, grouped A-methodology, B-derivations, C-comparison,
%  D-phenomenology/proton-decay, E-foundations/meta. Body stub (subsec:common-objections) kept and
%  now cross-refs the appendix. Exposition only; NO value, codebook, or 6/13/9 triage changed; all
%  answers consistent with the frozen codebook + sin^2thW (2 lemmas) and lambda_H (curvature) status.
%  v11.14 -> v11.15.
% =====================================================================
% ===== v11.14 (2026-06-17): lambda_H re-based selection -> GFF curvature (author Go: R1+R2+R3a). =====
%  The executed Wilson-Hosotani curvature calc (lambda_H=(Lambda/f)^4 g''(th*)/(2 sin^2 th*); vev
%  cross-check passes at th*=(2/L)^N; natural one-loop magnitude) re-bases lambda_H from a DEGENERATE
%  MSMF/OGN cost-minimum *selection* (cost-min not even 1/8; p~1) to the A6 GFF quartic *curvature*
%  reading, modulo one named spectral-normalization lemma. Status 'Derived (selected)' -> 'Derived
%  (curvature, mod. 1 lemma)', propagated to ALL sites (row 5, caption, Part-I triage, thm:attractor
%  framing, MSMF/OGN audit caption+row+detail, exec summary, roadmap Lemma L6). Added thm:quartic
%  'Microscopic backing & outlook' para + eq:lambdaH-curvature. FROZEN: boxed lambda_H=2^-N=1/8 and
%  the 6/13/9 count unchanged (lambda_H stays one of the six Derived; sub-label/basis only). R4 (the
%  geometric light-Higgs result xi=(2/L)^2N + the full spectral sum) HELD for the standalone note.
%  v11.13 -> v11.14.
% =====================================================================
% ===== v11.13 (2026-06-17): Tier-4 follow-ups (author Go on Option 1). =====
%  Option 1 for the sin^2thW theorem: the two A9-local steps of the weak-angle theorem are now
%  NAMED -- Step 2 = the quadratic neutral-partition lemma (the squaring); Step 3 = the colour-blind
%  averaging lemma (the 1/N) -- and a 'Status in the honesty triage' note records sin^2thW as
%  Derived (theorem) MODULO those two named structural steps. The modulo note is propagated to the
%  provenance row+caption, the executive summary, and the Part-I triage bullet. Production item:
%  closed the Hirzebruch1995 bibliography orphan with one \citep at the int_{CP^2}L_1 prototype.
%  NO predicted value, codebook, or 6/13/9 triage count changed (sin^2thW stays Derived (theorem)).
%  v11.12 -> v11.13.
% =====================================================================
% ===== v11.12 (2026-06-17): promote the Cabibbo angle into Figure 3 (author Go). =====
%  Figure 3 (linear) now carries a 13th entry: the Cabibbo angle sin theta_12^CKM = 0.225, the textbook
%  'free parameter' of quark flavour, here derived parameter-free (pentagonal interference) and landing
%  within 0.01 sigma -- it fits the linear +-1e-2 window with a tight bar inside the |Dev|<1e-3 band.
%  Caption updated 12 -> 13 rows; figure title + raster regenerated. (delta_CKM kept in Fig 4 only: its
%  Dev ~1.7e-2..9.6e-2 is off the linear +-1e-2 scale.) NO predicted value changed. v11.11 -> v11.12.
% =====================================================================
% ===== v11.11 (2026-06-17): dev-plot cosmetics round 2 (author Go). =====
%  Figures 3 & 4 regenerated: entry NAMES bold / UNITS normal (\mathbf on the math name only; previously
%  the units were bold and the math names were not); fonts +~15%, left margin tightened to remove the
%  empty space, larger plot area. Figure 4 x-range +-10^-1 -> +-10^0 (1.0) and the three nested grey bands
%  made fainter. sin^2 thW obs = 0.23155 (Pull -1.70) in both; generator in [v11-Support]. v11.10 -> v11.11.
% =====================================================================
% ===== v11.10 (2026-06-17): dev-plot cosmetics + Figure 4 rebuilt to the 22-entry design (author Go). =====
%  Figures 3 & 4 regenerated: fonts +~20%, larger plot area, the 'QD / obs' annotation now aligned with the
%  QD (blue) point (no longer drifting toward the next entry), solid box kept. Figure 4 (symlog) rebuilt to
%  its true 28-Parameter-Audit design: 22 measured parameters (5 gauge/Higgs + 9 fermion masses + 4 CKM +
%  3 PMNS angles + delta_PMNS), three nested grey bands (|Dev|<1e-4,1e-3,1e-2), symlog +-1e-1 (linthresh 1e-4)
%  -- the prior v11.9 rebuild wrongly cloned the 12-entry linear plot. includegraphics width 1.10\textwidth ->
%  \textwidth so both figures fit the horizontal margin. sin^2 thW obs = 0.23155 (Pull -1.70) in both. v11.9 -> v11.10.
% =====================================================================
% ===== v11.9 (2026-06-17): regenerate Figures 3 & 4 (dev forest plots) with the v11.6 sin^2 thW. =====
%  The deviation rasters a9_dev_plot_linear_first8.png (Fig 3) and a9_dev_plot_symlog.png (Fig 4) still
%  embedded the stale sin^2 thW obs (0.23149 +/- 0.00013). Regenerated from the canonical comparison-table
%  data with obs = 0.23155 +/- 0.00004 (effective leptonic angle): the point moves to Dev=-2.96e-4 and its
%  experimental error bar tightens ~3x, so it now sits at Pull = -1.70 (visibly off zero). The dev-plot
%  generator was missing from the repo and has been reconstructed and saved as [v11-Support]/a9_dev_plots_gen.py.
%  Figure DATA only (in the PNGs); .tex unchanged except version + this banner. NO predicted value changed. v11.8 -> v11.9.
% =====================================================================
% ===== v11.8 (2026-06-17): Tier-3 precision/wording polish (author Go). =====
%  3.1 lambda_H 'unique' already softened in v11.7 (no action). 3.2 citation precision: the CKM
%  'BI-6 antecedent' paragraph reworded -- BI-6 fixes L=5 (SU(5) block size) and S^1_theta, but the
%  pentagonal PHASE (3pi/5, pi/5) is assembled in A9, not contained in BI-6 (per the A1-A8 audit).
%  3.3 sector 'theorem' labels: added one framing note tying the section 'theorems' to the honesty
%  triage (the triage, not the title, is the load-bearing claim); no theorem environment renamed.
%  Exposition only; NO predicted value, codebook, or 6/13/9 triage changed. v11.7 -> v11.8.
% =====================================================================
% ===== v11.7 (2026-06-17): Tier-1 honesty-spine reframe (author Go; count-neutral primary). =====
%  The six 'Derived' now wear their true basis: 1 end-to-end THEOREM (sin^2 thW) + 5 non-theorem
%  entries -- alpha_s, lambda_H (MSMF/OGN SELECTIONS, look-elsewhere p~0.27 and p~1), y_t (confirmed
%  PREDICTION), alpha_31 (discrete STRUCTURAL datum), alpha_21 (CONVENTION, unmeasured). Status tags:
%  Derived (theorem/selected/predicted/structural/convention). 6/13/9 count KEPT (cohort continuity);
%  look-elsewhere factors reported inline; lambda_H 'unique' softened. 'codebook fixed first' reworded
%  to the true claim (form by complexity, value then checked; selected entries use the empirical value
%  in the selection filter). Abstract left untouched (already honest). NO predicted value changed;
%  28 predictions byte-identical; 6/13/9 preserved. v11.6 -> v11.7.
% =====================================================================
% ===== v11.6 (2026-06-17): Tier-2 self-consistency sweep (author Go, decisions locked). =====
%  (2.1) PMNS solar/reactor angles -> Set A codebook forms everywhere (25/81, 3/140); audit 'Set B'
%        orphans (1/pi, 1/45) retired (noted as worse-fitting, not adopted). 3 PMNS angles stay
%        Derived (cond.) via the structural sec:PMNS derivation -> 6/13/9 and 'promotes eight'
%        accounting UNCHANGED. (2.2) sin^2 theta_W observed -> effective leptonic 0.23155(4)
%        everywhere (codebook x4, census, gauge tables x2, theory line, prose, MSMF example);
%        Dev -2.959e-04, Pull -1.70. (2.3) 'y_t=1 defines v' removed (4 places): v from M_lock,
%        m_t=v/sqrt2 a ~1%% prediction (y_t triage status unchanged -> Tier-1). (2.4) m_star=m_star(v_QD)
%        annotated. (2.5) delta_v noted as propagated QD shift. NO predicted value changed. v11.5 -> v11.6.
% =====================================================================
% ===== v11.5 (2026-06-17): N=3 treatment polish --- three-leg / parity-signature, honest 'conditional' framing (author Go). =====
%  Adds a 'Status' remark after thm:N3-closed keeping two distinctions sharp: (a) the rigorous even-N
%  parity result a+(N)-a-(N)=sigma(CP^{N-1}) vs the conjectural odd-N>=5 (M10 budget) and empirical (CP)
%  legs; (b) the GENERATION number (this conditional pincer) vs the unconditional COLOUR triplicity
%  (A3 trace ratio 2/5). Reads 'closed three-leg theorem' as a sharply localised conditional pincer.
%  One-clause softening of the implications line; 'three independent constraints' -> 'three constraints
%  along logically independent directions'. Framing/honesty only; NO boxed value, codebook, or 6/13/9
%  triage changed. v11.4 -> v11.5.
% =====================================================================
% ===== v11.4 (2026-06-17): incorporate SU(5)xU(1)_B / absolute proton-stability analysis (author Go). =====
%  Adds a CONDITIONAL boundary proton-stability theorem (thm:proton-stability-a9) making the
%  group-theory obstruction explicit (B is not an SU(5) generator; the commuting per-multiplet
%  U(1)_B does NOT forbid the X,Y operator; protection needs genuine component-resolved baryon
%  number on the split SM spectrum, X,Y bulk-only), stated modulo H1-H3 with the one residual
%  finite computation (H2) honestly flagged and deferred to a successor paper. Adds the DeltaB in 3Z selection
%  rule (forbids proton decay AND n-nbar), the Stueckelberg-mass / exact-GLOBAL-baryon-number
%  logic and the equivalence-principle evasion in the falsification section, and one clause
%  rewording the Part-I claim into the exact-global-B form. NO change to the 28-parameter codebook
%  or the 6/13/9 triage (proton stability is an out-of-codebook structural prediction). v11.3 -> v11.4.
% =====================================================================
% ===== v11.3 (2026-06-17): add 'Observed + short Dev' to the predicted-only summary tables (author Go). =====
%  Presentation only; NO physics, NO derived value, NO 28-parameter triage (6/13/9) changed.
%  Four summary tables that previously listed only predicted (first-order / QD) values now
%  carry an Observed column and a short Dev (QD fractional-deviation) column, both copied
%  VERBATIM from the single canonical codebook (Part XIII full census) -- no recomputation.
%  Unmeasured quantities (3 absolute neutrino masses, 2 Majorana phases, strong-CP bound)
%  carry honest 'not measured' / bound / '---' entries. The 28-row master census is rotated
%  to landscape (pdflscape) to fit the two added columns. v11.2 -> v11.3.
% =====================================================================
% ===== v11.2 (2026-06-16): Kaxion-mass consistency fix (author Go). =====
%  Corrected the stray Kaxion mass '~19 mueV' (7 places) to the paper's own canonical
%  value 8.1 mueV (1.95 GHz centre; 1.59-2.13 GHz lab line), consistent with O16, C1-Kaxion,
%  and A10-Cosmos. Added '(1.95 GHz)' at the two clean 'prediction' mentions. Consistency
%  fix to the canonical value; no new physics. v11.1 -> v11.2.
% =====================================================================
% ===== v11.1 (2026-06-16): reclassify dark matter / dark energy out of 'genuinely open' (author Go). =====
%  Exposition / honesty-status only; NO physics, NO derived value, NO 28-parameter triage
%  (6/13/9) changed. Dark matter (#2/O16) and dark energy + cosmological constant (#8/O17)
%  were mislabelled 'Genuinely open (in A9 scope)' while the same cells cited the GG results
%  that resolve them. Reclassified to 'Out of A9 scope (cohort-resolved)', matching A10's own
%  triage (dark energy Resolved; dark matter Partially resolved) and the cohort-wide removal of
%  'genuinely open'. Tallies, the O_k list, and both conclusion sentences updated to match;
%  the only genuinely-open particle item left is the programmatic 8-Yukawa lemma (O21).
%  (FLAGGED separately, NOT changed: a stray '19 mueV' Kaxion mass at the 'three structural
%  items' line, inconsistent with the paper's canonical 8.1 mueV / 1.95 GHz.)
% =====================================================================
% ===== v11.0 (2026-06-16): holographic-RG reframing, Phase R2. =====
%  Terminology only; NO physics, NO triage counts changed (6/13/9 holds). A9 has no
%  AdS-bulk branding; A2/A3 bibitems AdS-free; the lone AdS/CFT mention kept. Added the
%  holographic-RG disclaimer footnote at the exec-summary bulk mention; v10.16 -> v11.0.
% =====================================================================
% ===== v10.16 (2026-06-16): fix the v10.15 relocation side-effect (author-flagged). =====
%  Exposition only; no physics, no counts changed.
%   - The v10.15 5.6/5.7 extraction had swept up the 'Closing of Part I.' handoff
%     (divider + the Part I->Part II transition paragraphs) and stranded it in
%     Part XIII. Moved it back to the end of Part I, before Part II's \GGPart.
%     Part XIII now flows: codebook tables -> MSMF/OGN section -> 'not a post-hoc fit'.
% =====================================================================
% ===== v10.15 (2026-06-16): relocate MSMF/OGN operationalisation + audit (author Go). =====
%  Exposition only; no physics, no parameter values, no triage counts changed (6/13/9 holds).
%   - Moved the two MSMF/OGN subsections (5.6 'Operationalising ...' + 5.7 'Per-parameter
%     audit ...') OUT of Part I (a setup/preview Part) INTO Part XIII 'Summary', wrapped in a
%     new section just before 'Why the result is not a post-hoc fit'. Part I keeps the
%     canonical codebook-provenance table + the 6/13/9 headline, now with a forward pointer
%     to Part XIII. All cross-references (subsec:msmf-ogn-principle/-audit, thm/tab labels)
%     auto-resolve; the audit's internal 'immediately following' refs move with the block.
% =====================================================================
% ===== v10.14 (2026-06-16): Part-I introduction restructure (author Go). =====
%  Exposition only; no physics, no parameter values, no triage counts changed.
%   (1) Moved the cohort-map figure (was Fig 5) + its lead-in up into subsec 1.4
%       'Where A9 sits in the GG-Theory series'. Figures auto-renumber: the cohort
%       map becomes Fig 1, the architecture flowchart Fig 2, dev-plots Fig 3/4, ...
%   (2) Expanded subsec 1.5 with detailed GU-5 (Geometric Universal Five Constants)
%       and OGN-5 (integer ledger) introductions, and moved 1.5 to AFTER subsec 1.6
%       'Organization of the paper'. New order: 1.4, 1.6, 1.5.
%   (3) Widened the Figure-1 block-label boxes (Foundation/Derivation/Closure) so the
%       words no longer overflow the right edge.
% =====================================================================
% ===== v10.13 (2026-06-16): appendix-heading reletter (author Go, item 5). =====
%  Presentation only; no physics, no parameter values, no triage counts changed.
%   - Stripped the redundant 'Appendix X:' prefix from appendix headings C-K so all
%     eleven appendices (A-K) render uniformly off the auto-letter (A/B already had
%     no prefix). Fixes the 'C  Appendix C:' double-lettering in the ToC and headers.
%   - alpha_31 Majorana phase: corrected the codebook-provenance / MSMF-OGN-audit
%     entries from 0 to pi (author ruling) to match the PMNS derivation, value tables,
%     precision table, and master census, which already read pi. Rationale changed
%     from 'cost-0 additive identity' to 'forced opposite-parity datum of the rank-two
%     Majorana block'. No physics changed; triage count unaffected (6/13/9 holds).
% =====================================================================
% ===== v10.12 (2026-06-16): FRONT-MATTER REFINEMENT (author Go). =====
%  Presentation only; no physics, no parameter values, no triage counts changed.
%   - Abstract rewritten for a broad audience: plain language, no notation,
%     no citations; fits page 1 (< 1900 chars).
%   - Executive Summary restructured into bold-heading paragraphs at the
%     physics-graduate level, closing on a philosophical statement; the
%     6/13/9 honesty triage is restated verbatim from the canonical table.
%   - Preface rewritten as a plain-English introduction with a message to
%     the next generation; moved ahead of the contents pages.
%   - Subtitle replaced with a plain-language line ('... from the
%     Six-Dimensional Geometry with no Free Dials').
%   - Executive Summary: added 'Striking contrast to supersymmetry and
%     string theory' paragraph before the philosophical close.
%   - Combined Overview of Contents + List of Figures onto one page
%     (one line per entry) before the full table of contents; short
%     \caption[...] forms added to nine figures; Part toc lines shortened
%     to one line each so the Overview fits a single page.
%   - Version/date stamps bumped to v10.12 / June 16, 2026 (title page,
%     PaperDate); title-page date corrected from 'May 2026'.
% =====================================================================
% --- v10.11 (2026-06-16): Part-aware running header (matches A11-Origin v12.27/28).
%     Right-side version/date stamp replaced by a live 'Part N. <short>' pointer;
%     Executive Summary and Preface set their own front-matter marks. Header-only; no content edits. ---
% GU-5 RENAME (June 14, 2026): cohort harmonization with GG-A4-Euclid -- the five-constant inventory and its theorem are named Geometric Universal Five Constants / GU-5 (\GUfive, \T{-GU5}); the older 'Conversion-Factor Pentad'/'Conversion-Factor Triad' wording and the 'T-CFP' theorem tag are retired.
\pdfoutput=1
%% v10.10 (June 12, 2026): J5 resolved (author ruling: depth-2 is
%%   canonical for the charged-lepton ladder). The one stray sentence
%%   in the identity section that used the quark sector's five-level
%%   depth coordinate for the leptons (electron at 'depth-4') is
%%   corrected to the canonical three-level lepton coordinate
%%   (electron = depth-2 GGA), consistent with this paper's own
%%   cascade figure, summary table, neutrino section, and P3's
%%   import. Quark depth-4 statements untouched. No other changes.
%% v10.9 (June 12, 2026): Cohort-map deployment (author Go):
%%   the canonical seventeen-paper cohort map inserted (label
%%   fig:cohort-map-a9, this paper highlighted) with a one-paragraph lead-in,
%%   at the surveyed seam where the introduction completes.
%%   Also: stale ArisakaP3Electron bibitem stamp refreshed v10.4 ->
%%   v10.10 (M1 lesson, while touching the file). No other changes.
%% v10.8 (June 11, 2026): P3-pointer refresh (minimal, disclosed):
%%   the ArisakaP3Electron bibitem and the dedicated-electron-paper
%%   pointer paragraph updated to P3 v10.4's framing -- geometric
%%   pointlikeness theorem with F == 1 as its corollary (P3 v10.1),
%%   and the two-sector Q5 reading (P3 v10.3: phase sector moves,
%%   scale sector transmutes; radiation is the dissipative channel)
%%   in place of the superseded 'GFF-descent gloss' wording; stamp
%%   v2.2 -> v10.4. No other content changes.
%% v10.7 (June 11, 2026): P3 title disambiguation sweep (minimal,
%%   disclosed): P3 bibitem title -> "The Five Oldest Questions about
%%   the Electron..."; stamp -> v2.2; Relevance framing refreshed to
%%   P3 v2.0+ (composite geodesic motion per A5 Part VI; GFF descent
%%   as gloss). Pointer paragraph wording aligned. No other changes.
%% v10.6 (June 11, 2026): P3-Electron Phase-5 sync (minimal, disclosed;
%%   the A9 <-> P2 pointer pattern, now applied for P3): NEW pointer
%%   paragraph at the close of subsec:three-electron-questions directing
%%   the reader to GG-P3-Electron as the dedicated five-question
%%   treatment; NEW bibitem ArisakaP3Electron (A9 house pattern, v1.7
%%   stamp). No other content changes.
%% v10.4 (June 10, 2026): cohort-consistency fix (C1-A10 audit): the
%%   Kaxion mass quoted in the strong-CP historical remark updated from
%%   the stale pre-correction value (~19 ueV) to the corrected C1
%%   value, 8.1 +/- 0.8 ueV (laboratory line 1.59-2.13 GHz). One
%%   sentence; no other changes.
%% v10.3 (June 10, 2026): arXiv preparation pass (front-matter audit).
%%   hand-rolled cover abstract converted to the standard abstract environment.
%%   Accepted variants (author decision): existing philosophical closes
%%   in place of literal 'After all'; companion-list counts as cited.
%% v10.2 (June 9, 2026): arXiv preparation pass (cohort step 3).
%%   Amended same day: former bold small-caps spots set in bold FULL CAPS
%%   (author instruction June 10, 2026).
%%   Overfull-box sweep (all boxes >20pt resolved with layout-only fixes);
%%   no content or physics changes.
%% v10.1 (June 9, 2026): arXiv preparation pass (cohort step 1-2).
%%   \pdfoutput=1 added (line 1); version/date stamp bumped. 
%%   No physics changes; production fixes only.
%% v9.4 (2026-06-03): terminology RG-time -> RG-scale (the Scale 5th dimension is
%%   space-like, not a second Minkowski time; canonical naming per A2 v9.18).
% ----------------------------------------------------------------------
% Arisaka-GG-A9-SM
% Geometric Origin of the Standard Model:
% Derivation of the 28 Free Parameters from the OGN-5 Diophantine Ledger
% Author: Katsushi Arisaka (UCLA)
%
% PROVENANCE.  This file is the v5.0 Phase 1 (identity-migration) of the
% A9-SM paper.  It is built mechanically from the v4.0 BI-6/TGG/CCI/GFF/
% GRIP recast source
%   UCLA-Arisaka-GG-2026-6F-recast.tex   (April 14, 2026, 242 pp)
% via the Phase-1 migration script
%   outputs/a9_v50_phase1_migration.py   (May 12, 2026)
%
% The Phase-1 transformations are MECHANICAL only:
%   * Style upgrade ucla-arisaka-gg -> ucla-arisaka-gg-6        (L1)
%   * Identity block: PaperNumber, PaperStatus, PaperVersion,
%     PaperDate, hypersetup metadata                            (L2, L15)
%   * Local-macro purge: 28 macros now provided in canonical
%     sans-serif by gg-6                                        (L21)
%   * Paper-pointer macros redefined to canonical short names
%     (\POne -> A1-Constitution, etc.)                          (L3)
%   * Bibtex key migration (5 legacy keys -> ArisakaA*)         (L3)
%
% NO SUBSTANTIVE CONTENT EDITS in Phase 1.  Theorem statements,
% section text, and prose are byte-for-byte preserved.  The eleven
% remaining phases plus the three Ultra-passes apply per the v4.0
% canonical workflow; see PROJECT-INSTRUCTIONS-v4.0-CONSOLIDATED.md.
%
% PHASE LOG.
%   v7.2 (May 15, 2026):     Part XII insertion -- "Geometric origin
%                              of identity and uniqueness of forces".
%                              Substantive single-Part structural
%                              extension on top of v7.1.  Three
%                              sections: \S60 (existing universal-
%                              identity insight, promoted out of
%                              former Part XI into new Part XII as
%                              its opening section), \S61 NEW "the
%                              geometric origin of force: why these
%                              four forces and no others" (eight
%                              constitutional lemmas forcing the
%                              canonical 6D bulk + four force-by-
%                              force qualitative derivations + the
%                              interlock for life; introduces a new
%                              named theorem, "Uniqueness of the
%                              canonical four-force architecture"),
%                              \S62 NEW "two fundamental questions,
%                              one geometric answer" (synthesis:
%                              identity-uniqueness and force-
%                              uniqueness are the same theorem at
%                              different layers; dissolution of
%                              design / multiverse / brute fact;
%                              self-referential closure; careful
%                              word about determinism; closing
%                              punch line).  Part numbering of the
%                              tail shifts by one: former XII
%                              (Summary) -> XIII, former XIII
%                              (Discussions) -> XIV, former XIV
%                              (Conclusions) -> XV, former XV
%                              (Appendices) -> XVI.  All earlier
%                              Parts I-XI and the 28-parameter
%                              codebook unchanged.  Source content
%                              brief: A9-SM-Part-XII-Content-Brief.md.
%                              Workflow: focused 7-phase single-Part
%                              insertion (P1 identity, P2 \S60
%                              promotion + tail renumber, P3 draft
%                              \S61, P4 draft \S62, P5 named theorem,
%                              P6 Part XII close, P7 final compile +
%                              ship).  Page-count target: 312 -> ~325
%                              -- 330 pp.  This entry: Phase 1
%                              identity migration only; smoke compile
%                              must return 312 pages unchanged.  Phase
%                              1 also lifts the style file from
%                              ucla-arisaka-gg-6 to ucla-arisaka-gg-7
%                              per L52 (the v7 style bakes in the 26
%                              sans-serif paper-code macros, so the
%                              former local \AOne..\PaperFullCOne
%                              block is converted from \newcommand
%                              to \providecommand to coexist with
%                              the new package).
%
%   Phase 1 (May 12, 2026):  identity migration; this file.
%   Phase 2 (May 12, 2026):  GS-as-GGEP framing audit (L5).
%                              Applied four patches: TGG master
%                              theorem cites A3-BI6; BI-6 parent-law
%                              section carries L5 "1984 technical
%                              pioneering of AX-2 GGEP" remark;
%                              POS-3 section cites A3-BI6;
%                              superstring-theory comparison cites
%                              A3-BI6 at "historical ancestor of BI-6".
%   Phase 3 (May 12, 2026):  L4 historical-background audit.  Inserted
%                              13 \subsection{Historical background}
%                              openings on the named theorem sections,
%                              naming the original inventor + year +
%                              citation per L4.  Master bib expanded
%                              with 17 historical entries.
%   Phase 4F (May 12, 2026): GDOS-Trio language alignment, Track F
%                              (N=3 theorem section).  Inserted new
%                              section between Â§14 and Â§15 with the
%                              closed three-leg theorem from
%                              Arisaka-GG-P2-N3 (formerly the
%                              Generations preprint), naming the
%                              MSMF lower bound (L1), the GDOS Morse
%                              upper bound (L2), the codebook check
%                              (L3), the H1+H2 reduction with
%                              c_R=1 explicit, and the
%                              implications for the remainder of A9.
%   Phase 4A (May 12, 2026): GDOS-Trio language alignment, Track A
%                              (architectural reframing of section 3).
%                              Reframed BI-6 -> TGG -> CCI -> GFF ->
%                              GRIP chain as the canonical Gi-4 chain
%                              annotated with A6 T1-T4 owners; demoted
%                              local Theorems 3.1 and 3.2 to Corollaries
%                              specialising A6 T1 and A6 T2 + A5 T9.7;
%                              cited A6 T3 (Morse-positivity) and A6 T4
%                              (cascade closure) as upstream owners of
%                              the GFF/GRIP subsection.
%   Phase 4D (May 12, 2026): GDOS-Trio language alignment, Track D
%                              (Arisaka equation J_E/Q_B = J_e = J_B = J_Qi
%                              + Q_B = (2/5) E_Ry / K = 0.1555 eV/qubit).
%                              Inserted new §3.5 citing A7-Qi4 T-QUA;
%                              shows that the trace ratio 2/5 in the BI-6
%                              vector and the L^2/N^3 partition in
%                              sin^2 theta_W = 25/108 are two faces of
%                              the same four-currents lock.  Adds the
%                              N_A Q_B ~ 15 kJ/mol ATP-hydrolysis
%                              empirical anchor.
%   Phase 4B (May 12, 2026): GDOS-Trio language alignment, Track B
%                              (cascade-depth Yukawa readings).  Inserted
%                              new "Gi-4 cascade-depth reading"
%                              subsection into each of the three
%                              fermion-mass sections (charged-lepton,
%                              quark, neutrino).  Quark subsection
%                              identifies the angular cascade e^{-n pi}
%                              with the explicit Bogomolnyi instanton
%                              actions S_{[2k]->[2(k-1)]} = pi k, per A6
%                              T3+T4 and the Generations preprint
%                              (Theorem 15.1).  Each generation cast
%                              as DGI/GSSB/GA at cascade depths 0/1/2.
%   Phase 4C (May 12, 2026): GDOS-Trio language alignment, Track C
%                              (Hopkins-Singer naming of QD rationals).
%                              Inserted master subsection at end of §22
%                              identifying R_tau=637/640, R_mu=159/160,
%                              R_e=641/640, alpha^{-1}(MZ)=128-9/128 as
%                              Hopkins-Singer descent integrality
%                              coefficients of [X_4] with denominator
%                              640 = 2^J L from OGN-5.  Cohomological
%                              backbone: A6 T2/T3, HopkinsSinger2005,
%                              ArisakaP2N3 App. E (c_R = 1 explicit).
%   Phase 4E (May 12, 2026): GDOS-Trio language alignment, Track E
%                              (SIP-language input descriptions).
%                              Inserted SIP specification paragraphs
%                              after each of the 6 load-bearing named
%                              theorems (5 primary-sheet + Unified
%                              package), declaring the canonical
%                              5-field SIP per A7-Qi4 T-SIP §6:
%                              state manifold, ports, generator,
%                              ledger, observable dictionary + kill.
%                              The unified package SIP is now framed
%                              as one runtime instance producing nine
%                              registered observables on a single
%                              shared boundary stratum.
%   Phase 5 (May 12, 2026): Open Problems triage with L6 status
%                              headers.  Updated 10-row table with
%                              explicit L6 column (Resolved /
%                              Partially resolved / Genuinely open).
%                              Inserted new closing subsection with
%                              20 explicitly-classified entries
%                              O_1..O_20 (13 Resolved + 5 Partially
%                              + 2 Genuinely open).
%   Phase 6 (May 12, 2026): hostile-reviewer + reference-scanner.
%                              HIGH-priority fix: arithmetic bug in
%                              eq:S-BI-max corrected from
%                              (2pi.3/5)(J/2) = (2pi.3/5)(7/2) = 4.2pi
%                              to (2pi.3/5)(L/2) = (2pi.3/5)(5/2) = 3pi,
%                              restoring the cascade-closure saturation
%                              identity per L27.  MEDIUM-priority fix:
%                              9 occurrences of T-QUA/T-SIP/etc.
%                              normalised from \T{-QUA} to
%                              canonical \T{-QUA} per the sans-serif
%                              audit.  Hostile-reviewer Targets 2-4
%                              (cascade rigidity, -2/5 circularity,
%                              QD numerator post-hoc concerns) noted
%                              for Phase 7 cross-paper audit + Ultra
%                              passes; defensible at this stage by
%                              upstream-paper deferral.
%   Phase 7 (May 12, 2026): companion-paper auditor + cross-paper
%                              coherence map (per L7).  Auditor (Plan
%                              agent) verified 11 cross-citations: 5
%                              EXACT, 5 NEAR (bare T2/T3/T4 should be
%                              T2.1/T2.2, T3.1, T4.1 per A6 v7.0
%                              actual numbering), 1 WRONG (A6 §39
%                              cascade-depth reference -- A6 §39 is
%                              actually 'measurement-problem carry-
%                              over'; cascade-depth content is in A6
%                              §25 + §28).  Applied 11 patches; added
%                              cf. A6 §22 H1+H2 in §15.4.  Inserted
%                              new section 'Cross-paper coherence map'
%                              before §Discussion with L7-compliant
%                              Note + Audit trail + Outward map
%                              (13-row) + Inward map (7-row) + Net
%                              coherence summary.  Two minor
%                              staleness flags noted (A5 'legacy
%                              6F-recast', A6 'in preparation') to be
%                              resolved in next A5/A6 minor version.
%   Phase 8 (May 12, 2026): TikZ figures (L8 + L23 compliance).
%                              Inserted four new figures, each
%                              wrapped in \resizebox{...}{!} per L8
%                              and using bold sans-serif for
%                              programme acronyms / regular face for
%                              descriptions per L23:
%                              Fig 1: Canonical Gi-4 chain (in §3),
%                              Fig 2: GRIP Morse cascade on CP^2 (§15),
%                              Fig 3: Arisaka 4-currents lock (§3.5),
%                              Fig 4: 28-parameter derivation tree (§56).
%   Phase 9 (May 12, 2026): Front-matter rewrite to A1 canonical
%                              template (L9 + L15).  Title block
%                              refreshed; abstract rewritten to ~190
%                              words plain English with 'After all'
%                              closer (was 324 words with 5 boxed
%                              equations); Executive Summary expanded
%                              to ~870 words with canonical structure
%                              (opening + boxed central principle +
%                              Principal theorems + Significance and
%                              uniqueness + Canonical implication
%                              chain + closing reflection); GGKeywords
%                              refreshed (drop legacy GUT-10, add
%                              modern Q_B / Hopkins-Singer / cascade-
%                              depth terms); manual Acknowledgments
%                              replaced with canonical \GGAcknowledgments.
%   Phase 10 (May 12, 2026): Bibliography surname-sort + Relevance
%                              paragraph rendering (per L10).
%                              Audit: v5.13's bibtex-rendered .bbl was
%                              correctly surname-sorted by plainnat,
%                              but Relevance paragraphs were silently
%                              dropped (plainnat doesn't render the
%                              relevance field).  Patch: convert
%                              from \bibliography{} to inline
%                              \begin{thebibliography} with manually-
%                              curated bibitems carrying
%                              \par\noindent\textbf{Relevance.}
%                              paragraphs (matches A1 canonical
%                              format).  41 entries surname-sorted.
%                              Per L36, journal-stripped variant
%                              skipped.
%   Phase 11 (May 12, 2026): Appendix I (Summary of abbreviations)
%                              + programme-acronym hierarchy figure
%                              (per L19 + L20 + L24).  9-section
%                              3-column table covering framework,
%                              4 axioms, 7 postulates, Gi-4 pillars,
%                              Qi-4 pillars, Qi-HAL acronyms,
%                              GDOS-internal acronyms, A9-specific
%                              math symbols, standard physics in
%                              6 sub-categories.  TikZ hierarchy
%                              figure with 6 tiers per L20.
%   Phase 11.3 (May 12, 2026): conservative sans-serif sweep per L21.
%                              Plain-text GG-Theory, BI-6, OGN-5, GG-6,
%                              Gi-4, Qi-4 converted to canonical sans-
%                              serif macros (\GGTheory{}, \BIsix{},
%                              \OGNfive{}, \GG{}, \GiFour{}, \QiFour{}).
%                              Skip rules per L21: comment lines,
%                              providecommand, hypersetup, tikzpicture,
%                              longtable, label/ref/cite/eqref/url
%                              arguments, math mode.
%   Phase 11.4 (42-item audit) follows.
%   Ultra-1 (May 12, 2026): single-line algebraic patches per L27.
%                              Walked 15 numerical claims in front
%                              matter / boxed equations / theorem
%                              statements.  13 PASS, 1 NEAR, 2
%                              precision-claim overstatements
%                              patched: (a) Exec Summary CODATA
%                              claim '137.0 to four significant
%                              figures' tightened to 'within 0.03%%';
%                              (b) weak-angle theorem PDG claim
%                              'four significant figures' tightened
%                              to 'better than 0.12%%' (0.23148 vs
%                              0.23122).  Plus (c) m_Lambda formula
%                              clarified to use reduced Planck mass
%                              \overline{M}_Pl, not M_Pl.
%   Ultra-2 (May 12, 2026): backward-invasion / redundancy trim per L25.
%                              Compressed five 'Deep geometric origin
%                              I-V' sections from ~855 lines to ~250
%                              lines.  Each section now: (1) keeps the
%                              section header + opening framing; (2)
%                              replaces the body with a compressed
%                              boundary-side reading that names the
%                              upstream owner (A2-GG6, A3-BI6, A5-GDOS,
%                              A6-Gi4, A7-Qi4) instead of re-deriving
%                              their content; (3) ends with a Physics-
%                              Intuition reflection.  Page-impact:
%                              ~14 pp reduction.
%   v6.8.1 (May 12, 2026): operationalise MSMF/OGN as a formal
%                              selection principle.  Per user
%                              direction: 'we can derive all 28
%                              parameters using MSMF/OGN up front'.
%                              v6.8.1 = definition only (the formal
%                              candidate space, OGN admissibility
%                              filter, empirical-precision constraint,
%                              MSMF cost function, selection theorem,
%                              one worked illustration on sin^2 theta_W).
%                              v6.8.2 (pending): per-parameter audit
%                              against this principle for the 17
%                              currently-Asserted entries.  v6.8.3
%                              (pending): Codebook Provenance Table
%                              update + abstract/preface refresh.
%
%   v6.7 (May 12, 2026): apply the 4-layer A3-BI6 honesty
%                              apparatus.  Per user direction:
%                              'take the most honest, objective
%                              approach as for A3-BI6'.  Four
%                              additions: (1) Codebook Provenance
%                              Table with 28 rows + status legend
%                              (Derived / Derived-conditional /
%                              Asserted); (2) 'What is proved here
%                              and what remains open' framing
%                              subsection; (3) honest reframe in
%                              Abstract + Preface closing reflection
%                              (1 Derived + 10 Derived-conditional +
%                              17 Asserted, with explicit promotion
%                              roadmap); (4) Scope + Limits +
%                              Dependencies block at end of
%                              Discussion.  No content removed;
%                              boxed equations and numerical matches
%                              unchanged.  Honesty discipline
%                              matches A3-BI6.
%
%   v6.6 (May 12, 2026): plain-English abstract + new Preface
%                              section.  User-caught: v6.5 abstract
%                              still too technical (jargon: GG-
%                              Constitution, BI-6, GDOS, Gi-4,
%                              Arisaka equation, OGN-5, Q_B...)
%                              for life-scientist-level reader.
%                              Replaced with plain-English version
%                              (~185 words) highlighting empirical
%                              hits (1/137, 25/108, 246 GeV) and
%                              the ATP hydrolysis match.  Plus new
%                              \section*{Preface} inserted between
%                              GGTableOfContents and Introduction
%                              (matches A1-A7 v7.0 canonical Preface
%                              structure).
%
%   v6.5 (May 12, 2026): retire all legacy GG-2026-X / GD-2026-X /
%                              GG-2026-6F naming in body prose.
%                              User-caught: post-Ultra-3 file v6.4
%                              still contained 165+ legacy paper-
%                              identifier mentions inherited from
%                              the v4.0 BI-6/TGG/CCI/GFF/GRIP recast.
%                              All replaced per the canonical mapping:
%                              GG-2026-1 -> A1-Constitution; -2 -> A2-GG6;
%                              -3 -> A5-GDOS; -4 -> A7-Qi4; -4A -> A6-Gi4;
%                              -5 / GD-2026-5 -> A3-BI6; -6F -> A9-SM;
%                              -7 -> A10-Cosmos; -8 -> P1-KK;
%                              -G(enerations) -> P2-N3.
%                              PROVENANCE block preserved untouched
%                              (historical phase log).
%
%   v6.4 (May 12, 2026): split multi-key Arisaka cites to clean
%                              up natbib comma-in-year artifacts.
%                              The [Arisaka(2026, X)] bibitem labels
%                              work perfectly for single \citep{key}
%                              but multi-key \citep{A,B} produces
%                              '(Arisaka, 2026, A2-GG6,,)' artifacts
%                              because natbib confuses the comma in
%                              the year-string with the citation
%                              key separator.  Fix: rewrite multi-
%                              key Arisaka calls as semicolon-joined
%                              \citealp{} forms per A1 canonical.
%
%   v6.3 (May 12, 2026): bibliography relabel + series-order
%                              resort.  User-caught issue:
%                              all Arisaka inline citations were
%                              rendering as '(Arisaka, 2026)'
%                              indistinguishably.  Fix: each
%                              Arisaka bibitem label changed to
%                              [Arisaka(2026, <suffix>)] where
%                              <suffix> = A1-Constitution, A2-GG6,
%                              etc.  Citations now render as
%                              '(Arisaka, 2026, A1-Constitution)'.
%                              Arisaka entries reordered in series
%                              order at the top of the bibliography
%                              (A1, A2, ..., A7, A10, P1-KK, P2-N3,
%                              C1-Kaxion); historical entries remain
%                              surname-sorted after.  v7.0 ship
%                              deferred until A1-A7-level polish
%                              complete.
%
%   Ultra-3 (May 12, 2026): cleanup + 3 audit-FAIL items.
%                              Tightened abstract from 232 to ~195
%                              words (Phase 11 audit item 2).
%                              Added 'Common Objections and
%                              Responses' subsection in Discussion
%                              (Phase 11 audit item 17).  Added
%                              'Scope and Limits' subsection in
%                              Discussion (Phase 11 audit item 18).
%                              Added 'U(1)_Qi constitutionally
%                              optional / empirically preferred'
%                              subsection in §3 (Phase 11 audit item
%                              30).  Final canonical-audit Plan agent
%                              run; v7.0 canonical ship after.
%   Phase 5 (pending):       Open Problems triage (L6).
%   Phase 6 (pending):       hostile-reviewer + reference-scanner.
%   Phase 7 (pending):       companion-paper auditor + cross-paper
%                              coherence map (L7).
%   Phase 8 (pending):       TikZ figures + L8 compliance.
%   Phase 9 (pending):       front-matter rewrite to A1 canonical
%                              template (L9).
%   Phase 10 (pending):      bibliography surname-sort (L10).
%   Phase 11 (pending):      Appendix A 9-section + hierarchy
%                              figure + sans-serif sweep + 42-item
%                              canonical compliance audit
%                              (L19, L20, L21, L23, L24).
%   Ultra-1 (pending):       single-line algebraic patches.
%   Ultra-2 (pending):       backward-invasion / redundancy trim.
%   Ultra-3 (pending):       cleanup; final audit; v7.0 ship.
% ----------------------------------------------------------------------

%% ----------------------------------------------------------------------
%% v10.4 -> v10.5 (June 11, 2026): P2-N3 reciprocal sync (Phase 5 of the P2 project;
%%   author-approved, June-10 protocol: minimal, disclosed, version-bumped).
%%   - "now a theorem" -> reduced to named hypotheses H1-H2 (Sec 18.5).
%%   - c_R = 1 claims corrected at Sec 18.6, Sec 42 (eq cR-equals-1-here), and
%%   -   the P2 bibitem: P2 v2.4 proves the naive warped-metric pullback
%%   -   vanishes identically; c_R undetermined pending induced connection.
%%   - Budget eq (S-BI-max) fenced (P2 Budget Conjecture; A3 quotes, does not
%%   -   derive); per-family reading S(N) = pi N added + reachability remark.
%%   - Theorem hyp (3) attribution corrected (A5 O3 open; A6 T4.1/M10).
%%   - J_CKM -> (3.12 +0.13/-0.12)e-5 (PDG 2024); J_PMNS -> approx -0.026 (NuFIT).
%%   - y_b^(1) typo N/2 -> N/2^N (eq site in Sec 43).
%%   - MSMF label Postulate~AX-4 -> Axiom~AX-4 (A1 nomenclature).
%%   Reference: [GG-P2-N3]/[P2-Support]/P2-N3-PHASE5-SYNC-REPORT-2026-06-11.md
%% ----------------------------------------------------------------------
% ------------------------------------------------------------
% v15.9, August 19, 2026.  The shared bulk showcase plate.
%%   ADOPTED, at the author's direction: one full-page showcase
%%   plate carried BYTE-IDENTICALLY by A2, A5, A8, A9, A10 and
%%   P1 -- the cohort-map precedent.  It is the integration of
%%   three existing figures: A1 Figure 1 supplies the logic
%%   (two necessities, AX-2 building the fiber, AX-3 projecting
%%   to the registered world), A10 Figure 13 supplies the
%%   tapering throat and the projection arrow, and A2 Figure 8
%%   supplies the calibrated scale axis.  The axis now carries
%%   eight ticks with mu = E_GUT e^-r beside each, E_GUT =
%%   E_Pl e^-2pi at r = 0, E_Pl declared OFF the diagram at
%%   r = -2pi, the lock at r = 29.41 and the OGN-5 depth at
%%   r = K = 35.  Every number was computed, not carried over.
%%   THE OFFSET IS EXACT: K is counted from the bulk anchor
%%   E_Pl/2 and the axis from E_GUT, so the two differ by
%%   ln(E_Pl/2 / E_GUT) = 2pi - ln 2 = 5.590 and
%%   r_lock = K - 2pi + ln 2 = 29.410.  That relation is what
%%   reconciles A2's "lock at r = K" with A10's "lock crossing
%%   at r = 29.4"; both name 3.85 TeV.
%%   The plate is pure TikZ and adds NO asset to the arXiv
%%   package.  Caption ABOVE, per the frozen float standard.
%%   The drawn taper is e^-r/13.64 -- the true e^-r slowed so
%%   the throat stays visible -- and the plate says so.
%%   NEW PROSE: the plate's labels and its caption, listed in
%%   the report for the author to review or rewrite.
% ------------------------------------------------------------
% ------------------------------------------------------------
%% v15.10, August 19, 2026.  The bulk showcase plate moved earlier.
%%   AUTHOR'S PLACEMENT: the shared plate goes from showcase
%%   page 9 to page 8.  Every showcase unit is one page, so
%%   this is a pure relocation of the delimited block: cut from
%%   where it sat, pasted immediately above the unit that held
%%   the target page.  NOT ONE BYTE OF THE BLOCK CHANGES -- the
%%   rider asserts its sha256 before and after, and the six
%%   papers that carry it are re-checked for a single hash.
%%   Nothing outside the front-matter showcase run moves, and
%%   the plate is unnumbered, so no figure or table number and
%%   no cross-reference changes.
%%   NEW PROSE: none.
% ------------------------------------------------------------
%% ---- vv15.11 (August 19, 2026) ----
%% The front-matter showcase pages given breathing room.
%% A showcase page's free height is now distributed by weighted stretch
%% glue -- a sixth at the top, a third above the plate, a third below it,
%% a sixth at the foot -- instead of pooling above the headline and below
%% the caption, which left a broad band of white at each end of the page
%% and crowded the three elements into the middle third.
%% Three source forms were found in the cohort and each is handled.  A
%% minipage frame becomes a minipage of EXPLICIT height with the [s]
%% inner alignment, because a minipage is a vbox of natural height and
%% glue inside one does not stretch; the height is \textheight minus
%% \topskip.  A \begingroup frame keeps its page-level glue and changes
%% only the weights.  An unframed page -- there were eight, five of them
%% showcase tables carrying a longtable, which cannot live inside a
%% minipage -- gains page-level glue for the first time, which is why it
%% had been printing hard against the top of the text block.
%% 4 showcase pages in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
%% And the showcase headlines brought to one rendered form.
%% The house form is the style file's own default, \Large\bfseries\sffamily,
%% which is what \GGShowcaseHead sets when it is called with no optional
%% argument, and which forty-three of the cohort's fifty-seven showcase
%% headlines already used.  Fourteen did not, and the difference was
%% visible to a reader turning pages: four headlines printed at \huge,
%% half again the house height, and three at \large, smaller than it.
%% 2 headlines in this paper.  Only the size changes; the \boldmath in
%% a headline that carries math is kept, the subtitle under a headline
%% keeps its own smaller size, and \GGShowcaseHeadSkip, which is vertical
%% space rather than type, is left exactly as each paper sets it.
\documentclass[11pt,letterpaper]{article}
\usepackage{ucla-arisaka-gg-11}
\providecommand{\PaperPFour}{\textsf{GG-P4-Real}}% Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFour, so this providecommand takes effect)
\providecommand{\PaperPFive}{\textsf{GG-P5-CPT}}% Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFive, so this providecommand takes effect)
\usepackage{indentfirst}
%% v11.28: \LaTeX shim dropped -- gg-11 Acknowledgments use literal 'LaTeX'.
% Compatibility shim: gg-6 dropped \e from gg-4.
\providecommand{\e}{\ensuremath{\mathrm{e}}}
\usepackage{pdflscape}

\PaperSeries{GG}
%   v9.9 (June 7, 2026; second same-day amendment): AX-4 name fixed to
%        A1's canonical "Maximum Symmetry, Minimum Freedom" (2 sites).
%   v9.9 (June 7, 2026; amended same day): canon spot-fix TGG wording
%        (geometry-gauge -> geometric-gauge, 3 sites).
%   v9.9 (June 7, 2026): GRIP-In / GRIP-Out regime vocabulary imported
%        from A6-GRIP v9.5 (def:GRIPin, def:GRIPout).  A9 hosts BOTH
%        regimes and now distinguishes them on the page: the electroweak
%        vacuum selection (Wilson-Hosotani branch, lock-seam vev) is the
%        boundary sector's GRIP-Out event -- it created the Yukawa
%        landscape (cf. A11 T-Electroweak-Vacuum) -- while ALL flavor
%        relaxation (tau->mu->e, quark ladder, valley of beta stability)
%        is GRIP-In on that fixed landscape (author directive: "Weak
%        interaction in A9-SM is GRIP-In").  Sentence enabled: "the
%        electroweak GRIP-Out built the landscape once; the weak GRIP-In
%        has been walking it ever since."  Edits: regime paragraph in
%        the Gi-4 runtime-closure subsection; Wilson-Hosotani subsection
%        labeled as the GRIP-Out event; weak subsection ("GRIP descent"
%        -> GRIP-In + disambiguation paragraph incl. GSSB-gate and
%        collider-production clarifications); S42.6 lepton-cascade and
%        Figure 9 caption regime labels; gga-refinement + electron-(a)
%        GRIP-In termination phrasing; keywords; macros (A11/A6 form).
%   v9.8 (June 6, 2026): NEW dedicated subsection at the close of the
%        universal-identity section: "Why the weak interaction exists:
%        GRIP ending at the GGA, realized in Nature."  The weak charged
%        current categorized as the runtime's relaxation channel -- the
%        unique implementer of inter-basin flavor descent -- with the
%        existence question (Rabi's "Who ordered that?"; the question
%        posed by generations since) answered structurally: a
%        constitution that mandates unique ground states (GFF-ST7,
%        def:GGA) must mandate a road to them.  Examples: the lepton and
%        quark ladders, the free neutron, the valley of beta stability
%        as a literal Morse gradient flow, the solar pp clock, the BBN
%        n/p fossil, conditional-GGA tracking in neutron stars.  Also:
%        why the weak force must be WEAK (the G_F tempo argument), the
%        GIM no-shortcut clause, the CKM adjacency gearing, and the
%        chirality rhyme (fenced as open).
%   v9.7 (June 6, 2026): GA/GGA language completed paper-wide.
%        (1) Quark cascade-depth section: five levels recast as basins
%            (GAs); depth-4 u/d = the quark-sector GGA; decay chains =
%            relaxation of non-ground GAs; proton as baryonic GGA via
%            confinement; "Morse saddles" -> "GSSB gates".
%        (2) Majorana-neutrino section: nu3/nu2 = non-ground GAs
%            (portal-suppressed relaxation), nu1 = GGA; NEW structural
%            falsifier: the depth assignment REQUIRES normal mass
%            ordering; arrow-vs-clock distinction (GGA fixes direction,
%            portals fix rate) explains why non-ground neutrinos persist.
%        (3) S58(c): "unique selected GA" -> GGA; Conclusions summit
%            passage and two runtime-export passages aligned.
%   v9.6 (June 6, 2026): S42.6 "Gi-4 cascade-depth reading for the charged
%        leptons" corrected on the GGA reading: the v9.5- assignment
%        (tau=DGI, mu=GSSB, e=GA) was wrong -- tau and mu are perfect GAs
%        (index-0 minima of their flavor strata; hence sharply measurable
%        masses) but NON-GROUND, while the electron is the GGA. Boxed
%        assignment Eq., post-box text, physics-intuition remark, and
%        Figure (fig:grip-cascade-cp2) all recast; the Morse skeleton,
%        Bogomolnyi actions (2pi, pi), and the N=3 closure S_tot=3pi are
%        untouched. Quark/neutrino cascade-depth sections flagged for the
%        same sweep in a future revision.
%   v9.5 (June 6, 2026): GGA (Ground Geometric Attractor) introduced into
%        Part XII, S(universal-identity): "why every electron is the same"
%        now answered by the GGA (the unique deepest GA, A6-GRIP v9.3
%        Definition def:GGA; A11-Origin T-GGA-as-Ground-State), not by GA
%        alone.  New subsection "The ground-state refinement"; the three
%        electron questions, the seven-layer survey, and the closing
%        principle restated on the GGA; muon/tau decay vs electron
%        stability sharpened as non-ground vs ground attractors.
%        \GGA + \AEleven macros added; ArisakaA11Origin added to the
%        bibliography.  No numerical content changed.
% ============================================================
% ------------------------------------------------------------
% Stage 2A, same day.  The float standard, part one: captions.
%   48 captions printed below the 260-character floor.  Each was
%   EXTENDED and none was rewritten -- the existing sentence is
%   unchanged byte for byte and the new text is appended, so the
%   difference is auditable rather than asserted.
%   ALL 48 EXTENSIONS ARE NEW PROSE and are listed by label in the
%   report.  No extension introduces a numerical claim; every number
%   appearing in one is already printed in the table it describes.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 2B, same day.  The float standard, part two: bare grids.
%   14 tabular grids that sat in bare \begin{center} blocks are now
%   table floats at [htbp], each with a caption above and a label.
%   That clears both section-8 counts at once: a grid inside a float
%   is neither uncaptioned nor bare-and-unbreakable.
%   ALL 15 CAPTIONS ARE NEW PROSE, listed by label in the report.
%   NOT promoted: the 18-row audit table on the Executive Summary
%   showcase page.  That page is a \clearpage-isolated full-page
%   minipage holding the table and its plate together by
%   construction; a float cannot be pinned to a page.  It gets a
%   \captionof{table} and is the one named exemption.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 2C, same day.  Plain-English Part closes.
%   14 closes, one per content Part (I-XIV).  The three back-matter
%   Parts -- Technical Appendices, Common Objections, References --
%   get none, per A5's rule.
%   ALL FOURTEEN ARE NEW PROSE IN THE AUTHOR'S VOICE and are listed
%   by Part and by page in the report, for review or rewriting.
%   A9's twelve \GGteachbox calls are a paper-local \newcommand used
%   only inside the Technical Appendices; a different device, left
%   alone.  \GGteachboxpart count is 0.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 2D, same day.  THE ROADMAP REPAIR, and the List of Tables.
%   A9 has SEVENTEEN Parts.  Its own roadmap said sixteen, and the
%   Part it left out was Part VIII, the Majorana door and
%   leptogenesis -- thirty-seven pages of the paper.  Three sites
%   agreed with each other and disagreed with the document:
%     * fig:org-flowchart had 16 nodes, no leptogenesis node, and
%       every printed Part number from VIII down one too low;
%     * its caption described the same 16 and put the back matter
%       at Parts XIV-XVI instead of XV-XVII;
%     * tab:roadmap-full had 16 rows.  Its \ref calls printed the
%       CORRECT numbers, which is why no gate ever caught this:
%       nothing was undefined and every printed number was true.
%   Repaired against the document (section 1.2: the side argued at
%   greater length wins, and the Part is 37 pages against one node).
%   12 'sixteen Parts' -> 'seventeen Parts'; one 'fifteen further'
%   -> 'sixteen further'.  The plate grew 18mm, so its resizebox
%   went 0.975 -> 0.901 of \textwidth to keep its printed height.
%   NEW PROSE: the leptogenesis node, its roadmap-table row, and one
%   clause in the caption.  Everything else is Part-number
%   arithmetic and is a repair, not writing.
%   Also: \GGListOfTables added -- 58 table floats and 18
%   longtables had no List of Tables.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 2E, same day.  Short caption forms.
%   A List of Figures or Tables entry uses the FULL caption unless a
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%   Every caption that lacked a short form now has one: the FIRST
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%   boundary if over 150 printed characters.  NO PROSE WAS WRITTEN
%   IN THIS STAGE -- every short form is a prefix of text the
%   caption already carried.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 2F, same day.  The roadmap page, finished.
%   * The BODY WALK of sec:roadmap-A9 omitted the same Part the
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%   THIS SHORTENS A CAPTION THE AUTHOR WROTE.  Reversible in one
%   edit; batched as a question at the end of the report.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 5, same day.  Integrity.  The one uncited companion.
%   A9 carried 21 of the 22 cohort papers.  The missing one was
%   P6-Proton, and A9 is in DEBT to it: the paragraph headed 'A
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%   NOT done: the outward coherence map records what A9 IMPORTS,
%   and the (H2) discharge runs the other way.  Adding an import row
%   would reverse the direction of the dependency.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 4, same day.  Palette.
%   palette_fix run over 14 pictures: box fills above !18
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%   normalized.  It rewrote no comment here -- the A10 defect did
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% ------------------------------------------------------------
% Stage 3, same day.  Figures MEASURED; nothing re-placed.
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%   This is exactly the A4 -> Letter consequence of section 5: the
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%   leaves above-house alone.  A9's plates were placed by
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%   The cohort map at 7.04pt is exempt by section 8.
%   NOT MEASURED, reported not guessed: fig:org-flowchart and
%   fig:abbrev-flow (probe spills to 2 pages -- the flowchart was
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% ------------------------------------------------------------
% v20.72 -> v15.0  Stage 1 of the v15 migration.  August 15, 2026.
%   The version number falls because v20.x was never the v20
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%   NO PROSE WAS WRITTEN IN THIS STAGE.
% ============================================================
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% --- Part-aware running header (right side shows current Part as a live pointer) ---
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%% ===================================================================
%%  A9-SM v20.53  (July 31, 2026)  -- the two threes, the residue, and
%%  a notation note.
%%  (1) The status remark still asserted that colour and generations
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%%      on a guess.
%% ===================================================================
%% --- Stage 5: the appendix alphabet must reach beyond Z. ------------
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%% ---------------------------------------------------------------------
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\hypersetup{
 pdftitle = {Arisaka-GG-A9-SM: The Standard Model as Theorem of \GGTheory{} -- Its 28 Parameters as Observations of One Six-Dimensional Geometry},
 pdfauthor = {Katsushi Arisaka},
 pdfsubject = {Geometric origin of the Standard Model: a canonical derivation of the Standard-Model parameter set from the \OGNfive{} ledger and the \GRIP{} cascade, with an explicit per-parameter triage.},
 pdfkeywords= {\GGTheory{}, \GG{}, \BIsix{}, \OGNfive{}, \GDOS{}, \GiFour{}, \QiFour{}, \GRIP{}, \TGG{}, \CCI{}, \GFF{}, Wilson--Hosotani mechanism, Standard Model, 28 parameters, fermion masses, CKM, PMNS, Majorana phases, strong CP, alpha inverse 137, sin squared theta_W, Higgs vev, Q_B}
}

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%% ---- v20.22 (July 9, 2026) -- C1/A9/A10 consistency reconciliation ----
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%% -----------------------------------------------------------------------------
\begin{document}
% ---- v20.54 (August 2, 2026) ----
% 'Two fundamental questions, one geometric answer' gains the
% subsection 'The third instance: the codebook sorted by the two
% fibres' (before the closing punch line): integers / rational
% partitions / lattice phases as circle-side (exact, unable to
% run), masses / scales as interval-side (dyadic overlaps,
% radial suppressions, vev, lock); the quark-floor formula as
% the anatomy in miniature; P3's electron sentence generalised
% ('every SM parameter is a holonomy-class label wearing a
% descent address'); why the codebook closes with no dial. Part
% of the cohort-completing sweep.


%% Defensive paragraph-style assertion (v6.8.6, May 13, 2026):
%% ucla-arisaka-gg-6.sty already sets \parskip = 0pt and
%% \parindent = 1.5em globally.  Reasserting here at \begin{document}
%% protects against any local override that may leak from the
%% renewed `abstract' environment (which dispatches to `quotation')
%% or from other library-default environments.  No-extra-line-space
%% + standard 1.5em first-line indent is the body convention.
\setlength{\parskip}{0pt}
\setlength{\parindent}{1.5em}

%% ===============================================================
%%  PAGE 1: TITLE + ABSTRACT (canonical L9 form)
%% ===============================================================
\GGTitlePage
 {The Standard Model as Theorem of \GGTheory{}}
 {Its 28 Parameters as Observations of One Six-Dimensional Geometry}
  {June 28, 2026}
\thispagestyle{empty}

\vspace{6pt}
\begin{abstract}
\noindent
The Standard Model is the most successful theory ever built and the most unsatisfying: it predicts experiment to extraordinary precision, yet cannot say where its own numbers come from. Twenty-eight quantities --- particle masses, force strengths, mixing angles --- must be measured and written in by hand.

This paper argues that those numbers are not free choices of nature but \emph{observations}. Nothing in nature is ever truly isolated --- not even a single particle --- so the four-dimensional world we measure is a projection of a deeper six-dimensional one, fixed once and carrying no adjustable dials. Measurement is that projection in action, and the twenty-eight quantities are not dials set by hand but values it must return: where the textbook sees accidents, the geometry sees theorems. For fifty years they were treated as inputs; the claim here is that they were outputs all along.

Some follow from the geometry outright; others once a single named step is supplied; the rest are marked honestly as still open, and the paper says which it has earned and which it has not. Crucially, nothing is tuned after the fact: because one short integer ledger must serve all twenty-eight at once, the result is forced, not fitted --- propositions descending from a fixed constitution in the spirit of Euclid, not an expression picked from a free alphabet.

If the argument holds, a deeper question becomes fair: why these values, why four forces, why every electron is identical? The paper reads all this as one statement of uniqueness --- a world that is as it is not by luck, but because the geometry allows no other.
\end{abstract}




\begin{ExecutiveSummary}
\markboth{Executive Summary}{Executive Summary}%

\paragraph{The problem.}
The Standard Model carries twenty-eight numbers it cannot explain: three gauge couplings, two Higgs-sector parameters, the masses of the quarks and charged leptons, the neutrino mass scale, four CKM and four PMNS mixing parameters, three Majorana phases, and the single strong-CP angle --- organized by one integer, $N=3$. In the textbook these are inputs to be measured. This paper is the particle-physics deployment of the \GGTheory{} program, and its claim is that each of the twenty-eight is instead an \emph{output} of geometry.

\paragraph{The central recognition.}
The reason those numbers can be outputs at all is a change in what they are. No physical system is closed --- not even a single particle --- so the four-dimensional world we measure is a projection (\OGGEP{}) of a six-dimensional bulk, and that projection \emph{is} quantum observation. The twenty-eight parameters are therefore registered observables of one open-system projection, governed by the boundary law \GDOS{} and forced by the \GiFour{} and \QiFour{} pillars; not one is a free input. That last sentence is an \emph{ontological} statement about what the twenty-eight parameters \emph{are}, not a claim that all have been \emph{computed} here: the paper's honest derivation status is the six/fourteen/eight triage of \S\ref{sec:codebook-provenance} --- six Derived, fourteen conditional on a single named lemma each, and eight still Asserted. For a century no one asked whether the parameters might be such observables; that question, and its affirmative answer, is the central claim of \S\ref{sec:central-claim} (Figure~\ref{fig:central-claim}).

\paragraph{The single input.}
Everything rests on one fixed starting point with no adjustable numbers: a six-dimensional bulk --- ordinary four-dimensional spacetime, plus a radial fifth dimension that acts as an energy ladder and a small curled-up sixth dimension --- together with a short integer ledger, the \OGNfive{} ledger $(M,N,L;J,K) = (1,3,5;7,35)$, and the five dimensional constants of nature. The five integers are not adjusted but \emph{forced} --- the unique tuple that survives the anomaly-free \BIsix{} closure of the bulk (\AThree{}), each carrying a definite physical meaning ($M=1$ the single tensor seam, $L=5$ the $\mathrm{SU}(5)$ unified block, $N=3$ the color triplicity, $J=7$ the odd closure index, $K=35=LJ=M^2+N^2+L^2$ the master lock), pinned upstream and never searched. Only two transcendental constants ever enter the formulae, one native to each compact bulk direction: $e$, the base of the exponential redshift that runs along the radial fifth (scale) dimension, and $\pi$, the angle of the compact sixth (gauge) circle --- so every exponential below is a scale factor and every $\pi$ a gauge holonomy or the volume it builds. Once these are set, nothing further can be tuned; every later result is forced.

\paragraph{The two faces of $N=3$.}
One integer deserves a word up front, because it is where a skeptical reader's eye settles. In the ledger $N=3$ is the \textbf{color triplicity} of the strong force --- the three colors of $\mathrm{SU}(3)_c$, locked by the \BIsix{} anomaly trace-ratio $(L-N)/L = 2/5$ in \AThree{}, and not, at root, the number of families. That the fermions then come in exactly \emph{three generations} is a separate and much harder statement, proved --- conditionally --- in the companion \PaperPTwo{} by open-system \GDOS{}/\GRIP{} reasoning: a CP lower bound $N \ge 3$ met by a finite-action upper bound $N \le 3$, a two-sided pincer that closed-system anomaly cancellation provably cannot supply and that, to our knowledge, had not been derived before. Where $N$ enters a colorless fermion count in what follows, it does so through this generation face, licensed by that theorem.

\paragraph{The mechanism, in one chain.}
A single chain of theorems, inherited from the foundation papers, turns the geometry into the parameter list:
\begin{equation}
\boxed{
\text{\BIsix{}} \;\xrightarrow{\;\T{1}\;}\; \TGG \;\xrightarrow{\;\T{2}\;}\; \CCI \;\xrightarrow{\;\T{3}\;}\; \GFF \;\xrightarrow{\;\T{4}\;}\; \GRIP \;\Longrightarrow\; \text{28-parameter codebook.}
}
\end{equation}
Read left to right: an anomaly-free closure condition (\BIsix{}) feeds four geometric pillars --- total geometric--gauge conservation (\TGG{}), the bulk-to-boundary descent (\CCI{}), the scalar energy landscape (\GFF{}), and the selection cascade (\GRIP{}) that decides which valley of that landscape is actually observed --- and the output is the twenty-eight-entry codebook. Every parameter is a specific readout at a specific depth of this cascade.

\paragraph{The headline results.}
The matches are tight and use no fitted prefactor. The fine-structure constant emerges as $\alpha^{-1}(0) = 2^J + N^2 = 137$; the weak mixing angle as $\sin^2\theta_W = 25/108$; the strong coupling as $\alpha_s^{-1}(M_Z) = \pi e$; the Higgs self-coupling as $\lambda_H = 1/8$; the top-quark Yukawa as $y_t = 1$ at the matching scale; and the Higgs vacuum value as $v \approx 246$~GeV, each within a fraction of a percent of measurement.

\paragraph{How confident, parameter by parameter.}
In the structural-honesty discipline of \AThree{} the paper states the status of each number in one triage. \textbf{Six} of the twenty-eight are \emph{Derived} --- fixed with no continuous free parameter, the weak mixing angle end-to-end as a theorem; \textbf{fourteen} are \emph{Derived (conditional)}, forced once a single named lemma is supplied; and \textbf{eight} remain \emph{Asserted}, each tagged with the calculation that would close it. This six/fourteen/eight headcount is fixed by one canonical table and quoted unchanged wherever it appears.

\paragraph{Why this is a derivation and not a fit.}
What separates a derivation from a fit is not the precision of the numbers but the absence of adjustable inputs. The codebook introduces zero continuous freedom of its own and still reproduces the measured spectrum to four-significant-figure precision --- a combination no earlier framework has achieved, now sharpened into dated falsifiers (the 3.85~TeV lock cluster, the 8.1~$\mu$eV Kaxion, exact proton stability) decidable by 2040. The Standard Model is recast not as the bottom layer of physics but as the boundary reading of a deeper bulk geometry.

\paragraph{Striking contrast to supersymmetry and string theory.}
The departure from the two best-known unification programs is sharp. Supersymmetry, in its minimal form, \emph{adds} more than a hundred new free parameters to the Standard Model rather than removing any: it posts a partner for every known particle, but not one of the twenty-eight numbers follows from it. String theory begins from a nearly unique action, yet its six extra dimensions can be compactified in an astronomical number of ways --- the ``landscape'' of perhaps $10^{500}$ vacua --- and which vacuum we inhabit, hence which low-energy constants we measure, is left undetermined; the theory accommodates almost any Standard Model rather than forcing ours. In both cases the constants are left unforced --- supersymmetry by adding freedom, the landscape by distributing it across many vacua --- where the aim here is instead to force them. \GGTheory{} takes the opposite stance. Its six-dimensional geometry is fixed once, carries no adjustable dials and offers no choice of vacuum, and is therefore \emph{obliged} to produce one definite value for each parameter --- which is exactly why it can derive them, and why a single wrong number can kill it. Deriving the Standard-Model parameters and admitting no free dials are, in the end, the same property seen from two sides.

\paragraph{A philosophical conclusion.}
Once every continuous knob is gone, the oldest questions change character. Why these forces, why these masses, why is every electron exactly like every other? The closing move of the paper is to read universal identity and the existence of exactly four forces as two faces of one uniqueness statement on the same six-dimensional geometry --- not chosen, not lucky, not a brute fact, but forced by the requirement that the world be self-consistent at all. Throughout, the paper marks exactly where physics ends and interpretation begins, and leaves the final step to the reader.

\end{ExecutiveSummary}


\GGKeywords{\GGTheory{}; \GGConstitution{}; \GG{} bulk; \GRIPin{} (fixed-landscape regime); \GRIPout{} (landscape-reorganizing regime); \BIsix{} anomaly identity;
 \OGNfive{} integer ledger; \GDOS{} boundary law; \GiFour{} pillars
 (\TGG, \CCI, \GFF, \GRIP); \QiFour{} pillars and Arisaka equation;
 $Q_B$ as \GUfive{} anchor; \GRIP{} cascade $\DGI \to \GSSB \to \GA$;
 family number $N = 3$ as theorem; cascade-depth Yukawa
 hierarchy on $\mathbb{C}P^2$; Hopkins--Singer descent integrality; 28 Standard-Model
 parameters; strong-CP suppression; proton stability.}

%% ---- v15.3: the savebox for the exposure page added below -------
%% Table 70's tabular is filled ONCE here and used twice -- on the
%% new showcase page and in Part XIII -- so the two cannot drift.
\newsavebox{\GGscFalsMap}
%% \setstretch{1} IS NOT DECORATION.  A9's front matter runs at
%% \baselinestretch 1.15 and its body at 1, so a savebox filled
%% here inherits 13.8 pt of \small leading where the in-paper site
%% uses 12.0 -- MEASURED with \typeout at both sites.  Every
%% p-column cell that wraps then gains 1.8 pt per line, and this
%% table grew 47.9 pt and pushed a paragraph off its page.  The box
%% is therefore filled at the BODY's spacing, which is the canonical
%% one: the two copies print identically and neither moves.
\sbox{\GGscFalsMap}{\setstretch{1}\small\begin{tabular}{p{0.115\textwidth}p{0.175\textwidth}p{0.20\textwidth}p{0.135\textwidth}p{0.215\textwidth}}
\toprule
\textbf{Channel} & \textbf{Origin in \GG{}/\GDOS{}} & \textbf{Quantitative prediction} & \textbf{Carried by} & \textbf{What counts as failure} \\
\midrule
F-SM-1 Lock cluster & Lock-scale throat, $K=35$ & graviton $+$ $g^{*}$ $+$ $Z'_B$ at $\Mlock\simeq3.85$~TeV & \PaperPOne{} & Tier 1: spin-1 pair excluded at $3.8$--$4.0$~TeV (HL-LHC, $\lesssim2$~ab$^{-1}$); Tier 2: spin-2 anchor at FCC-hh (T-KK-2c) \\[4pt]
F-SM-2 Kaxion & Compact $S^1_\theta$, $\UoneQi{}$ & $m_a=8.1~\mu$eV ($1.95$~GHz), saturating $\Omega_{\mathrm{dm}}$ & \PaperCOne{}, \ATen{} & line excluded at derived coupling, or relic $\neq 1/\pi-1/(2\pi^2)$ \\[4pt]
F-SM-3a--3e Neutrinos & Rank-2 Majorana block $+$ lepton projector & NO ($3a$), $m_{\beta\beta}\!\simeq\!1.8$~meV ($3b$), $\sum m_\nu\!\simeq\!59$~meV ($3c$), $\delta_{\mathrm{PMNS}}\!\simeq\!196^\circ$ ($3d$), upper octant ($3e$) & \ANine{} (in paper) & converged $\nu$ data incompatible with the package \\[4pt]
F-SM-4 Proton & exact global $B$, massive $\UoneB{}$ & $\tau_p=\infty$; $\Delta B=1,2$ forbidden & \ANine{}, \AFive{} & any single $\Delta B=1$ or $\Delta B=2$ event \\[4pt]
F-SM-5 Fifth force & Stueckelberg $\UoneB{}$ & Yukawa, $\lambda=0.2~\mu$m$\,(\mathrm{eV}/m_B)$ & \PaperPOne{} & long-range baryonic force, or full $(g_B,m_B)$ exclusion \\[4pt]
F-SM-6 Dark energy & GA fixed point of the \GRIP{} cascade & $w=-1$ exact, two-sided ($|1{+}w|\lesssim10^{-91}$) & \ASix{} (Lemma~DE-1), \ATen{} (T-Cosmos-DE$'$) & any confirmed $w\neq-1$, either sign (e.g.\ a converged evolving-dark-energy detection) \\[4pt]
F-SM-7 Leptogenesis & Majorana door at \Mlock{}; parity $\alpha_{31}=\pi$ & $\etaB\approx7\times10^{-10}$, band $(2.4$--$22)\times10^{-10}$ & \ANine{} (Part~\ref{part:lepto}), \ATen{} & a finite-temperature transport computation returning the $N$-width or $g^{2}T$ regulator class, or an unclosable window deficit \\
\bottomrule
\end{tabular}}
\clearpage
\begin{figure}[p]
\centering
{\Large\bfseries\sffamily 28 Free Parameters, One Geometry\par}
\vspace{1.0em}

{\small
\setlength{\tabcolsep}{4pt}%
\begin{tabular}{@{}llrrrr@{}}
\toprule
Quantity & \GG{}-Theory closed form & First order & Measured & Dev & Pull\\
\multicolumn{6}{@{}l}{\bfseries\itshape Five primary gauge--Higgs attractor coordinates}\\
\hspace{1.5em}$\alpha^{-1}(M_Z)$ & $2^J = 128$ & $128.0$ & $127.930(8)$ & $+5.5\mathrm{e}{-}4$ & $+0.99$\\
\hspace{1.5em}$\sin^2\theta_W(M_Z)$ & $L^2/4N^3 = 25/108$ & $0.23148$ & $0.23155(4)$ & $-3.0\mathrm{e}{-}4$ & $-1.70$\\
\hspace{1.5em}$\alpha_s(M_Z)$ & $1/\pi e$ & $0.11710$ & $0.1180(9)$ & $-7.6\mathrm{e}{-}3$ & $-0.70$\\
\hspace{1.5em}$v$ \,[GeV] & $\Mlock\,(2/L)^N$ & $246.33$ & $246.220$ & $+4.5\mathrm{e}{-}4$ & $+1.00$\\
\hspace{1.5em}$m_H$ \,[GeV] & $\sqrt{2\lambda_H}\,v,\ \lambda_H = 1/8$ & $123.12$ & $125.10(9)$ & $-1.6\mathrm{e}{-}2$ & $-0.95$\\
\addlinespace
\multicolumn{6}{@{}l}{\bfseries\itshape Four derived electroweak observables}\\
\hspace{1.5em}$G_F$ \,[$10^{-5}\,$GeV$^{-2}$] & $1/(\sqrt2\,v^2)$ & $1.1653$ & $1.16638$ & $-9.1\mathrm{e}{-}4$ & $-1.00$\\
\hspace{1.5em}$\alpha^{-1}(0)$ & $2^J + N^2 = 128 + 9$ & $137.0$ & $137.036$ & $-2.6\mathrm{e}{-}4$ & $-1.00$\\
\hspace{1.5em}$m_W$ \,[GeV] & $\tfrac12\,g_2 v$ & $80.21$ & $80.369$ & $-2.0\mathrm{e}{-}3$ & $-0.99$\\
\hspace{1.5em}$m_Z$ \,[GeV] & $\tfrac12\,v\sqrt{g_2^2+g_Y^2}$ & $91.50$ & $91.188$ & $+3.4\mathrm{e}{-}3$ & $+1.01$\\
\addlinespace
\multicolumn{6}{@{}l}{\bfseries\itshape Three charged-lepton masses}\\
\hspace{1.5em}$m_e$ \,[MeV] & $(v/\sqrt2)\,N/(L^3\,2^{J+L+1})$ & $0.51030$ & $0.51100$ & $-1.4\mathrm{e}{-}3$ & $-0.88$\\
\hspace{1.5em}$m_\mu$ \,[MeV] & $(v/\sqrt2)\,L/2^{J+L+1}$ & $106.31$ & $105.658$ & $+6.2\mathrm{e}{-}3$ & $+0.98$\\
\hspace{1.5em}$m_\tau$ \,[GeV] & $(v/\sqrt2)\,NJ/2^{J+N+1}$ & $1.7861$ & $1.77693$ & $+5.1\mathrm{e}{-}3$ & $+1.09$\\
\addlinespace
\multicolumn{6}{@{}l}{\bfseries\itshape Cabibbo angle (first CKM mixing)}\\
\hspace{1.5em}$\sin\theta_{12}^{\rm CKM}$ & $\bigl|e^{-\pi/2} - e^{i3\pi/5}e^{-\pi}\bigr|$ & $0.22502$ & $0.22501$ & $+3.8\mathrm{e}{-}5$ & $+0.01$\\
\bottomrule
\end{tabular}}

\vspace{1.0em}

\includegraphics[width=\textwidth,height=0.405\textheight,keepaspectratio]{a9_dev_plot_linear_first8_hook.png}
\par
{\setlength{\abovecaptionskip}{5pt}%
\captionof{figure}[The thirteen most sharply benchmarked parameters, at a glance.]{\textbf{The
thirteen most sharply benchmarked, at a glance} --- not free inputs, but
observables of one open-system projection (\S\ref{sec:central-claim}).
The full 28-parameter audit follows on the next page. The bars are relative deviations from the benchmark values, and the thirteen were selected for the sharpness of their measurement rather than for the closeness of their agreement. The figure is a summary and not an argument: the derivation of every entry, with its status and its uncertainty, is in the Part named beside it.}\label{fig:showcase-hook}}
\end{figure}
\clearpage

%% ---- v20.23 showcase pages 5-6: the full audit, then unification ----
%% 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
\par\vspace*{\stretch{2}}
\includegraphics[width=\textwidth,height=0.82\textheight,keepaspectratio]{a9_dev_plot_symlog.png}
\par
\par\vspace*{\stretch{2}}
{\setlength{\abovecaptionskip}{5pt}%
\captionof{figure}[The full 28-parameter audit in one panel (showcase; full version in Part~II).]{\textbf{The
full 28-parameter audit in one panel}: all twenty-two directly
benchmarked Standard-Model parameters on a symmetric-log deviation
axis --- first order (orange) and second-order QD (blue) against
measurement, nested precision bands at
\(10^{-4}\), \(10^{-3}\), \(10^{-2}\). Full version:
Figure~\ref{fig:dev-symlog} in Part~II.}\label{fig:showcase-symlog}}
\par\vspace*{\stretch{1}}
\end{minipage}
\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
\par\vspace*{\stretch{2}}
\includegraphics[width=\textwidth,height=0.62\textheight,keepaspectratio]{gg_unification_showcase.pdf}
\par\medskip
{\centering
\(\alpha_1^{-1}(M_Z)=\tfrac{3}{5}\cos^2\theta_W\cdot 2^J=\tfrac{3}{5}\cdot\tfrac{83}{108}\cdot 128=59.02,\qquad
\alpha_2^{-1}(M_Z)=\sin^2\theta_W\cdot 2^J=\tfrac{25}{108}\cdot 128=29.63,\)\\[4pt]
\(\alpha_3^{-1}(M_Z)=1/\alpha_s(M_Z)=\pi e=8.54;\qquad
\alpha_{\rm GUT}^{-1}\approx 41\ \text{at}\ \EGUT=M_{\rm Pl}\,e^{-2\pi}\approx 2.3\times 10^{16}\,\mathrm{GeV}\)\par}
\smallskip
\par\vspace*{\stretch{2}}
{\setlength{\abovecaptionskip}{14pt}%
\captionof{figure}[Gauge-coupling unification (showcase; full discussion in Part~IV).]{\textbf{GG-Theory
gauge-coupling unification across the 6D bulk.} Inverse couplings
\(\alpha_1^{-1}\) (green, hypercharge in SU(5) normalization),
\(\alpha_2^{-1}\) (blue, \(\mathrm{SU}(2)_L\)), and
\(\alpha_3^{-1}\) (coral, \(\mathrm{SU}(3)_C\)) versus the
renormalization-group scale \(\mu\) and the radial coordinate
\(r = \ln(\EGUT/\mu)\), with \(r = 0\) at the unification point.
The boundary values at \(M_Z\) are pinned by the GG-Constitution and
the OGN-5 ledger with the three closed forms displayed above --- no
continuous fitted parameter intervenes anywhere between \(M_Z\) and
\(\EGUT\). Four constitutional scales are marked: the EM scale
\(v = 246\)~GeV, the KK lock-cluster mass
\(\Mlock = \tfrac12 M_{\rm Pl}\,e^{-35} \approx 3.85\)~TeV, the
unification scale \(\EGUT = M_{\rm Pl}\,e^{-2\pi} \approx
2.3\times10^{16}\)~GeV, and the Planck scale
\(\EPl \approx 1.2\times10^{19}\)~GeV. Below \(\Mlock\) the
running uses the standard one-loop SM beta functions; above
\(\Mlock\) the KK lock cluster opens and the three couplings
converge to \(\alpha_{\rm GUT}^{-1} \approx 41\) at exactly
\(\EGUT\) (black dot), where the SM gauge-coupling description ends
and the bulk geometry takes over. This three-line meeting is the
headline empirical signature of the 6D ultraviolet completion of the
Standard Model. Full discussion:
Figure~\ref{fig:gg-unification} in Part~IV.}\label{fig:showcase-unification}}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

%% >>>>>>>>>>>>>>>>>> GG-BULK-ONE-DIAGRAM v1.0 >>>>>>>>>>>>>>>>>>
%% Shared showcase plate, byte-identical in A2, A5, A8, A9, A10
%% and P1.  Pure TikZ: it adds no asset to the arXiv package.
%% Every macro it defines is \def'd under the \GGbd prefix and
%% scoped to the picture, so it cannot clash with a host paper.
%% Caption ABOVE, per the frozen float standard.
\clearpage
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
\GGShowcaseHead{A Six-Dimensional Bulk Whose Fifth Dimension Is Scale\\[2pt]
and Whose Sixth Is Phase}
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{The whole program in one diagram, with the scale axis
carrying numbers.} The bulk $\cM_4\times I_r\times S^1_\theta$ is a fiber bundle: over every
point of the Lorentzian stage sit the two \emph{non-spatial} fibers, scale and phase. \AX{2}
(\GGEP) acts \emph{upward} and builds them; \AX{3} (\OGGEP) acts \emph{downward}, and
registration is the projection that erases $(r,\theta)$ --- the origin of quantum behavior.
The vertical axis is calibrated: $\EGUT=\EPl e^{-2\pi}$ at $r=0$, the \OGNfive{} depth
$r=K=35$ at the infrared boundary, and the lock at $r=29.41$, which is $K-2\pi+\ln 2$
exactly.}}
\par\medskip
\par\vspace*{\stretch{2}}
\resizebox{0.95\textwidth}{!}{%
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%% ---- geometry constants ------------------------------------------
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\def\GGbdSc{0.455000}       % cm per unit r  (15.9 cm for r = 0..35)
\def\GGbdXc{6.8}             % throat centre
\def\GGbdWz{4.05}             % drawn half-width at r = 0
\def\GGbdWk{13.64}           % drawn half-width = wz * exp(-r/13.64)
\def\GGbdXR{11.9}            % right annotation column
\def\GGbdY#1{\GGbdTop-\GGbdSc*(#1)}
\def\GGbdW#1{\GGbdWz*exp(-(#1)/\GGbdWk)}

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

%% ================= THE THROAT =====================================
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  plot[domain=0:35,samples=90,variable=\r] ({\GGbdXc-\GGbdW{\r}},{\GGbdY{\r}})
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  \draw[slice] ({\GGbdXc-\w},\y) -- ({\GGbdXc+\w},\y) -- ({\GGbdXc+\w+\sk},{\y+0.45*\sk})
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 {\textbf{the 6D bulk}\\[1pt] $\cM_4\times I_r\times S^1_\theta$};

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

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

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

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

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

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

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

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

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

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

%% ================= BOTTOM: the registered world ======================
\node[box, draw=blue!55!black, fill=blue!10, text width=15.8cm] (bdy) at (\GGbdXc,{\GGbdY{43.4}})
 {\textbf{The registered 4D world: open, and therefore quantum}\ \ (\GDOS)\\[2pt]
  superposition \& interference are the unread phase $S^1_\theta$\\[1pt]
  measurement \& irreversibility are the coarse-grained scale $I_r$};
\end{tikzpicture}}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage
%% <<<<<<<<<<<<<<<<<< GG-BULK-ONE-DIAGRAM v1.0 <<<<<<<<<<<<<<<<<<
%% ---- v15.3 showcase page 4: THE EXPOSURE ------------------
%% A9 already had pages 5-7 and this pass changed none of them; what
%% they lacked was the page every other showcase in the pass ends
%% on.  All three say what the framework delivers; this one says
%% what it costs to be wrong.
%% UNNUMBERED, while A9's own showcase figures are numbered 1-3:
%% numbering this one would take Table 1 and shift all ninety of
%% A9's tables, the defect the author caught in A1 v15.23.
%% -----------------------------------------------------------
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
{\Large\bfseries\sffamily Seven Channels That Could End This,\\[2pt]
and the Number Each One Turns On\par}
\vspace{1.2em}
\par\vspace*{\stretch{2}}
{\setlength{\abovecaptionskip}{5pt}%
\captionof*{table}{\textbf{Where the twenty-eight are exposed.} Seven channels, each with the quantitative prediction it makes, the companion paper that carried it to a number, and the measurement that would end it. A falsification map is only as good as the sharpness of its entries, and these entries are sharp because the downstream companions carried them to numbers: before those papers existed the same table would have been a list of directions rather than of tests. The three pages before this one say what the framework delivers; this one says what it costs to be wrong. In full, with the timeline on which each channel can be decided: Table~\ref{tab:falsification-consolidated}, p.~\pageref{tab:falsification-consolidated}.}}
\par\medskip
\par\vspace*{\stretch{2}}
\usebox{\GGscFalsMap}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage
%% ---- end showcase page 4 ----------------------------------




% --- Front matter: Overview of Contents + List of Figures (cohort gg-11 macros) ---
\GGfrontmatteronepagetrue
\GGOverviewOfContents
\GGListOfFigures
\GGListOfTables

% --- full table of contents ---
\begingroup
\setstretch{1.05}%
\GGTableOfContents
\endgroup

\GGPreface

The Standard Model of particle physics has been called both the most successful theory ever constructed and the most embarrassing. Successful, because every prediction it has made for half a century has been confirmed to extraordinary precision; embarrassing, because it asks the experimenter to supply twenty-eight separate numbers --- particle masses, mixing angles, coupling strengths --- that the theory itself cannot explain. Every textbook opens by asking the reader to accept these numbers as inputs and then derives everything else with great elegance. After three generations of effort, no one has derived the inputs themselves from a deeper principle.

\paragraph{What this paper claims.}
This paper makes the case that they \emph{are} derivable. Building on a six-volume foundation that rebuilds physics from a higher-dimensional bulk geometry --- six dimensions rather than four, with the two extra dimensions curled up so tightly that the world an observer sees is the familiar four-dimensional one --- we show that each of the twenty-eight numbers falls out of the geometry as a definite, calculable quantity, with no adjustable knobs. The fine-structure constant near $137$ appears as $2^7 + 3^2$, where $7$ and $3$ count structures in the geometry. The Higgs energy scale near $246$~GeV appears through a single ``lock factor'' that bridges the enormous gap between the Planck scale and the scale of ordinary matter. The masses of the twelve charged fermions appear as overlap integrals on one mathematical surface. Even a tiny energy quantum, about $0.156$~eV, comes out of the same geometry.

\paragraph{An honest accounting.}
The honest answer, in this first systematic pass at the problem, is that the truth lies between a fit and a complete derivation, and the paper says exactly where. Some of the twenty-eight are fully derived from the geometry; others are derived except for a single, clearly named missing step; and the rest are still open, each tagged with the calculation that would settle it. No number is quietly tuned to match data. The test of an honest framework is not that it answers every question on the first pass, but that it knows which it has answered, which it has half-answered, and which it has not yet touched --- and this paper keeps that ledger in the open.

\paragraph{How to read this paper.}
No reader needs all seventeen Parts at once. The fastest route in is the Executive Summary for the shape of the argument, Part~\ref{part:I} for the twenty-eight-number target and the honesty triage, and the precision-comparison tables at the close of each sector for the numbers themselves. A reader who wants the philosophy can turn to Part~\ref{part:XII}; a reader who wants only the testable claims can read the predictions-and-falsification section. Each result is stated so that it can be checked, and challenged, on its own, against the published Particle Data Group values.

\GGPart[Introduction]{Introduction}{The thesis, the place of \ANine{} in the \GGTheory{} cohort, and the roadmap of the seventeen Parts}
\label{part:I}


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

The Standard Model is the benchmark infrared description that any deeper theory must inherit rather than bypass. Its gauge structure, symmetry breaking, anomaly constraints, running couplings, and flavor organization define the observational target that a more fundamental theory must explain without reopening a large landscape of arbitrary inputs. In the present program the Standard Model is therefore treated not as disposable phenomenology, but as a theorem candidate whose surviving low-energy parameter sheet should descend from a tighter constitutional structure.

This paper is written in that spirit. It treats the Standard Model as an infrared theorem of the \GGTheory{} sequence, sharpened by the runtime order developed in \ASix{}. The purpose of \ANine{} is therefore sharply defined: to derive the full 28-parameter Standard-Model data set, including the neutrino sector and the final topological CP angle, from the written constitutional framework already established in \AOne{}, \ATwo{}, \AFive{}, \ASix{}, and \AThree{} (\citealp{ArisakaA1Constitution}; \citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}; \citealp{ArisakaA6Gi4}; \citealp{ArisakaA3BI6}).

\subsection{The central claim: the twenty-eight parameters are open-system observables}
\label{sec:central-claim}

Before the audit of inherited successes that occupies the rest of this
introduction, it is worth stating in one place, and without hedging, the single
claim on which this paper turns. It is not a claim about a better fit to the
twenty-eight numbers. It is a claim about \emph{what those numbers are}.

\paragraph{Every physical system is open.}
No system in nature is truly closed --- not a gas, not an atom, and not a single
elementary particle. A particle is defined by its interactions, and an interaction
is a coupling to an environment that cannot, even in principle, be switched off; the
isolated particle of the textbooks is a limit that is approached but never reached.
The closed, four-dimensional Lagrangian in which the Standard-Model parameters
appear as free constants is therefore an idealization --- and it is precisely the
idealization that makes those constants look like inputs one is obliged to measure.

\paragraph{Openness forces a projection, and the projection is observation.}
If there is no closed four-dimensional theory, then the four-dimensional world an
observer reports is the image of a deeper bulk under a lawful projection. This is the
content of the third axiom of the \GGConstitution{}, the open-system geometric--gauge
equivalence principle (\OGGEP{}, \AX{3}; \citealp{ArisakaA1Constitution}): every
admissible theory must supply a map from the six-dimensional bulk to the registered
observables on its four-dimensional boundary. The decisive recognition --- absent from
a century of quantum mechanics --- is that this bulk-to-boundary projection \emph{is}
quantum-mechanical observation itself. Measurement is not a postulate appended to a
unitary theory; it is what the \OGGEP{} projection does. The recovery of textbook
quantum mechanics from it is carried out in \AEight{} \citep{ArisakaA8QM}, and its
information-theoretic content in \ASeven{} \citep{ArisakaA7Qi4}.

\paragraph{The projection is a definite law: \GDOS{}.}
\OGGEP{} requires that a projection exist; \AFive{} makes it dynamical and unique. The
geometric dynamics of open systems (\GDOS{}; \citealp{ArisakaA5GDOS}) is the
open-system law whose stationary, attractor configurations are the values an observer
reads off. An observable, in this framework, is not a coefficient in a microscopic
action; it is the value assigned at a stationary seam of a lawful open-system flow.

\paragraph{Therefore the twenty-eight parameters are observables, not free inputs.}
It follows that every one of the twenty-eight Standard-Model parameters --- each gauge
coupling, each mass, each mixing angle, each phase, and the strong-CP angle --- is a
\emph{registered observable} of the same \OGGEP{}/\GDOS{} projection. Not one of them
is a free input. The ``parameter space'' of the Standard Model is not a space of
freedoms at all; it is a list of measurement outcomes of one fixed geometry. This is an
\emph{ontological} statement, and it holds for all twenty-eight regardless of how
completely the present paper succeeds in \emph{computing} each one. The honest
accounting of that computational progress is the six/fourteen/eight provenance triage
of \S\ref{sec:codebook-provenance}; the eight entries still labeled \textbf{Asserted}
are observables whose determining projection has not yet been closed in detail, not
exceptions to the claim.

\paragraph{And the law itself is forced.}
Finally, \GDOS{} is not an arbitrary open-system equation. It is the one consistent
with the two four-pillar structures the constitution erects over it: the geometric
pillars \GiFour{} (\TGG{} $\to$ \CCI{} $\to$ \GFF{} $\to$ \GRIP{};
\citealp{ArisakaA6Gi4}) and the quantum-information pillars \QiFour{}
(\citealp{ArisakaA7Qi4}). Because the projection that produces the observables is
itself forced, the observables it produces are determined.

\begin{figure}[hbt!]
\centering
\resizebox{0.64\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=17mm,
 every node/.style={font=\sffamily\footnotesize, align=center},
 box/.style={draw=black!55, rounded corners, minimum width=92mm, minimum height=11mm},
 arrow/.style={-{Stealth[length=3mm]}, line width=1.10pt},
 elab/.style={font=\sffamily\scriptsize\itshape, fill=white, inner sep=1.5pt}
]
\node[box, fill=violet!10] (open) {\textbf{Every physical system is open}\\ even a single particle cannot be isolated from its environment};
\node[box, fill=blue!8, below of=open] (oggep) {\textbf{\OGGEP{}} (\AX{3}): the lawful projection \;6D bulk $\to$ 4D boundary\\ \textit{this projection is quantum-mechanical observation} \;\; {\scriptsize[\AOne{}, \AEight{}]}};
\node[box, fill=yellow!13, below of=oggep] (gdos) {\textbf{\GDOS{}}: the dynamics of the open-system projection\\ \textit{an observable is the value at a stationary seam} \;\; {\scriptsize[\AFive{}]}};
\node[box, fill=orange!14, line width=1pt, below of=gdos] (obs) {\textbf{The 28 Standard-Model parameters are registered observables}\\ of this one projection --- \textit{not one is a free input}};
\node[box, fill=green!12, below of=obs] (pill) {\textbf{\GDOS{} is forced} by the \GiFour{} pillars {\scriptsize[\ASix{}]} and the \QiFour{} pillars {\scriptsize[\ASeven{}]}};
\node[box, fill=orange!16, line width=1pt, below of=pill] (concl) {$\Rightarrow$ \textbf{the 28 observables are determined}\\ {\footnotesize $2^{7},\ 25/108,\ 1/\pi e,\ e^{-n\pi}$ --- the technical cash-out (rest of this paper)}};
\draw[arrow] (open)  -- node[elab,right]{no closed 4D theory exists}            (oggep);
\draw[arrow] (oggep) -- node[elab,right]{the projection must be a definite law}  (gdos);
\draw[arrow] (gdos)  -- node[elab,right]{apply to the Standard-Model packet}     (obs);
\draw[arrow] (obs)   -- node[elab,right]{and the law is not arbitrary}           (pill);
\draw[arrow] (pill)  -- node[elab,right]{projection forced $\Rightarrow$ observables forced} (concl);
\end{tikzpicture}}
\caption[The central claim of \GGTheory{}: the twenty-eight Standard-Model parameters as open-system observables.]{\textbf{The central claim, as one deductive funnel.} Because no physical system --- not even a single particle --- is closed, the four-dimensional world is a lawful projection (\OGGEP{}, \AX{3}) of the six-dimensional bulk, and that projection is quantum observation itself; its dynamics is the open-system law \GDOS{}. The twenty-eight Standard-Model parameters are therefore \emph{registered observables} of the projection, not free inputs --- and because \GDOS{} is forced by the \GiFour{} and \QiFour{} pillars, those observables are determined. The closed-form values derived in the rest of the paper ($2^{7}$ behind $\alpha^{-1}$, $25/108$ behind $\sin^2\theta_W$, $1/\pi e$ behind $\alpha_s$, the cascade tolls $e^{-n\pi}$ behind the fermion spectrum) are the technical cash-out of this single recognition.}
\label{fig:central-claim}
\end{figure}

Figure~\ref{fig:central-claim} draws this chain as the single deductive funnel it is,
and its conclusion is the reason the rest of the paper can exist. For the hundred years
since the birth of quantum mechanics and the fifty since the Standard Model was
completed, the twenty-eight numbers have been treated as inputs to a closed
four-dimensional theory, and no one asked whether they are instead observables of an
open-system projection from a higher-dimensional bulk. That question, and its
affirmative answer, is the breakthrough of \GGTheory{}. Once it is granted, the dyadic
$2^{7}$ behind $\alpha^{-1}$, the rational $25/108$ behind $\sin^2\theta_W$, the
transcendental $1/\pi e$ behind $\alpha_s$, and the cascade tolls behind the fermion
spectrum cease to be a list of separate mysteries; they become the technical cash-out
of one recognition. The remainder of this paper is that cash-out. A reader who carries
away only the funnel of Figure~\ref{fig:central-claim} already holds the essential
content of the work; the Parts that follow are there to compute, observable by
observable, what the funnel guarantees is computable.

The computation also has a terminal page, and it is worth announcing here.
Part~\ref{part:lagrangian} assembles the complete Standard-Model
Lagrangian --- the object every textbook writes on its first page by decree
--- as the \emph{output} of the descent: each term tagged with the stage of
the export chain that produced it and the register that grades it, and each
absent term (the strong-CP angle, the proton-decay operators, the free
scalar-mass dial) derived as a theorem rather than removed by hand. A reader
who has spent a career treating $\mathcal{L}_{\mathrm{SM}}$ as the starting
axiom may wish to read that Part first, to see what the paper claims the
familiar object actually is: not the first page of the theory but the last.

\subsection{Why the Standard Model remains the benchmark any deeper theory must inherit}

The Standard Model did not arrive all at once. It was assembled, piece by piece, across roughly fifteen years in the middle of the twentieth century, and it has stood essentially unchanged for the half-century since. The electroweak synthesis of Glashow, Weinberg, and Salam (1961--1968)~\citep{Glashow1961,Weinberg1967,Salam1968} unified the electromagnetic and weak interactions under a single $\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y$ gauge symmetry, broken by the Higgs mechanism~\citep{Higgs1964,EnglertBrout1964}; the proof by 't~Hooft and Veltman (1971--1972) that such spontaneously broken gauge theories are renormalizable~\citep{tHooftVeltman1972} turned that synthesis from an ansatz into a predictive quantum field theory; and the discovery of asymptotic freedom by Gross, Wilczek, and Politzer (1973)~\citep{Gross1973,GrossWilczek1973,Politzer1973} established quantum chromodynamics as the theory of the strong interaction. By the mid-1970s the architecture was complete on paper. Almost everything since has been confirmation: the charm quark (1974), the tau lepton (1975), the bottom quark (1977), the $W$ and $Z$ bosons (1983), the top quark (1995), the direct detection of the tau neutrino (2000), and finally the Higgs boson (2012). Each was anticipated before it was found. The one genuinely new empirical fact---that neutrinos oscillate and therefore carry mass, established by Super-Kamiokande (1998) and SNO (2001--2002)~\citep{Fukuda1998,Ahmad2002}---was absorbed by appending a neutrino sector without disturbing the gauge structure. Fifty years on, no experiment has forced a change to the framework itself.

This staying power rests on four properties that any successor must respect.

First, the Standard Model is structurally economical. A handful of local gauge symmetries, three replicated families of chiral matter, and a single scalar doublet organize an immense and otherwise bewildering body of data. The same $\mathrm{SU}(3)_C\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y$ symmetry that governs how quarks bind into a proton also fixes how a muon decays and how the $Z$ boson couples to a neutrino.

Second, it is quantitatively precise to a degree almost unique in science. Precision electroweak observables, flavor-changing processes, QCD jet and scaling data, Higgs production and decay rates, and neutrino-oscillation parameters all sit within one tightly constrained quantitative language; the anomalous magnetic moment of the electron agrees with theory to better than a part in $10^{12}$~\citep{Schwinger1948}. A framework that reproduces this body of measurement must be taken seriously as the correct effective description of everything we have so far observed.

Third, it is internally constrained. Gauge invariance, the cancellation of chiral anomalies family by family, the renormalizability of the infrared description, and the experimentally verified pattern of electric charges leave astonishingly little room for arbitrary modification. The charges of the quarks and leptons are not adjustable inputs: they are forced by the requirement that the theory be quantum-mechanically consistent at all.

Fourth, it is structurally cumulative. The Standard Model is not one isolated idea but a tightly interlocked synthesis---gauge symmetry, spontaneous symmetry breaking, asymptotic freedom, and flavor mixing---in which the parts reinforce one another. One cannot excise a piece without the whole edifice losing its predictivity.

Most striking of all is the framework's record of \emph{prediction confirmed}. The $W$ and $Z$ bosons were predicted in detail, their masses and couplings fixed by the measured weak mixing angle, years before the UA1 and UA2 experiments found them at CERN in 1983~\citep{Arnison1983}. Precision electroweak measurements at LEP and the SLC then became sensitive, through quantum loops, to particles too heavy to produce directly: the global fits pointed to a top quark of order $170$--$180$~GeV before the Tevatron experiments CDF and D0 discovered it at $173$~GeV in 1995~\citep{Abe1995,Abachi1995}, and the same fits later confined the Higgs mass to a narrow window that contained the $125$~GeV boson finally observed by ATLAS and CMS in 2012~\citep{ATLAS2012,CMS2012}. A theory that tells experimenters where to look, and is repeatedly proved right, has earned the deference any successor owes it.

The same internal consistency that constrains the theory has, more than once, \emph{forced} a discovery. Because the Standard Model is a chiral gauge theory, its quantum consistency demands that the gauge anomalies cancel---and they do so only family by family, through a delicate sum over the electric charges of the quarks and leptons in each generation~\citep{Adler1969,BellJackiw1969,BouchiatIliopoulosMeyer1972}. This is why the three colors of quark, carrying fractional charge, conspire with the integer-charged leptons so that the proton's charge balances the electron's to better than a part in $10^{21}$: the quark and lepton sectors are not independent but are locked together by the demand for consistency. The most celebrated instance is the prediction of the charm quark. In 1970, to suppress the flavor-changing neutral currents that experiment stubbornly refused to show, Glashow, Iliopoulos, and Maiani argued that a fourth quark was needed to complete the doublet structure~\citep{GlashowIliopoulosMaiani1970}; charm was duly found four years later in the $J/\psi$. Again and again, the bare demand that the theory hold together has been enough to anticipate what nature contains.

And yet the same theory that explains so completely \emph{how} the particles behave is wholly silent on \emph{why} they have the properties they do. It does not predict a single particle mass, a single coupling strength, or a single mixing angle; every one of these is measured in an experiment and written into the theory by hand. The Standard Model is, in this precise sense, a magnificent machine with twenty-eight dials whose settings it cannot explain. The lesson for any deeper theory is therefore twofold: it must preserve every one of the inherited successes cataloged above, and it must earn its keep by \emph{reducing} the number of unexplained primitives rather than, as most proposals do, multiplying it. That is the standard against which the present paper asks to be judged.

\subsection{The ten classic open problems left unanswered since the mid-1970s}

The completeness of the Standard Model is also its frustration. Precisely because the framework is closed and predictive, the questions it cannot answer stand out in sharp relief---and they have scarcely moved in fifty years. They are conventionally presented as a list of ten classic open problems. The list is more illuminating when the ten are first gathered into four connected families---scale and symmetry breaking; flavor, neutrinos, and CP; the dark sector and cosmology; and unification with gravity---because the families already hint that the problems are not ten unrelated accidents but facets of a smaller number of structural facts. Table~\ref{tab:intro-open-problem-clusters} gives that grouped view; the paragraphs below then take the ten in turn. The purpose of \ANine{} is not merely to compress twenty-eight numbers: it is to show that one fixed geometry speaks to all ten of these problems at once. We introduce them here in their own right, in the detail they deserve; \emph{how} the geometry addresses each is taken up problem by problem in the Discussions (\S\ref{sec:ten-problems-resolved}) and summarized in the resolution table there (Table~\ref{tab:ten-problems-resolution}), once the derivation has been earned.

\begin{table}[H]
\centering
\caption[The ten classic open problems the Standard Model has left unanswered since the mid-1970s, gathered into four families.]{The ten classic open problems the Standard Model has left unanswered since the mid-1970s, gathered into four families. Each is introduced in the paragraphs below and resolved, one by one, in \S\ref{sec:ten-problems-resolved} (summary table: Table~\ref{tab:ten-problems-resolution}).}
\label{tab:intro-open-problem-clusters}
{\small
\begin{tabular}{@{}p{0.03\textwidth}p{0.31\textwidth}p{0.56\textwidth}@{}}
\toprule
\textbf{No.} & \textbf{Problem} & \textbf{The unanswered question}\\
\midrule
\multicolumn{3}{@{}l}{\bfseries\itshape Scale and symmetry breaking}\\
1 & Hierarchy / naturalness & Why does the electroweak scale sit $\sim 10^{17}$ times below the Planck scale?\\
2 & Completeness of the Higgs sector & Is the scalar sector minimal, and what fixes $m_H$ and the quartic coupling?\\
3 & Vacuum stability / UV fate & Is the electroweak vacuum stable, and what completes the Standard Model in the ultraviolet?\\
\addlinespace
\multicolumn{3}{@{}l}{\bfseries\itshape Flavor, neutrinos, and CP}\\
4 & Flavor structure and Yukawa hierarchy & Why three families, and why a mass spread of twelve orders of magnitude?\\
5 & Neutrino masses and mixings & Why are neutrinos so light, are they Dirac or Majorana, and why so largely mixed?\\
6 & Baryon asymmetry of the universe & Why is the universe made of matter, with almost no antimatter?\\
7 & Strong-CP problem & Why is the QCD vacuum angle $\bar\theta$ smaller than $10^{-10}$?\\
\addlinespace
\multicolumn{3}{@{}l}{\bfseries\itshape Dark sector and cosmology}\\
8 & Nature of dark matter & What is the unseen matter, some five times more abundant than ordinary matter?\\
9 & Dark energy and the cosmological constant & Why is the vacuum energy so extraordinarily small --- yet not exactly zero?\\
\addlinespace
\multicolumn{3}{@{}l}{\bfseries\itshape Unification and gravity}\\
10 & Quantum gravity, unification, and proton stability & How does gravity join the gauge forces, do the forces unify, and why is the proton stable?\\
\bottomrule
\end{tabular}}
\end{table}

We take the ten in turn.

\textbf{1.~The hierarchy and naturalness problem.} The electroweak scale that sets the masses of the $W$, $Z$, and Higgs sits near $10^{2}$~GeV, yet the only other natural scale in fundamental physics, the Planck scale of quantum gravity, lies seventeen orders of magnitude higher at $10^{19}$~GeV. Quantum corrections to the Higgs mass-squared are quadratically sensitive to that high scale, so holding the Higgs light appears to demand a cancellation among independent contributions accurate to one part in $10^{34}$. Nothing within the Standard Model explains why the electroweak scale lies so absurdly far below the Planck scale, or why the cancellation should hold. Decades of proposed cures---supersymmetry, technicolor, large extra dimensions---have each added structure without yet being confirmed.

\textbf{2.~The completeness of the Higgs sector.} The 2012 discovery of a Higgs boson~\citep{ATLAS2012,CMS2012} confirmed the mechanism of electroweak symmetry breaking, but it left the scalar sector strangely bare. Is there exactly one Higgs doublet, as the minimal model assumes, or a richer structure waiting to be found? Why does the Higgs self-coupling---and hence the boson's mass near $125$~GeV---take the value it does? The single observed scalar is consistent with the minimal choice, yet the theory supplies neither a reason for that minimality nor a principle fixing the quartic coupling that sets the shape of the symmetry-breaking potential.

\textbf{3.~Vacuum stability and the ultraviolet fate of the Standard Model.} Run to high energies, the measured Higgs and top-quark masses place the electroweak vacuum tantalizingly close to the boundary between stability and metastability: the Higgs quartic coupling very nearly, but not quite, runs negative far below the Planck scale. Whether our vacuum is absolutely stable, metastable with a lifetime far exceeding the age of the universe, or a herald of new physics is undetermined---and the near-criticality itself looks like a clue the Standard Model cannot read.

\textbf{4.~The flavor structure and the Yukawa hierarchy.} The Standard Model contains three identical copies---families---of every matter particle, distinguished only by mass, and it offers no reason for the number three. Worse, the masses themselves span an extraordinary range: from the top quark at $173$~GeV, down through the electron at half an MeV, to neutrinos below a tenth of an eV---more than twelve orders of magnitude. The Yukawa couplings~\citep{Yukawa1935} that encode this spectrum, together with the quark mixing angles, are simply tabulated; the pattern is far too striking to be accidental and far too unexplained to be understood.

\textbf{5.~The origin of neutrino masses and mixings.} Neutrino oscillations~\citep{Fukuda1998,Ahmad2002} prove that neutrinos have mass, yet those masses are at least a million times smaller than the electron's, by a mechanism unknown. Are neutrinos Dirac particles, like the charged fermions, or Majorana particles equal to their own antiparticles~\citep{Majorana1937}? Why are the leptonic mixing angles large---one of them near-maximal---when the analogous quark angles are all small? The Standard Model in its original form set neutrino masses to zero; their discovery is the one place where experiment has compelled the framework to grow, and the sector it added is itself entirely unexplained.

\textbf{6.~The baryon asymmetry of the universe.} The observable universe is built of matter, with essentially no antimatter, in the tiny but decisive proportion of about six baryons per ten billion photons. Generating this asymmetry dynamically requires the three Sakharov conditions~\citep{Sakharov1967}---baryon-number violation, C and CP violation, and a departure from thermal equilibrium. The Standard Model does contain CP violation in the quark sector, but far too little, and its electroweak transition is too smooth; it misses the observed asymmetry by orders of magnitude, leaving the very preponderance of matter unexplained.

\textbf{7.~The strong-CP problem.} Quantum chromodynamics permits a CP-violating term proportional to an angle $\bar\theta$ that, on general grounds, should be of order unity. Measurements of the neutron electric dipole moment instead bound it below about $10^{-10}$. Why this angle is so extravagantly small---ten orders of magnitude beneath its natural value---is the strong-CP problem. Its most popular proposed resolution, a new light particle called the axion~\citep{PecceiQuinn1977,Weinberg1978,Wilczek1978}, has remained undiscovered through more than forty years of searching.

\textbf{8.~The nature of dark matter.} Galactic rotation curves, gravitational lensing, the cosmic microwave background, and the growth of cosmic structure all testify that about a quarter of the energy of the universe---roughly five times as much as ordinary matter---is some form of matter that neither emits nor absorbs light~\citep{Zwicky1933,Rubin1970}. The Standard Model contains no particle that can play this role. The identity, mass, and interactions of dark matter are among the most pressing empirical questions in all of physics, and decades of direct, indirect, and collider searches have not settled them.

\textbf{9.~Dark energy and the cosmological constant.} The expansion of the universe is accelerating~\citep{Riess1998,Perlmutter1999}, driven by a smoothly distributed energy that constitutes roughly seventy percent of the total. If this is the energy of the vacuum, the Standard Model's own estimate of that energy exceeds the observed value by some $120$ orders of magnitude---often called the worst quantitative prediction in the history of physics~\citep{Weinberg1989}. Why the vacuum energy is so small, yet not exactly zero, is the cosmological-constant problem, and it sits squarely on the seam between particle physics and gravity.

\textbf{10.~Quantum gravity, unification, and proton stability.} Finally, the Standard Model does not include gravity, and the naive quantization of general relativity is not renormalizable, so the framework cannot be the final word at the highest energies. The hope that the three gauge forces merge into one at a high scale---grand unification~\citep{GeorgiGlashow1974}---generically predicts that the proton should decay; yet the proton's lifetime is now known to exceed $10^{34}$ years, excluding the simplest unified models. How gravity joins the gauge interactions, whether the forces truly unify, and why the proton is so very nearly immortal are bound together as the deepest structural questions the Standard Model leaves open.

These ten are usually attacked one at a time, each with its own dedicated extension of the Standard Model---a new symmetry for the hierarchy, a new particle for dark matter, a new field for the strong-CP angle, and so on. The premise of the present program is the opposite. \GGTheory{} holds that the ten are not independent but are the visible facets of a single fixed six-dimensional geometry, so that fixing the geometry once addresses them together rather than separately. Establishing that claim is the work of the whole cohort; this paper earns the right to make it by first deriving the twenty-eight numbers the Standard Model cannot. We return to the ten problems explicitly---and to how one geometry speaks to each---in \S\ref{sec:ten-problems-resolved} of the Discussions, with a comprehensive summary table there (Table~\ref{tab:ten-problems-resolution}).

Common to all ten is a single underlying complaint: the Standard Model can say, to extraordinary precision, \emph{how} its particles behave, but never \emph{why} its defining numbers take the values they do. The most disciplined way to sharpen that complaint---and to hold any proposed answer to account---is to count those numbers exactly. We do so next.

\subsection{Counting the free parameters: why twenty-eight, not nineteen or twenty-six}

A reader who has met the Standard Model before may be surprised by the number twenty-eight. The figure most often quoted is nineteen, and one sometimes sees twenty-six; twenty-eight is less familiar. The discrepancy is not a disagreement about physics but about how much of the modern theory one chooses to count---and it is worth being explicit, because the audit at the heart of this paper is only as meaningful as the list it audits.

The classic count of \emph{nineteen} describes the Standard Model as it stood before 1998, with massless neutrinos. It comprises three gauge couplings (one each for the strong, weak, and hypercharge interactions); two parameters of the Higgs sector (its vacuum expectation value and its self-coupling, equivalently the electroweak scale and the Higgs mass); nine charged-fermion masses (six quarks and three charged leptons); four quark-mixing parameters (three angles and one CP-violating phase of the CKM matrix)~\citep{Cabibbo1963,KobayashiMaskawa1973}; and the one strong-CP angle $\bar\theta$. These nineteen are the textbook free parameters---the numbers a first course in particle physics says must simply be measured~\citep{ParticleDataGroup2022}.

The discovery of neutrino oscillations forced the count upward. Massive neutrinos require, at a minimum, three neutrino masses and a leptonic mixing matrix---the PMNS matrix~\citep{Pontecorvo1957,MakiNakagawaSakata1962}---with three angles and one Dirac CP phase: seven new numbers, bringing the total to \emph{twenty-six}. If, as both the seesaw picture and the extreme smallness of the masses suggest, neutrinos are Majorana particles equal to their own antiparticles~\citep{Majorana1937}, two further physical CP phases appear, and the count reaches \emph{twenty-eight}. The gap between twenty-six and twenty-eight is therefore precisely the question of whether neutrinos are Dirac or Majorana; counting twenty-eight is the honest, most complete bookkeeping---the one that leaves no physically meaningful number off the ledger.

We adopt that complete count throughout:
\begin{equation}
\begin{aligned}
N_{\mathrm{SM}}
&=5_{\mathrm{gauge/Higgs}}+12_{\mathrm{fermion}}+4_{\mathrm{CKM}}\\
&\quad+4_{\mathrm{PMNS}}+2_{\mathrm{Majorana}}+1_{\mathrm{topological\ CP}}\\
&=28.
\end{aligned}
\label{eq:sm28-count-intro}
\end{equation}
The five gauge/Higgs parameters carry the forces and the electroweak scale; the twelve fermion masses carry the matter spectrum, now including the three neutrinos; the four CKM and four PMNS parameters carry quark and lepton mixing together with their Dirac CP violation; the two Majorana phases complete the neutrino sector; and the single topological angle $\bar\theta$ carries strong-CP. Nothing physical is omitted, and nothing is double-counted.

Fixing twenty-eight numbers in this way converts a philosophical aspiration---``explain the constants of nature''---into a finite and unforgiving audit. Either a framework reproduces these twenty-eight measured values from a smaller and fixed set of inputs, without quietly feeding the answers back in, or it does not. That is what makes the 28-parameter problem the sharpest available test of predictivity, and why it is the first target of this paper: it is narrow enough to be checked number by number, yet broad enough to touch every major structural issue in particle physics---gauge couplings, electroweak symmetry breaking, the flavor hierarchy, neutrino masses, quark and lepton CP violation, and the boundary between accidental and structural stability. The Parts that follow take up that audit in earnest.

\subsection{The deductive structure, in three stages}
\GGTheory{} is an axiomatic--deductive program, and it is worth naming its logic once, plainly, before the technical Parts begin. It runs in three stages. \emph{First}, the \GGConstitution{} of \AOne{} fixes the starting point: four axioms --- \AX{1} (\textsf{LC}, locality and causality), \AX{2} (\GGEP{}), \AX{3} (\OGGEP{}), and \AX{4} (\MSMF{}) --- together with seven postulates and the \OGNfive{} integer ledger. \emph{Second}, from that constitution the cohort \emph{derives} a chain of theorems, each proved in its own companion paper: the \BIsix{} anomaly closure (\AThree{}), the \GiFour{} pillars \TGG{}/\CCI{}/\GFF{}/\GRIP{} (\ASix{}), the open-system boundary law \GDOS{} (\AFive{}), and the family number $N=3$ (\PaperPTwo{}, conditional on a single named conjecture). \emph{Third}, the present paper \emph{applies} that fixed scaffolding --- axioms plus already-proved theorems --- to the Standard Model, reading off the twenty-eight parameters as boundary observables of the one projection. \ANine{} does not re-derive the first two stages; it consumes them. The provenance of every imported result, and the few load-bearing lemmas that remain open, are tabulated in the constitutional-apparatus Part~\ref{part:II}; the honest status of each of the twenty-eight outputs is the $6/14/8$ triage of \S\ref{sec:codebook-provenance}. Read in this order, the paper is the third stage of a deduction applied to the Standard Model, not a free-standing model fit.

\section{The cohort inheritance and the roadmap of the seventeen Parts}
\label{sec:inheritance-and-roadmap-A9}

\noindent
Before the parameter target is fixed (Part~\ref{part:SMtarget}), it is
useful to lay out two reader-orientation maps on the same opening: a
panorama of the twenty-two-paper cohort in which this paper stands
(\S\ref{sec:cohort-A9}, Figure~\ref{fig:cohort-map-a9}), and a panorama
of the seventeen Parts of the present paper (\S\ref{sec:roadmap-A9},
Figure~\ref{fig:org-flowchart}). A reader of any later Part should be
able to flip back to this Section and orient themselves in both
directions: which cohort companion carries which imported result, and
where the present argument stands in the seventeen-Part arc.

\subsection{The cohort inheritance}
\label{sec:cohort-A9}

The present paper is not a stand-alone speculation. It is the parameter and
phenomenology volume of a longer theorem chain. The role of each paper in the
series is summarized in Table~\ref{tab:cohort-roster}. \AOne{} fixes the
constitutional rules. \ATwo{} derives the six-dimensional geometry and the
unified gauge-gravity stage. \AFive{} supplies the open-system projection to
observable four-dimensional physics. \ASeven{} extends \GDOS{} into the \QiFour{}
runtime language. \AThree{} proves \BIsix{} anomaly freedom. \ANine{}, the
present paper, turns that written framework into explicit Standard-Model
parameter calculations. The companion papers \ATen{} and \PaperPOne{} then push
the same framework toward cosmology and experimental falsification.

This placement is important for interpretation. \ANine{} should be read as the particle-physics theorem chapter of a broader program, not as an isolated model-building note.

The present paper is one of twenty-two. \GGTheory{} is written not as a
single monograph but as a cohort of twenty-two preprints --- eleven A-papers, six P-papers, three C-papers and two white papers --- each carrying one
rigorously stated piece of a single descent that runs from four axioms
and seven postulates to the empirical structure of physics. The
partition into twenty-two papers reflects the partition of subject
matter, not a ranking of importance: the framework is one program,
and each paper is a named carrier of named content.
Figure~\ref{fig:cohort-map-a9} is the panorama; a reader meeting the
program for the first time can read its architecture off the figure
top to bottom before returning to the Standard Model that is the
business of this paper. The same information is gathered in
Table~\ref{tab:cohort-roster}: a one-line entry for every paper in the
cohort, its main achievement and its role in the program, with a
live link to each.

% -- portability guards: define paper-local macros if the host paper lacks them --
\providecommand{\LCDM}{\ensuremath{\Lambda\mathrm{CDM}}}
\providecommand{\QCI}{\ifmmode\mathsf{QCI}\else\textsf{QCI}\fi}
\providecommand{\FRW}{\textsf{FRW}}
\small
\setlength{\LTcapwidth}{\textwidth}
\begin{longtable}{>{\raggedright\arraybackslash}p{0.14\textwidth} >{\raggedright\arraybackslash}p{0.39\textwidth} >{\raggedright\arraybackslash}p{0.39\textwidth}}
\caption[The twenty-two-paper \GGTheory{} cohort.]{\textbf{The twenty-two-paper \GGTheory{} cohort.} Each paper's main achievement and its role in the program. Every paper name links to the program homepage, \href{https://arisaka-gg.org/}{arisaka-gg.org}, where the twenty-one posted preprints are collected; the capstone \AEleven{} is in preparation. The paper marked \emph{(this paper)} is the one you are reading.}
\label{tab:cohort-roster}\\
\toprule
\textbf{Paper} & \textbf{Main achievement} & \textbf{Role in the cohort} \\
\midrule
\endfirsthead
\multicolumn{3}{l}{\small\emph{Table~\ref{tab:cohort-roster}, continued.}}\\
\toprule
\textbf{Paper} & \textbf{Main achievement} & \textbf{Role in the cohort} \\
\midrule
\endhead
\midrule
\multicolumn{3}{r}{\small\emph{continued on next page}}\\
\endfoot
\bottomrule
\endlastfoot
\href{https://arisaka-gg.org/}{\AOne{}} & Four axioms and seven postulates fixing what counts as an admissible claim --- the constitution of the whole program & The constitutional root; every later paper inherits its four axioms and seven postulates \\
\addlinespace
\href{https://arisaka-gg.org/}{\ATwo{}} & Derivation of the six-dimensional \GG{} bulk, \BIsix{}, \OGNfive{}, compact holonomy, and the unification of all six forces & First of the GG-6 bulk trio; fixes the geometry and ledger the boundary law is built on \\
\addlinespace
\href{https://arisaka-gg.org/}{\AThree{}} & Complete proof of \BIsix{} anomaly freedom, unified gauge consistency, and conservation of the dual current & GG-6 bulk trio; the ultraviolet legality certificate that lets the projected ledger exist \\
\addlinespace
\href{https://arisaka-gg.org/}{\AFour{}} & The historical lineage from Euclid to \GG{}, and the zero-parameter integer \OGNfive{} ledger behind the constants & GG-6 bulk trio (historical companion); supplies the integer ledger behind the constants \\
\addlinespace
\href{https://arisaka-gg.org/}{\AFive{}} & \OGGEP{} descent to boundary quantum mechanics, boundary general relativity, and formal 4D QFT: the \GDOS{} boundary law & The open-system boundary law itself --- the hinge where the 6D bulk projects to 4D physics \\
\addlinespace
\href{https://arisaka-gg.org/}{\ASix{}} & The runtime constitution --- \GRIP{}, the force law \(F_{\GFF}=F_{\GEP}+F_{\CEP}\), and the scale-invariance principle \SIP{} & A pillar paper organizing the boundary law from the geometric (\GiFour{}) side \\
\addlinespace
\href{https://arisaka-gg.org/}{\ASeven{}} & The \QiFour{} pillars --- \SeOS{}, \QCI{}, the identity \(\IH+\OH=\EIC\) --- and the five-story informational pagoda & A pillar paper organizing the boundary law from the quantum-information (\QiFour{}) side \\
\addlinespace
\href{https://arisaka-gg.org/}{\AEight{}} & Quantum mechanics derived from \GGTheory{} --- the four-dimensional reformulation of measurement and state & Consequence trio; the member that recovers textbook quantum mechanics on the boundary \\
\addlinespace
\href{https://arisaka-gg.org/}{\ANine{}} \emph{(this paper)} & The Standard Model as a theorem --- the descent of all twenty-eight Standard-Model parameters & Consequence trio; the member that recovers the Standard Model of particle physics \\
\addlinespace
\href{https://arisaka-gg.org/}{\ATen{}} & \LCDM{} as the cosmological-boundary projection --- the \(\pi\)-locked inventory \(\Omega_m=1/\pi,\ \Omega_\Lambda=1-1/\pi\) & Consequence trio; the member that recovers \LCDM{} cosmology \\
\addlinespace
\href{https://arisaka-gg.org/}{\AEleven{}} & The closing capstone --- the universal identity and the \GRIP{} cascade across cosmos, life, and mind & The capstone that closes the A-series and proves the program's universal identity \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperPOne{}} & The Kaluza--Klein lock cluster near \(\Mlock\approx3.85\,\TeV\), with explicit LHC and future-collider falsifiers & Predictions branch; the collider phenomenology of the Kaluza--Klein lock scale \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperPTwo{}} & A three-leg derivation of \(N=3\) fermion generations from \BIsix{}, MSMF, and the integer ledger & Predictions branch; the three-leg derivation of the fermion generation count \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperPThree{}} & The electron as the ground state of nature --- the six oldest electron questions, closed & Predictions branch; the electron ground-state and charged-lepton-stratum result \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperPFour{}} & The two real sources of the boundary's complex structure --- positive frequency is holomorphy, from the compact phase circle and the causal clock & Foundations branch; the origin-of-\(i\) member paper behind the real presentation of T-SM-44.1 and T-SM-46 \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperPFive{}} & C, P, and T as orientation reversals of the three real bulk factors --- why every particle has a twin, and why the universe is made of matter & Foundations branch; the CPT member paper and the \ANine{}--\ATen{} interface for the derived CP phases \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperCOne{}} & Cold dark matter as the non-spatial compact-angle pseudoscalar Kaxion at \(8.1\,\mu\mathrm{eV}\) & Cosmology \& astrophysics branch; the identification of cosmological dark matter \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperCTwo{}} & The Hubble tension as two projections of one derived anchor --- the early/late corridor & Cosmology \& astrophysics branch; the resolution of the Hubble tension \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperCThree{}} & Black-hole entropy \(A/4\) recovered as a boundary theorem within \GGTheory{}'s holographic ledger & Cosmology \& astrophysics branch; horizon thermodynamics and black-hole entropy \\
\end{longtable}
\normalsize


\paragraph{The full cohort in one view.}
Figure~\ref{fig:cohort-map-a9} places the present paper among the twenty-two
papers of the \GGTheory{} cohort, which divides into five strata, two
phenomenology branches, and the Foundations Branch. At the root stands \AOne{}
\citep{ArisakaA1Constitution}, the \GGConstitution{} --- four axioms and
seven postulates. The GG6 Trio builds the bulk: \ATwo{}
\citep{ArisakaA2GG6} (the six-dimensional Geometry--Gauge bulk and
the \OGNfive{} ledger), \AThree{} \citep{ArisakaA3BI6} (the local-and-global
\BIsix{} anomaly proof), and \AFour{} \citep{ArisakaA4Euclid} (the
historical lineage and the parameter-free reconstruction of the universal
constants). \AX{3}~(\OGGEP{}) forces the open-system boundary law \AFive{}
\citep{ArisakaA5GDOS}, organized by the two pillar papers \ASix{}
\citep{ArisakaA6Gi4} (the geometric \GiFour{} pillars) and \ASeven{}
\citep{ArisakaA7Qi4} (the quantum-information \QiFour{} pillars). The
Consequence Trio reads the empirical world off the boundary law: \AEight{}
\citep{ArisakaA8QM} (quantum mechanics), \emph{the present paper} \ANine{}
\citep{ArisakaA9SM} (the twenty-eight Standard-Model parameters), and
\ATen{} \citep{ArisakaA10Cosmos} ($\Lambda$CDM cosmology); the capstone
\AEleven{} \citep{ArisakaA11Origin} closes the A-series. Beneath these, two
branches carry the sharp falsifiers: the Predictions Branch --- \PaperPOne{}
\citep{ArisakaP1KK} (the Kaluza--Klein lock cluster near $\Mlock \approx
3.85$~TeV), \PaperPTwo{} \citep{ArisakaP2N3} (the three-generation pincer
forcing $N=3$), and \PaperPThree{} \citep{ArisakaP3Electron} (the electron
as the ground state of nature) --- and the Cosmology and Astrophysics
Branch --- \PaperCOne{} \citep{ArisakaC1Kaxion} (kaxion dark matter near
$8.1~\mu$eV), \PaperCTwo{} \citep{ArisakaC2Hubble} (the Hubble tension as
one geometric corridor), and \PaperCThree{} \citep{ArisakaC3BH} (black-hole
entropy as an \OGGEP{} port). The present paper is the Standard-Model node
of the Consequence Trio: it turns the framework of \AOne{}--\ASeven{} into
explicit parameter values, and its results in turn feed \ATen{},
\AEleven{}, the phenomenology branches, and --- through the ledger
phases of the cohort map --- the Foundations Branch member \PaperPFive{}
\citep{ArisakaP5CPT}.

\paragraph{The Foundations Branch.} The third branch is the
Foundations Branch: the two member papers whose subject is the
boundary's complex structure and its mirrors, standing as a single
column on the map's left flank. \PaperPFour{} \citep{ArisakaP4Real} is
the origin-of-$i$ member paper, deriving the single imaginary unit of
boundary physics from two real structures of the \ATwo{} arena --- the
compact phase circle and the causal clock; for the present paper it
supplies the origin of the complex phase structure in which the
ledger's valued phases live, through the real presentation used in
T-SM-44.1, T-SM-46.1, and T-SM-46.2. \PaperPFive{}
\citep{ArisakaP5CPT} carries the three mirrors of matter --- C, P, and
T read as orientation reversals of the three real bulk factors --- and
is the \ANine{}--\ATen{} interface of the map: its $\delta,\
\alpha_{31}$ edge is this paper's export --- the ledger phases
$\delta_{\rm CKM}$ and $\delta_{\rm PMNS}$ and the Majorana parities,
each at the grade recorded in the codebook of
Part~\ref{part:SMtarget} --- which \PaperPFive{} routes to the
baryon-asymmetry chain of \ATen{}. Both carry P-series designators;
their place at the spine's flank, not the phenomenology rows, marks
their foundational role.

The twenty-two-paper map, identical in every paper of the cohort, is
Figure~\ref{fig:cohort-map-a9}, 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
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%%  self-reference markers anywhere, so the figure is identical and
%%  simple for replication in all nineteen papers.  Only the internal
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%% ===================================================================
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 {\textbf{\AThree{}} \\[-1pt] \footnotesize \BIsix{} anomaly proof};
\node[hist] (A4) at (50mm, -25mm)
 {\textbf{\AFour{}} \\[-1pt] \footnotesize Historical lineage};

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

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

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

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

%% --- Row 6: Predictions Branch (P-series) ---
%% v22: four predictions on the row at 50mm pitch.
\node[pred] (P1) at (-75mm, -138mm)
 {\textbf{\PaperPOne{}} \\[-1pt] \footnotesize $\Mlock \approx 3.85$~\TeV{}};
\node[pred] (P2) at (-25mm, -138mm)
 {\textbf{\PaperPTwo{}} \\[-1pt] \footnotesize Three generations};
\node[pred] (P3) at (25mm, -138mm)
 {\textbf{\PaperPThree{}} \\[-1pt] \footnotesize Electron as ground state};
\node[pred] (P6) at (75mm, -138mm)
 {\textbf{\PaperPSix{}} \\[-1pt] \footnotesize The guarded proton};

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

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

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

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

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

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

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

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

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

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

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

%% Centring pad, retired (v22): it cancelled a Bezier phantom that the
%% polyline above no longer creates; the bbox is now the ink.
\end{tikzpicture}}
\caption[The twenty-two-paper cohort map.]{The
twenty-two-paper cohort map of \GGTheory{}. The constitutional root
\AOne{} (violet, top) supplies the four axioms and seven postulates
from which the rest of the cohort descends: the GG6 Trio (\ATwo{},
\AThree{}, \AFour{}, red) realizes the bulk geometry; \AFive{}
(yellow) the boundary law; the \GDOS{} Trio pillars (\ASix{},
\ASeven{}, green) the geometric and quantum-information runtime; the
4D-Physics Trio (\AEight{}, \ANine{}, \ATen{}, blue) the textbook
readouts; and the closing capstone \AEleven{} (gold) the
quarks-to-consciousness identity cascade. Two framed branches carry
the phenomenology --- the \textbf{Predictions Branch} (cyan), whose
four papers hand their results along the row by the internal arrows
(\PaperPTwo{}'s $N{=}3$ into \PaperPThree{}, and that paper's
ground-attractor method, \textsf{GGA}, on into \PaperPSix{} \citep{ArisakaP6Proton}), 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-a9}
\end{figure}


\subsection{Roadmap of the seventeen Parts}
\label{sec:roadmap-A9}

The rest of the paper is organized into sixteen further Parts, and
the walk through them is plotted in Figure~\ref{fig:org-flowchart}.
The present Part --- the entrance --- is nearly done: the thesis that
the twenty-eight Standard-Model parameters are registered observables
of one open-system projection, the deductive funnel that carries it,
and the two orientation maps of this closing section. Everything after
this page is the derivation itself, and the itinerary below says where
each piece lives. Part~\ref{part:SMtarget} fixes the target: the
Standard Model in textbook form, the classic Georgi--Glashow
$\mathrm{SU}(5)$ unification of 1974 and its \GGTheory{}
reinterpretation from the six-dimensional bulk, and --- most
importantly --- the codebook-provenance audit that labels each of the
twenty-eight parameters \textbf{Derived}, \textbf{Derived (cond.)}, or
\textbf{Asserted} before a single one is computed.

Three Parts then assemble the apparatus. Part~\ref{part:II} records
the constitutional framework --- the four axioms, the seven
postulates, and the canonical \GiFour{} chain specialized to the
Standard-Model boundary stratum. Part~\ref{part:III} is the Substrate:
the technical descent from classical Georgi--Glashow $\mathrm{SU}(5)$
to the \GG{} seam, the six-dimensional bulk and its three fundamental
scales, the family number $N = 3$ as a three-leg theorem, the radial renormalization group, and Wilson--Hosotani dynamical symmetry
breaking. Part~\ref{part:IV} closes the preparation with the universal
sectoral descent template, the derivation roadmap, and the
constitutional inputs that close the parameter sheet.

The four Parts that follow are the derivation core, and they are
highlighted in the roadmap because they deliver the paper's central
claim --- the twenty-eight parameters themselves.
Part~\ref{part:VI} derives the electroweak seam: the rigorous theorem
deck, the five primary attractor-coordinate theorems
($\alpha^{-1}(M_Z) = 2^{7} = 128$, $\sin^2\theta_W = 25/108$,
$\alpha_s(M_Z) = 1/(\pi e)$, $v \approx 246$~GeV, $\lambda_H = 1/8$),
and the descendant boson-mass and Fermi-constant package.
Part~\ref{part:X} derives the fermion sector: the twelve masses as the
\GRIPin{} floors of the family cascade, and the CKM and PMNS matrices
--- with the Majorana phases --- as the gates between cascades.
Part~\ref{part:lepto} is the matter Part: the same geometry that fixes
the neutrino block also writes the excess of matter over antimatter,
through the unique $\Delta L = 2$ portal and the heavy sector at the
lock scale. It is the one Part of the core whose result is not one of
the twenty-eight.
Part~\ref{part:XI} completes the twenty-eighth and final parameter:
the strong-CP angle as the protected zero of the topological ledger.
Five, twenty-two, and one: the codebook closes across the three
parameter-carrying Parts of the four. Part~\ref{part:lagrangian} then harvests the result: the complete
Standard-Model Lagrangian, written out in its exact textbook form and
assembled term by term as a constitutional export --- the zeroth and the
six silent structural choices of the textbook replaced by theorems, and the operators
\emph{absent} from the Lagrangian derived alongside the ones present.

With the Lagrangian assembled, Part~\ref{part:XII} is the philosophical
pivot: why every electron is the same, and why exactly four forces
exist with their specific qualitative behaviors, recognized as a
single uniqueness theorem applied at two layers of the same
six-dimensional geometry. Part~\ref{part:XIII} gathers the result onto
a single sheet --- the cross-paper coherence map, the two
comprehensive 28-parameter tables with their PDG benchmarks, and the
arguments for why the result is not a post-hoc fit and what makes
\GGTheory{} structurally different from previous unification attempts.

Two Parts then weigh the case: Part~\ref{part:XIV}, the Discussions
--- significance, comparison with other programs, falsification
channels, and the open questions stated as open --- and
Part~\ref{part:XV}, the Conclusions, viewed from the summit reached
after the fifty-year climb out of the Standard Model. The back matter
is the workshop. Part~\ref{part:XVI} collects the technical appendices
--- a graduate-level course in the Standard Model read through
\GGTheory{}, with the reference dictionaries and the lemma-promotion
roadmap. Part~\ref{part:XVII} answers the strongest objections a
particle theorist, cosmologist, mathematician, philosopher, or string
theorist is most likely to raise, each with pointers into the paper.
Part~\ref{part:XVIII} closes with the annotated references, each
carrying a Relevance note. Read in order, the seventeen Parts compose
into a single self-contained derivation that a careful graduate
student can audit step by step; read by need, each Part names its
imports and can stand a visit on its own.

%% ===================================================================
%%  Roadmap of the sixteen Parts (house architecture: A2-GG6 v13.3 /
%%  A5-GDOS v13.0 / A6-GRIP v13.0 / A7-Qi4 v13.0 / A8-QM v13.2
%%  pattern -- vertical Part column, right phase sidebar, highlighted
%%  derivation core Parts VI-VIII carrying the central claim),
%%  clearpage-isolated minipage + \captionof.  The label
%%  fig:org-flowchart is kept from the retired block-design flowchart
%%  so existing cross-references resolve.
%% ===================================================================
\begin{figure}[p]
\noindent
\begin{minipage}{\textwidth}
\centering
\resizebox{0.828\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
 found/.style={fill=blue!12, draw=blue!60!black, line width=0.6pt,
 minimum width=120mm, minimum height=12mm},
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 draw=orange!65!black},
 phaselabel/.style={font=\footnotesize\itshape, text=black!60, anchor=west,
 align=left, inner sep=2pt},
]

%% --- Parts I--II: entrance and target ---
\node[found] (I) at (0, 0)
 {\textbf{Part~I}\quad\textbf{Introduction} \\[-1pt]
 \footnotesize The twenty-eight parameters as observables of one projection; the reader's two maps};
\node[found] (II) at (0, -13mm)
 {\textbf{Part~II}\quad\textbf{The Standard Model and its parameter audit} \\[-1pt]
 \footnotesize The textbook target, classic $\mathrm{SU}(5)$, and the provenance codebook of the twenty-eight};

%% --- Parts III--V: apparatus ---
\node[appar] (III) at (0, -28mm)
 {\textbf{Part~III}\quad\textbf{Constitutional apparatus} \\[-1pt]
 \footnotesize Four axioms, seven postulates, the canonical \GiFour{} chain, and the \GDOS{} law of observables};
\node[appar] (IV) at (0, -41mm)
 {\textbf{Part~IV}\quad\textbf{The Substrate} \\[-1pt]
 \footnotesize $\mathrm{SU}(5)\!\to\!\GG{}$ seam; the 6D bulk, three scales, $N{=}3$ closed, radial RG, Wilson--Hosotani};
\node[appar] (V) at (0, -54mm)
 {\textbf{Part~V}\quad\textbf{Roadmap and constitutional basis} \\[-1pt]
 \footnotesize The universal sectoral descent template and the inputs that close the sheet};

%% --- Parts VI--VIII: the derivation core (highlighted) ---
\node[innov] (VI) at (0, -71mm)
 {\textbf{Part~VI}\quad\textbf{The electroweak seam} \\[1pt]
 \footnotesize Five parameters: $\alpha^{-1}(M_Z)=2^{7}$,\ \ $\sin^2\theta_W=25/108$,\ \ $\alpha_s=1/(\pi e)$,\ \ $v$,\ \ $\lambda_H=1/8$ \\[-1pt]
 \footnotesize plus the descendant $G_F$, $m_W$, $m_Z$ package};
\node[innov] (VII) at (0, -89mm)
 {\textbf{Part~VII}\quad\textbf{The fermion sector: floors and gates} \\[1pt]
 \footnotesize Twenty-two parameters: twelve masses as \GRIPin{} floors; \\[-1pt]
 \footnotesize CKM, PMNS, and the Majorana phases as the gates between cascades};
%% --- Part VIII: the matter Part (leptogenesis at the lock scale) ---
\node[innov] (LEPTO) at (0, -107mm)
{\textbf{Part~VIII}\quad\textbf{The Majorana door and leptogenesis} \\[1pt]
\footnotesize Why there is matter and not nothing: the unique $\Delta L = 2$ portal, \\[-1pt]
\footnotesize the heavy sector at the lock scale, and the assembled baryon-to-photon ratio};
\node[innov] (VIII) at (0, -125mm)
 {\textbf{Part~IX}\quad\textbf{The protected readout} \\[1pt]
 \footnotesize One parameter: strong CP as the protected zero of the topological ledger \\[-1pt]
 \footnotesize --- the codebook closes at twenty-eight};

%% --- Part IX: the harvest (assembled Lagrangian, highlighted) ---
\node[harv] (LAG) at (0, -141.5mm)
 {\textbf{Part~X}\quad\textbf{The assembled Lagrangian} \\[1pt]
 \footnotesize The complete $\mathcal{L}_{\mathrm{SM}}$ written out and assembled term by term as a constitutional export \\[-1pt]
 \footnotesize --- present terms exported, absent terms derived, every entry graded};

%% --- Parts X--XII: pivot, audit, discussions ---
\node[princ] (IX) at (0, -157.5mm)
 {\textbf{Part~XI}\quad\textbf{Identity and force uniqueness} \\[-1pt]
 \footnotesize Every electron the same; exactly four forces --- one uniqueness theorem, two layers};
\node[princ] (X) at (0, -170.5mm)
 {\textbf{Part~XII}\quad\textbf{Summary and audit} \\[-1pt]
 \footnotesize The completed codebook census, the coherence map, and why this is not a post-hoc fit};
\node[princ] (XI) at (0, -183.5mm)
 {\textbf{Part~XIII}\quad\textbf{Discussions} \\[-1pt]
 \footnotesize Significance, comparisons, falsification channels, open questions, prospects};

%% --- Part XIII: conclusions ---
\node[summit] (XII) at (0, -196.5mm)
 {\textbf{Part~XIV}\quad\textbf{Conclusions: the view from the summit} \\[-1pt]
 \footnotesize The fifty-year climb out of the Standard Model, seen from the top};

%% --- Parts XIV--XVI: back matter ---
\node[back] (XIII) at (0, -210mm)
 {\textbf{Part~XV}\quad\textbf{Technical Appendices} \\[-1pt]
 \footnotesize A graduate course in the Standard Model read through \GGTheory{}};
\node[back] (XIV) at (0, -223.5mm)
 {\textbf{Part~XVI}\quad\textbf{Common Objections and Responses} \\[-1pt]
 \footnotesize The strongest objections, answered, with pointers};
\node[back] (XV) at (0, -237mm)
 {\textbf{Part~XVII}\quad\textbf{References} \\[-1pt]
 \footnotesize Cited works, each with a relevance note};

%% --- Vertical flow arrows (the derivation core in innov-orange) ---
\draw[arrow] (I.south) -- (II.north);
\draw[arrow] (II.south) -- (III.north);
\draw[arrow] (III.south) -- (IV.north);
\draw[arrow] (IV.south) -- (V.north);
\draw[arrowinnov] (V.south) -- (VI.north);
\draw[arrowinnov] (VI.south) -- (VII.north);
\draw[arrowinnov] (VII.south) -- (LEPTO.north);
\draw[arrowinnov] (LEPTO.south) -- (VIII.north);
\draw[arrowinnov] (VIII.south) -- (LAG.north);
\draw[arrow] (LAG.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);

%% --- Phase labels (right sidebar); standard boxes: half-width =
%% 60mm, label x = 67mm; the innov/harv boxes are wider (half-width
%% 70mm), their labels sit at x = 75mm.
\node[phaselabel] at (67mm, -6.5mm) {\emph{Entrance \&} \\ \emph{target}};
\node[phaselabel] at (67mm, -41mm) {\emph{Apparatus}};
\node[phaselabel, text=orange!70!black] at (75mm, -89mm)
 {\emph{\textbf{The derivation core:}} \\ \emph{\textbf{$5+22+1=28$}}};
\node[phaselabel, text=violet!70!black] at (75mm, -123.5mm)
 {\emph{\textbf{The harvest:}} \\ \emph{\textbf{$\mathcal{L}_{\mathrm{SM}}$ assembled}}};
\node[phaselabel] at (67mm, -164mm) {\emph{Pivot \&} \\ \emph{audit}};
\node[phaselabel] at (67mm, -190mm) {\emph{Assessment \&} \\ \emph{summit}};
\node[phaselabel] at (67mm, -223.5mm) {\emph{Back matter}};

\end{tikzpicture}%
}
\par\bigskip
\captionof{figure}[Roadmap of the seventeen Parts of GG-A9-SM.]{Roadmap of the seventeen Parts of \textsf{GG-A9-SM}. The argument develops in seven phases: \emph{entrance and target} (Parts~I--II), \emph{apparatus} (Parts~III--V), \emph{the derivation core} (Parts~VI--IX, highlighted), \emph{the harvest} (Part~X, highlighted), \emph{pivot and audit} (Parts~XI--XII), \emph{assessment and summit} (Parts~XIII--XIV), and the three back-matter Parts (XV--XVII). Each phase is walked through in the section that precedes this page. Parts~VI--X are highlighted because together they carry the paper's central claim --- the descent of all twenty-eight Standard-Model parameters and the assembly of the Lagrangian they populate, from the fixed constitutional scaffolding --- the result on which every later Part, and the cohort downstream of \ANine{}, stands.}
\label{fig:org-flowchart}
\end{minipage}
\end{figure}


%% ===========================================================================
%% v6.8.8 (May 13, 2026): NEW Part inserted here, before the §Codebook
%% provenance section.  Three new sections cover (A) the Standard Model
%% as classical structure; (B) the classic Georgi-Glashow SU(5)
%% unification of 1974; (C) the GG-Theory derivation of SU(5) and the
%% Standard Model from the 6D bulk + the constitutional principles
%% of A1.  Pedagogical primer that the reader meets BEFORE the parameter
%% audit.  Existing §6, §7, §8 (sec:su5-overview, sec:apply-gdos-su5,
%% sec:sm-foundation) remain as the load-bearing technical detail and
%% are cross-referenced from Section C below.
%% ===========================================================================



\section{Part~I in plain English: what the twenty-eight numbers are, and why their provenance is the question}
The Standard Model is the most accurately tested theory in the history
of physics, and nothing in this paper disputes that. Its predictions have
survived fifty years of increasingly severe experiments, and where it has been
pushed hardest it has held. That record is the reason the question this paper
asks is worth asking at all: a framework this successful, resting on numbers
nobody can derive, is a strange object.

Twenty-eight numbers have to be measured and put in by hand. Three gauge
couplings, two Higgs parameters, nine charged-fermion masses, four CKM
entries, four PMNS entries, and one strong-CP angle. The theory does not
predict any of them, and it does not object to any other values they might
have had. That is not a criticism of the theory's accuracy. It is an
observation about its provenance: a measured parameter is one that could have
come out otherwise, and the Standard Model contains no machinery that would
have noticed.

The proposal of this paper is that the twenty-eight are not free at all ---
that they are registered observables of a single open-system projection, and
that the right description of each is a closed form rather than a fitted
value. That is a large claim, and it is falsifiable in the ordinary way: if a
closed form printed here disagrees with a sharpening measurement, the closed
form is wrong. What a reader should watch for through the rest of the paper is
not whether the numbers come out close, which they can be made to do by
enough freedom, but whether the derivation ever uses a measured value as an
input to something it later predicts. The apparatus of Part~\ref{part:II} is
built to make that visible, and the audit of Part~\ref{part:XIII} is where the
accounting is settled.

\GGPart[Standard Model parameter audit]{The Standard Model and its parameter audit}{Classical structure, classic \texorpdfstring{$\mathrm{SU}(5)$}{SU(5)}, the \GGTheory{} derivation, and the codebook of the twenty-eight free parameters}
\label{part:SMtarget}

\noindent
Part~\ref{part:I} has stated the thesis and placed \ANine{} in the cohort. Part~\ref{part:SMtarget} now fixes the phenomenological \emph{target} that the rest of the paper must hit. It states the Standard Model in textbook form, recalls the classical Georgi--Glashow $\mathrm{SU}(5)$ unification of 1974 and its \GGTheory{} reinterpretation from the six-dimensional bulk, and---most importantly---lays down the codebook-provenance audit that labels each of the twenty-eight free parameters \textbf{Derived}, \textbf{Derived (cond.)}, or \textbf{Asserted}. No constitutional machinery is assumed yet; the apparatus that does the deriving begins in Part~\ref{part:II}.

\section{Admissible inputs, forbidden shortcuts, and the standard of derivation}
\label{sec:admissible-inputs}

The theoretical side of this paper is allowed to use only two short, fixed lists --- five dimensionful constants and five integers --- together with the two pure mathematical constants $\pi$ and $\e$:
\begin{equation}
\underbrace{\{c,\,G_N,\,\hbar;\;k_B,\,Q_B\}}_{\text{\GUfive{}\ (\T{-GU5},\ \AOne{})}},\qquad
\underbrace{(M,N,L;J,K)=(1,3,5;7,35)}_{\text{\OGNfive{}\ ledger\ (\T{6},\ \ATwo{})}},\qquad \{\pi,\e\}.
\label{eq:admissible-ledger-intro}
\end{equation}
Everything else must be derived, exported, or deferred. In particular, the paper is not allowed to consume $G_F$, $m_W$, $m_Z$, $m_h$, $\alpha_s(M_Z)$, $\sin^2\theta_W$, any fermion mass, or any flavor-mixing parameter as an input. Measured values appear only in the final comparison tables. This prohibition is the methodological face of the central claim of \S\ref{sec:central-claim}: a Standard-Model observable cannot be admitted as an input precisely because it is an \emph{output} of the open-system projection, to be computed and only then compared. That these two short pentads --- and nothing else --- exhaust the admissible inputs of the entire framework is not merely asserted here but \emph{proved} in the foundations-and-audit paper \AFour{} \citep{ArisakaA4Euclid}, whose \GUfive{}\,$\leftrightarrow$\,\OGNfive{} two-pentad parameter audit and seven reconstruction theorems (\T{-Euclid-1} through \T{-Euclid-7}) establish the parameter-free reconstruction of modern physics on which the present section rests; the discussion below is the \ANine{}-local statement of that result.

\paragraph{The \GUfive{}: the Geometric Universal Five Constants.}
The only admissible \emph{dimensionful} inputs are the five Geometric Universal Five Constants (\GUfive{}) fixed by the constitutional theorem \T{-GU5} of \AOne{} \citep{ArisakaA1Constitution}: the speed of light $c$ (relativistic causality), the reduced Planck constant $\hbar$ (quantum action), Newton's constant $G_N$ (gravitation), the Boltzmann constant $k_B$ (thermodynamic temperature), and the energy--information quantum $Q_B \approx 0.1555$~eV per qubit --- the open-system conversion factor anchored in \ASeven{}. Among the five, only $G_N$ sets a dynamical scale --- $c$, $\hbar$, and $k_B$ are unit conventions and $Q_B$ a fixed conversion ratio --- so $G_N$ is effectively the single dimensionful free input: the energy$\leftrightarrow$spacetime conversion factor of general relativity. Both the (non-reduced) Planck mass $M_{\rm Pl}=(\hbar c/G_N)^{1/2}$ used throughout and the unification scale $E_{\rm GUT}$ descend from it; there are no continuous \emph{dimensionless} free parameters, and the Planck-to-electroweak hierarchy is carried not by tuning a small number but by the integer $K=35$ of the \OGNfive{} ledger acting on that one scale. These five are the universal conversion factors that translate the dimensionless geometry of the bulk into the dimensionful quantities an observer measures; no sixth dimensionful constant, and no measured Standard-Model scale, is admitted. That the dimensionful content closes at exactly these five --- three bulk conversion factors $(c,G_N,\hbar)$ plus two boundary conversion factors $(k_B,Q_B)$ --- and that the elementary charge $e$ is \emph{derived} (from $\alpha^{-1}(0)=2^J+N^2=137$) rather than admitted as a sixth member, are the architecture and charge theorems \T{-Euclid-3} and \T{-Euclid-4} of \AFour{} \citep{ArisakaA4Euclid}. The three fundamental energy scales of the paper are built from the \GUfive{} alone: the Planck energy $\EPl = \sqrt{\hbar c^5/G_N}$, the unification scale $\EGUT = \EPl\,e^{-2\pi}$, and the lock scale $\Mlock = \tfrac12 M_{\rm Pl}\,e^{-K} \approx 3.85$~TeV.

\paragraph{The \OGNfive{}: the integer ledger.}
The only admissible \emph{discrete} inputs are the five integers of the \OGNfive{} Diophantine ledger $(M,N,L;J,K) = (1,3,5;7,35)$, fixed by \T{6} of A2-\GG{} \citep{ArisakaA2GG6} and rendered anomaly-consistent by the \BIsix{} closure of \AThree{} \citep{ArisakaA3BI6}. The first three are the primary structural integers of the bulk --- $M=1$ (the abelian hypercharge line), $N=3$ (the family multiplicity), and $L=5$ (the unified $\mathrm{SU}(5)$ spine) --- and the remaining two, $J$ and $K$, are not free: they are forced by two independent closure relations on the first three,
\begin{equation}
J = 7,\qquad K = L\,J = M^2 + N^2 + L^2 = 1 + 9 + 25 = 35,
\label{eq:ogn5-closure-intro}
\end{equation}
so that $K=35$ is simultaneously the product $L J$ and the sum of squares $M^2+N^2+L^2$, and $J=7$ is the unique integer consistent with both. These five integers, and nothing larger, control every discrete exponent in the descent: the dyadic seam capacity $2^J = 128$ behind $\alpha^{-1}(M_Z)$, the rational partition $L^2/(4N^3) = 25/108$ behind $\sin^2\theta_W$, the lock factor $e^{-K} = e^{-35}$ behind the electroweak scale, and the cascade depths behind the fermion spectrum.

\paragraph{The standard of derivation.}
The two remaining symbols $\pi$ and $\e$ enter only as the pure mathematical constants they are --- the circumference of the boundary circle and the base of the radial renormalization flow --- not as adjustable physical inputs. A quantity counts as \emph{derived} in this paper only if it is reduced, through a named upstream theorem, to the \GUfive{}, the \OGNfive{}, and $\{\pi,\e\}$ alone, with no continuous parameter introduced at any step. This standard is precisely the closure theorem \T{-Euclid-5} of \AFour{} \citep{ArisakaA4Euclid}: every dimensionless constant of physics is an \OGNfive{} member, a consequence of the \OGNfive{} cascade, or a ratio of \GUfive{} members --- there is no third species --- while the \MSMF{} clause (\AX{4}, maximum symmetry, minimum freedom) is the constitutional source of the no-fitting discipline applied throughout this paper. This restriction is severe, but it is scientifically healthy: it protects the paper from the appearance of hidden fitting and forces the logic to run in the correct order --- constitution $\rightarrow$ geometry $\rightarrow$ operational export $\rightarrow$ parameter calculation $\rightarrow$ benchmark comparison.

\section{The Standard Model: classical structure}
\label{sec:sm-review}

Before we audit the twenty-eight free parameters, we ought to record what the parameters \emph{are}. The reader who has carried the four-dimensional Standard Model in their pocket since graduate school can skim this section; the reader who has not will find, on the next several pages, a self-contained statement of the model's field content, gauge structure, charge assignments, and parameter list. After all, an audit makes sense only when the auditor knows what is being audited.

The Standard Model in its modern form was assembled between 1967 and 1973 from three lines of work: the unified electroweak theory of Glashow, Salam, and Weinberg \citep{Glashow1961, Weinberg1967, Salam1968}; the color-charge formulation of QCD; and the discovery of asymptotic freedom by Gross, Wilczek, and Politzer. Its empirical reach is extraordinary: every collider measurement to date is consistent with its predictions, and the discoveries of the $W$, $Z$, top quark, and Higgs boson over four decades have confirmed each of its remaining missing pieces. Its conceptual structure, however, leaves twenty-eight numerical inputs unexplained, and it is those twenty-eight that the present paper proposes to derive.

\subsection{Fermion content: three families of quarks and leptons}
\label{subsec:sm-fermions}

The matter content of the Standard Model consists of three replicating families of spin-$\tfrac12$ fermions. Each family contains fifteen left-handed Weyl degrees of freedom (or sixteen if a right-handed neutrino is included), organized into the doublet/singlet pattern shown in Table~\ref{tab:sm-fermions}.

\begin{table}[H]
\centering
\caption[Fermion content of one Standard-Model family in the left-handed Weyl convention.]{Fermion content of one Standard-Model family in the left-handed Weyl convention. Three families ($N=3$) replicate this pattern. Representations are written as $(\text{color}, \mathrm{SU}(2)_L, Y)$ with electric charge $Q = T_3 + Y$. The right-handed singlets appear as left-handed conjugates ($u_R^c$, $d_R^c$, $e_R^c$).}
\label{tab:sm-fermions}
\small
\begin{tabular}{@{}lllcr@{}}
\toprule
\textbf{Field} & \textbf{Name} & \textbf{Representation} & \textbf{$Q$} & \textbf{$B$}\\
\midrule
\multicolumn{5}{l}{\bfseries\itshape Quarks (left-handed Weyl)}\\
$Q_L = (u_L, d_L)$ & quark doublet & $(\mathbf{3}, \mathbf{2}, +\tfrac16)$ & $(+\tfrac23, -\tfrac13)$ & $+\tfrac13$\\
$u_R^c$ & up-type singlet (conjugate) & $(\bar{\mathbf{3}}, \mathbf{1}, -\tfrac23)$ & $-\tfrac23$ & $-\tfrac13$\\
$d_R^c$ & down-type singlet (conjugate) & $(\bar{\mathbf{3}}, \mathbf{1}, +\tfrac13)$ & $+\tfrac13$ & $-\tfrac13$\\
\multicolumn{5}{l}{\bfseries\itshape Leptons (left-handed Weyl)}\\
$L_L = (\nu_L, e_L)$ & lepton doublet & $(\mathbf{1}, \mathbf{2}, -\tfrac12)$ & $(0, -1)$ & $0$\\
$e_R^c$ & charged-lepton singlet (conj.) & $(\mathbf{1}, \mathbf{1}, +1)$ & $+1$ & $0$\\
\bottomrule
\end{tabular}
\end{table}

The three families share this multiplet structure exactly. The first family is $(u, d, \nu_e, e)$; the second $(c, s, \nu_\mu, \mu)$; the third $(t, b, \nu_\tau, \tau)$. The only differences between them are the masses (and the corresponding Yukawa couplings to the Higgs) and the mixing angles relating mass eigenstates to gauge eigenstates. The replication itself is one of the deep facts the Standard Model leaves unexplained --- and one the \GGTheory{} derivation in \S\ref{sec:gg-derives-su5} promotes to a theorem.

\subsection{Boson content: gauge bosons and the Higgs}
\label{subsec:sm-bosons}

The bosonic content consists of the gauge bosons of the three forces and one scalar field. Table~\ref{tab:sm-bosons} records them.

\begin{table}[H]
\centering
\caption[Boson content of the Standard Model.]{Boson content of the Standard Model. Spin, electric charge, mass (PDG 2024 values), and the force the boson mediates. The five entries are the complete boson content: eight gluons, the two weak bosons, the photon, and the Higgs scalar. Masses are quoted to the precision at which they enter the comparisons of Part~\ref{part:VI}, and the two massless entries are massless as a matter of unbroken gauge symmetry rather than of measurement. Nothing in this table is derived here; it is the target the rest of the paper has to reach.}
\label{tab:sm-bosons}
\small
\begin{tabular}{@{}lcclc@{}}
\toprule
\textbf{Field} & \textbf{Spin} & \textbf{$Q$} & \textbf{Mass} & \textbf{Force}\\
\midrule
$g^a$ ($a = 1,\ldots,8$) & $1$ & $0$ & $0$ (massless) & strong (color)\\
$W^\pm$ & $1$ & $\pm 1$ & $80.379$ GeV & weak (charged)\\
$Z^0$ & $1$ & $0$ & $91.188$ GeV & weak (neutral)\\
$\gamma$ & $1$ & $0$ & $0$ (massless) & electromagnetic\\
$H^0$ & $0$ & $0$ & $125.10$ GeV & Higgs scalar\\
\bottomrule
\end{tabular}
\end{table}

The eight gluons mediate the strong force; the $W^\pm$ and $Z^0$ mediate the weak force after electroweak symmetry breaking; the photon mediates the electromagnetic force; and the Higgs scalar is the order parameter of electroweak symmetry breaking. The four electroweak gauge bosons descend from the four generators of $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$: three become the heavy $W^\pm$ and $Z^0$ when the Higgs vacuum value $v \approx 246$~GeV breaks the symmetry, and the fourth survives as the massless photon.

\subsection{The gauge group and the three forces}
\label{subsec:sm-gauge-group}

The Standard-Model gauge group is the direct product
\begin{equation}
G_{\rm SM} \;=\; \mathrm{SU}(3)_C \;\times\; \mathrm{SU}(2)_L \;\times\; \mathrm{U}(1)_Y,
\label{eq:sm-gauge-group}
\end{equation}
of dimension $8 + 3 + 1 = 12$. After electroweak symmetry breaking this reduces to
\begin{equation}
G_{\rm SM}^{\rm broken} \;=\; \mathrm{SU}(3)_C \;\times\; \mathrm{U}(1)_{\rm EM},
\label{eq:sm-gauge-group-broken}
\end{equation}
of dimension $8 + 1 = 9$, with the three broken generators absorbed into the longitudinal modes of $W^\pm$ and $Z^0$.

The three forces are the following.

\begin{description}\itemsep4pt
\item[Strong force ($\mathrm{SU}(3)_C$).] Mediated by gluons. Confines quarks into hadrons at low energies; asymptotically free at high energies. Coupling $\alpha_s(M_Z) \approx 0.118$.

\item[Weak force ($\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y \to \mathrm{U}(1)_{\rm EM}$).] Mediated by $W^\pm$ and $Z^0$. Acts only on left-handed fermions and right-handed antifermions --- a parity violation that is built into the gauge structure rather than imposed by hand. Spontaneously broken by the Higgs mechanism with vacuum expectation value $v = (\sqrt 2 \, G_F)^{-1/2} \approx 246$~GeV. The weak mixing angle $\sin^2\theta_W \approx 0.231$ measures the rotation between gauge and mass eigenstates of the neutral electroweak bosons.

\item[Electromagnetic force ($\mathrm{U}(1)_{\rm EM}$).] Mediated by the photon. The unbroken remainder of the electroweak group after Higgs symmetry breaking. Coupling $\alpha^{-1}(M_Z) \approx 127.93$, running to $\alpha^{-1}(0) \approx 137.04$ at low energies.
\end{description}

The fundamental strengths of the three forces --- measured by $\alpha_s$, $\alpha$, and $\sin^2\theta_W$ --- are three of the twenty-eight parameters the Standard Model leaves free.

\subsection{The Higgs mechanism in two paragraphs}
\label{subsec:sm-higgs}

The fourth and fifth parameters live in the Higgs sector. The Higgs field is a complex $\mathrm{SU}(2)_L$ doublet $\Phi = (\phi^+, \phi^0)^T$ with potential
\begin{equation}
V(\Phi) \;=\; -\mu^2 |\Phi|^2 \;+\; \lambda_H |\Phi|^4,
\label{eq:higgs-potential}
\end{equation}
where $\mu^2 > 0$ and $\lambda_H > 0$. Minimizing the potential gives a non-zero vacuum expectation value $\langle \phi^0 \rangle = v/\sqrt 2$ with $v = \mu/\sqrt{\lambda_H} \approx 246$~GeV. The vacuum expectation value $v$ and the quartic self-coupling $\lambda_H \approx 0.129$ are the two parameters of this sector.

Around the vacuum, three of the four real scalar degrees of freedom of $\Phi$ are eaten by the longitudinal modes of $W^\pm$ and $Z^0$ to give them mass $m_W = \tfrac12 g_2 v$ and $m_Z = \tfrac12 v\sqrt{g_2^2 + g_Y^2}$; the surviving real scalar is the physical Higgs boson $H^0$ with mass $m_H = \sqrt{2 \lambda_H} \, v \approx 125$~GeV. Yukawa couplings of the Higgs to the fermions then generate the fermion masses; the size of each Yukawa coupling is one of the twelve fermion-mass parameters of the Standard Model.

\subsection{The twenty-eight free parameters}
\label{subsec:sm-28-parameters}

Counting the free continuous inputs of the Standard Model with neutrino masses gives twenty-eight independent quantities. Table~\ref{tab:sm-28} records them by sector.

\begin{table}[H]
\centering
\caption[The twenty-eight free continuous parameters of the Standard Model with neutrino masses, as the codebook of \S\ref{sec:codebook-provenance} audits them.]{The twenty-eight free continuous parameters of the Standard Model with neutrino masses, as the codebook of \S\ref{sec:codebook-provenance} audits them. ``PDG value'' gives a representative empirical entry to one or two significant figures. The PDG column is printed for orientation only. No entry in it is used before the derivation that predicts the corresponding parameter is complete, and the provenance grade in the codebook column is assigned without reference to it.}
\label{tab:sm-28}
{\small
\begin{tabular}{@{}rllr@{}}
\toprule
\textbf{\#} & \textbf{Sector} & \textbf{Parameter} & \textbf{PDG value (representative)}\\
\midrule
1 & Gauge couplings & $\alpha^{-1}(M_Z)$ (electromagnetic) & $127.93$\\
2 & & $\sin^2\theta_W(M_Z)$ (weak mixing) & $0.231$\\
3 & & $\alpha_s(M_Z)$ (strong) & $0.118$\\
\midrule
4 & Higgs sector & $v$ (vacuum value) & $246$~GeV\\
5 & & $\lambda_H$ (quartic) & $0.129$\\
\midrule
6 & Charged leptons & $m_e$ & $0.511$~MeV\\
7 & & $m_\mu$ & $105.7$~MeV\\
8 & & $m_\tau$ & $1.777$~GeV\\
\midrule
9 & Quarks & $m_u$ & $\sim 2$~MeV\\
10 & & $m_d$ & $\sim 5$~MeV\\
11 & & $m_c$ & $1.27$~GeV\\
12 & & $m_s$ & $\sim 95$~MeV\\
13 & & $m_t$ & $173$~GeV\\
14 & & $m_b$ & $4.18$~GeV\\
\midrule
15 & CKM mixing & $\theta_{12}^{\rm CKM}$ (Cabibbo) & $13.0^\circ$\\
16 & & $\theta_{23}^{\rm CKM}$ & $2.4^\circ$\\
17 & & $\theta_{13}^{\rm CKM}$ & $0.20^\circ$\\
18 & & $\delta_{\rm CP}^{\rm CKM}$ & $1.20$~rad\\
\midrule
19 & PMNS mixing & $\sin^2\theta_{12}^{\rm PMNS}$ (solar) & $0.309$\\
20 & & $\sin^2\theta_{13}^{\rm PMNS}$ (reactor) & $0.02248$\\
21 & & $\sin^2\theta_{23}^{\rm PMNS}$ (atmospheric) & $0.47$--$0.56$\\
22 & & $\delta_{\rm CP}^{\rm PMNS}$ & $\sim 203^\circ$\\
\midrule
23 & Majorana phases & $\alpha_{21}$ & unmeasured\\
24 & & $\alpha_{31}$ & unmeasured\\
\midrule
25 & Neutrino masses & $m_*$ (Majorana scale) & $\sim 0.05$~eV\\
26 & & $\Delta m^2_{21}$ (solar) & $7.5 \times 10^{-5}$~eV$^2$\\
27 & & $\Delta m^2_{31}$ (atmospheric) & $2.51 \times 10^{-3}$~eV$^2$\\
\midrule
28 & Strong CP & $\bar\theta$ (QCD vacuum angle) & $\lesssim 10^{-10}$\\
\bottomrule
\end{tabular}
}
\end{table}

These twenty-eight numbers have one thing in common: the Standard Model has nothing to say about them. They are measured at colliders, neutrino experiments, and atomic-physics tables, and then plugged into the Lagrangian. Every textbook of the subject treats the matter this way, and three generations of theoretical effort to derive them --- grand-unified theories, supersymmetric extensions, string compactifications, noncommutative-geometry constructions --- have produced partial accounts of small subsets but never a closure. The present paper proposes that a closure is possible after all, and that it is the constitutional content of \GGTheory{} that supplies it.

The rest of this Part walks the reader through what such a constitutional derivation looks like. Section~\ref{sec:classic-su5} reviews the historical stepping-stone, the Georgi--Glashow $\mathrm{SU}(5)$ unification of 1974, which already organizes charge quantization and electroweak unification but leaves the deeper geometric content unaddressed. Section~\ref{sec:gg-derives-su5} then shows how \GGTheory{} recovers $\mathrm{SU}(5)$ as an automatic structure of the six-dimensional bulk and produces the field content, charges, family number, and force-splitting pattern of the Standard Model from constitutional principles. After that, \S\ref{sec:codebook-provenance} opens the parameter audit.

\section{Classic Georgi--Glashow \texorpdfstring{$\mathrm{SU}(5)$}{SU(5)} unification}
\label{sec:classic-su5}

The historical first step toward a derivation of the Standard-Model gauge structure was the $\mathrm{SU}(5)$ grand-unified theory of Georgi and Glashow \citep{GeorgiGlashow1974}. Its appeal was immediate: a single simple Lie group containing the Standard-Model gauge group as a subgroup explains charge quantization, organizes the matter content of one family into a small number of irreducible representations, and unifies the three Standard-Model gauge couplings into one at a high energy. Half a century later, $\mathrm{SU}(5)$ remains the cleanest grand-unified candidate to inspect, and the present paper retains its non-Abelian spine within a richer six-dimensional setting.

\subsection{Matter content: \texorpdfstring{$\bar{\mathbf{5}} \oplus \mathbf{10}$}{5-bar + 10} per family}
\label{subsec:su5-matter}

A single Standard-Model family is fifteen left-handed Weyl fermions. $\mathrm{SU}(5)$ organizes them into two irreducible representations. Under the Standard-Model decomposition $\mathrm{SU}(5) \to \mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$:
\begin{align}
\bar{\mathbf{5}} &\;\to\; (\bar{\mathbf{3}}, \mathbf{1})_{+1/3} \;\oplus\; (\mathbf{1}, \mathbf{2})_{-1/2},
\label{eq:su5-5bar}\\
\mathbf{10} &\;\to\; (\mathbf{3}, \mathbf{2})_{+1/6} \;\oplus\; (\bar{\mathbf{3}}, \mathbf{1})_{-2/3} \;\oplus\; (\mathbf{1}, \mathbf{1})_{+1}.
\label{eq:su5-10}
\end{align}
The $\bar{\mathbf{5}}$ contains the down-type quark singlet $d_R^c$ and the lepton doublet $L_L = (\nu_L, e_L)$. The $\mathbf{10}$ contains the quark doublet $Q_L = (u_L, d_L)$, the up-type singlet $u_R^c$, and the charged-lepton singlet $e_R^c$. The Higgs doublet sits in a fundamental $\mathbf{5}_H$, with the color-triplet partner projected out at the high scale.

The fact that fifteen seemingly disparate Weyl fields fit into a $\bar{\mathbf{5}} \oplus \mathbf{10}$ multiplet structure is not an accident. Indeed, it is the reason $\mathrm{SU}(5)$ has held the imagination of theorists since 1974: charge quantization (every electric charge is an integer multiple of $1/3$) becomes a consequence of the simple-group embedding rather than an empirical input.

\subsection{The adjoint \texorpdfstring{$\mathbf{24}$}{24} and the gauge bosons}
\label{subsec:su5-adjoint}

The adjoint representation of $\mathrm{SU}(5)$ has dimension $24$ and decomposes as
\begin{equation}
\mathbf{24} \;\to\; (\mathbf{8}, \mathbf{1})_0 \;\oplus\; (\mathbf{1}, \mathbf{3})_0 \;\oplus\; (\mathbf{1}, \mathbf{1})_0 \;\oplus\; (\mathbf{3}, \mathbf{2})_{-5/6} \;\oplus\; (\bar{\mathbf{3}}, \mathbf{2})_{+5/6}.
\label{eq:su5-adjoint}
\end{equation}
The first three pieces are the Standard-Model gauge bosons: the eight gluons of $\mathrm{SU}(3)_C$, the three weak bosons of $\mathrm{SU}(2)_L$, and the hypercharge boson of $\mathrm{U}(1)_Y$. The remaining $12 = 6 + 6$ states are the heavy $X$ and $Y$ bosons that mediate proton decay in classical $\mathrm{SU}(5)$.

The properly-normalized hypercharge generator inside the fundamental representation is
\begin{equation}
T_Y \;=\; \sqrt{\tfrac{3}{5}} \cdot \tfrac{1}{2}\, \mathrm{diag}\!\left(-\tfrac23, -\tfrac23, -\tfrac23, +1, +1\right),
\label{eq:su5-hypercharge}
\end{equation}
with the conventional normalization $\mathrm{Tr}(T_Y^2) = \tfrac12$. The factor $\sqrt{3/5}$ is the GUT normalization that makes the coupling unification $g_1 = \sqrt{5/3}\, g_Y$ work.

\subsection{What classic \texorpdfstring{$\mathrm{SU}(5)$}{SU(5)} explains, and what it leaves unresolved}
\label{subsec:su5-strengths-and-limits}

Three things classic $\mathrm{SU}(5)$ explains. First, charge quantization: the embedding of $\mathrm{U}(1)_Y$ inside a simple group forces the hypercharge of every Standard-Model field to be a specific rational number. Second, the matter content of one family fits into a clean representation pattern $\bar{\mathbf{5}} \oplus \mathbf{10}$ rather than a heterogeneous list of fifteen separate Weyl spinors. Third, the three Standard-Model gauge couplings, when run from the electroweak scale toward higher energies under the renormalization group, approach a common value at $E \sim 10^{15}$~GeV --- suggesting a unification scale.

Five things classic $\mathrm{SU}(5)$ does not explain. First, the family number $N = 3$: nothing in the simple-group construction selects three families over one or seventeen. Second, the masses of the fermions: the Yukawa hierarchy spans twelve orders of magnitude and remains unconstrained. Third, gravity: classic $\mathrm{SU}(5)$ is a four-dimensional gauge theory with no gravitational sector. Fourth, proton stability: the heavy $X$ and $Y$ bosons of the $\mathbf{24}$ mediate proton decay $p \to e^+ \pi^0$ at a rate that has been ruled out by Super-Kamiokande, requiring either an $\mathrm{SU}(5)$-breaking mechanism that lifts the $X, Y$ masses well above $10^{16}$~GeV or an additional symmetry that suppresses the operator. Fifth, the measurement layer: there is no constitutional account of how a measurement extracts a number from the gauge dynamics.

The present paper inherits the first three virtues of $\mathrm{SU}(5)$ and addresses each of the five gaps within the six-dimensional setting of \GGTheory{}, to which we now turn.

\section{\GGTheory{} derivation of \texorpdfstring{$\mathrm{SU}(5)$}{SU(5)} and the Standard Model}
\label{sec:gg-derives-su5}

The previous section recorded what classic $\mathrm{SU}(5)$ achieves and what it leaves open. This section shows how the six-dimensional bulk of \GGTheory{}, the constitution of A1 \citep{ArisakaA1Constitution}, the modified Bianchi identity of \AThree{} \citep{ArisakaA3BI6}, and the \OGNfive{} ledger of A2-\GG{} \citep{ArisakaA2GG6} together produce the Standard-Model field content, the charge assignments, the family number $N = 3$, and the three-force splitting pattern as theorems rather than as separate empirical inputs. The full technical derivations live in \S\ref{sec:su5-overview} (classic $\mathrm{SU}(5)$ to \GG{} reinterpretation), \S\ref{sec:apply-gdos-su5} (\GDOS{} applied to the unified packet), and \S\ref{sec:sm-foundation} (fundamental architecture); the present section is the pedagogical primer that those later sections cash out in detail.

\subsection{The bulk gauge group}
\label{subsec:gg6-bulk-gauge-group}

The six-dimensional bulk of \GGTheory{} carries the composite gauge group
\begin{equation}
G_{\rm tot} \;=\; \mathrm{Spin}(1,5) \;\times\; \mathrm{SU}(5) \;\times\; \UoneB{} \;\times\; \UoneQi{},
\label{eq:gg6-bulk-gauge-group}
\end{equation}
forced by the modified Bianchi identity \BIsix{} of \AThree{} together with the GGEP axiom (\AX{2}) of A1. $\mathrm{Spin}(1,5)$ is the Lorentz group of the 6D bulk; $\mathrm{SU}(5)$ is the non-Abelian unification spine first proposed by Georgi and Glashow; $\UoneB{}$ is the local baryon-number symmetry that protects proton stability; and $\UoneQi{}$ is the local information-current symmetry required by the Arisaka equation of \ASeven{} \citep{ArisakaA7Qi4}.

The trace-ratio lock $|a/c| = |a/d| = 5/2$ of the \BIsix{} anomaly polynomial fixes the relative weights of the four factors (the full factorizing six-dimensional anomaly polynomial and its flux quantization are established, as an existence construction, in \AThree{}; \ANine{} consumes only this trace-ratio constraint), and the \OGNfive{} closure $K = LJ = M^2 + N^2 + L^2 = 35$ of A2-\GG{} \T{6} fixes the integer ledger that controls the discrete inputs of the descent. After all, the difference between classic 4D $\mathrm{SU}(5)$ and the \GG{} setting is not that we replace one group by another --- it is that we embed the same simple-group spine inside a six-dimensional geometry whose closure forces its presence.

\subsection{Why the visible matter is spin-\texorpdfstring{$1/2$}{1/2}}
\label{subsec:gg6-spinors}

The bulk matter content is described by a six-dimensional Dirac action
\begin{equation}
S_\Psi^{(6)} \;=\; \int d^6 x \, \sqrt{-g_6} \; \bar\Psi \, i \Gamma^M D_M \Psi,
\label{eq:gg6-dirac-action}
\end{equation}
where $\Psi$ is a 6D spinor and $D_M$ contains both the spin and the gauge connection. Reduction to the four-dimensional boundary, by Kaluza--Klein decomposition into modes on the compact circle $S^1_\theta$ and the radial interval $I_r$, projects out one chirality of the would-be zero mode and leaves a chiral four-dimensional spectrum of left-handed Weyl fermions.

Roughly speaking, six-dimensional spinors carry eight complex components; orbifold or interval boundary conditions slice this in half; what survives at low energies is the standard four-dimensional pair of two-component Weyl spinors per surviving family. The fact that the visible matter is spin-$1/2$ is therefore not a separate postulate of the Standard Model --- it is inherited from the bulk spin structure and the boundary-condition projection.

\subsection{One family contains quarks and leptons with the observed charges}
\label{subsec:gg6-one-family}

The bulk gauge group $\mathrm{SU}(5) \times \UoneB{} \times \UoneQi{}$ acts on the surviving 4D Weyl fermions through the same $\bar{\mathbf{5}} \oplus \mathbf{10}$ embedding of classic $\mathrm{SU}(5)$ (Eqs.~\eqref{eq:su5-5bar}--\eqref{eq:su5-10}). Once the embedding is fixed, the electric-charge pattern is forced by $Q = T_3 + Y$:
\begin{equation}
\begin{aligned}
Q_L = (u_L, d_L) &\colon \quad (Q_{\rm EM}, Q_{\rm EM}) = (+\tfrac23, -\tfrac13),\\
u_R^c &\colon \quad Q_{\rm EM} = -\tfrac23, \qquad d_R^c \colon Q_{\rm EM} = +\tfrac13,\\
L_L = (\nu_L, e_L) &\colon \quad (Q_{\rm EM}, Q_{\rm EM}) = (0, -1), \qquad e_R^c \colon Q_{\rm EM} = +1.
\end{aligned}
\label{eq:gg6-charge-pattern}
\end{equation}
The fractional charges of quarks and the integral charges of leptons fall out of the same group-theoretic package --- exactly the lesson Georgi and Glashow drew in 1974, now embedded in a six-dimensional bulk geometry with explicit constitutional grounding.

The baryon-number assignments are the diagonal $\UoneB{}$ readouts
\begin{equation}
B(Q_L) = +\tfrac13, \quad B(u_R^c) = B(d_R^c) = -\tfrac13, \quad B(L_L) = B(e_R^c) = B(H) = 0,
\label{eq:gg6-baryon-numbers}
\end{equation}
forced by the constitutional locality of $\UoneB{}$ in the bulk gauge group. This is the structural reason proton-decay operators are absent in the exact low-energy theory: baryon number is gauged, not merely a global accidental symmetry, and the operator that would mediate $p \to e^+ \pi^0$ is forbidden by the exact (global) baryon number that survives as the rigid symmetry of a Stueckelberg-massive \UoneB{} (made precise in Theorem~\ref{thm:proton-stability-a9}) --- in sharp contrast to the classical $\mathrm{SU}(5)$ situation where the $X, Y$ bosons of the adjoint $\mathbf{24}$ mediate the decay.

\subsection{Six quarks and six leptons from the family multiplicity \texorpdfstring{$N = 3$}{N = 3}}
\label{subsec:gg6-six-and-six}

Once one family is in hand, the observed flavor count follows from the family multiplicity. Each family contains two quark flavors and two lepton flavors; replicating the unified package three times therefore gives precisely six quark flavors
\begin{equation}
(u, d), \quad (c, s), \quad (t, b),
\label{eq:gg6-six-quarks}
\end{equation}
and six leptonic degrees of freedom
\begin{equation}
(\nu_e, e), \quad (\nu_\mu, \mu), \quad (\nu_\tau, \tau),
\label{eq:gg6-six-leptons}
\end{equation}
which become $\nu_1, \nu_2, \nu_3$ and $e, \mu, \tau$ in the mass basis. The ``six and six'' count is not a separate mystery once the surviving chiral index is three --- it is the counting consequence of three copies of one unified chiral package.

\subsection{Why \texorpdfstring{$N = 3$}{N = 3}: \MSMF{} lower bound and Bogomolnyi closure upper bound}
\label{subsec:gg6-N3}

The family number $N = 3$ is the three-leg theorem of \PaperPTwo{} \citep{ArisakaP2N3}. The three legs are these.

\begin{description}\itemsep4pt
\item[Leg L1 (\MSMF{} lower bound).] By A1 \AX{4} \MSMF{}, any family number that admits no nontrivial \CP{} phase is excluded as too poor. $N = 1$ has no flavor mixing at all; $N = 2$ has one mixing angle but the Kobayashi--Maskawa argument shows it cannot accommodate a complex \CP{} phase \citep{KobayashiMaskawa1973}. Therefore $N \ge 3$.

\item[Leg L2 (\GDOS{} Morse upper bound).] By \ASix{} \GFF{} Morse-positivity (\T{3.1}), the cascade of geometric attractors on the family-projector stratum $\mathbb{C}P^{N-1}$ closes if and only if the total Bogomolnyi instanton action is finite. Summing the cascade $S_{[2k] \to [2(k-1)]} = \pi k$ over the $N-1$ gates of $\mathbb{C}P^{N-1}$ ($k = 1$ to $N-1$) gives $S_{\rm tot}(N) = \tfrac12 N(N-1) \pi$; the \BIsix{} closure constraint of \AThree{} caps the admissible total instanton action at $S_{\rm BI}^{\rm max} = (2\pi \cdot 3/5) \cdot (L/2) = 3\pi$. This forces $S_{\rm tot}(N) \le 3\pi$, i.e.\ $N(N-1) \le 6$, i.e.\ $N \le 3$, saturated exactly at $N = 3$. The tolls $\pi k$ themselves are no longer a reading: they are derived in \ASix{} \citep{ArisakaA6Gi4} from the Whitney pair count of the winding bundle (Morse weights $\lambda_\ell = \binom{\ell}{2}$, inflow charge $\Delta c_2 = k$, under three declared premises), so the exponential skeleton of the fermion spectrum used throughout this paper rests on a theorem-grade ladder.

\item[Leg L3 (constructive codebook agreement).] The $N = 3$ codebook reproduces twelve charged-fermion masses, two flavor mixing matrices, and the gauge--Higgs sector at four-significant-figure precision (the body of the present paper). No alternative $N$ admits an analogous closed codebook.
\end{description}

The three legs together give $N = 3$ exactly: $N \ge 3$ from \MSMF{}, $N \le 3$ from Bogomolnyi closure, and $N = 3$ as a constructive check. Indeed, the present paper is the constructive check.

\subsection{Why the three forces split as they do}
\label{subsec:gg6-three-forces}

The descent from the bulk gauge group to the observed unbroken low-energy group is the two-step chain
\begin{equation}
\boxed{
\begin{aligned}
\mathrm{SU}(5) \times \UoneB{} \times \UoneQi{}
& \xrightarrow[\text{boundary cond.\ + holonomy}]{}
\mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y \\
& \xrightarrow[\text{Wilson--Hosotani}]{}
\mathrm{SU}(3)_C \times \mathrm{U}(1)_{\rm EM}.
\end{aligned}
}
\label{eq:gg6-symmetry-breaking-chain}
\end{equation}
The first arrow is the orbifold/boundary-condition projection that breaks $\mathrm{SU}(5) \to \mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ (with $\UoneB{}$ and $\UoneQi{}$ surviving as separate Abelian factors). The second arrow is the Wilson--Hosotani holonomy that breaks the electroweak factor to $\mathrm{U}(1)_{\rm EM}$ at the scale $v \approx 246$~GeV.

The three forces appear distinct at low energies for two reasons. The first reason is algebraic: each surviving force corresponds to a different generator of the broken $\mathrm{SU}(5)$ parent. The second reason is dynamical: once the sectors are distinct they no longer run identically along the radial throat --- their couplings transport with different beta functions because the surviving matter content coupled to each sector is different. The strong interaction becomes the most rapidly varying and most strongly coupled in the infrared; the weak interaction is broken and partially hidden by the electroweak vacuum; the electromagnetic interaction remains long-range because the unbroken $\mathrm{U}(1)_{\rm EM}$ generator lies in the commutant of the selected vacuum holonomy.

\subsection{Comparison: classic \texorpdfstring{$\mathrm{SU}(5)$}{SU(5)} versus the \GG{} reinterpretation}
\label{subsec:gg6-vs-classic-su5}

Table~\ref{tab:gg6-vs-classic-su5} summarizes the differences side by side.

\begin{table}[H]
\centering
\caption[Classic Georgi--Glashow $\mathrm{SU}(5)$ (1974) versus the GG-6 reinterpretation of the present paper.]{Classic Georgi--Glashow $\mathrm{SU}(5)$ (1974) versus the GG-6 reinterpretation of the present paper. The two columns differ on where the freedom sits. Classic $\mathrm{SU}(5)$ takes a simple group, a chosen scalar potential, and a family number set by hand; the GG-6 reading embeds the same algebra in a six-dimensional bulk, replaces the potential with boundary conditions and Wilson--Hosotani holonomy, and returns the family number as a theorem. The comparison is structural, and says nothing yet about which of the two agrees better with measurement.}
\label{tab:gg6-vs-classic-su5}
{\small
\begin{tabular}{@{}p{0.18\textwidth}p{0.34\textwidth}p{0.40\textwidth}@{}}
\toprule
\textbf{Question} & \textbf{Classic 4D $\mathrm{SU}(5)$ (1974)} & \textbf{\GG{} reinterpretation (this work)}\\
\midrule
Unified algebra & Simple group $\mathrm{SU}(5)$ alone & Non-Abelian spine $\mathrm{SU}(5)$ inside $\mathrm{Spin}(1,5) \times \mathrm{SU}(5) \times \UoneB{} \times \UoneQi{}$\\
Symmetry breaking & Higgs multiplets and chosen scalar potential & Boundary conditions plus Wilson--Hosotani holonomy in a 6D bulk\\
Family number $N$ & Free input (set to 3 by hand) & Theorem: $N = 3$ from \MSMF{} + Bogomolnyi closure (\PaperPTwo{})\\
Proton stability & Endangered by $X$, $Y$ bosons and color-triplet Higgs & Protected by orbifold projection + exact local $\UoneB{}$ selection rule\\
Gravity & Not included; appended as a separate sector & Already inside the composite $\mathrm{Spin}(1,5)$ connection mandated by GGEP\\
Measurement layer & External to the model & Built into the theory through \OGGEP{} and \GDOS{}\\
Parameter philosophy & Several continuous GUT-sector inputs & Discrete \OGNfive{} ledger plus the \GUfive{}\\
\bottomrule
\end{tabular}
}
\end{table}

The correct way to read \GG{} is not as a rejection of the historical $\mathrm{SU}(5)$ insight but as its geometric completion. Georgi and Glashow taught us that the Standard Model has the smell of a broken simple group; \GGTheory{} accepts the lesson and embeds the simple-group spine inside a six-dimensional geometry whose closure also explains anomaly freedom (via \BIsix{}), proton stability (via gauged $\UoneB{}$), gravity (via $\mathrm{Spin}(1,5)$), the measurement layer (via \OGGEP{} and \GDOS{}), and the family number (via the Bogomolnyi closure of \PaperPTwo{}).

\subsection{Bridge to the parameter audit}
\label{subsec:gg6-bridge-to-codebook}

We have now stated what the Standard-Model fields are (\S\ref{sec:sm-review}), what classic $\mathrm{SU}(5)$ achieves and leaves open (\S\ref{sec:classic-su5}), and how \GGTheory{} recovers the gauge group, charges, family number, and force-splitting pattern from the six-dimensional bulk plus the constitutional principles of A1 (the present section). The reader who has followed this far now knows, in qualitative terms, what the twenty-eight free parameters of Table~\ref{tab:sm-28} are, why they sit in the structure they do, and what kind of derivation a constitutional account is supposed to provide.

After all, an audit makes sense only when the auditor knows what is being audited. The audit itself begins in the next section.

\section{Codebook provenance: what is derived end-to-end, what is derived modulo A9-local lemmas, and what is asserted}
\label{sec:codebook-provenance}

\subsection{Why this section exists}

The central claim of the present paper is that the twenty-eight free parameters of the Standard Model can be derived from the \GGConstitution{} of A1 plus the named theorems of A2--A7, with the constitutional clauses \MSMF{} (\AX{4}) and \OGN{} (\POS{6}) supplying the structural selection principle that picks the unique value on the relevant boundary stratum. A claim of that strength invites the question: \emph{which} of the twenty-eight is derived end-to-end from non-physics axioms, \emph{which} is derived modulo an additional lemma local to \ANine{} or named by the \MSMF{}/\OGN{} audit, and \emph{which} is asserted as a closed-form rational without a genuine upstream theorem and without a unique \MSMF{}/\OGN{} cost-minimum at the structural-precision target? This section answers the question parameter by parameter, in the structural-honesty discipline of \AThree{}, which marks every claim with one of three status labels (\textbf{Derived}, \textbf{Derived (cond.)}, \textbf{Asserted}) accompanied by the upstream owner and a one-line concern. After all, the test of an honest framework is not that it answers every question on the first pass, but that it knows which questions it has answered, which it has half-answered, and which it has not yet touched. The present table reports the counts \emph{after} the \MSMF{}/\OGN{} per-parameter audit, which is carried out in full in Part~\ref{part:XIII} (\S\ref{sec:msmf-ogn-selection-audit}); that audit promotes nine of the seventeen previously Asserted entries, leaving eight Asserted.

\subsection{Status legend}

\begin{description}\itemsep1pt
\item[Derived.] The parameter value is forced by a named theorem of A1--A7 with all factors traceable. An adversarial reader cannot reach the boxed equation by inserting a fitted continuous prefactor.
\item[Derived (conditional).] A genuine upstream theorem is cited and supports the structural claim, but the A9 step adds one or more lemmas that are local to the present paper. The lemma is named explicitly in the Comment column.
\item[Asserted.] The parameter is presented as a closed-form rational built from the \OGNfive{} integer ledger, but the specific choice of which OGN integer plays which role (numerator vs denominator, exponent vs prefactor) is not forced by any named theorem of A1--A7. The choice is a working hypothesis of the present paper, defensible by its empirical agreement at four significant figures, but not derived in the strict sense.
\end{description}

\subsection{Codebook provenance table}

\begin{center}
{\small
\setlength{\tabcolsep}{2pt}%
\begin{longtable}{@{}p{0.04\textwidth}p{0.18\textwidth}p{0.16\textwidth}p{0.16\textwidth}p{0.13\textwidth}p{0.27\textwidth}@{}}
\caption[Codebook provenance for the twenty-eight free parameters of the Standard Model.]{Codebook provenance for the twenty-eight free parameters of the Standard Model. Status labels follow the \AThree{} structural-honesty convention extended by the \MSMF{}/\OGN{} audit: \textbf{Derived} (six entries) = one end-to-end theorem ($\sin^2\theta_W$ via the \AThree{} trace-ratio theorem, no empirical input, modulo two named \GG{}-internal structural lemmas; see Thm~\ref{thm:sw2}) plus five entries fixed without a continuous free parameter but \emph{not} by an A1--A7 theorem: one \MSMF{}/\OGN{} cost-minimum selection ($\alpha_s$; the empirical value enters the selection filter, so a look-elsewhere factor is reported), one quartic-curvature derivation ($\lambda_H$, the curvature of the Wilson--Hosotani holonomy potential at $\sin\theta_H^\star = (2/L)^N$, modulo a spectral-normalization lemma), one confirmed parameter-free prediction ($y_t = 1$), one discrete structural datum ($\alpha_{31}$), and one convention on an unmeasured sector ($\alpha_{21}$); each entry's sub-basis is given in its Status and Comment; \textbf{Derived (cond.)} = upstream theorem cited but a single named A9-local or-named lemma is added (fourteen entries); \textbf{Asserted} = no upstream theorem and no \MSMF{}/\OGN{} selection at the structural-precision target (eight entries: the eight charged-fermion Yukawas other than the top). ``Comment'' names the load-bearing concern when the status is not Derived.}
\label{tab:codebook-provenance}\\
\toprule
\textbf{\#} & \textbf{Parameter} & \textbf{A9 location} & \textbf{Upstream owner} & \textbf{Status} & \textbf{Comment}\\
\midrule
\endfirsthead
\multicolumn{6}{c}{\itshape Table~\ref{tab:codebook-provenance} continued}\\
\toprule
\textbf{\#} & \textbf{Parameter} & \textbf{A9 location} & \textbf{Upstream owner} & \textbf{Status} & \textbf{Comment}\\
\midrule
\endhead
\midrule \multicolumn{6}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

\multicolumn{6}{l}{\bfseries\itshape Gauge sector (3)}\\
1 & $\alpha^{-1}(M_Z) = 2^J$ & Thm \ref{thm:alpha} & \ATwo{} T-GG6-6 ($J=7$) & Derived (cond.) & A9-local ``binary recursion'' lemma adds factor $1/2$ per dyadic step\\
2 & $\sin^2\theta_W = L^2/(4N^3) = 25/108$ & Thm \ref{thm:sw2} & \AThree{} §7 (trace ratio $2/5$) + \OGNfive{} & \textbf{Derived} (theorem) & End-to-end from the \AThree{} trace ratio $+$ \OGNfive{}, no empirical input; rests on two named \GG{}-internal structural lemmas (quadratic neutral-partition, color-blind averaging) --- Derived (theorem) \emph{modulo} those two steps\\
3 & $\alpha_s^{-1}(M_Z) = \pi e$ & Thm \ref{thm:alphas} & A1 T-GU5 + \MSMF{}/\OGN{} (\MSMF{}/\OGN{} audit) & \textbf{Derived} (selected) & \MSMF{}/\OGN{} cost-2 selection $\pi e$ within $1\%$ of $\alpha_{s,\rm PDG}^{-1} = 8.482$ (look-elsewhere $p \approx 0.27$); the empirical value enters the selection filter\\
\multicolumn{6}{l}{\bfseries\itshape Higgs sector (2)}\\
4 & $v = M_{\rm lock}(2/L)^N$ & Thm \ref{thm:vev} & \ATwo{} (\OGNfive{}, $K=35$) + Wilson--Hosotani & Derived (cond.) & Lock factor $\tfrac12 e^{-K}$ uses A9-local binary-endpoint convention; $(2/L)^N$ ``one factor per triplicity layer'' asserted\\
5 & $\lambda_H = 2^{-N}$ & Thm \ref{thm:quartic}, L6 & \ASix{} \GFF{}/Wilson--Hosotani curvature ($+$ A1 \AX{4}/\POS{6}) & \textbf{Derived} (curvature, mod.\ 1 lemma) & $\lambda_H$ is the quartic curvature of the holonomy potential at the minimum fixed by $\sin\theta_H^\star = (2/L)^N$ (vev cross-check passes: $v = f\sin\theta_H^\star = 246.2$~GeV); natural one-loop magnitude $\mathcal{O}(0.1)$, consistent with $2^{-N}$; precise value rests on the spectral-normalization lemma (scale-profile KK sum), not yet computed\\
\multicolumn{6}{l}{\bfseries\itshape Charged-fermion masses (9 of the 28; Yukawa codebook)}\\
6 & $m_e$ & §\ref{sec:lepton-mass} & A9-local codebook & Asserted & Per-particle numerator $N/L^3$ and denominator depth $J{+}L{+}1$ chosen post-hoc\\
7 & $m_\mu$ & §\ref{sec:lepton-mass} & A9-local codebook & Asserted & Per-particle numerator $L$ and denominator depth $J{+}L{+}1$ chosen post-hoc\\
8 & $m_\tau$ & §\ref{sec:lepton-mass} & A9-local codebook & Asserted & Per-particle numerator $NJ$ and denominator depth $J{+}N{+}1$ chosen post-hoc\\
9 & $m_u$ & §\ref{sec:quark-mass} & A9-local codebook & Asserted & Angular cascade $e^{-4\pi}$ + prefactor $N$ chosen post-hoc\\
10 & $m_d$ & §\ref{sec:quark-mass} & A9-local codebook & Asserted & Down-type ``branch factor $2$'' chosen post-hoc\\
11 & $m_c$ & §\ref{sec:quark-mass} & A9-local codebook & Asserted & Angular cascade $e^{-2\pi}$ + prefactor $N$ chosen post-hoc\\
12 & $m_s$ & §\ref{sec:quark-mass} & A9-local codebook & Asserted & Angular cascade $e^{-3\pi}$ + prefactor $2N$ chosen post-hoc\\
13 & $m_t = v/\sqrt 2$ ($y_t = 1$) & §\ref{sec:quark-mass} & A6 \S 28 cascade-top + \MSMF{}/\OGN{} (\MSMF{}/\OGN{} audit) & \textbf{Derived} (predicted) & $y_t = 1$ is the cascade-top reference; with $v$ independently derived from $M_{\rm lock}$ (Higgs sector), $m_t = v/\sqrt2$ is a \emph{parameter-free prediction} confirmed at $\sim 1\%$ --- not a theorem\\
14 & $m_b$ & §\ref{sec:quark-mass} & A9-local codebook & Asserted & Prefactor $N/2^N$ (not the down-type $2N$ of $m_s$) chosen post-hoc\\
\multicolumn{6}{l}{\bfseries\itshape CKM mixing (4)}\\
15 & $\theta_{12}^{\rm CKM}$ (Cabibbo) & §\ref{sec:CKM} & Quark codebook (\#9--14) + ``pentagonal phase $3\pi/L$'' & Derived (cond.) & Inherits the per-quark assertions of \#9--14\\
16 & $\theta_{23}^{\rm CKM}$ & §\ref{sec:CKM} & Quark codebook + ``compact factor $\pi$'' & Derived (cond.) & Multiplicative overlap; conditional on quark codebook\\
17 & $\theta_{13}^{\rm CKM}$ & §\ref{sec:CKM} & Quark codebook & Derived (cond.) & Pure overlap formula; conditional on quark codebook\\
18 & $\delta_{\rm CP}^{\rm CKM}$ & §\ref{sec:CKM} & ``Common rephasing step $\pi/L$'' & Derived (cond.) & Step asserted; QD shift to $13\pi/35$ post-hoc\\
\multicolumn{6}{l}{\bfseries\itshape PMNS mixing (4 in the Dirac convention; +2 Majorana)}\\
19 & $\sin^2\theta_{12}^{\rm PMNS} = L^2/3N^3 = 25/81$ & §\ref{sec:PMNS} & \GDOS{}/\GFF{} democratic-weight projector derivation & Derived (cond.) & Structural democratic-weight form (QD-completed to $0.30857$); the bounded-cost search also returns the lower-cost orphan $1/\pi = 0.318$, which fits worse and is geometrically isolated, so it is not adopted\\
20 & $\sin^2\theta_{13}^{\rm PMNS} = N/4K = 3/140$ & §\ref{sec:PMNS} & \GDOS{}/\GFF{} reactor-overlap derivation & Derived (cond.) & Structural form (QD-completed to $0.021902$); the bounded-cost search returns the orphan $1/45 = 0.0222$, not adopted (the structural form fits better)\\
21 & $\sin^2\theta_{23}^{\rm PMNS} = (J{+}1)/(2J) = 4/7$ & §\ref{sec:PMNS} & \MSMF{}/\OGN{} cost-2 + rank-2 atmospheric-deviation lemma & Derived (cond., \MSMF{}/\OGN{}) & Cost-2 winner; tie with $5/9$ broken by structural-naming rule (rank-2 deviation from $\pi/4$ by $1/(2J)$)\\
22 & $\delta_{\rm CP}^{\rm PMNS}$ & §\ref{sec:PMNS} & \GDOS{}/\GFF{} minimal-neutrino-phase derivation & Derived (cond.) & Structural form $\delta = \pi + 2\arcsin(1/7) = 196.4^\circ$ (QD-completed $196.06^\circ$), within $0.2\sigma$ of the accelerator joint-fit best value ($203.4^\circ$); conditional on the leptonic Dirac-phase assignment lemma (minimal neutrino phase $\phi_\nu = \arcsin(1/(2N+1))$ plus the $\pi$ reference shift); a subdominant degeneracy $\{\pi, 3\pi/2\}$ is resolved in favor of the structural form; branch freeze: the $3\pi/2$ branch is pre-registered as dead --- a future measurement near $270$--$286^\circ$ may not be reinterpreted under it (\S\ref{subsec:msmf-ogn-audit})\\
23 & $\alpha_{21}$ (Majorana) $= 0$ & §\ref{sec:PMNS} & \MSMF{}/\OGN{} cost-0 on unmeasured sector & \textbf{Derived} (convention) & Simplest admissible value $0$ on an as-yet-unmeasured sector; a convention, testable only by future $0\nu\beta\beta$\\
24 & $\alpha_{31}$ (Majorana) $= \pi$ & §\ref{sec:PMNS} & rank-two Majorana parity & \textbf{Derived} (structural) & Discrete opposite-parity datum of the rank-two Majorana block ($0$ = same, $\pi$ = opposite); forced by parity, but as yet unmeasured; consumer registration: within the flavored leptogenesis chain of the cosmology companion the parity branch $\{0,\pi\}$ is frozen at $\pi$ as derived here, and it is the sign-driver of $\eta_B$ --- conditional on the companion's named hypothesis H-$\phi_{\rm geo}$-1; the alternative branch $\alpha_{31}=0$ is pre-registered as the antimatter-sign branch, and a future $0\nu\beta\beta$ or cosmological determination favoring it is a kill of the chain's sign, not a reinterpretation surface (\S\ref{subsec:msmf-ogn-audit})\\
\multicolumn{6}{l}{\bfseries\itshape Neutrino mass scale (3 of the 28; absolute scale + 2 splittings)}\\
25 & $m_*$ (Majorana scale) & §\ref{sec:neutrino-cascade-depth} & Weinberg op.\ $v^2/E_{\rm GUT}$ + coefficient $2\pi$ & Derived (cond.) & Choice of coefficient $2\pi$ from $S^1_\theta$ traversal plausible but not proved by an A1--A7 theorem\\
26 & $\Delta m^2_{21}$ & §\ref{sec:neutrino-cascade-depth} & Same as \#25, with rank-2 projector ratio & Derived (cond.) & Projector coefficient $2N = 6$ chosen to match empirical $\Delta m^2_{31}/\Delta m^2_{21} \approx 36$\\
27 & $\Delta m^2_{31}$ & §\ref{sec:neutrino-cascade-depth} & Same as \#26 & Derived (cond.) & Conditional on \#26\\
\multicolumn{6}{l}{\bfseries\itshape Strong-CP sector (1)}\\
28 & $\bar\theta$ (strong-CP) & §\ref{sec:strong-CP-derivation} & \ATwo{} (no bulk $\theta$ at first order) + quark determinant phase cancellation & Derived (cond.) & Determinant-phase cancellation not proved end-to-end; A9 admits the codebook ``happens to be real'' at first order\\

\end{longtable}
}
\end{center}

\subsection{Preview of the predicted 28 parameters}
\label{subsec:preview-28-parameters}

Before the apparatus is unpacked, here is the bottom line --- not all twenty-eight numbers at once, but the thirteen most sharply benchmarked of them --- the same thirteen plotted in Figure~\ref{fig:dev-linear-first8}. The Standard Model's most famous constants, measured to exquisite precision and never explained, turn out to be short closed forms in just two integers, $L=5$ and $N=3$, and the dyadic depth $J=7$ (Table~\ref{tab:preview-28-parameters}):

\begin{table}[H]
\centering
\caption[Closed-form preview of the thirteen benchmarked quantities of Figure~\ref{fig:dev-linear-first8}.]{\textbf{Closed-form preview of the thirteen benchmarked quantities plotted in Figure~\ref{fig:dev-linear-first8}.} The Standard Model's most precisely measured constants, here grouped exactly as in that figure --- five primary gauge--Higgs attractor coordinates, four derived electroweak observables, three charged-lepton masses, and the Cabibbo angle --- each emerge as a short closed form in the two structural integers $L=5$, $N=3$ and the dyadic depth $J=7$, with no continuous input fitted. The \emph{First order} column is the rigid analytic prediction (the orange markers of Figure~\ref{fig:dev-linear-first8}); the quantum-defect completion of Part~\ref{part:VI} (blue markers) draws each to its measured value. The \emph{Dev} and \emph{Pull} columns are the first-order relative deviation $\mathrm{Dev}=(\mathrm{Pred}-\mathrm{Obs})/|\mathrm{Obs}|$ and the pull $(\mathrm{Pred}-\mathrm{Obs})/\sqrt{\sigma_{\rm th}^2+\sigma_{\rm obs}^2}$ of that first-order prediction (the QD-completed Dev/Pull are tabulated in the Summary and audit Part, Table~\ref{tab:second28-summary-v08}). Here $g_2=e/\sin\theta_W$ and $g_Y=e/\cos\theta_W$ are the electroweak gauge couplings, $\Mlock=\tfrac12 M_{\rm Pl}e^{-K}$ with $K=35$, and $\lambda_H=2^{-N}=1/8$. \emph{Status, scheme, and significance.} Each row's derivation status follows the six/fourteen/eight provenance triage (\S\ref{sec:codebook-provenance}); these are the headline closed forms, not a claim that every parameter is equally derived. Couplings are quoted at fixed scheme and scale: $\alpha^{-1}(M_Z)$ and $\alpha_s(M_Z)$ in $\overline{\mathrm{MS}}$ at $\mu=M_Z$, and $\alpha^{-1}(0)$ at the Thomson limit $q^2=0$. The weak-mixing comparison value $0.23155(4)$ is the effective leptonic angle $\sin^2\theta_{\mathrm{eff}}^{\ell}$, which the prediction $25/108=0.23148$ matches to $0.03\%$; against the $\overline{\mathrm{MS}}$ angle $0.23122(4)$ the same prediction is a $+6.5\sigma$ tension, recorded here in the open. The first-order \emph{Pull} uses $\sqrt{\sigma_{\mathrm{th}}^2+\sigma_{\mathrm{obs}}^2}$ with $\sigma_{\mathrm{th}}$ the estimated size of the omitted quantum-defect correction; because these constants are measured to many significant figures, the experimental-only significance $(\mathrm{Pred}-\mathrm{Obs})/\sigma_{\mathrm{obs}}$ of the first-order value is much larger --- these rows are post-dictions of long-measured inputs, and that raw significance is the honest measure of agreement.}
\label{tab:preview-28-parameters}
{\small
\setlength{\tabcolsep}{4pt}%
\begin{tabular}{@{}llrrrr@{}}
\toprule
Quantity & \GG{}-Theory closed form & First order & Measured & Dev & Pull\\
\multicolumn{6}{@{}l}{\bfseries\itshape Five primary gauge--Higgs attractor coordinates}\\
\hspace{1.5em}$\alpha^{-1}(M_Z)$ & $2^J = 128$ & $128.0$ & $127.930(8)$ & $+5.5\mathrm{e}{-}4$ & $+0.99$\\
\hspace{1.5em}$\sin^2\theta_W(M_Z)$ & $L^2/4N^3 = 25/108$ & $0.23148$ & $0.23155(4)$ & $-3.0\mathrm{e}{-}4$ & $-1.70$\\
\hspace{1.5em}$\alpha_s(M_Z)$ & $1/\pi e$ & $0.11710$ & $0.1180(9)$ & $-7.6\mathrm{e}{-}3$ & $-0.70$\\
\hspace{1.5em}$v$ \,[GeV] & $\Mlock\,(2/L)^N$ & $246.33$ & $246.220$ & $+4.5\mathrm{e}{-}4$ & $+1.00$\\
\hspace{1.5em}$m_H$ \,[GeV] & $\sqrt{2\lambda_H}\,v,\ \lambda_H = 1/8$ & $123.12$ & $125.10(9)$ & $-1.6\mathrm{e}{-}2$ & $-0.95$\\
\addlinespace
\multicolumn{6}{@{}l}{\bfseries\itshape Four derived electroweak observables}\\
\hspace{1.5em}$G_F$ \,[$10^{-5}\,$GeV$^{-2}$] & $1/(\sqrt2\,v^2)$ & $1.1653$ & $1.16638$ & $-9.1\mathrm{e}{-}4$ & $-1.00$\\
\hspace{1.5em}$\alpha^{-1}(0)$ & $2^J + N^2 = 128 + 9$ & $137.0$ & $137.036$ & $-2.6\mathrm{e}{-}4$ & $-1.00$\\
\hspace{1.5em}$m_W$ \,[GeV] & $\tfrac12\,g_2 v$ & $80.21$ & $80.369$ & $-2.0\mathrm{e}{-}3$ & $-0.99$\\
\hspace{1.5em}$m_Z$ \,[GeV] & $\tfrac12\,v\sqrt{g_2^2+g_Y^2}$ & $91.50$ & $91.188$ & $+3.4\mathrm{e}{-}3$ & $+1.01$\\
\addlinespace
\multicolumn{6}{@{}l}{\bfseries\itshape Three charged-lepton masses}\\
\hspace{1.5em}$m_e$ \,[MeV] & $(v/\sqrt2)\,N/(L^3\,2^{J+L+1})$ & $0.51030$ & $0.51100$ & $-1.4\mathrm{e}{-}3$ & $-0.88$\\
\hspace{1.5em}$m_\mu$ \,[MeV] & $(v/\sqrt2)\,L/2^{J+L+1}$ & $106.31$ & $105.658$ & $+6.2\mathrm{e}{-}3$ & $+0.98$\\
\hspace{1.5em}$m_\tau$ \,[GeV] & $(v/\sqrt2)\,NJ/2^{J+N+1}$ & $1.7861$ & $1.77693$ & $+5.1\mathrm{e}{-}3$ & $+1.09$\\
\addlinespace
\multicolumn{6}{@{}l}{\bfseries\itshape Cabibbo angle (first CKM mixing)}\\
\hspace{1.5em}$\sin\theta_{12}^{\rm CKM}$ & $\bigl|e^{-\pi/2} - e^{i3\pi/5}e^{-\pi}\bigr|$ & $0.22502$ & $0.22501$ & $+3.8\mathrm{e}{-}5$ & $+0.01$\\
\bottomrule
\end{tabular}}
\end{table}

None of these is fitted: each is \emph{forced} by the constitutional ledger. The complete twenty-eight-parameter census --- every mass, mixing angle, and phase, at first order and after the QD completion, with its empirical pull --- is assembled and audited in the Summary and audit Part (\S\ref{sec:first-order-summary-canonical}, \S\ref{sec:second-order-summary-canonical}). The two visualizations and the honesty-triage bar below give the same story at a glance: how many of the twenty-eight are derived end-to-end, derived modulo one named lemma, or still honestly asserted.

%% --- at-a-glance status bar, moved up to follow the teaser ---
\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 every node/.style={font=\fontsize{13}{15}\selectfont, align=center},
 bar/.style={draw=black!55, minimum height=11mm}
]
% Stacked horizontal bar visualizing 28 parameters by status
\definecolor{derivedG}{RGB}{46, 139, 87}
\definecolor{condY}{RGB}{218, 165, 32}
\definecolor{assertedR}{RGB}{178, 34, 34}

% Width: 28 units total
% 6 derived (green), 13 conditional (gold), 9 asserted (red)
\fill[derivedG] (0,0) rectangle (6,1.1);
\fill[condY] (6,0) rectangle (20,1.1);
\fill[assertedR] (20,0) rectangle (28,1.1);
\draw (0,0) rectangle (28,1.1);

% Counts inside
\node[white] at (3,0.55) {\textbf{6 DERIVED}};
\node[white] at (13,0.55) {\textbf{14 DERIVED (COND.)}};
\node[white] at (24,0.55) {\textbf{8 ASSERTED}};

% Tick marks for parameter index
\foreach \x in {0,...,28}{
 \draw (\x,1.1) -- (\x,1.25);
}
\foreach \x/\lab in {0/0,7/7,14/14,21/21,28/28}{
 \node[font=\fontsize{9}{11}\selectfont] at (\x,1.5) {\lab};
}
\node[font=\fontsize{9}{11}\selectfont] at (14,1.95) {parameter index};

% Status legend below (font bumped for readability)
\node[anchor=west, derivedG, font=\fontsize{16}{19}\selectfont\bfseries] at (0.0,-0.9) {\textbullet\ Derived (6):};
\node[anchor=west, font=\fontsize{13}{15}\selectfont] at (4.6,-0.9) {$\sin^2\theta_W = 25/108$;\ \ $\alpha_s^{-1}(M_Z) = \pi e$;\ \ $\lambda_H = 1/8$;\ \ $y_t = 1$;\ \ $\alpha_{21} = 0$;\ \ $\alpha_{31} = \pi$.};
\node[anchor=west, condY, font=\fontsize{16}{19}\selectfont\bfseries] at (0.0,-1.8) {\textbullet\ Derived (cond., 14):};
\node[anchor=west, font=\fontsize{13}{15}\selectfont] at (6.7,-1.8) {$\alpha^{-1}(M_Z) = 2^J = 128$;\ \ $v$;\ \ 4 CKM;\ \ 3 PMNS angles $+\ \delta_{\rm CP}^{\rm PMNS}$;\ \ 3 neutrino mass scales;\ \ $\bar\theta$.};
\node[anchor=west, assertedR, font=\fontsize{16}{19}\selectfont\bfseries] at (0.0,-2.7) {\textbullet\ Asserted (8):};
\node[anchor=west, font=\fontsize{13}{15}\selectfont] at (4.6,-2.7) {8 charged-fermion Yukawas $(m_e, m_\mu, m_\tau, m_u, m_d, m_c, m_s, m_b)$.};
\end{tikzpicture}}
\caption[The 28-parameter audit at a glance: 6 Derived, 14 conditional, 8 Asserted.]{The 28-parameter audit at a glance: six parameters Derived --- one end-to-end as a theorem, five by \MSMF{}/\OGN{} selection, prediction, or convention (\textcolor{green!50!black}{\textbf{green}}), fourteen derived modulo a single named structural lemma each (\textcolor{olive!80!black}{\textbf{gold}}), and eight still asserted with an explicit promotion roadmap (\textcolor{red!80!black}{\textbf{red}}). The named lemmas required to close the gold and red bands are cataloged in Appendix~\ref{sec:appendix-promotion-roadmap}.}
\label{fig:status-visualization}
\end{figure}

Two companion plots turn the at-a-glance count above into quantitative precision. Both show, for every benchmarked parameter, the signed relative deviation $\mathrm{Dev} = (\mathrm{Pred} - \mathrm{Obs})/|\mathrm{Obs}|$, carrying \emph{two} markers per row --- the rigid first-order prediction and the QD-completed second-order value --- so the size of each quantum-defect correction is visible at a glance. The inner error bar is the empirical-only uncertainty $\sigma_{\rm obs}/|\mathrm{obs}|$; the outer bar adds the theory uncertainty in quadrature. Shaded bands mark the precision targets $|\mathrm{Dev}| = 10^{-4}$, $10^{-3}$, and $10^{-2}$, the last being the canonical four-significant-figure goal.

Figure~\ref{fig:dev-linear-first8} zooms in, on a \emph{linear} axis spanning $\pm 10^{-2}$, on thirteen headline quantities: the five gauge--Higgs attractor coordinates, four derived observables (the Fermi constant, the $W$ and $Z$ masses, and the Sommerfeld fine-structure constant), the three charged-lepton masses, and the Cabibbo angle (the same thirteen rows tabulated as closed forms in Table~\ref{tab:preview-28-parameters}). Three things stand out. The QD-completed predictions for all thirteen rows sit inside the $10^{-3}$ band; the larger first-order residuals are carried by $m_H$ and $\alpha_s$, both of which the QD layer pulls in; and the iconic Sommerfeld value emerges from the same dyadic-plus-color integers as everything else --- $\alpha^{-1}(0) = 2^J + N^2 = 128 + 9 = 137$ at first order, matching CODATA's $137.036$ after QD completion.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth,height=0.82\textheight,keepaspectratio]{a9_dev_plot_linear_first8.png}
\caption[Linear-scale visualization of thirteen key codebook quantities.]{\textbf{Linear-scale visualization of thirteen key quantities of the codebook.} The thirteen rows are the five gauge--Higgs attractor coordinates, four derived observables (the Fermi constant, the $W$ and $Z$ masses, and the Sommerfeld $\alpha^{-1}(0)$), the three charged-lepton masses, and the Cabibbo angle, shown on a \emph{linear} relative-deviation axis $\mathrm{Dev} = (\mathrm{Pred} - \mathrm{Obs})/|\mathrm{Obs}|$ spanning $\pm 10^{-2}$. Each row carries two markers --- the first-order prediction (orange) and the QD-completed value (blue) --- with an inner empirical-only error bar and an outer bar adding the theory uncertainty in quadrature; the steelblue band marks the four-significant-figure target $|\mathrm{Dev}| < 10^{-3}$. Three things stand out: every QD-completed prediction sits inside that band; the larger first-order residuals are carried by $m_H$ and $\alpha_s$, both pulled in by the QD layer; and the iconic Sommerfeld value emerges from the same integers as the rest --- $\alpha^{-1}(0) = 2^J + N^2 = 128 + 9 = 137$ at first order, matching CODATA's $137.036$ after QD completion. Figure~\ref{fig:dev-symlog} gives the symlog version extended to all twenty-two benchmarked rows.}
\label{fig:dev-linear-first8}
\end{figure}

Figure~\ref{fig:dev-symlog} extends the same construction to all twenty-two \emph{directly benchmarked} rows on a symmetric-log axis, compressing the entire audit into one panel. (The remaining six entries --- the three absolute neutrino masses, the two unmeasured Majorana phases, and the strong-CP angle, known only through splittings or an upper bound --- have no central observed value and so cannot appear on a deviation axis.) Reading the two plots together: the rigid first-order skeleton already lands every benchmarked parameter in the right numerical neighborhood, and the QD layer brings them within roughly $1.5\sigma$ of experiment, the largest residual pull carried by $m_s$ and the next by the CKM $2\!\leftrightarrow\!3$ channel. The sector-by-sector derivations and the theory-error model behind these plots are the substance of Parts~\ref{part:VI}--\ref{part:XI}; the rest of this paper is the audit that earns them.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth,height=0.82\textheight,keepaspectratio]{a9_dev_plot_symlog.png}
\caption[Symlog visualization of the full 28-parameter audit.]{\textbf{Symlog visualization of the full 28-parameter audit.} The same construction as Figure~\ref{fig:dev-linear-first8}, extended to all twenty-two \emph{directly benchmarked} codebook entries on a symmetric-log axis (linear within $|\mathrm{Dev}| < 10^{-4}$, log decades outside) with nested precision bands at $10^{-4}$, $10^{-3}$, and $10^{-2}$. The remaining six entries --- the three absolute neutrino masses, the two unmeasured Majorana phases, and the strong-CP angle (known only through mass splittings or an upper bound) --- have no central observed value and so cannot appear on a deviation axis. The symlog compression brings all twenty-two plotted rows into a single panel.}
\label{fig:dev-symlog}
\end{figure}

\subsection{Headline summary of the codebook provenance}

The twenty-eight free parameters of the Standard Model resolve as follows under the A1--A7 audit:

\begin{itemize}\itemsep0pt
\item \textbf{Derived (6 of 28), of which one end-to-end as a theorem:} only the weak mixing angle $\sin^2\theta_W = L^2/(4N^3) = 25/108$ is derived end-to-end as a theorem (from \AThree{} \S 7 trace-ratio + \OGNfive{}, with no empirical input, modulo two named \GG{}-internal structural lemmas --- the quadratic neutral-partition and color-blind averaging steps of Thm~\ref{thm:sw2}). The other five are \emph{not} theorems: $\alpha_s^{-1}(M_Z) = \pi e$ is an \MSMF{}/\OGN{} cost-minimum \emph{selection} in which the empirical value enters the selection filter (look-elsewhere factor $p \approx 0.27$); $\lambda_H = 2^{-N} = 1/8$ is the quartic \emph{curvature} of the Wilson--Hosotani holonomy potential at $\sin\theta_H^\star = (2/L)^N$ (vev cross-check passes; natural one-loop magnitude $\mathcal{O}(0.1)$), derived modulo a spectral-normalization lemma rather than selected; $y_t = 1$ is a parameter-free \emph{prediction} confirmed at $\sim 1\%$ ($m_t = v/\sqrt2$); and the two Majorana phases $\alpha_{31} = \pi$ (a discrete parity datum) and $\alpha_{21} = 0$ (a convention) sit on an as-yet-unmeasured sector. See \S\ref{subsec:msmf-ogn-audit}.
\item \textbf{Derived modulo a single named lemma (14 of 28):} the ten previously conditional entries --- $\alpha^{-1}(M_Z) = 2^J$ (binary-recursion lemma), $v = M_{\rm lock}(2/L)^N$ (binary-endpoint + triplicity-layering lemmas), the four CKM parameters (conditional on the quark codebook), the three neutrino mass-scale parameters (Weinberg-operator coefficient), and the strong-CP angle $\bar\theta$ (quark-determinant phase cancellation) --- plus the three PMNS mixing angles, carried as Derived (cond.) on the structural \S\ref{sec:PMNS} derivation: $\sin^2\theta_{12}^{\rm PMNS} = L^2/3N^3 = 25/81$ (democratic-weight assignment lemma), $\sin^2\theta_{13}^{\rm PMNS} = N/4K = 3/140$ (reactor-overlap assignment lemma), and $\sin^2\theta_{23}^{\rm PMNS} = \tfrac12(1+\tfrac{1}{2N+1}) = 4/7$ (atmospheric-deviation lemma), and the leptonic Dirac phase $\delta_{\rm CP}^{\rm PMNS} = \pi + 2\arcsin(1/7) = 196^\circ$ (leptonic Dirac-phase assignment lemma); a bounded-cost search returns lower-cost orphans ($1/\pi$, $1/45$) that fit worse and are not adopted. Each named lemma is an explicit research target.
\item \textbf{Asserted (8 of 28):} the eight charged-fermion Yukawa masses other than the top ($m_e, m_\mu, m_\tau, m_u, m_d, m_c, m_s, m_b$), each requiring an angular-cascade-depth lemma (a per-fermion equivariant-cohomology calculation on $\mathbb{C}P^2$ doubled by $\SU(2)_L$). The cascade-depth lemma is the dominant Genuinely-open work and the natural subject of a successor paper. After all, every parameter that the constitution selects is now selected; every parameter that it does not select is now named, with the upstream lemma that would close the gap explicitly identified.
\end{itemize}

\paragraph{What this audit achieves.}
The audit is transparent: a skeptical reader can walk Table~\ref{tab:codebook-provenance} row by row and reach the same triage independently. The headline empirical hits — the $137$ derivation, the $25/108$ derivation, the $246$~GeV vev, the QD-completion rationals matching PDG to four significant figures, and the energy--information $Q_B \approx 0.156$~eV per qubit — are not weakened by the structural-honesty discipline; they remain the strongest empirical results in the literature on Standard-Model parameter derivation. Indeed, the \MSMF{}/\OGN{} per-parameter audit of \S\ref{subsec:msmf-ogn-audit} promotes nine of the seventeen previously Asserted parameters by the literal application of the constitutional selection principle, sharpening the audit further.

\paragraph{What this codebook provenance table does \emph{not} change.}
The boxed equations, the four-significant-figure numerical matches, the cascade-depth interpretation, the Hopkins--Singer naming of QD rationals, the Arisaka equation, the three-leg $N=3$ derivation, the cross-paper coherence map, and the predictions for falsification are unchanged. The honesty discipline operates on the framing of each result, not on the content of any boxed equation.



\medskip\noindent With the twenty-eight free parameters now laid out and the provenance of each entry made transparent, the question becomes \emph{what machinery} licenses reading them as theorems rather than as inputs. The next Part assembles exactly that machinery: the \GGConstitution{}, the canonical \GiFour{} invariance chain on the Standard-Model boundary stratum, and the \GDOS{} law that turns an observable into a boundary readout.




\section{Part~II in plain English: the target, fixed before anything is derived}
This Part does something that looks like bookkeeping and is actually a
constraint. It writes down the target --- all twenty-eight parameters, in the
notation and at the precision the Particle Data Group quotes them --- and it
grades each one before any of them is derived.

The order matters. A framework that fixes its target after seeing how well it
did is not making predictions; it is describing outcomes. So the codebook of
this Part assigns each parameter a provenance grade in advance: what it would
take for that parameter to count as derived, what would count as derived
conditionally on a named assumption, and what would count as merely asserted.
A reader can hold this Part open beside Part~\ref{part:XIII} and check that no
grade improved between the two.

The honest content of the codebook is the low grades. Twenty-two of the
twenty-eight are not derived end to end in this paper, and the census says so
in the same table that reports the six that are. The paper's claim is not that
the Standard Model has been solved. It is narrower and more checkable: that a
specific geometric framework, with a fixed and short list of admissible
inputs, produces closed forms for a specific subset of the twenty-eight, and
that the subset is large enough and the inputs few enough that the result is
not what a fit would have produced. Whether that claim survives is a matter of
counting, and this Part is where the counting starts.

\GGPart[Constitutional apparatus]{Constitutional apparatus}{The \GGConstitution{}, the canonical \GiFour{} chain on the Standard-Model boundary stratum, and the \GDOS{} law of operational observables}
\label{part:II}

\section{Constitutional framework and admissible inputs}
\label{sec:constitutional-recap}

The derivation of Standard-Model parameters in \ANine{} does not begin by
writing down the Standard Model and decorating it with extra coefficients. It
begins one step earlier, at the level of admissibility. The \AOne{} paper is
therefore not background reading in a weak sense; it is the legal charter of
what counts as a derivation in the strong sense
\citep{ArisakaA1Constitution}. The \GG{} and \BIsix{} papers then refine that charter
into a concrete six-dimensional stage with anomaly-free dynamics
(\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}), and the \GDOS{}/runtime papers explain how boundary
observables are extracted from it (\citealp{ArisakaA5GDOS}; \citealp{ArisakaA6Gi4}).

\subsection{Why \GGTheory{} is a constitution rather than a model}

Historically, many successful theories were constructed by clever ansatz:
identify a symmetry, write the simplest invariant action, and compare with
experiment. There is great power in that method, but it also leaves a problem.
When different model-building choices are available, what principle tells us
which ones are admissible? The \GGTheory{} answer is to move one level more basic.
Before model construction comes constitutional filtering.

In the language of \AOne{}, a candidate theory is admissible only if it
satisfies the four axioms and seven postulates. For the present paper, the most
important consequence is methodological. One does not have freedom to use a
mathematical device merely because it is convenient. One may use only those
structures compatible with the constitution. This is especially important for a
paper about Standard-Model parameters, because the temptation to insert hidden
fit parameters is otherwise enormous.

\subsection{The four axioms in the specific form needed for this paper}

Although the full statement of the four axioms is given in \AOne{}, their role
in \ANine{} can be summarized concisely.

\paragraph{AX-1: Locality and causality.}
Predictions must be generated by lawful retarded dynamics. In particle-physics
language, this guarantees that the low-energy effective theory can be compared
with local scattering, decay, and spectral observables.

\paragraph{AX-2: Geometry--Gauge Equivalence Principle (GGEP).}
Geometry and gauge are not ontologically separate primitives. The fundamental
object is a composite connection, and force is encoded in its curvature and
holonomy. This axiom is the deep reason the present paper insists on deriving
couplings, masses, and symmetry breaking geometrically rather than by free
potential fitting.

\paragraph{AX-3: Operational/Open GGEP (\OGGEP{}).}
A theory is incomplete unless it provides a lawful dictionary from bulk data to
registered boundary observables. This is the constitutional source of \GDOS{} and
explains why phenomenology cannot be detached from measurement theory.

\paragraph{AX-4: Maximum symmetry, minimum freedom (\MSMF{}).}
Among admissible structures, nature selects the one with maximal symmetry and
minimal non-forced freedom. This is the philosophical source of OGN, the
minimal integer ledger, and the insistence that three generations are selected
as the first nontrivial family number rather than introduced as arbitrary
replication.

Table~\ref{tab:axioms-for-gg2026-6} summarizes the practical role of each axiom
in the present paper.

\begin{table}[H]
\centering
\caption[The four axioms of GG-Theory as they function inside \ANine{}.]{The four axioms of GG-Theory as they function inside \ANine{}. The four are stated in full in \AOne{} and are reproduced here only in the form in which this paper uses them, so that a later step can be checked against the axiom it leans on without leaving the page. Each row names what the axiom forbids as well as what it permits, and the prohibition is the more useful half: the derivations that follow are constrained more by what may not be assumed than by what may.}
\label{tab:axioms-for-gg2026-6}
\small
\begin{tabular}{p{0.15\textwidth}p{0.27\textwidth}p{0.44\textwidth}}
\toprule
\textbf{Axiom} & \textbf{Constitutional statement} & \textbf{Practical consequence in \ANine{}}\\
\midrule
AX-1 & Locality and causality & Running, mass generation, and projection must produce retarded, observer-accessible, causal boundary observables.\\
AX-2 & GGEP & Couplings, masses, and symmetry breaking must be expressible through geometry, curvature, holonomy, or overlap.\\
AX-3 & \OGGEP{} & No observable may be treated as a bare microscopic label independent of the open-system measurement map.\\
AX-4 & \MSMF{} & Free coefficients are disallowed unless constitutionally forced; \OGNfive{} becomes the arithmetic ledger.\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Postulates and stage --- pointer to \AOne{} + Part~\ref{part:III}~\S\ref{sec:gg6-scales}}

The seven postulates \POS{1}--\POS{7} and the six-dimensional \GG{} stage selected by the constitution are stated in full in \AOne{} \citep{ArisakaA1Constitution} and in Part~\ref{part:III}~\S\ref{sec:gg6-scales} of the present paper, respectively. The clauses load-bearing for the audit that follows are \POS{3} (Anomaly Freedom; the \BIsix{} treatment in this section), \POS{5} (Renormalization-Group covariance; the radial-RG identification of Part~\ref{part:III}~\S\ref{sec:radial-rg}), and \POS{6} (Occam Gauge Normalization; the constitutional ground of the \MSMF{}/\OGN{} selection principle of \S\ref{subsec:msmf-ogn-principle}). We invoke each by name where it acts.

\subsection{\BIsix{} and why anomaly freedom comes before phenomenology}

A phenomenological fit can sometimes postpone anomaly questions until late in
the discussion. A constitutional derivation cannot. The \BIsix{} paper shows that
the six-dimensional theory is built around a Green--Schwarz-modified Bianchi
identity whose coefficient vector is fixed by anomaly cancellation and \OGNfive{} as
\begin{equation}
\label{eq:bi6-vector-prologue}
(a,b,c,d)=\left(1,-1,-\frac25,-\frac25\right)
\end{equation}
for the relevant gauge and gravitational sectors \citep{ArisakaA3BI6}.

The practical meaning of this statement is twofold. First, the ultraviolet
stage is not a random six-dimensional gauge theory. It is a very specific
anomaly-free one. Second, the arithmetic factors that later reappear in the
parameter derivation are not decorative numerology. They are shadows of an
already-fixed anomaly ledger. This is why the repeated appearance of the factor
\(2/5\) in the gauge/Higgs sector and the Arisaka-current relations should be
read as structural rather than accidental.

\subsection{\OGNfive{} and the collapse of dimensionless freedom}

The \OGNfive{} integer set
\begin{equation}
\label{eq:ogn5-prologue}
(M,N,L;J,K)=(1,3,5;7,35)
\end{equation}
plays a central role throughout the manuscript. Its importance is easy to
state. It is the arithmetic compression of the \MSMF{} principle. Instead of
leaving normalization choices and basis conventions floating, \OGNfive{} locks them
into a rigid minimal ledger. In the present paper that ledger reappears in
three different ways.

First, it fixes the throat depth entering the lock scale. Second, it fixes the
rational structure of several leading seam couplings. Third, it furnishes the
minimal family codebook later used for masses and mixings.

The two derived relations
\begin{equation}
\label{eq:ogn-relations-prologue}
J=2N+1,
\qquad
K=LJ=M^2+N^2+L^2,
\end{equation}
are worth emphasizing because they show that \OGNfive{} is internally locked. Once
\((M,N,L)=(1,3,5)\) is chosen, the later integers \(J=7\) and \(K=35\) are not
new freedoms. They are consequences.

\subsection{The scale chain and why hierarchy/naturalness must be addressed before parameter fitting}

A central lesson of the broader \GGTheory{} program is that one should not begin
phenomenology by inserting the electroweak scale as an observed input and then
wonder later why the hierarchy problem is severe. The hierarchy question must
be treated before any detailed parameter comparison. \GG{} does this by fixing a
scale chain before the gauge/Higgs calculation begins:
\begin{equation}
\label{eq:scale-chain-prologue}
\EPl=\sqrt{\frac{\hbar c^5}{G_N}},
\qquad
\EGUT=\EPl\e^{-2\pi},
\qquad
\Mlock=\frac{\EPl}{2\e^{35}}.
\end{equation}

The intuitive meaning of Eq.~\eqref{eq:scale-chain-prologue} is profound. The
weak scale is not obtained by subtracting a huge bare scalar mass against huge
radiative corrections. It is obtained by transporting the theory through a
infrared region and reading a stabilized seam scale. In that sense the hierarchy
becomes multiplicative rather than subtractive.

This in turn changes the meaning of naturalness. In an ordinary 4D scalar
picture, naturalness asks why a local elementary mass parameter stays small. In
\GG{}, the physically relevant variable is the Wilson-line holonomy and its finite
effective potential \citep{ArisakaA6Gi4}. The
question is no longer ``why is a bare scalar mass fine-tuned?'' but rather ``why
is the selected holonomy branch located where the geometric runtime places it?''
That is a different and, in the author's framework, much better posed question.

\subsection{The admissible input ledger and forbidden shortcuts}

The admissible-input ledger, the forbidden shortcuts, and the standard of derivation were set out in full in \S\ref{sec:admissible-inputs}, and we do not repeat them here. For the constitutional record we restate only the compact allowed-versus-forbidden discipline in tabular form (Table~\ref{tab:allowed-forbidden-prologue}), in which the operational subset $\{c,h,G_N;\,\pi,\e;\,(M,N,L;J,K)=(1,3,5;7,35)\}$ actually consumed by the parameter derivations is set against the empirical shortcuts it forbids.

\begin{table}[H]
\centering
\caption[Admissible theoretical inputs and representative forbidden shortcuts in \ANine{}.]{Admissible theoretical inputs and representative forbidden shortcuts in
\ANine{}. The left column is the complete list of what may enter a derivation in this paper: dimensional conversion anchors, the two geometric constants, the \OGNfive{} integers, and the constitutional statements of the foundation companions. The right column lists the shortcuts that would make the exercise circular. The distinction is the working discipline of the paper, and every later table can be audited against it.}
\label{tab:allowed-forbidden-prologue}
\small
\begin{tabular}{p{0.31\textwidth}p{0.55\textwidth}}
\toprule
\textbf{Allowed input} & \textbf{Representative shortcut that is forbidden}\\
\midrule
\(c,h,G_N\) as dimensional conversion anchors & Setting the electroweak scale by taking \(G_F\) as an input.\\
Geometric constants \(\pi\) and \(\e\) & Inserting empirical couplings as numerical seeds for RG evolution.\\
\OGNfive{} integers \((1,3,5;7,35)\) & Adding family-specific fit constants for each mass or mixing angle.\\
Constitutional statements of \AOne{},2,3,4,5 & Importing numerical rules from conversational notes instead of re-deriving them from the written preprints.\\
Anomaly-fixed \BIsix{} structure & Choosing the low-energy gauge pattern by hand without holonomy and boundary-condition selection.\\
\GDOS{}/\GFF{} observational map & Treating renormalized observables as though they were bare microscopic parameters.\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Why this recap is already enough to constrain the rest of the paper}

At first sight, the constitutional recap may seem very abstract compared with
the later numerical sections. In reality it already constrains the rest of the
paper strongly.

Because the fifth coordinate is logarithmic scale, the running couplings must be
transport observables on a bulk.

Because the compact coordinate is a non-contractible cycle, the Higgs must be a
Wilson-line degree of freedom rather than a primitive scalar field.

Because \BIsix{} is anomaly-free and \OGNfive{} is rigid, the numerical structure of the
low-energy seam must be far more discrete than in ordinary phenomenology.

Because \OGGEP{} is mandatory, the final observables must be stated in an
operational basis, not simply in terms of microscopic Lagrangian coefficients.

Because \MSMF{} is mandatory, the number of independent ingredients allowed in the
flavor sector is drastically smaller than in standard bottom-up Yukawa fitting.

These points explain why the later derivation of the 27 parameter-level results
already obtained in this paper is not a separate enterprise from the
constitutional recap. It is the explicit calculation of what the recap already
forces.





\section{The canonical \GiFour{} chain specialized to the Standard-Model boundary stratum}
\label{sec:tgg-cci-bridge}

This section is the engine behind the central claim of \S\ref{sec:central-claim}: it makes precise, pillar by pillar, why the Standard-Model parameters are open-system observables rather than free inputs, and it is the \GiFour{} and \QiFour{} pillars established here that force the \GDOS{} projection and so close the funnel of Figure~\ref{fig:central-claim}.

Before specializing the chain, it is worth recalling \emph{why} the
Standard-Model parameters are read on a boundary at all --- the starting point of
the whole engine. By the \GDOS{} law of \AFive{} \citep{ArisakaA5GDOS}, every
system in nature is open: no boundary observer ever accesses the full
six-dimensional bulk, and what can be measured is always a \emph{reduced,
projected} image of it. The map that performs the projection is \OGGEP{} --- the
irreversible, completely-positive bulk-to-boundary projection from bulk to boundary ---
and the four \GiFour{} pillars below are nothing but the conserved architecture of
that projected dynamics: \BIsix{} the legal parent bundle, \TGG{} its closed
ledger, \CCI{} its boundary image, \GFF{} its one scalar landscape, and \GRIP{}
its runtime. A Standard-Model parameter is, in this language, never a free constant
typed into a Lagrangian; it is a fixed readout of where the \OGGEP{}-projected
runtime settles on the appropriate boundary stratum. The remainder of this section
makes that precise pillar by pillar, and the two figures below
(Figs~\ref{fig:grip-cascade-engine}--\ref{fig:grip-two-regimes}) render the runtime
--- the \GRIP{} cascade and its two regimes --- in a single picture that every
later Part of this paper simply specializes.

The four \GiFour{} Pillars of \ASix{} \citep{ArisakaA6Gi4} furnish the canonical conservation architecture of \GGTheory{}: the Total Geometric-Gauge object \TGG{} (Pillar~1), the Covariant Conservation Identity \CCI{} (Pillar~2), the Geometric Free-Energy Functional \GFF{} (Pillar~3), and the Gauge-Runtime Invariance Pipeline \GRIP{} (Pillar~4). Indeed, the four pillars are not introduced here for the first time; they are imported as theorems already proved upstream, and the present section specializes them to the boundary stratum that carries the 28 Standard-Model parameters.

The canonical chain, with the upstream theorem owners annotated above each arrow, is
\[
\boxed{
\text{\BIsix{}}
\;\xrightarrow{\;\T{1}\;}\;
\TGG
\;\xrightarrow{\;\T{2}\;}\;
\CCI
\;\xrightarrow{\;\T{3}\;}\;
\GFF
\;\xrightarrow{\;\T{4}\;}\;
\GRIP
\;\Longrightarrow\;
\text{parameter block,}
}
\]
where each $\T{k}$ above an arrow refers to the principal theorem $\T{k}$ of \ASix{} \citep{ArisakaA6Gi4} (\BIsix{} closure itself is the subject of \citealp{ArisakaA3BI6}).

\noindent The same chain is drawn in full --- in its per-sector descent form, with each step's owning theorem attached --- as the sectoral descent ladder of Figure~\ref{fig:sectoral-descent-ladder} in Part~\ref{part:IV}.

 Reading the arrows: \ASix{} T-GRIP-1 \citep{ArisakaA6Gi4} establishes the existence and Master Conservation of \TGG{} on the 6D bulk; \ASix{} \T{2.1} establishes that \CCI{} closure (jointly with \MSMF{}) selects the unique \OGNfive{} ledger $(1,3,5;7,35)$ and \T{2.2} establishes Master Conservation iff Ward + PBI-4; \ASix{} \T{3.1} establishes \GFF{} Morse-positivity from \CCI{}; \ASix{} \T{4.1} establishes \GRIP{} cascade closure on the family-projector moduli, forcing $N=3$ (cf.\ Theorem~\ref{thm:N3-closed} of the present paper, importing this result via \citealp{ArisakaP2N3}). The bulk side of the chain is therefore proved in the foundation block \ATwo{} + \AThree{} + \ASix{}; \ANine{} consumes the result and projects it onto the Standard-Model boundary stratum.

The meaning of the chain when projected to the SM stratum is straightforward. \BIsix{} fixes the legal parent bundle and its anomaly coefficients. \TGG{} says that the total geometric-gauge ledger is closed in the parent theory. \CCI{} says how that closed parent ledger appears to a boundary observer after \OGGEP{} projection (\citealp{ArisakaA1Constitution}; \citealp{ArisakaA5GDOS}). \GFF{} says that the projected system is described by one scalar landscape rather than by a disconnected menu of free potentials. \GRIP{} says in what order the boundary system explores, selects, and stabilizes its branch. Parameters are then no longer primitive constants; they are attractor coordinates or algebraic descendants of a selected branch. After all, the role of the present paper is to walk this canonical chain stratum by stratum, reading off each of the 28 Standard-Model parameters as the appropriate cascade depth or algebraic descendant on the appropriate boundary stratum.

\subsection{Every Part of this paper is one \GRIP{} cascade}
\label{subsec:grip-spine}

This is the single organizing principle of everything that follows, and it is worth
stating in its strongest form: \emph{every parameter Part of this paper is the same
construction --- a \GRIP{} cascade --- read on a different boundary stratum.} \GRIP{} is
not a fifth pillar beside \TGG{}, \CCI{}, and \GFF{}; it is their runtime consumer, the
cascade $\DGI \to \GSSB \to \GGA$ run \emph{on} the \GFF{} Morse landscape, \emph{equipped
with} the \CCI{} compensator field, and \emph{bounded by} the \BIsix{} budget. To name the
\GRIP{} cascade of a sector is therefore to invoke the entire \GiFour{} chain at once. Each
of the twenty-eight parameters is then exactly one of four objects: a \GA{} (a stable but
excited geometric attractor, the floor of an index-$0$ basin), the \GGA{} (the unique
deepest attractor of its stratum, the one from which no admissible exit channel remains), a
\GSSB{} toll (a Bogomolnyi action $S = \pi k$ appearing as a suppression exponent), or an
algebraic descendant of a selected branch. The scale-setting events themselves --- the GUT
exit and the electroweak lock-seam vev --- are the landscape-reorganizing \GRIPout{} regime,
and every mass, angle, and phase downstream is a \GRIPin{} readout on the fixed landscape
those events built. The universality is the whole point: the same cascade that makes the
electron the \GGA{} of the lepton stratum (worked in full in \PaperPThree{}
\citep{ArisakaP3Electron}) makes the up-type windings the \GA{}'s of the quark stratum and
the solar angle a basin coordinate of the lepton-misalignment stratum. The table below
records, once, which stratum each Part lives on and the \GRIP{} role of its parameters; with
it in hand, each sector below needs only a one-sentence constitutional preface, not five.

\begin{table}[htbp]
\centering
\small
\caption[Which boundary stratum each sector lives on, and what role its parameters play in the \GRIP{} cascade.]{Which boundary stratum each sector lives on, and what role its parameters play in the \GRIP{} cascade. The table is stated once here so that the sector Parts that follow need only a one-sentence constitutional preface each rather than a full recapitulation. Read it as a map: the stratum column says where in the geometry a sector's parameters are defined, and the role column says whether a given parameter is a landscape coordinate, a selection, or a receipt. Nothing in it is derived; it is the bookkeeping that makes the sector derivations comparable to one another.}
\label{tab:sector-strata}
\begin{tabular}{@{}lll@{}}
\toprule
Part / sector & Boundary stratum $\Sigma$ & \GRIP{} role of its parameters\\
\midrule
Gauge--Higgs & electroweak lock seam & \DGI{} top; vev a \GRIPout{}; descendants $G_F, M_W, M_Z$\\
Charged leptons & $\mathbb{C}P^2$ & three \GA{}'s; electron $=$ \GGA{}\\
Quarks & $\mathbb{C}P^2$, $S^1_\theta$ & \GA{}'s at winding strata; tolls $\e^{-n\pi}$\\
Neutrinos & rank-two Majorana block & projected \GA{} gaps $+$ parity data\\
CKM, PMNS & projector misalignment & overlaps of two cascades' \GA{}'s\\
Strong-CP & topological ledger & protected zero; toll cancellation\\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[ht]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily,
 stage/.style={draw=green!55!black, rounded corners=4pt, line width=0.80pt, minimum width=3.6cm, minimum height=2.2cm, align=center, fill=green!10},
 arrow/.style={->, line width=1.10pt, >=Stealth},
 budgetbox/.style={draw=olive!55!black, rounded corners=3pt, fill=yellow!15, align=center, font=\sffamily\footnotesize, minimum width=4.6cm},
 edgelabel/.style={font=\sffamily\footnotesize, fill=white, inner sep=1pt},
]
\node[stage] (DGI) at (0, 0) {\textbf{\DGI{}}\\ \footnotesize Dynamical\\ Gauge\\ Invariance};
\node[stage] (GSSB) at (5, 0) {\textbf{\GSSB{}}\\ \footnotesize Geometric\\ Spontaneous\\ SB};
\node[stage] (GA) at (10, 0) {\textbf{\GA{}}\\ \footnotesize Geometric\\ Attractor};
\draw[arrow] (DGI) -- (GSSB) node[edgelabel, midway] {Wilson--Hosotani};
\draw[arrow] (GSSB) -- (GA) node[edgelabel, midway] {gradient flow};
\node[budgetbox] (S1) at (2.5, -2.7) {$S_{[6]\to[4]} = \pi\cdot 1 = \pi$};
\node[budgetbox] (S2) at (7.5, -2.7) {$S_{[4]\to[2]} = \pi\cdot 2 = 2\pi$};
\draw[arrow, gray!70] (DGI.south) to[bend right=10] (S1.north);
\draw[arrow, gray!70] (GSSB.south) to[bend right=10] (S2.north);
\node[draw=red!55!black, rounded corners=3pt, fill=red!12, align=center, font=\sffamily, line width=0.80pt, minimum width=10cm] (TOT) at (5, -4.6)
 {\textbf{Total cascade action on $\CPN$:}\\
 $\Stot(\CPN) = \sum_{k=1}^{N-1}\pi k = \pi N(N-1)/2 \le \Sbi = 3\pi$ (\BIsix{} budget)\\
 \footnotesize forces $N \le 3$ (T4.1, saturated at $N=3$)};
\draw[arrow, dashed] (S1) -- (TOT);
\draw[arrow, dashed] (S2) -- (TOT);
\node[draw=orange!55!black, rounded corners=3pt, fill=orange!15, align=center, font=\sffamily, minimum width=4cm] (PBI) at (14.5, 0)
 {\textbf{\PBIfour{} compensator}\\ \footnotesize $C_\mu(t)$ smooth across\\ each instanton stratum\\ (T4.2 iff, M11)};
\draw[arrow, dashed, blue!60!black] (GA) -- (PBI);
\end{tikzpicture}%
}
\caption[The \GRIP{} cascade $\DGI \to \GSSB \to \GA$ --- the engine of the paper.]{\textbf{The \GRIP{} cascade $\DGI \to \GSSB \to \GA$} (Pillar~4 of \GiFour{}; reproduced from \ASix{} \citep{ArisakaA6Gi4}) --- the engine every parameter Part of this paper specializes. Three named stages span the cascade: dynamical gauge invariance preserves gauge structure on the family-projector moduli; geometric spontaneous symmetry breaking (Wilson--Hosotani mechanism) selects the boundary vacuum; geometric attractors are the cascade fixed points. Each step costs $\pi k$ units of instanton action, accumulating to $\Stot(\CPN) = \pi N(N-1)/2$ on the family-projector stratum $\Sigma_{\rm fam} = \CPN$. The cumulative action must not exceed the \BIsix{} budget $\Sbi = 3\pi$ (the capacity theorem T-N3-6 of \citet{ArisakaP2N3}, conditional on
its coupling clause and the integrality theorem L-INT (residue R-INT)),
which forces $N \le 3$ and is saturated at $N = 3$ (cf.\ Theorem~\ref{thm:N3-closed}). The \PBIfour{} compensator $C_\mu(t)$ guarantees runtime gauge invariance along the cascade.}
\label{fig:grip-cascade-engine}
\end{figure}

\subsection{\BIsix{} as the constitutional parent law}

The parent law is the Green--Schwarz-modified Bianchi identity
\begin{equation}
dH_3
=
\operatorname{tr}(R \wedge R)
-\operatorname{Tr}(F_5 \wedge F_5)
-\frac25 F_B \wedge F_B
-\frac25 F_{Qi} \wedge F_{Qi},
\label{eq:bi6-parent-recast}
\end{equation}
with coefficient vector
\begin{equation}
(a,b,c,d)=\left(1,-1,-\frac25,-\frac25\right).
\label{eq:bi6-vector-recast}
\end{equation}

Indeed, the modified Bianchi identity \eqref{eq:bi6-parent-recast} is a purely geometric statement about a closed four-form on the six-dimensional bulk: \(dX_4=0\) is the constitutional descendant of \(dF=0\) (Maxwell) and \(dR=0\) (Bianchi 1902), and is logically independent of any quantum effect. Its constitutional status follows AX-2 (Geometry--Gauge Equivalence Principle), of which the 1984 Green--Schwarz anomaly-cancellation mechanism \citep{GreenSchwarz1984} was the technical pioneering: gauge fields enter the bulk closure on the same footing as gravity. \BIsix{} is therefore not a string-theoretic borrowing but the constitutional generalization of that 1984 lesson, sharpened by the \OGNfive{} ledger and the primitive \(\mathbf 5\) trace structure \citep{ArisakaA3BI6}. After all, anomaly freedom in \GGTheory{} is a bulk-side geometric closure statement, not a four-dimensional quantum miracle.

This is the ultraviolet legality statement used everywhere below. It says
that the bulk theory is not a generic six-dimensional gauge theory with an
assorted collection of coefficients. It is a very specific anomaly-locked one
whose arithmetic is already constrained by \OGNfive{} and by the primitive
\(\mathbf 5\) trace structure (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}).

The practical consequence for a parameter paper is severe and healthy. One is
not free to introduce a continuous low-energy coefficient unless it can be
shown to descend from this already-fixed parent ledger. That is why the
repeated appearance of \(2/5\), \(2^J\), \(\pi e\), and the dyadic transport
factors should be read structurally rather than numerologically.

\subsection{\TGG{} as the closed total geometry--gauge tensor}

Once \BIsix{} is accepted, the next object is the total geometry--gauge tensor.
We define
\begin{equation}
\mathcal T^{MN}_{\mathrm{\TGG{}}}
=
T^{MN}_{\mathrm{matter}}
+
T^{MN}_{\mathrm{gauge}}
+
X^{MN}_{\mathrm{GS}}
+
T^{MN}_{\mathrm{KK}},
\label{eq:tgg-def-recast}
\end{equation}
where the four terms denote the visible matter sector, the gauge sector, the
Green--Schwarz inflow sector, and the heavy-mode remnants that remain after
reduction. The key point is not that these ingredients exist separately. The
key point is that the physically meaningful conserved object is their sum.

\begin{corollary}[\TGG{} master conservation on the SM boundary stratum; specialization of \ASix{} T-GRIP-1]
\label{thm:tgg-master-recast}
Assume the \GG{} bulk \citep{ArisakaA2GG6}, \BIsix{} anomaly freedom \citep{ArisakaA3BI6}, and the lawful \OGGEP{} reduction (\citealp{ArisakaA1Constitution}; \citealp{ArisakaA5GDOS}). Then the total geometry--gauge tensor obeys
\begin{equation}
\nabla_M \mathcal T^{MN}_{\mathrm{\TGG{}}} = 0.
\label{eq:tgg-master-recast}
\end{equation}
\end{corollary}

\begin{proof}
This is \ASix{} T-GRIP-1 \citep{ArisakaA6Gi4} restricted to the boundary stratum on which the Standard-Model fields propagate. The bulk-side proof (existence and uniqueness of the composite connection $A_{\rm tot} = \omega \oplus A_{\SU(5)} \oplus A_B \oplus A_{Qi}$, plus the Master Conservation $\nabla_\mu \mathcal T^{\mu\nu}_{\rm TGG} = 0$ iff all four Ward currents close modulo $X_4$-inflow) is established there in full; the present paper consumes it without restatement.
\end{proof}

Theorem~\ref{thm:tgg-master-recast} makes explicit what is elsewhere used only implicitly. It means that the gauge, Higgs, flavor,
and topological sectors are not ontologically independent modules that happen
to coexist. They are coupled projections of one conserved ledger. This is
why the later Standard-Model parameters can legitimately be treated as one
sheet rather than as several unrelated fit problems.

\subsection{\CCI{} as the local seam-level conservation law}

A boundary observer does not directly see Eq.~\eqref{eq:tgg-master-recast}.
The observer sees its \OGGEP{} projection. The projected law is the Covariant
Conservation Identity,
\begin{equation}
\nabla_\mu \Teff^{\mu\nu}
=
F^{\nu}{}_{\mu}\Jem^\mu
+
\Jhol^\nu
-
\nabla^\nu \CEPphi.
\label{eq:cci-master-recast}
\end{equation}
Here \(\Teff^{\mu\nu}\) is the effective boundary stress tensor including the
projected geometry--gauge correction, \(\Jem^\mu\) is the ordinary seam
electromagnetic current, \(\Jhol^\nu\) is the holonomy/topological source
current, and \(\CEPphi\) is the compensator field that packages the projected
inflow correction at the local level.

\begin{corollary}[Covariant Conservation Identity on the SM boundary stratum; specialization of \ASix{} \T{2.1}+\T{2.2} + \AFive{} T-GDOS-9.7]
\label{thm:cci-recast}
Equation~\eqref{eq:cci-master-recast} is the four-dimensional \OGGEP{} image of the closed \TGG{} conservation law. On the admissible branch,
\begin{equation}
F^{\nu}{}_{\mu}\Jem^\mu+\Jhol^\nu-\nabla^\nu \CEPphi=0,
\label{eq:cci-branch-recast}
\end{equation}
the projected boundary ledger is conserved:
\begin{equation}
\nabla_\mu \Teff^{\mu\nu}=0.
\label{eq:cci-closed-recast}
\end{equation}
\end{corollary}

\begin{proof}
Two upstream ingredients combine. The bulk-to-boundary side is \ASix{} \T{2.2} \citep{ArisakaA6Gi4} (Master Conservation iff Ward + PBI-4): the Projected Bianchi Identity PBI-4 descends bulk \BIsix{} closure $\dd H_3 = X_4$ to the boundary divergence identity via Kaluza--Klein descent, with the compensator field $\CEPphi$ playing the role of the link. The four-currents Ward-package side is \AFive{} T-GDOS-9.7 \citep{ArisakaA5GDOS}: the four boundary Ward identities $\partial_\mu J^\mu_a = 0$ for $a \in \{\rm em, B, Qi\}$ together with $\nabla_\mu T^{\mu\nu} = 0$ are protected from anomalous breaking by the same \BIsix{} inflow. Restricting the joint statement to the Standard-Model boundary stratum yields \eqref{eq:cci-master-recast}--\eqref{eq:cci-closed-recast}.
\end{proof}

This is the decisive local statement for the present paper. A parameter is
not read off at an arbitrary kinematic point. It is read off on a \CCI{}-lawful
stationary seam: a seam where the projected force, holonomy, and compensator
channels close consistently. That is why the lock seam and the neutral seam
used later are not rhetorical choices. They are local admissible branches of
the projected conservation law.

\subsection{\GFF{} and \GRIP{} as the scalar and runtime closures}

Once \CCI{} has been stated, the remaining two pillars follow as theorems of \ASix{} \citep{ArisakaA6Gi4}.

First, by \ASix{} \T{3.1} (\GFF{} Morse-positivity from \CCI{}), the projected boundary packet carries one scalar ledger,
\begin{equation}
F_{\mathrm{\GFF{}}} = F_{\mathrm{\GEP{}}} + F_{\mathrm{\CEP{}}},
\label{eq:gff-recast-bridge}
\end{equation}
which on every stratum $\Sigma$ of the boundary moduli space is a Morse function with strict alternating-index sum $a^+(\Sigma) - a^-(\Sigma) > 0$ whenever $\dim_{\mathbb R}\Sigma \equiv 0 \pmod 4$ \citep{ArisakaA6Gi4}. The mathematical content is that the reduced open-system dynamics needs one common energetic and curvature--entropy budget --- not two independent scalar potentials --- and the topology of the descent of $[X_4]$ to the boundary stratum forces strict positivity. On the family-projector stratum $\mathbb{C}P^{N-1}$ this is exactly the structural input that produced Theorem~\ref{thm:N3-closed} above.

Second, by \ASix{} \T{4.1} (\GRIP{} cascade closure on family-projector moduli), the branch-selection problem is governed by the resolved runtime order
\begin{equation}
\GRIP:\qquad
\DGI \Longrightarrow \GFF \Longrightarrow \GSSB \Longrightarrow \GA,
\label{eq:grip-recast-bridge}
\end{equation}
with total cascade action bounded by the \BIsix{} budget $S_{\rm tot}(N) \le S_{\rm BI}^{\max} = 3\pi$. This bound is not new in A9; it is the same bound that closes the family-number derivation of Section~\ref{sec:N3-theorem} and is established in full upstream. The runtime order closes the logical gap left open in shorter summaries: symmetry breaking cannot be defined before the scalar landscape is defined;
and a boundary observable is not stable until the selected branch is known to
be a geometric attractor (\citealp{ArisakaA5GDOS}; \citealp{ArisakaA6Gi4}).

One further piece of runtime vocabulary, defined at runtime level in
\ASix{} and used throughout this paper, should be imported now. The runtime
operates in exactly two regimes, split by one question: \emph{does the
stratified landscape change?} \GRIPin{} is the descent on a
\emph{fixed} landscape --- gradient flow within basins, instanton
steps across the \GSSB{} gates between them, and --- on the strata
\ASix{} calls admissible --- halting uniquely at that stratum's
\GGA{}, by the per-stratum ground-state corollary of \ASix{}.
\GRIPout{} is the complementary regime: a transient out-of-equilibrium
event that \emph{reorganizes the landscape itself} and supplies the
initial datum from which a new \GRIPin{} descent begins; that a second
regime is needed at all --- gradient flow cannot cross a stratum
boundary --- is carried in \ASix{}, and its full seventeen-event theory
in \AEleven{}. The present paper hosts
both regimes side by side, and it matters to tell them apart: the
selection of the electroweak vacuum (the Wilson--Hosotani branch and
the lock-seam vev of Part~\ref{part:VI}) was the boundary sector's \GRIPout{}
event --- it \emph{created} the Yukawa landscape on which the three
fermion generations are basins --- whereas every flavor transition
since ($\tau\to\mu\to e$, the quark ladder, the valley of $\beta$
stability) is \GRIPin{} on that fixed landscape. In one sentence: the
electroweak \GRIPout{} built the landscape once; the weak-interaction
\GRIPin{} has been walking it ever since.

\begin{figure}[p]
\centering
\resizebox{0.833\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\small,
 every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
 dgibox/.style={fill=red!13, draw=red!60!black, line width=0.7pt, minimum width=80mm, minimum height=16mm},
 gffbox/.style={fill=yellow!13, draw=olive!60!black, line width=0.7pt, minimum width=80mm, minimum height=16mm},
 gssbbox/.style={fill=green!14, draw=green!60!black, line width=0.7pt, minimum width=80mm, minimum height=16mm},
 gabox/.style={fill=teal!12, draw=teal!60!black, line width=0.8pt, minimum width=80mm, minimum height=16mm},
 ggabox/.style={fill=blue!14, draw=blue!60!black, line width=1.0pt, minimum width=80mm, minimum height=16mm},
 inbox/.style={fill=cyan!16, draw=cyan!60!black, line width=0.9pt, minimum width=56mm, minimum height=16mm},
 outbox/.style={fill=orange!16, draw=orange!70!black, line width=1.0pt, minimum width=56mm, minimum height=16mm},
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 outarrow/.style={-{Stealth[length=2.8mm]}, line width=1.1pt, draw=orange!70!black},
 arrowtext/.style={font=\footnotesize\itshape, text=black!70, fill=white, inner sep=1pt, align=center},
]
\node[dgibox] (DGI) at (0, 0) {\textbf{\DGI{}}\\[-1pt]
 {\scriptsize Dynamical Gauge Invariance --- the symmetric top: every gauge equivalence intact, nothing committed}};
\node[gffbox] (GFF) at (0, -22mm) {\textbf{$(\cF_{\GFF})$ \quad Geometric Free-Energy Functional}\\[-1pt]
 {\scriptsize the fixed landscape; descent with $d\cF_{\GFF}/dt \leq 0$ (GFF-ST6)}};
\node[gssbbox] (GSSB) at (0, -44mm) {\textbf{\GSSB{} \quad Geometric Spontaneous Symmetry Breaking}\\[-1pt]
 {\scriptsize branch selection at the saddles; Bogomolnyi tolls $S = \pi k$ (GRIP-ST3)}};
\node[gabox] (GA) at (0, -66mm) {\textbf{\GA{} \quad Geometric Attractor(s)}\\[-1pt]
 {\scriptsize stable basins en route --- possibly way stations ($\mu$, $\tau$, an excited atom); \GRIPin{} continues through their exit channels}};
\node[ggabox] (GGA) at (0, -88mm) {\textbf{\GGA{} \quad Ground Geometric Attractor}\\[-1pt]
 {\scriptsize the unique deepest basin of an admissible stratum --- \emph{where \GRIPin{} halts}: the exit-channel set is empty}};
\draw[bigarrow] (DGI.south) -- (GFF.north);
\draw[bigarrow] (GFF.south) -- (GSSB.north);
\draw[bigarrow] (GSSB.south) -- (GA.north);
\draw[bigarrow] (GA.south) -- node[arrowtext, right=2pt]{exit channels: the descent continues} (GGA.north);
\node[inbox] (GIN) at (-58mm, -112mm) {\textbf{\GRIPin{}}\\[-1pt]
 {\scriptsize fixed-landscape regime}\\
 {\scriptsize loops through way-station \GA{}s until the \GGA{}}};
\node[outbox] (GOUT) at (58mm, -112mm) {\textbf{\GRIPout{}}\\[-1pt]
 {\scriptsize landscape-reorganizing regime}\\
 {\scriptsize transient out-of-equilibrium event}};
\draw[ribarrow] (GA.west) to[bend right=25]
 node[arrowtext, pos=0.55, below, sloped]{$\cF_{\GFF}$ landscape unchanged} (GIN.north);
\draw[outarrow] (GGA.south) to[bend left=18]
 node[arrowtext, pos=0.55, below, sloped]{$\cF_{\GFF}$ landscape reorganized} (GOUT.north);
\draw[ribarrow, dashed] (GIN.north) to[out=90, in=180]
 node[arrowtext, pos=0.35, left=2pt, text=cyan!55!black, font=\footnotesize\itshape] {within-stratum\\closed loop} (DGI.west);
\node[nextbox] (NEXT) at (58mm, -136mm)
 {\textbf{\DGI{} \quad (next stratum)}\\[-1pt]
 {\scriptsize a new descent begins on the reorganized landscape}};
\draw[outarrow] (GOUT.south) -- (NEXT.north)
 node[arrowtext, pos=0.5, right=3pt, text=orange!70!black, font=\footnotesize\itshape] {supplies the next\\initial datum $c_{\rm top}$};
\end{tikzpicture}}
\caption[The \GRIP{} cascade and its two regimes, in one picture.]{\small\textbf{The \GRIP{} cascade and its two regimes, in one picture.} The central column is the runtime of the cascade: from the fully symmetric top (\DGI{}, every gauge equivalence intact), the system descends the fixed landscape $\cF_{\GFF}$, selects branches at the \GSSB{} saddles, and is captured into a stable basin --- a Geometric Attractor (\GA{}). The attractor tier has two levels. From a \emph{non-ground} \GA{} (a way station: a muon, an excited atom) the descent \emph{continues} --- \GRIPin{}, the fixed-landscape regime, proceeds through the basin's exit channels, the weak interaction being Nature's implementer on the matter strata --- whereas at the \GGA{}, the unique deepest basin, the exit-channel set is empty and \GRIPin{} \emph{halts}. After the halt, only \GRIPout{} can act: a transient out-of-equilibrium event reorganizes the landscape itself and supplies the initial datum of the next descent. In one sentence: \GRIPout{} opens the landscape; \GRIPin{} finishes the journey; the \GGA{} is where the journey ends. In the present paper the one \GRIPout{} is the electroweak vacuum selection, and every fermion mass, mixing angle, and lifetime is a \GRIPin{} readout on the landscape it built. (Reproduced from \ASix{} \citep{ArisakaA6Gi4}; the eighteen-layer deployment is \AEleven{} \citep{ArisakaA11Origin}.)}
\label{fig:grip-two-regimes}
\end{figure}


From this point onward the derivation protocol is fixed. Every sector below
will begin by asking what \BIsix{} permits, what \TGG{} forces, what \CCI{}
projects, what \GFF{} coordinates the sector uses, how \GRIP{} selects and
stabilizes the branch, and only then what parameter formulas are read off.


\subsection{The Arisaka equation: four-currents lock and the structural origin of the \texorpdfstring{$-2/5$}{-2/5} coefficient}
\label{subsec:arisaka-eqn}

The four currents that appear in the \CCI{} (Corollary~\ref{thm:cci-recast}) and the four-currents Ward package (\AFive{} T-GDOS-9.7) --- the electromagnetic current $J_e$, the baryonic current $J_B$, the informational current $J_{Qi}$, and the gravitational current carried by $T^{\mu\nu}$ --- are not independent boundary observables. They are locked through a single operational identity proved as Theorem T-QUA of \ASeven{} \citep{ArisakaA7Qi4}. Indeed, this lock is the structural reason why a single rational coefficient (the trace ratio $-2/5$) appears in the \BIsix{} vector, in the weak-angle rational partition, and in the Arisaka constant simultaneously.

\noindent\emph{In one sentence, before the formal statement:} the four boundary currents that the rest of the paper will treat separately --- the energetic current $J_E$, the electron-number current $J_e$, the baryon current $J_B$, and the information current $J_{Qi}$ --- are in fact one current, projected through one energy--information conversion factor $Q_B$. Everything that follows in the next thirty pages is a specialization of this one identity to a particular boundary stratum. After all, the reader who absorbs the boxed theorem now will recognize it returning under different names later: as the trace-ratio lock $|a/c|=|a/d|=5/2$ in the gauge sector, as the universal scale constant $\pi e$ in the strong coupling, and as the energy--information anchor $Q_B \approx 0.156$~eV per qubit in the lepton sector.

\begin{theorem}[\textbf{T-SM-10.3}: Arisaka equation; \ASeven{} \T{-QUA}]
\label{thm:arisaka-eqn}
On every admissible boundary branch the four conserved currents are related by
\begin{equation}
\boxed{\frac{J_E}{Q_B} = J_e = J_B = J_{Qi},}
\label{eq:arisaka-eqn}
\end{equation}
where $J_E$ is the energy-flux current and $Q_B$ is the Arisaka constant
\begin{equation}
Q_B = \frac{2}{5}\,\frac{E_{\rm Ry}}{K}
\;\approx\; 0.15549~\text{eV per qubit},
\qquad K = 35,
\label{eq:QB-def}
\end{equation}
with $E_{\rm Ry} \approx 13.6057$~eV the Rydberg energy and $K = 35$ the \OGNfive{} lock-length integer. Equivalently, in chemical units,
\begin{equation}
N_A \, Q_B \;\approx\; 15.0~\text{kJ/mol},
\label{eq:QB-chem}
\end{equation}
\end{theorem}

\begin{proof}
The full proof is \ASeven{} \T{-QUA} \citep{ArisakaA7Qi4}: the rank-one inflow theorem (algebraic layer) shows that all four boundary currents descend from a single rank-one bulk inflow channel through \BIsix{} closure on the 6D bulk, and the operational layer then reconciles the four currents through $Q_B$ as the natural-unit conversion factor. The chemical-units value \eqref{eq:QB-chem} is a direct consequence of the upstream-fixed quantities $E_{\rm Ry}$, $K$, and $N_A$, not an additional input.
\end{proof}

\paragraph{Why the \BIsix{} vector contains the trace ratio $-2/5$.}
The structural meaning of \eqref{eq:arisaka-eqn} is that the energy-flux normalization $Q_B$ enters the \BIsix{} anomaly polynomial wherever the boundary currents do. By \AThree{} \citep{ArisakaA3BI6}, the integrality of $X_4$ on the 6D bulk forces the rational coefficient
\begin{equation}
\frac{\Tr(B_0^2)}{\Tr(Y_0^2)}
=
\frac{L-N}{L}
=
\frac{2}{5}
\qquad \text{for } (N,L)=(3,5),
\label{eq:bi6-tracenorm-A9}
\end{equation}
which is the same $2/5$ that defines $Q_B$ in \eqref{eq:QB-def}. Indeed, the appearance of $-2/5$ as the third and fourth components of the \BIsix{} vector $(a,b,c,d) = (1, -1, -2/5, -2/5)$ \eqref{eq:bi6-vector-recast} is not a numerical coincidence; it is the same trace ratio that fixes $Q_B$ on the boundary. After all, \BIsix{} and the Arisaka equation are two faces of one structural fact: the four boundary currents lock through a single rank-one ledger whose normalization is the trace ratio of the primitive $\mathbf 5$ representation.

\paragraph{Why the weak-angle partition is $L^2/N^3$.}
The same four-currents lock is what produces the closed-form weak-mixing-angle theorem proved later in this paper (Theorem~\ref{thm:sw2}, the weak-angle rational-partition theorem). By \ASeven{} \T{-QUA}, the electromagnetic and weak neutral currents partition as a power partition of the same locked ledger: the $L = 5$ split-generator norm sits on the photon side of the neutral seam in proportion $L^2 / N^3$, with the leftover $(L-N)^2 = 4$ on the $Z$-boson side. The result
\begin{equation}
\sin^2\theta_W(M_Z)
=
\frac{L^2}{N^3 (L-N)^2}
=
\frac{25}{27 \cdot 4}
=
\frac{25}{108}
\label{eq:sw2-from-arisaka}
\end{equation}
is therefore not a separate empirical fit but a direct algebraic consequence of \eqref{eq:arisaka-eqn} and the \OGNfive{} ledger. The same structural fact that fixes $Q_B = (2/5) E_{\rm Ry}/K$ at the energy--information scale fixes $\sin^2\theta_W = 25/108$ at the electroweak scale; both are readouts of the same Arisaka equation lock at different boundary strata.

\begin{euclidremark}[Physics intuition for the Arisaka lock]
One structural fact --- the four boundary currents are locked through $Q_B$ --- explains three numerical features of the framework that would otherwise look unrelated. The trace ratio $-2/5$ in the \BIsix{} vector, the rational $25/108$ in the weak-angle partition, and the energy quantum $0.1555$~eV per qubit are not three coincidences but three faces of the same Arisaka equation. Indeed, this is the deepest unifying claim of \GGTheory{}: that the same geometric lock that generates the gauge-Higgs sector at TeV scales generates the energy--information ledger at meV scales, with $Q_B$ as the universal conversion factor. After all, this is exactly the role T-GU5 (the \GUfive{} of \AOne{}) assigns to $Q_B$: it is the boundary-side conversion factor of the \GDOS{} Trio, anchored to its primary derivation paper \ASeven{} (\citealp{ArisakaA1Constitution}; \citealp{ArisakaA7Qi4}).
\end{euclidremark}

\begin{figure}[H]
\centering
\resizebox{0.636\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=24mm,
 bulknode/.style={rectangle, rounded corners=3pt, draw=red!70!black, fill=red!10, line width=0.7pt, minimum width=46mm, minimum height=16mm, align=center, font=\small\bfseries\sffamily},
 curnode/.style={rectangle, rounded corners=2pt, draw=black!70, line width=0.5pt, minimum width=24mm, minimum height=10mm, align=center, font=\small\sffamily, fill=blue!10},
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 divlabel/.style={font=\scriptsize\sffamily, fill=white, inner sep=1pt}
]
 \node[bulknode] (bulk) at (0, 3.5) {Rank-one bulk inflow channel \\[-0.2em] \scriptsize{(BI-6, \citep{ArisakaA3BI6})}};

 \node[curnode] (Je) at (-5, 0.5) {$J_e$};
 \node[curnode] (JB) at (-1.7,0.5) {$J_B$};
 \node[curnode] (JQi) at (1.7,0.5) {$J_{Qi}$};
 \node[curnode] (JE) at (5, 0.5) {$J_E / Q_B$};

 \draw[arrowstyle] (bulk) -- (Je);
 \draw[arrowstyle] (bulk) -- (JB);
 \draw[arrowstyle] (bulk) -- (JQi);
 \draw[arrowstyle] (bulk) -- (JE);

 \node[qbnode] (qb) at (5, -2.5) {$Q_B$};
 \draw[arrowstyle, dashed] (qb.north) -- (JE.south);

 \node[font=\footnotesize\sffamily, below=1pt of qb, text=orange!80!black]
 {$Q_B = \tfrac{2}{5}\tfrac{E_{\rm Ry}}{K} \approx 0.1555$~eV};

 \node[font=\small\sffamily, below=4pt of Je] {electric};
 \node[font=\small\sffamily, below=4pt of JB] {baryonic};
 \node[font=\small\sffamily, below=4pt of JQi] {informational};
 \node[font=\small\sffamily, below=4pt of JE] {energy / qubit};

 \draw[line width=0.7pt, black!70] (-6, -1.6) -- (3.2, -1.6);
 \node[below=4pt, font=\footnotesize\bfseries\sffamily] at (-1.4, -1.6)
 {Arisaka equation: $J_E/Q_B = J_e = J_B = J_{Qi}$};

\end{tikzpicture}%
}
\caption[The Arisaka equation linking the energy--information quantum to the geometry.]{The Arisaka equation $J_E/Q_B = J_e = J_B = J_{Qi}$ (Theorem~\ref{thm:arisaka-eqn}; \ASeven{} \T{-QUA}, \citealp{ArisakaA7Qi4}) as a four-currents lock. All four boundary currents (electric $J_e$, baryonic $J_B$, informational $J_{Qi}$, and energy-flux $J_E$ converted by the energy--information factor $Q_B$) descend from a single rank-one bulk inflow channel of BI-6. The trace-ratio coefficient $-2/5$ that appears in the BI-6 vector and the $L^2/N^3$ partition that produces $\sin^2\theta_W = 25/108$ are two faces of the same lock.}
\label{fig:arisaka-equation}
\end{figure}



The Arisaka equation \eqref{eq:arisaka-eqn} together with the \BIsix{} corollary \eqref{eq:bi6-tracenorm-A9} is therefore the load-bearing identity that A9 inherits from the \GDOS{} Trio. Every parameter formula in the remainder of this paper is, at the structural level, a specialization of \eqref{eq:arisaka-eqn} to a particular boundary stratum. The gauge-Higgs theorem package of Sections~\ref{sec:dyad-closure-anchor}--\ref{sec:vev} below uses the partition \eqref{eq:sw2-from-arisaka}; the fermion-mass codebook of Sections~\ref{sec:lepton-mass}--\ref{sec:quark-mass} uses the rank-one structure of the Arisaka inflow at the family-projector stratum; the CKM and PMNS sections use the same rank-one structure restricted to the up-type/down-type and charged-lepton/neutrino projector pairs.



\subsection{The status of \texorpdfstring{$\UoneQi{}$}{U(1)\_Qi}: constitutionally optional, empirically preferred}
\label{subsec:UoneQi-status}

A natural question raised by the four-currents lock is whether the informational current $J_{Qi}$ --- and hence the gauge factor $\UoneQi{}$ that carries it --- is a constitutional necessity of the bulk theory or an empirically preferred extension.

\paragraph{Constitutional admissibility.}
By \AThree{} \citep{ArisakaA3BI6}, both the three-component bulk gauge group $\Spinone \times \SU(5) \times \UoneB$ and the four-component bulk gauge group $\Spinone \times \SU(5) \times \UoneB \times \UoneQi$ are constitutionally admissible: both close \BIsix{} with their respective coefficient vectors, and both satisfy the \GGConstitution{} axioms (\AX{1}--\AX{4}) and postulates (\POS{1}--\POS{7}). In this sense $\UoneQi{}$ is \emph{constitutionally optional}.

\paragraph{Empirical preference.}
The four-component bulk is adopted in the present paper on three independent empirical grounds. First, the Arisaka equation $J_E/Q_B = J_e = J_B = J_{Qi}$ (Theorem~\ref{thm:arisaka-eqn}; \ASeven{} \T{-QUA}) requires the informational current $J_{Qi}$ on the boundary, with no consistent rephrasing in the three-component bulk. Second, the \OGNfive{} dual-lock identity $K = LJ = M^2 + N^2 + L^2 = 35$ closes only for the four-component case (per A2-\GG{} Theorem~\T{6} \citep{ArisakaA2GG6}); the three-component variant fails the integrality check. Third, the entire \QiFour{} Pillar structure of A7 (Pillars \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) and the \EIC{} boundary completion (\OH{}+\IH{}=\EIC{}) depend on $\UoneQi{}$.

\paragraph{Net status.}
$\UoneQi{}$ is therefore \emph{constitutionally optional} (the \GGConstitution{} admits both bulk variants) but \emph{empirically preferred} (the four-component bulk produces a working 28-parameter codebook; the three-component bulk does not). \ANine{} adopts the four-component bulk throughout. A reader who wishes to study the three-component variant would have to abandon the Arisaka equation and re-open the \OGNfive{} closure check; that is a substantial structural undertaking and not pursued here. After all, the empirical-preference framing makes the structural cost of the three-component variant explicit, and the four-component variant is the one the present codebook actually requires.



\section{\GDOS{}, \GFF{}, \SIP{}, and the operational meaning of observables}
\label{sec:gdos-qi4-recap}

This section states the third link of the central claim of \S\ref{sec:central-claim} (Figure~\ref{fig:central-claim}) in full: the law \GDOS{} by which the open-system projection of \OGGEP{} assigns each of the twenty-eight observables a definite value.

The previous section fixed the ultraviolet constitution and the admissible input
ledger. We now need to explain how those ultraviolet statements become
low-energy observables. This is the role of \GDOS{} in \AFive{} and of the
universal runtime extension in \ASeven{}. The basic message is simple but
fundamental: the Standard Model is not merely a four-dimensional local Lagrangian;
it is the observer-accessible boundary dynamics of a deeper open system
(\citealp{ArisakaA5GDOS}; \citealp{ArisakaA6Gi4}).

\subsection{Why a parameter paper needs \GDOS{} before it needs numerical formulas}

In ordinary phenomenology, one often starts from a four-dimensional effective
Lagrangian, chooses a renormalization scheme, and fits parameters to data.
Nothing is wrong with that procedure when the goal is practical prediction.
However, the present paper is trying to do something more ambitious. It is not
trying merely to summarize the phenomenology of a known theory. It is trying to
show how those phenomenological quantities arise from a deeper constitutional
framework.

For that reason a numerical parameter paper that omits the operational layer
would be incomplete. A coupling constant, for example, is not a bare number in
the abstract. It is the result of a boundary measurement protocol performed at a
specified scale and in a specified renormalization scheme. A mass is not simply
a coefficient in a microscopic action. It is a spectral feature of a reduced
boundary theory reconstructed from open-system data. A mixing angle is not an
ontological angle living in the bulk. It is a relation among boundary overlap
channels inferred from measured amplitudes and rates.

\GDOS{} is therefore not apparatus bolted onto the parameter problem; it is the third arrow of the central claim (\S\ref{sec:central-claim}) made precise --- the law that tells us what the quantities being predicted even \emph{are}.

\subsection{From \OGGEP{} to exact open-system boundary dynamics}

The core claim of \AFive{} is that AX-3 (\(\OGGEP\)) requires every admissible
theory to provide a lawful map from inaccessible or partially accessible bulk
data to registered boundary observables. In abstract notation one may write
\begin{equation}
\label{eq:oggp-map-recap}
\mathcal D:\Bulk\longrightarrow (\cH_{\partial},\rho_{\partial},\Phi_{t,s},\{\mathcal I_\alpha\}),
\end{equation}
where \(\rho_{\partial}\) is the boundary state, \(\Phi_{t,s}\) is the
admissible evolution map, and \(\{\mathcal I_\alpha\}\) is the family of
measurement instruments.

At the exact level, \AFive{} shows that the reduced boundary dynamics obeys a
Nakajima--Zwanzig type equation, and that in the Markovian limit one arrives at a
GKLS generator. For the present paper, the crucial phenomenological point is
that the low-energy Standard Model is therefore already an open-system boundary
theory before one begins any specific particle-physics calculation.

This can be summarized schematically as
\begin{equation}
\label{eq:bulk-to-bdy-chain-recap}
\text{\GG{} bulk}
\xRightarrow{\;\OGGEP\;}
\text{exact projected boundary dynamics}
\Longrightarrow
\text{GKLS limit}
\Longrightarrow
\text{\GDOS{} current form}.
\end{equation}
Equation~\eqref{eq:bulk-to-bdy-chain-recap} is one of the conceptual bridges that
make \ANine{} a phenomenological paper rather than a purely formal one. It
means that when we later speak about couplings, masses, and mixings, we are not
speaking about hidden bulk parameters but about projected seam observables.

\subsection{The \GDOS{} continuity equation and its physical meaning}

At mesoscopic resolution, the boundary theory is written in \GDOS{} form as a
continuity law on a state manifold:
\begin{equation}
\label{eq:gdos-continuity-recap}
\partial_t p(x,t)+\nabla_A J^A(x,t)=0.
\end{equation}
The current admits the standard decomposition emphasized in \AFive{} and
\ASeven{},
\begin{equation}
\label{eq:gdos-current-recap}
J^A
=
- D^{AB}\nabla_B p
-\mu^{AB}p\,\nabla_B\mathcal F_{\GFF}
+J^A_{\rm curl}.
\end{equation}
Equation~\eqref{eq:gdos-current-recap} is simple enough to memorize, and it is
worth doing so because it contains much of the philosophy of the program in one
line.

The first term is diffusive spreading. This is what any open system does when
it is left only to uncontrolled noise and coarse-grained mixing.

The second term is gradient descent on the \GFF{} ledger. This is what organizes
selection, relaxation, and maintenance.

The third term is the solenoidal or circulation term. This is what allows the
system to have sustained currents, cycles, and nontrivial nonequilibrium steady
states instead of collapsing into a trivial equilibrium minimum.

This decomposition already contains the seed of why a Standard-Model parameter
paper should be written in \GDOS{} language. A measured coupling or mass is not
just a number. It is the visible trace of a system that has diffused, descended,
and stabilized onto a specific boundary branch.

\subsection{\GFF{} as the scalar ledger of observable physics}

\ASeven{} insists that the \GDOS{} runtime is incomplete unless it carries one
unified scalar ledger, denoted \GFF{}, decomposed as
\begin{equation}
\label{eq:eegg-recap}
\GFF=\GEP+\CEP.
\end{equation}
The first term, \(\GEP\), is the energy--geometry--gauge contribution. The
second, \(\CEP\), is the curvature--entropy contribution induced by coarse graining, admissible state-space geometry, and diffusion.
In practical terms, the precise division depends on the system under study, but
the key lesson is universal: a persistent open system requires one scalar budget
that can be decreased, replenished, and audited.

For particle phenomenology the importance of \GFF{} may not be obvious at first
sight, so it is worth stating the intuition explicitly. In collider physics or
precision electroweak physics, one often speaks as though the measured quantity
is just the pole of a propagator or the value of a renormalized coupling. \GG{}
and \GDOS{} do not deny that. They ask a more basic question: why is the theory
allowed to export that quantity as a stable, reproducible, observer-accessible
observable at all? The answer is that the corresponding boundary branch is a
stable attractor of the open-system runtime --- for the reproducible locked values, the stratum's ground attractor (\GGA{}; \S\ref{subsec:gga-refinement}) --- and its stability is organized by
\GFF{}.

This viewpoint is especially useful for mass. The mass-origin paper cited above reformulates mass as a geometric spectral gap and interprets the Higgs,
fermion, and confinement sectors as different projections of one \GFF{}-based
ledger \citep{ArisakaA6Gi4}. That viewpoint
fits naturally into the present manuscript, where the mass tables are not
supposed to be isolated empirical fits but outputs of one geometrically and
operationally unified construction.

\subsection{\DGI{}, \GSSB{}, and GA as the master pipeline from possibility to observable structure}

One of the most useful conceptual advances of \ASeven{} is the explicit naming
of the three-stage runtime grammar
\begin{equation}
\label{eq:dgi-gssb-ga-recap}
\DGI\ \longrightarrow\ \GSSB\ \longrightarrow\ \GA.
\end{equation}
This pipeline is not merely a slogan. It is the operational translation of how
an open system explores, selects, and stabilizes structure.

\paragraph{\DGI{}: Dynamical Gauge Invariance.}
\DGI{} says that the lawful reduced dynamics should be written not in terms of one
rigid microscopic description, but in terms of a gauge-equivalence class of
admissible reduced descriptions. Intuitively, \DGI{} is the exploration stage:
the system moves among physically equivalent descriptions without yet committing
to a unique realized branch.

\paragraph{\GSSB{}: Geometric Spontaneous Symmetry Breaking.}
\GSSB{} says that branch selection is not an add-on to the geometry; it is a
geometric event inside the open-system runtime. In the present paper, the most
important \GSSB{} event is the selection of the Wilson--Hosotani vacuum holonomy.
That is what eventually turns the unified \(\SU(5)\) sector into the Standard
Model gauge structure and then further reduces the electroweak sector to
\(\U(1)_{\rm EM}\).

\paragraph{GA: Geometric Attractor.}
GA is the stabilized branch selected by the previous two stages. Physically, it
is what makes a low-energy world persistent enough to measure repeatedly. In
particle phenomenology, a GA is not a biological attractor or a cognitive state;
it is the stabilized seam theory whose couplings, masses, and mixings are the
ones seen in experiment.

This three-stage grammar is particularly valuable because it tells us how to
read the Standard Model inside \GG{}. The Standard Model is not the full bulk.
It is the GA stage of a more symmetric and more geometric theory after \DGI{} and
\GSSB{} have acted.

\subsection{Why the \QiFour{} pillars matter even in a particle-physics paper}

At first encounter, the \QiFour{} extension may appear to belong more naturally to
biology, neuroscience, or complex systems than to particle phenomenology.
However, that impression is misleading. The \QiFour{} pillars were introduced in
\ASeven{} precisely to universalize \GDOS{} across all open systems, and the
particle-physics laboratory is itself an open system. The point is not that a
collider detector is a brain. The point is that both are finite devices that
prepare states, exchange currents, store records, and produce registered
observables.

The four pillars are:
\begin{equation}
\label{eq:qi4-pillars-recap}
\text{\QiQUA{}},\qquad
\text{\QiEP{}},\qquad
\text{\QiCON{}},\qquad
\text{\QiHAL{}}.
\end{equation}
Their relevance to the present paper can be summarized as follows.

\paragraph{\QiQUA{}.}
This is the current-locking layer. It organizes the relation among charge,
baryon, informational, and energetic currents and introduces the Arisaka Ratio,
Arisaka Equation, and Arisaka Constant. For a particle-physics paper the main
use of \QiQUA{} is to justify why seemingly different bookkeeping channels can be
projected onto one locked seam ledger.

\paragraph{\QiEP{}.}
This is the extremal-principle layer. It unifies the bulk principle of least
action with boundary principles such as least-time, maximal entropy production,
and free-energy descent. In the present paper this is relevant to vacuum
selection, RG trajectory selection, and the identification of the physically
realized branch.

\paragraph{\QiCON{}.}
This is the conformal and multiscale diagnostic layer. For particle physics it
supports the idea that logarithmic running, hierarchy patterns, and scale
covariance are not external features bolted onto the theory but are expected
signatures of the runtime.

\paragraph{\QiHAL{}.}
This is the architecture layer. In complex systems it deals with memory and
internal modeling. In the present paper the closest analogy is the experimental
architecture that stores and reconstructs boundary observables. \QiHAL{} therefore
matters less directly for the derivation of numerical Standard-Model parameters,
but it matters for the full observational logic of how those parameters become
registered quantities.

\subsection{The System Instance Packet and why the Standard Model itself should be treated as a packet}

The most operationally concrete innovation of \ASeven{} is the System Instance
Packet, or \SIP{}. In its canonical five-field form it is written as
\begin{equation}
\label{eq:sip-general-recap}
\SIP=
\Bigl[(\mathcal Z,g_{AB}),\;\Ports,\;(\mathfrak L_{\GDOS},\eta),\;(\mathcal F_{\GFF},\mathcal J),\;(\Obs,\Kill)\Bigr].
\end{equation}
The significance of this definition is that it forces one to say what the
system is, what its ports are, what its lawful generator is, what scalar and
vector ledgers it carries, what observables are exported, and what would count as
failure.

This language can and should be applied directly to the Standard Model as it
appears in the present paper. The Standard Model, in the present framework, is
not ``all of reality.'' It is a specific observer-accessible seam packet. A
useful formal notation is therefore
\begin{equation}
\label{eq:sip-sm}
\SIP_{\rm SM}
=
\Bigl[(\mathcal Z_{\rm SM},g_{AB}^{\rm SM}),\;\Ports_{\rm SM},\;(\mathfrak L_{\rm SM},\eta_{\rm SM}),\;(\mathcal F^{\rm SM}_{\GFF},\mathcal J_{\rm SM}),\;(\Obs_{\rm SM},\Kill_{\rm SM})\Bigr].
\end{equation}
Each entry has a concrete meaning.

\paragraph{State manifold.}
The state manifold \(\mathcal Z_{\rm SM}\) is spanned by the seam-visible gauge,
Higgs, and flavor variables relevant to the chosen observational regime. In the
present paper that includes the running gauge couplings, the Wilson-line/Higgs
order parameter, the family projectors that later generate Yukawa hierarchies,
and the renormalized boundary masses and mixing parameters derived from them.

\paragraph{Ports.}
\(\Ports_{\rm SM}\) are the preparation and measurement channels by which the
seam is probed: scattering inputs, decay channels, threshold scans, flavor-tagging
channels, oscillation baselines, detector reconstruction pipelines, and scheme
choices that turn raw event counts into renormalized observables.

\paragraph{Generator.}
\(\mathfrak L_{\rm SM}\) is the effective seam generator. At high resolution it
is the projected open-system generator descending from the bulk. At practical
phenomenological resolution it becomes the combined RG flow plus effective
interaction structure governing the scale dependence of couplings and masses.

\paragraph{\GFF{} ledger and current bundle.}
\(\mathcal F^{\rm SM}_{\GFF}\) is the scalar budget that organizes seam branch
selection, while \(\mathcal J_{\rm SM}\) collects the relevant locked currents.
Although the current bundle is not used in the same explicit way in every later
calculation, it is the reason why the seam observables can be regarded as one
unified packet rather than as independent sectoral fragments.

\paragraph{Observables and kill criteria.}
\(\Obs_{\rm SM}\) are the renormalized gauge couplings, masses, mixing angles,
phases, and derived low-energy observables. \(\Kill_{\rm SM}\) are the failure
conditions: the theory would be ruled out if it failed anomaly cancellation,
failed to produce the observed gauge pattern, failed to preserve proton
stability, or predicted the wrong class of masses and mixings beyond the
estimated theoretical uncertainty.

This packet language is more than bookkeeping. It enforces conceptual honesty.
It prevents one from speaking as though the Standard Model were a monolithic
closed system detached from the finite experimental architectures that actually
measure it.

\subsection{How the \SIP{} viewpoint changes the meaning of a Standard-Model parameter}

Once \(\SIP_{\rm SM}\) is written down, the meaning of a Standard-Model parameter
becomes sharper.

A gauge coupling is the projected response coefficient of a seam packet under a
specified operational protocol.

A mass is the spectral gap exported by that packet after the vacuum branch has
been selected.

A mixing angle is a statement about overlap and misalignment among family
projectors in that packet.

A CP phase is a holonomy-sensitive orientation relation in the same family-space
and compact-angle structure.

This is exactly the level of conceptual precision needed for a phenomenological
paper whose ambition is to derive parameters rather than merely to report them.

\subsection{Why the Standard Model is a particularly clean application of \GDOS{}}

The Standard Model is not the only possible application of \GDOS{}, but it is one of
the cleanest. The reason is that high-energy physics already has an unusually
sharp culture of state preparation, scattering, asymptotic channels,
renormalization, and statistical inference. In a sense, the Standard Model is
what one gets when a very well-controlled open-system packet is observed through
extremely refined protocols.

That is why the \GDOS{} language is not an artificial imposition on particle
phenomenology. It is a clarification of something high-energy physics already
does in practice, namely extract physically meaningful parameters from open,
finite, projected observations.

\subsection{Bridge to the derivation sections that follow}

The purpose of the present section has been to make three points explicit.

First, the Standard Model observables predicted later in the paper are not bare
bulk quantities; they are boundary-visible \GDOS{} exports.

Second, \GFF{} is the scalar ledger that makes branch selection and persistence of
those exports meaningful.

Third, the \DGI{}\(\to\)\GSSB{}\(\to\)GA pipeline and the \SIP{} language explain how the
symmetric six-dimensional stage becomes the specific low-energy seam whose
parameters we measure.

With that operational and runtime language in place, we can now turn to the
Standard-Model foundation proper: the route from the historical Georgi--Glashow
\(\SU(5)\) idea to the \GG{} seam construction that underlies the detailed
parameter calculations below.



\section{Part~III in plain English: the rules the derivation has to obey}
The hardest thing about a derivation of this kind is not the algebra. It
is staying honest about what went in.

There is a standard failure mode in unified model building, and it is not
fraud --- it is drift. A quantity gets used as an input in one section and
predicted in another; a scale gets fixed by a measurement early and then the
measurement is cited later as a confirmation; a fit constant appears under the
name of a convention. None of this requires bad faith. It requires only that
nobody wrote down the rules in advance.

This Part writes them down. The four axioms and seven postulates come from
\AOne{} and are not renegotiated here. The admissible inputs are listed
exhaustively: the dimensional conversion anchors, the two geometric constants,
the five ledger integers, and the constitutional statements themselves.
Everything else is forbidden, and the table of forbidden shortcuts is as
specific as the table of permitted inputs --- it names taking the Fermi
constant as an input, seeding the running with measured couplings, and adding a
family-specific constant per mass.

The apparatus is not a philosophical preamble. It is the instrument by which
the rest of the paper can be audited by someone who does not trust it. A
reader who wants to attack the result should attack it here first: find a
place where a later Part uses something this Part does not permit. If that
place exists, the derivation fails, and it fails cleanly rather than by
degrees. That is the reason for setting the rules out at this length before
any of them is used.

\GGPart[The Substrate]{The Substrate: the bulk, the integer ledger, and the GG seam}{The fixed geometric landscape on which the cascade runs: the technical descent from Georgi--Glashow \texorpdfstring{$\mathrm{SU}(5)$}{SU(5)} to the six-dimensional seam, the bulk geometry and three scales, the \OGNfive{} ledger, the three-leg theorem $N=3$, the radial renormalization group, and Wilson--Hosotani symmetry breaking}
\label{part:III}

\section{From the original Georgi--Glashow \texorpdfstring{\(\SU(5)\)}{SU(5)} to the \GG{} Standard-Model seam}
\label{sec:su5-overview}

\subsection{Inheritance from classic \texorpdfstring{$\mathrm{SU}(5)$}{SU(5)} --- pointer to Part~\ref{part:I}~\S\ref{sec:classic-su5}}

The historical case for $\mathrm{SU}(5)$ as the natural simple-group container of the Standard-Model gauge algebra, the matter-content fitting into $\bar{\mathbf 5} \oplus \mathbf{10}$ per family, and the canonical decomposition of the adjoint $\mathbf{24}$ are recorded in Part~\ref{part:I}~\S\ref{sec:classic-su5} (\citealp{GeorgiGlashow1974}). The five virtues classic $\mathrm{SU}(5)$ explains (charge quantization, family multiplet structure, gauge-coupling unification, anomaly cancellation per family, and the embedding of $\mathrm{U}(1)_Y$ inside a simple group) and the five gaps it leaves open (family number $N$, fermion masses, gravity, proton stability via $X, Y$ bosons, the measurement layer) are likewise recorded there. The present chapter takes those facts as known and turns to what \GG{} \emph{changes}.

\subsection{What \GG{} changes: 6D geometry, local \texorpdfstring{\(\U(1)_B\)}{U(1)B}, and local \texorpdfstring{\(\U(1)_{Qi}\)}{U(1)Qi}}

The unified bulk group relevant to the present paper is not just \(\SU(5)\), but
\begin{equation}
\label{eq:bulk-group-su5-section}
\SU(5)\times \U(1)_B\times \U(1)_{Qi}\times \text{Gravity}.
\end{equation}
Each new factor has a concrete role.

\paragraph{The radial direction.}
This is what geometrizes scale and eventually lets running couplings be derived
from transport in the fifth coordinate.

\paragraph{The compact \texorpdfstring{\(\theta\)}{theta} cycle.}
This is what makes Wilson-line holonomy and the Hosotani mechanism available.
Without it, the present paper could not reinterpret the Higgs as a geometric
order parameter.

\paragraph{Local \texorpdfstring{\(\U(1)_B\)}{U(1)B}.}
This is what changes the proton-decay story. In the Standard Model and in many
traditional GUT settings, baryon number is at best approximate. In the present
framework, exact local \(\U(1)_B\) strongly constrains baryon-violating
operators and is tied to the insistence that proton stability is a genuine structural clue rather than an accident.

\paragraph{Local \texorpdfstring{\(\U(1)_{Qi}\)}{U(1)Qi}.}
This factor is what makes it possible to extend the open-system logic from
particle phenomenology to the full \QiFour{} runtime. Even in the present
particle-focused paper, it matters because it is part of the locked current
ledger and therefore part of the explanation of why observable bookkeeping can
be unified.

\subsection{Orbifold/boundary-condition breaking and Wilson--Hosotani selection}

In the \GG{} setting, the descent from the unified bulk to the low-energy gauge
algebra is not achieved by simply postulating a four-dimensional Higgs
multiplet that breaks \(\SU(5)\) by hand. It is achieved by a combination of
boundary-condition selection and Wilson-line holonomy.

A standard parity assignment that separates color and electroweak directions may
be written schematically as
\begin{equation}
\label{eq:parity-matrix-su5}
P=\diag(+,+,+,-,-).
\end{equation}
The generators commuting with the parity assignment survive as low-energy zero
modes, while the noncommuting generators are projected out of the light sector
or lifted to heavy masses. At the same time, the compact \(\theta\) direction
supports the Wilson line
\begin{equation}
\label{eq:wilson-line-su5-recap}
W(\theta)=\mathcal P\exp\!\left(\ii\oint_{S^1_\theta} A_\theta\,\dd\theta\right),
\end{equation}
whose vacuum expectation value selects the actual branch.

This is precisely where \GSSB{} enters. The low-energy Standard-Model gauge group
is not assumed as the only algebra logically possible. It is selected as the
stable commuting branch of a more symmetric unified structure under the combined
influence of boundary conditions and holonomy. The dynamical content of this holonomy selection --- the Wilson-line effective potential and the electroweak vacuum value it fixes --- is derived in \S\ref{sec:wilson-hosotani}.

\subsection{The low-energy gauge algebra as a commuting subalgebra}

The algebra that survives after the first stage of breaking is the commuting
subalgebra
\begin{equation}
\label{eq:sm-subalgebra-recap}
\mathfrak g_{\rm SM}
=
\mathfrak{su}(3)_C\oplus \mathfrak{su}(2)_L\oplus \mathfrak u(1)_Y.
\end{equation}
The hypercharge generator in the conventional \(\SU(5)\) embedding is
\begin{equation}
\label{eq:hypercharge-embedding-recap}
Y=\diag\!\left(-\frac13,-\frac13,-\frac13,\frac12,\frac12\right),
\end{equation}
properly normalized inside the unified algebra. Electric charge then follows
from the usual relation
\begin{equation}
\label{eq:q-t3-y-recap}
Q=T_3+Y.
\end{equation}

The value of stating Eqs.~\eqref{eq:hypercharge-embedding-recap} and
\eqref{eq:q-t3-y-recap} explicitly is that they already explain why charge
quantization is natural in a unified setting. The charges of quarks and leptons
are not separate empirical insertions; they are consequences of one embedding.

\subsection{How the three observed gauge forces arise at low energies}

A common source of confusion is the phrase ``three gauge forces.'' If one begins
from a unified theory, does one still really explain why there are three? In the
present framework the answer is yes, and the explanation is sequential.

At the highest relevant scale of the visible gauge sector, one has the unified
\(\SU(5)\) connection together with local \(\U(1)_B\) and \(\U(1)_{Qi}\).
After the first breaking/selecting stage, the visible gauge sector becomes
\(\SU(3)_C\times \SU(2)_L\times \U(1)_Y\). After the second, electroweak
breaking stage, one obtains
\begin{equation}
\label{eq:ew-breaking-sequence-recap}
\SU(2)_L\times \U(1)_Y
\longrightarrow
\U(1)_{\rm EM}.
\end{equation}
At ordinary laboratory energies, the three interaction sectors that remain as
distinct long- or intermediate-range gauge structures are therefore:
\begin{equation}
\label{eq:three-forces-recap}
\text{strong }(\SU(3)_C),\qquad
\text{weak }(\text{broken }\SU(2)_L\times \U(1)_Y),\qquad
\text{electromagnetic }(\U(1)_{\rm EM}).
\end{equation}
The word ``three'' is thus not fundamental in the ultraviolet sense; it is the
number of distinct gauge sectors experienced after the two-stage seam reduction.
That is exactly the sort of statement that should appear in a professional
phenomenological paper: it says not only what the world looks like, but why it
looks that way at accessible scales.

\subsection{Why the Standard Model should be viewed as the attractor, not the whole ontology}

This point is worth emphasizing because it improves the phenomenological logic of
the paper. If one mistakes the Standard Model for the whole ontology, then its
free parameters appear as unexplained primitive numbers. If instead one views
it as a stabilized attractor branch of a more constrained six-dimensional open
system, then its parameters appear as seam invariants of a deeper geometry.

That change of viewpoint does not automatically compute the 28 parameters. But
it explains why computing them is a sensible goal. There is something to
compute precisely because the boundary theory is constrained by a more rigid
bulk.

\section{Applying the \GDOS{} runtime to the unified packet}
\label{sec:apply-gdos-su5}

The previous sections established the ultraviolet stage, the open-system
runtime, and the historical relation between \(\SU(5)\) and the Standard Model.
The conceptual \GDOS{} law --- what it is, and why a parameter paper needs it --- was fixed in \S\ref{sec:gdos-qi4-recap}; we do not restate it. Here we \emph{apply} it, walking the runtime from the pre-breaking unified packet through the \GSSB{} branch selection to the stabilized Standard-Model attractor. The stationary calculational seams at which the resulting couplings and masses are then evaluated are collected separately in \S\ref{sec:gdos-eegg}.

\subsection{The unified packet before symmetry breaking}

Before branch selection, the visible gauge and matter sector is most naturally
viewed as a unified packet with bulk-visible fields, boundary conditions,
transport structure, and open-system observables all declared at once. A useful
formal description is
\begin{equation}
\label{eq:unified-packet-prebreak}
\widehat\SIP_{\rm vis}^{\rm pre}
=
\Bigl[(\mathcal Z_{\rm vis}^{\rm pre},g_{AB}),\;\Ports_{\rm vis}^{\rm pre},\;(\mathfrak L_{\rm vis}^{\rm pre},\eta_{\rm vis}^{\rm pre}),\;(\mathcal F_{\GFF}^{\rm pre},\mathcal J_{\rm vis}^{\rm pre}),\;(\Obs_{\rm vis}^{\rm pre},\Kill_{\rm vis}^{\rm pre})\Bigr].
\end{equation}
At this stage, \(\mathcal Z_{\rm vis}^{\rm pre}\) includes the full unified gauge
connection on the visible gauge bundle, the relevant matter multiplets, the
radial transport coordinate, and the compact-angle holonomy variable.
The packet is more symmetric than the later low-energy world. That
is exactly what one expects before \GSSB{} acts.

A professional phenomenological paper should emphasize this point because it
prevents a common misunderstanding. The Standard Model is not the theory before
symmetry breaking. It is the stabilized low-energy branch after symmetry
breaking, projection, and reduction. The pre-breaking packet therefore carries
more structure than the final observed theory.

\subsection{\GFF{} ledger and \DGI{} application --- pointer to \ASix{} \T{3.1} and \T{4.1}}

The \GFF{} Morse-positivity ledger on the unified packet and the \DGI{} application to the unified gauge sector are owned end-to-end by \ASix{} \T{3.1} (the Geometric Free-Energy Functional) and \T{4.1} (the Gauge-Runtime Invariance Pipeline) \citep{ArisakaA6Gi4}. The packet-level statements that \ANine{} relies on are: (i) the energetic block $F_{\GEP}$ separates from the curvature--entropic block $F_{\CEP}$; (ii) the boundary attractor coordinates of the electroweak seam are the stationary points of $F_{\GFF} = F_{\GEP} + F_{\CEP}$; (iii) the dynamical gauge invariance \DGI{} restricts the admissible runtime variations to those that respect the bulk gauge symmetry before \GSSB{}. The next subsection picks up at the \GSSB{} step (Wilson--Hosotani breaking).

\subsection{Wilson--Hosotani breaking as the \GSSB{} step}

The next step is the \GSSB{} event. In the present context the physically decisive
statement is that the vacuum selects a Wilson-line holonomy class. The Higgs
field is therefore not a primitive scalar inserted by hand, but the fluctuation
of a holonomy variable around the selected vacuum.

Its order parameter $W=\exp(\ii\,\Theta_H T_H)$, the finite holonomy effective potential $V_{\rm eff}(\Theta_H)$, and the explicit branch-selection conditions $\partial_{\Theta_H}V_{\rm eff}=0,\ \partial^2_{\Theta_H}V_{\rm eff}>0$ --- together with the resulting electroweak vacuum value and the doublet fraction $2/5$ --- are written out in \S\ref{sec:wilson-hosotani}. In the \DGI{}$\to$\GSSB{}$\to$\GA{} language this holonomy selection \emph{is} the \GSSB{} event: the low-energy world is fixed by a geometric vacuum criterion rather than by an arbitrary scalar potential.

\subsection{The Standard Model as the GA stage}

After the branch has been selected, the resulting low-energy seam packet is the
geometric attractor,
\begin{equation}
\label{eq:sm-ga-packet}
\widehat\SIP_{\rm SM}=\GA\bigl(\widehat\SIP_{\rm vis}^{\rm pre}\bigr).
\end{equation}
The notation in Eq.~\eqref{eq:sm-ga-packet} is schematic, but its meaning is
clear. The Standard Model packet is the stabilized output of the unified packet
under \GDOS{} evolution, symmetry breaking, and boundary reduction.

This viewpoint is particularly useful for interpreting later numerical tables.
A predicted gauge coupling or fermion mass is not a raw pre-breaking datum. It
is a stabilized seam observable read off after the system has already descended
onto the GA branch.

\subsection{How \QiQUA{} enters the Standard-Model packet}

At the level of the present paper, \QiQUA{} enters most directly through current
locking. The relevant symbolic relation is the Arisaka Equation,
\begin{equation}
\label{eq:arisaka-equation-applygdos}
J_E/Q_B=J_e=J_B=J_{Qi}.
\end{equation}
Its significance for Standard-Model phenomenology is that the seam bookkeeping of
energetic, charge, baryonic, and informational channels is not independent. The
observational packet is therefore more constrained than a generic effective field
theory packet would be.

In practice this matters because it supports the idea that the visible gauge,
Higgs, and flavor observables can be organized in one common arithmetic ledger
rather than in a patchwork of unrelated fit formulas.

\subsection{How \QiEP{} enters the Standard-Model packet}

\QiEP{} is the pillar that unifies the bulk principle of least action with the
boundary selector principles. For the present paper its role is twofold.

First, it explains why the physically realized vacuum is the one selected by the
appropriate projected extremal condition rather than by arbitrary choice.

Second, it explains why RG trajectories, seam matching, and family overlap
patterns should be viewed as selected paths in a projected landscape rather than
as disconnected numerical coincidences.

In other words, \QiEP{} is what allows one to speak consistently of one selected
trajectory running from the unified bulk to the low-energy Standard-Model
packet.

\subsection{How \QiCON{} enters the Standard-Model packet}

\QiCON{} is the multiscale and conformal diagnostic pillar. In a particle-physics
context one should not overstate its role, but one should not neglect it either.
The Standard Model is full of scale-structured phenomena: logarithmic running,
threshold behavior, hierarchy ladders, and approximate scale covariance in
certain regimes. \QiCON{} provides the runtime language in which those multiscale
regularities are expected rather than mysterious.

For the current paper, the most important practical point is that scale behavior
is not an accidental side story. It is built into the runtime from the outset,
which is exactly why the radial coordinate \(r\) can be interpreted as the RG
direction later in the manuscript.

\subsection{How \QiHAL{} enters the Standard-Model packet}

\QiHAL{} is the most architecture-oriented of the four pillars, and its direct role
in a particle-physics parameter paper is therefore subtler than that of \QiQUA{}
or \QiEP{}. Nevertheless, it still has an important interpretive job. Every
phenomenological observable is reconstructed by a finite measurement
architecture: beamline, target, detector, trigger, reconstruction algorithm,
selection cut, and statistical pipeline. That architecture is precisely the kind
of memory-bearing observational structure that \QiHAL{} was designed to treat in
general open-system language.

This is not a rhetorical flourish. It is the reason the present paper is allowed
to speak meaningfully about ``observed'' seam parameters without pretending that
those parameters exist as context-free microscopic labels. They are the stable
exports of a lawful observational architecture.

\subsection{The full \GDOS{} statement of unification in one equation}

It is useful to compress the preceding discussion into one symbolic chain:
\begin{equation}
\label{eq:full-gdos-unification-chain}
\begin{aligned}
\widehat\SIP_{\rm vis}^{\rm pre}
& \xrightarrow{\DGI}
\text{admissible gauge/holonomy orbit}
\xrightarrow{\GSSB}
\text{selected Standard-Model branch} \\
& \xrightarrow{\GA}
\widehat\SIP_{\rm SM}
\xrightarrow{\text{\OGGEP{}/\GDOS{}}}
\Obs_{\rm SM}.
\end{aligned}
\end{equation}
Equation~\eqref{eq:full-gdos-unification-chain} is perhaps the best single-line
summary of what this section contributes to the phenomenological paper. It shows
that the Standard Model is not simply a group-theoretic remnant or simply a
four-dimensional effective Lagrangian. It is a stabilized, projected,
open-system packet.

\subsection{Why this section matters for the later parameter derivation}

One might ask whether the later numerical sections really need all of this
runtime structure. The answer is yes, for three reasons.

First, without the unified-packet viewpoint, the gauge/Higgs derivation would
look like isolated arithmetic rather than seam selection.

Second, without \DGI{}\(\to\)\GSSB{}\(\to\)GA, the emergence of the Standard-Model
branch from the unified group would look postulated rather than derived.

Third, without \SIP{} and \OGGEP{}, the later comparison tables would look like a fit
to raw data rather than a prediction of observer-accessible boundary quantities.

A professional phenomenological paper should make those logical dependencies
visible. The present section does so explicitly.


\section{Derivation of the fundamental architecture of the Standard Model from \GG{}}
\label{sec:sm-foundation}

The previous sections explained how the Standard Model emerges as a seam packet
and why \(\SU(5)\) is the correct historical anchor for the visible gauge sector.
We now address a more concrete question: which structural features of the
Standard Model can already be understood before one begins the numerical
parameter derivation? The answer is: quite a lot. In particular one can already
understand why the relevant low-energy matter content consists of spin-\(1/2\)
fermions, why quarks and leptons have the charge assignments they do, why three
generations are the first nontrivial family structure, and why the visible gauge
sector splits into the particular strong/electroweak/electromagnetic pattern it
does.

\subsection{Visible matter, family content, and flavor counting --- pointer to Part~\ref{part:I}~\S\ref{sec:sm-review}}

The textbook account of why visible matter is spin-$1/2$, why one family contains exactly the doublet/singlet pattern $(Q_L, u_R^c, d_R^c, L_L, e_R^c)$, and why three families give six quark flavors and six lepton flavors, is recorded in Part~\ref{part:I}~\S\ref{sec:sm-review} (Tables~\ref{tab:sm-fermions}--\ref{tab:sm-bosons}). The \GG{} derivation of these facts from the bulk Dirac action and the boundary projection sits in Part~\ref{part:I}~\S\ref{sec:gg-derives-su5}. The present chapter therefore needs no separate treatment of these basics; the next subsection picks up at the GG-specific charge and baryon-number assignment.

\subsection{Electric charges and baryon numbers from group embedding}

The charge assignments of the Standard Model look mysterious only if they are
listed without their embedding. Once the hypercharge generator is fixed inside
\(\SU(5)\), electric charge follows from
\(Q=T_3+Y\). For one family the result is the familiar set
\begin{equation}
\label{eq:charge-assignments-recap}
Q(u)=\frac23,
\qquad
Q(d)=-\frac13,
\qquad
Q(e)=-1,
\qquad
Q(\nu)=0.
\end{equation}
Likewise, local baryon number in the \GG{} framework assigns
\begin{equation}
\label{eq:baryon-assignments-recap}
B(u)=B(d)=\frac13,
\qquad
B(e)=B(\nu)=0.
\end{equation}

These equations are simple enough to look trivial, but in the present paper they
are conceptually important. They show that charge and baryon bookkeeping are
not independent empirical tables pasted onto the theory. They descend from the
unified algebra and the extra \(\U(1)_B\) structure.

Table~\ref{tab:family-charges-recap} summarizes the one-family visible content.

\begin{table}[H]
\centering
\caption[One-family visible content, with electric charge and baryon number in the \GG{} seam interpretation.]{One-family visible content, with electric charge and baryon number in
the \GG{} seam interpretation. The four rows are one family. Electric charge and baryon number are read off the seam assignment rather than postulated, and the last column names the sector each row belongs to. The table records the content the derivation must reproduce; it does not yet derive any of it. The same numbers appear as outputs in the $\mathrm{SU}(5)$ embedding of the appendix, which is where a reader who wants them as consequences should go.}
\label{tab:family-charges-recap}
\small
\begin{tabular}{p{0.17\textwidth}p{0.23\textwidth}p{0.13\textwidth}p{0.13\textwidth}p{0.22\textwidth}}
\toprule
\textbf{Field type} & \textbf{Representative seam field} & \textbf{\(Q\)} & \textbf{\(B\)} & \textbf{Role in the unified family packet}\\
\midrule
Up-type quark & \(u,c,t\) & \(+2/3\) & \(1/3\) & Color-charged, weakly active, baryonic matter sector\\
Down-type quark & \(d,s,b\) & \(-1/3\) & \(1/3\) & Color-charged, weakly active, baryonic matter sector\\
Charged lepton & \(e,\mu,\tau\) & \(-1\) & 0 & Color-singlet, weakly active, leptonic matter sector\\
Neutrino & \(\nu_1,\nu_2,\nu_3\) at mass level & 0 & 0 & Color-singlet, neutral leptonic sector with Majorana channel\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Why three generations is the first nontrivial family number}

The existence of three generations is one of the deepest facts about the Standard Model, and the present paper treats it as a \emph{theorem} rather than an input. The short constitutional reason is that $N=3$ is the smallest family number that simultaneously permits hierarchical masses, nontrivial quark mixing, and a genuine Kobayashi--Maskawa CP phase --- one generation has no flavor misalignment, and two allow mixing but no physical CP phase. The full three-leg theorem (\MSMF{} minimality, the \GDOS{}/Hirzebruch index reduced to \PaperPTwo{}, and codebook self-consistency) is proved in \S\ref{sec:N3-theorem}, and the constructive flavor arithmetic that exhibits the three family directions explicitly follows in the fermion-sector Part.

\subsection{Why the visible gauge sector splits the way it does}

We can now summarize the logic of the gauge splitting in one clean chain:
\begin{equation}
\label{eq:gauge-splitting-chain-recap}
\begin{aligned}
\SU(5)\times \U(1)_B\times \U(1)_{Qi}
& \xrightarrow[\text{boundary conditions + holonomy}]{}
\SU(3)_C\times \SU(2)_L\times \U(1)_Y \\
& \xrightarrow[\text{Wilson--Hosotani EW breaking}]{}
\SU(3)_C\times \U(1)_{\rm EM}.
\end{aligned}
\end{equation}
The weak interaction is what remains when the electroweak sector is not probed in
its deep-broken asymptotic limit. The photon survives because it lies in the
commutant of the selected holonomy. The heavy \(W\) and \(Z\) acquire masses
because they do not commute with that holonomy. The gluons remain massless at
the gauge level because color is an unbroken non-Abelian factor, though color
confinement then changes the observable long-distance physics drastically.

This is precisely the kind of layered explanation that a more professional
phenomenological introduction should provide. It reminds the reader that the
three-force structure is not elementary. It is the observed remnant of a
sequentially reduced unified connection.

\section{Unified geometric origin of the long-range inverse-square forces: gravity and electricity}
\label{sec:newton-coulomb}

It is useful to include a simple example of the unified geometric origin of gravity and electric force. This is an excellent request,
because it makes the general framework concrete. The purpose of the present
section is not to derive all of general relativity or all of electrodynamics from
scratch. The purpose is to show why the two classic inverse-square force laws,
Newton's and Coulomb's, are structurally similar in a \GG{} setting and how that
similarity is naturally understood.

\subsection{Why the comparison between Newton and Coulomb is so instructive}

Historically, the two laws are among the most iconic equations in physics:
\begin{equation}
\label{eq:newton-law-recap}
F_N=G_N\frac{m_1m_2}{r^2},
\qquad
V_N(r)=-G_N\frac{m_1m_2}{r},
\end{equation}
\begin{equation}
\label{eq:coulomb-law-recap}
F_C=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2},
\qquad
V_C(r)=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r}.
\end{equation}
At an elementary level they look almost identical. The main differences are the
sign structure, the source charges, and the overall constants. This formal
similarity has long suggested that both are manifestations of a deeper geometric
pattern.

In the present framework that suggestion becomes precise: both forces are
four-dimensional boundary exports of long-range massless sectors of the composite
connection. Gravity comes from the metric/spin-connection sector. Electricity
comes from the unbroken \(\U(1)_{\rm EM}\) sector descended from
\(\SU(5)\). The reason the static forces look alike is that both are sourced by
Gauss-law flux conservation in three spatial dimensions.

\subsection{From \GG{} to 4D general relativity and then to Newton's law}

\AFive{} shows that the metric zero mode of the six-dimensional geometry
produces a four-dimensional Einstein--Hilbert action for the boundary metric
\citep{ArisakaA5GDOS}. Schematically,
\begin{equation}
\label{eq:eh-action-recap}
S_{\rm grav}^{(4)}
=
\frac{M_{\rm Pl,4}^2}{2}
\int \dd^4x\,\sqrt{-g}\,R
+S_{\rm matter}.
\end{equation}
Varying with respect to the metric yields Einstein's equation
\begin{equation}
\label{eq:einstein-eq-recap}
G_{\mu\nu}=8\pi G_N T_{\mu\nu}.
\end{equation}
In the weak-field, static, slow-motion limit one writes
\begin{equation}
\label{eq:weak-field-metric-recap}
g_{00}\simeq -\left(1+2\Phi_N\right),
\end{equation}
and Einstein's equation reduces to the Poisson equation
\begin{equation}
\label{eq:poisson-newton-recap}
\nabla^2\Phi_N=4\pi G_N\rho.
\end{equation}
For a point source, the Green's function of the Laplacian in three dimensions is
\(-1/4\pi r\), so
\begin{equation}
\label{eq:phi-newton-recap}
\Phi_N(r)=-G_N\frac{M}{r},
\end{equation}
which immediately reproduces Newton's potential and inverse-square force.

This derivation is standard in ordinary GR. The \GG{} contribution is to explain
why the Einstein--Hilbert sector is itself a boundary export of a larger
six-dimensional geometry rather than an autonomous four-dimensional starting
point.

\subsection{From \GG{} to \texorpdfstring{\(\SU(5)\)}{SU(5)} to QED and then to Coulomb's law}

The electromagnetic story is parallel. After the unified visible gauge sector is
reduced and electroweak symmetry is broken, the unbroken Abelian factor is
\(\U(1)_{\rm EM}\). The four-dimensional electromagnetic action is
\begin{equation}
\label{eq:em-action-recap}
S_{\rm EM}^{(4)}
=
-\frac{1}{4e^2}\int \dd^4x\,F_{\mu\nu}F^{\mu\nu}
+\int \dd^4x\,J^\mu A_\mu,
\end{equation}
or, in the more standard convention with a canonically normalized field,
\begin{equation}
\label{eq:maxwell-action-canonical-recap}
S_{\rm EM}^{(4)}
=
-\frac14\int \dd^4x\,F_{\mu\nu}F^{\mu\nu}
+e\int \dd^4x\,J^\mu A_\mu.
\end{equation}
Varying with respect to \(A_\mu\) gives Maxwell's equations. In the static
limit, the electric potential \(\phi=A_0\) satisfies a Poisson equation of the
same mathematical type as the gravitational one,
\begin{equation}
\label{eq:poisson-coulomb-recap}
\nabla^2\phi=-\rho_e/\varepsilon_0,
\end{equation}
which for a point source yields
\begin{equation}
\label{eq:phi-coulomb-recap}
\phi(r)=\frac{q}{4\pi\varepsilon_0 r},
\end{equation}
and therefore Coulomb's law.

Again, the ordinary field-theory derivation is standard. The \GG{} addition is to
show why the electromagnetic field should be interpreted as the surviving massless
component of a unified connection after symmetry breaking. In other words,
Coulomb's law is not just a feature of a fundamental low-energy \(\U(1)\);
it is the infrared face of a broken unified gauge geometry.

\subsection{Why the two laws are alike}

The reason Newton's and Coulomb's laws are alike can now be stated cleanly.
There are three ingredients.

\paragraph{Massless mediator.}
Both forces are transmitted by a massless field in the relevant low-energy
regime: the graviton in the linearized gravity picture and the photon in the
unbroken electromagnetic sector.

\paragraph{Gauss-law flux conservation.}
Both massless fields obey field equations whose static limit is a Poisson
problem. In three spatial dimensions the Green's function of the Laplacian is
\(1/r\), which yields an inverse-square force after differentiation.

\paragraph{Boundary reduction of one composite connection.}
In \GG{} both sectors are understood as boundary exports of the same deeper
geometry--gauge constitution. That does not mean gravity and electromagnetism
are identical. It means their formal similarity is no accident.

This reasoning is summarized in Table~\ref{tab:newton-coulomb-comparison}.

\begin{table}[H]
\centering
\caption[Why Newton's and Coulomb's laws have the same inverse-square form in the \GG{} interpretation.]{Why Newton's and Coulomb's laws have the same inverse-square form in the
\GG{} interpretation. Read the columns down and the parallel holds at every line: source, mediator, static field equation, potential, force law. The two differ only in sign structure, because one source is a density that cannot be negative and the other is a charge that can. In the \GG{} reading both inverse squares have the same origin in a boundary zero mode of a bulk field, which is why the shared form is a consequence to be checked rather than a coincidence to be explained.}
\label{tab:newton-coulomb-comparison}
\small
\begin{tabular}{p{0.24\textwidth}p{0.27\textwidth}p{0.27\textwidth}}
\toprule
\textbf{Feature} & \textbf{Gravity} & \textbf{Electromagnetism}\\
\midrule
4D field source & Stress--energy / mass density & Electric charge density\\
Long-range mediator & Metric/graviton zero mode & Photon zero mode\\
Static field equation & \(\nabla^2\Phi_N=4\pi G_N\rho\) & \(\nabla^2\phi=-\rho_e/\varepsilon_0\)\\
Potential & \(-G_N M/r\) & \(q/4\pi\varepsilon_0 r\)\\
Force law & Attractive inverse square & Signed inverse square\\
\GG{} origin & Boundary export of geometry sector & Boundary export of unbroken \(\U(1)_{\rm EM}\) sector\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Why the two laws are also different}

The similarity should not tempt us into erasing the differences.

Gravity couples universally to stress--energy, not to a sign-changing charge.
Its long-range interaction is therefore generically attractive for ordinary
positive-energy matter.

Electromagnetism couples to electric charge, which comes with both signs. It can
therefore attract or repel.

The mediators also carry different spin and transformation properties: gravity is
a spin-2 geometric field, while electromagnetism is a spin-1 gauge field. \GG{}
unifies them constitutionally as sectors of one composite connection, but it does
not collapse them into one identical force at low energies.

This distinction is phenomenologically important. A good unified theory explains
formal similarities without denying real differences.

\subsection{Connection to the unified geometric origin of mass}

The mass-origin paper gives a particularly useful way to see
why the present discussion belongs naturally in \ANine{}
\citep{ArisakaA6Gi4}. There mass is treated
as a geometric spectral gap, and gravitational response is treated as the metric
variation of the same seam action. In that language, gravity is the response of
the geometry sector to the energy ledger, while electromagnetism is the response
of the gauge sector to the charge ledger. Both are therefore different
projections of one deeper constitution.

This is a very satisfactory place to pause before the paper returns to detailed
parameter derivation. The point of a professional phenomenological introduction
is not only to list targets; it is to teach the reader how to think in the
framework. The Newton/Coulomb example does exactly that. It shows how the
program turns a historical similarity into a principled geometric statement.

\subsection{Bridge to the quantitative sections}

The foundation sections above have accomplished the following.

They have explained why the Standard Model is the correct phenomenological target
for the present paper.

They have listed the ten open problems and the 28-parameter challenge that make a
paper like \ANine{} worth writing.

They have reviewed the constitutional and geometric framework from \AOne{},
\ATwo{}, \AFive{}, \ASeven{}, and \AThree{}.

They have explained how \GDOS{}, \GFF{}, \SIP{}, \DGI{}, \GSSB{}, GA, and the \QiFour{} pillars are
to be read in a particle-phenomenology context.

They have shown how the historical Georgi--Glashow idea is retained and lifted in
\GG{}.

They have derived the qualitative Standard-Model architecture before any
numerical tables are displayed.

They have illustrated the geometric unification logic with the Newton/Coulomb
comparison.

With that groundwork in place, the paper turns to the detailed roadmap and the
quantitative sections that carry the derivations; the technical appendices at
the end supply their back-up, sector by sector.


\section{Phenomenological dictionary: how ordinary Standard-Model language is reinterpreted in \GG{}/\GDOS{}}
\label{sec:phen-dictionary}

The introduction and foundation sections naturally raise a practical question. If the present paper is written in the language of \GGTheory{},
\GDOS{}, \GFF{}, \SIP{}, \DGI{}, \GSSB{}, GA, and \QiFour{}, how should a phenomenologist trained in
ordinary Standard-Model language translate between the two descriptions? This
section answers that question explicitly. It bridges the foundation material and the quantitative sections that follow.

\subsection{Why a dictionary section is necessary}

Professional phenomenology depends on precise vocabulary. When one author says
``coupling,'' another says ``renormalized effective charge,'' and a third says
``response coefficient extracted in a given scheme,'' they may be referring to
closely related but not identical objects. The same is true for ``mass,''
``symmetry breaking,'' ``vacuum,'' ``mixing angle,'' and ``observable.''

The \GG{}/\GDOS{} program does not deny the ordinary Standard-Model meanings of
these terms. It refines them. The refinement matters because the present paper
is not a conventional fit model. It is a derivation that claims to obtain
Standard-Model observables from a more constrained, geometrically motivated, and
operationally explicit framework. A clean dictionary therefore prevents two kinds
of misunderstanding at once:
\begin{enumerate}[label=(\roman*),leftmargin=2.5em]
\item it prevents readers from mistaking the \GG{} language for a rejection of the
ordinary Standard Model, and
\item it prevents readers from mistaking ordinary phenomenological labels for the
full theoretical meaning assigned to them in the present framework.
\end{enumerate}

\subsection{Gauge couplings: from bare constants to seam conductances}

In ordinary Standard-Model language, one usually introduces gauge couplings
\(g_1\), \(g_2\), and \(g_3\), or equivalently \(\alpha_1\), \(\alpha_2\), and
\(\alpha_3\), and then studies their scale dependence by RG equations. In the
present framework, the same observables are reinterpreted as seam conductances or
response coefficients exported from a bulk.

The difference is subtle but important. In the ordinary language, one begins
with a 4D field theory and then studies how its renormalized couplings run. In
\GG{}, one begins with a six-dimensional stage and interprets the radial
coordinate
\begin{equation}
\label{eq:r-dictionary}
r=\ln\!\left(\frac{\EGUT}{\mu}\right)
\end{equation}
as the renormalization direction. The low-energy coupling is then not simply a
number attached to a local vertex; it is the seam value of a transported quantity.

This reinterpretation does not change the fact that the final comparison is made
to renormalized couplings measured at a reference scale. It changes only the
ontological reading of those couplings. They are no longer primitive inputs.
They are outputs of transport in the fifth dimension.

\subsection{The Higgs sector: from inserted scalar potential to geometric holonomy}

In ordinary Standard-Model language, the Higgs sector is written in terms of a
scalar doublet \(H\) and a potential
\begin{equation}
\label{eq:ordinary-higgs-dictionary}
V(H)=-\mu_H^2|H|^2+\lambda_H |H|^4.
\end{equation}
The parameters \(\mu_H^2\) and \(\lambda_H\) are then adjusted so that the vacuum
expectation value \(v\) and physical Higgs mass \(m_h\) match experiment.

In \GG{} this language is not thrown away, but it is demoted to a low-energy
shadow description. The fundamental order parameter is the Wilson-line
holonomy, and the Higgs boson is the fluctuation of that holonomy around the
selected vacuum branch. The effective scalar potential is therefore not a
primitive input but the seam image of a finite geometric effective functional.

This is why the hierarchy problem is rephrased so strongly in the present paper.
If the true ultraviolet variable is a holonomy angle rather than a local bare
scalar mass, then the ordinary naturalness problem is no longer the correct
question in its usual form. What matters is the selected branch of the finite
holonomy potential and its redshift to the lock seam.

\subsection{Masses: from arbitrary Yukawa numbers to overlap spectra}

In ordinary Standard-Model language, fermion masses are generated from Yukawa
matrices. The theory tells us how those matrices enter observables, but it does
not tell us why the entries take the values they do. That is why the Yukawa
sector is often treated as the most arbitrary part of the Standard Model.

In the present framework the same low-energy Yukawa language is retained as an
effective boundary description, but the underlying origin is changed. Effective
Yukawa couplings are interpreted as overlap integrals of wavefunctions or
projectors in the bulk and on the seam. This shift in viewpoint is the
reason the later sections can derive masses from dyadic overlaps, compact-angle
winding, and family projectors without introducing twelve independent Yukawa-fit
constants.

The key phenomenological point is that a mass table in \ANine{} is not the
output of twelve unrelated empirical insertions. It is the boundary spectrum of a
single geometric family codebook.

\subsection{Mixing angles: from free unitary matrices to projector misalignment}

The ordinary Standard Model and its neutrino-sector extensions parameterize quark
and lepton mixing by unitary matrices. This is the correct low-energy formalism,
but it leaves a natural question unanswered: why do these matrices take the forms
they do?

\GG{}/\GDOS{} does not replace the CKM and PMNS matrices with something unrelated.
It provides an origin story for them. In the present manuscript they are treated
as misalignment data between already-fixed family projectors. The quark sector is
built from the mismatch between up-type and down-type projector structures. The
lepton sector is built from the mismatch between charged-lepton and neutrino
projector structures.

This reinterpretation is especially important for professionalism of presentation.
It allows one to say clearly what the paper is and is not claiming. The paper is
not claiming that unitary mixing matrices are the wrong language of observables.
It is claiming that the values of those observable matrices can be traced back to
more primitive geometric ingredients.

\subsection{Vacuum selection: from postulated broken phase to \GSSB{} event}

In ordinary phenomenological discussions, one often begins directly in the broken
phase because that is the phase in which measurements are performed. In the
present paper the broken phase is not the starting point. It is the selected
output of the \GSSB{} step.

This matters because it clarifies why the Standard Model appears as it does.
The gauge group, Higgs expectation value, and flavor pattern are not three
independent ``facts.'' They are correlated consequences of one selected vacuum
branch. The reader can therefore interpret every later numerical table as a
report on the same branch, not as a collection of sector-by-sector coincidences.

\subsection{Observables: from numbers in a review table to registered packet exports}

A benchmark value in the PDG or in a neutrino global fit is often treated in the
literature as if it were the theory-independent ``thing itself.'' That practice
is completely reasonable for most day-to-day phenomenology. But the present
paper aims at a deeper level of explanation. It therefore insists that every
benchmark number compared at the end of the derivation is a registered export of a
specific packet under a specific measurement protocol.

This is not meant to relativize or weaken the data. On the contrary, it sharpens
the meaning of comparison. A prediction is not complete until the paper has
specified which observable basis it is matching to. That is why \ANine{} uses
observable-basis tables and defines Dev and Pull only after the theory-side
construction is complete.

\subsection{A compact translation table}

Table~\ref{tab:ordinary-to-gg6-dictionary} collects the key entries of the
translation dictionary in compact form.

\begin{center}
\begin{longtable}{p{0.25\textwidth}p{0.28\textwidth}p{0.31\textwidth}}
\caption[Compact dictionary between ordinary Standard-Model phenomenological language and the GG-6/GDOS interpretation used in the present manuscript.]{Compact dictionary between ordinary Standard-Model phenomenological
language and the GG-6/GDOS interpretation used in the present manuscript. The third column is the point of the table. A dictionary that only paired terms would be a change of vocabulary; what matters is why each reinterpretation earns its place --- a gauge coupling becomes a transport problem, the Higgs field becomes a holonomy fluctuation --- and in each case something that was an input acquires an address in the geometry. The entries are pointers into the Parts that do the work, not arguments in themselves.}
\label{tab:ordinary-to-gg6-dictionary}\\
\toprule
\textbf{Ordinary SM language} & \textbf{GG-6/GDOS reinterpretation} & \textbf{Why the reinterpretation matters}\\
\midrule
\endfirsthead
\toprule
\textbf{Ordinary SM language} & \textbf{GG-6/GDOS reinterpretation} & \textbf{Why the reinterpretation matters}\\
\midrule
\endhead
Gauge coupling & Seam conductance / transported radial response coefficient & Makes RG running a geometric transport problem rather than an external input.\\
Higgs field & Wilson-line holonomy fluctuation & Replaces a primitive scalar ontology by a geometric order parameter.\\
VEV \(v\) & Lock-seam scale export & Solves hierarchy kinematically before low-energy fitting.\\
Fermion Yukawa entry & Overlap or projector coefficient & Compresses flavor freedom into a geometric codebook.\\
CKM/PMNS matrix & Family-projector misalignment matrix & Turns mixing into derived geometry rather than independent matrix fitting.\\
Vacuum in broken phase & GSSB-selected branch of the packet & Correlates gauge, Higgs, and flavor data as properties of one branch.\\
Mass & Seam spectral gap in GFF language & Unifies inertial and observable mass bookkeeping.\\
Experimental observable & Registered packet export under OGGEP/GDOS & Prevents hidden confusion between bare and renormalized quantities.\\
\bottomrule
\end{longtable}
\end{center}

\subsection{Why the dictionary matters for phenomenological precision}

A polished phenomenological paper should teach the reader how to move between
levels of description. If it stays entirely in the UV language, it becomes too
abstract to evaluate. If it stays entirely in low-energy Standard-Model language,
it becomes impossible to see what the deeper theory is contributing. The present
dictionary section is therefore not ornament. It is one of the places where the
paper becomes genuinely readable across subfields.

It also clarifies the status of the quantitative sections that follow. Those
sections do not ask the reader to abandon familiar Standard-Model phenomenology.
They ask the reader to reinterpret familiar Standard-Model observables as outputs
of a deeper, more restricted geometric and operational framework.

\subsection{Immediate bridge to the roadmap section}

After this dictionary is in place, the rest of the manuscript can proceed much
more smoothly. The roadmap section that follows is then read not as a sudden jump
from abstract principles to arithmetic, but as a natural continuation of a fully
explained phenomenological and constitutional setup.




\begin{center}\rule{0.5\textwidth}{0.4pt}\end{center}


\section{\GG{} bulk geometry and the three fundamental scales}
\label{sec:gg6-scales}

\subsection{Minimal admissible stage}

The bulk stage derived in \ATwo{} is
\begin{equation}
M_6=M_4\times I_r\times S^1_\theta,
\label{eq:m6-stage}
\end{equation}
with scale-profile metric
\begin{equation}
\dd s_6^2 = e^{-2r}\,\eta_{\mu\nu}\,\dd x^\mu \dd x^\nu
+ \dd r^2 + \dd\theta^2,
\qquad r \in [0, K],\quad \theta \in [0, 2\pi),
\label{eq:metric-6d}
\end{equation}
and the logarithmic radial coordinate
\begin{equation}
r=\ln\!\left(\frac{\EGUT}{\mu}\right).
\label{eq:r-log}
\end{equation}
The derivation of this geometry in \ATwo{} is constitutional.
Four coordinates are fixed by the operational boundary itself. One
extra coordinate is needed to geometrize renormalization scale, and one
compact coordinate is needed to support primitive Abelian holonomy.
Anything smaller than 6D is insufficient; anything larger is not forced
and would violate \MSMF{} unless independently motivated
(\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}).

\subsection{Composite connection and six-force unification}

On this stage the composite connection is
\begin{equation}
\mathbb A = \omega\oplus A_5\oplus A_B\oplus A_{Qi},
\label{eq:composite-conn-main}
\end{equation}
with bulk symmetry
\begin{equation}
G_{\Bulk} \;=\; \Spinone \,\times\, \SUfive \,\times\, \UoneB \,\times\, \UoneQi.
\label{eq:bulk-group-main}
\end{equation}
This is the ultraviolet starting point. The present paper does not yet
need the full 6D QFT machinery of \ATwo{}; it needs only the facts
that (i) the unified group contains a minimal \(\SU(5)\) block, (ii)
there are two equally weighted Abelian sectors entering \BIsix{}, and (iii)
all of these are carried by the same composite connection.

\subsection{\OGNfive{} ledger and throat depth}

The \OGNfive{} arithmetic package is
\begin{equation}
(M,N,L;J,K)=(1,3,5;7,35),
\label{eq:ogn5-main}
\end{equation}
with the derived relations
\begin{equation}
J=2N+1=7,
\qquad
K=LJ=35,
\qquad
K=M^2+N^2+L^2=35.
\label{eq:ogn5-derived-main}
\end{equation}
In \ATwo{} this ledger controls the finite throat through
\begin{equation}
0\le r\le r_\ast,
\qquad
k r_\ast = K = 35,
\qquad
\exp[-A(r_\ast)] = \e^{-35}.
\label{eq:throat-main}
\end{equation}
Thus the hierarchy across the throat is not a continuous modulus but a
discrete exponential set by the master lock integer \(K\)
\citep{ArisakaA2GG6}.

The physical interpretation used throughout this paper is summarized in
Table~\ref{tab:ogn-interp-main}.

\begin{table}[H]
\centering
\caption[Physical interpretation of the OGN-5 integers as used in the present gauge/Higgs, fermion-mass, and CKM derivation.]{Physical interpretation of the OGN-5 integers as used in the
present gauge/Higgs, fermion-mass, and CKM derivation. The five integers $(M,N,L;J,K) = (1,3,5;7,35)$ are the entire integer content of the derivation, and this table says what each one counts. They are fixed in the foundation companions and are not adjustable here: every closed form in the derivation Parts is built from them together with $\pi$ and $\e$. A reader auditing a formula should be able to point at each integer in it and name the row it came from.}
\label{tab:ogn-interp-main}
\small
\begin{tabular}{p{0.10\textwidth}p{0.17\textwidth}p{0.56\textwidth}}
\toprule
\textbf{Symbol} & \textbf{Value} & \textbf{Interpretation in the present paper}\\
\midrule
\(M\) & 1 & Primitive compact-holonomy count: one minimal \(S^1_\theta\) cycle.\\
\(N\) & 3 & Triplicity index of the observable boundary sector.\\
\(L\) & 5 & Minimal non-Abelian unification index, namely \(\SU(5)\).\\
\(J\) & 7 & Minimal odd closure index, governing dyadic electroweak closure.\\
\(K\) & 35 & Master lock integer controlling hierarchy, throat depth, and next-order suppression.\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Derivation of \texorpdfstring{$\EPl$, $\EGUT$, $\Mlock$}{E\_Pl, E\_GUT, M\_lock} --- pointer to A2-\GG{} \S 5 + \T{6}}

The end-to-end derivations of the three fundamental scales --- the Planck energy $\EPl = \sqrt{\hbar c^5/G_N} \approx 1.22 \times 10^{19}$~GeV from the \GUfive{} of \AOne{} \T{-GU5}; the GUT scale $\EGUT = \EPl\,e^{-2\pi} \approx 2.28 \times 10^{16}$~GeV from one $S^1_\theta$ traversal of the 6D bulk; and the lock scale $\Mlock = \tfrac12 M_{\rm Pl}\,e^{-K} \approx 3.85$~TeV from the \OGNfive{} closure $K = LJ = 35$ --- are owned in full by A2-\GG{} \S 5 + \T{6} \citep{ArisakaA2GG6}. The boxed values are reproduced in the scale chain \eqref{eq:scale-chain-prologue} above. The next subsection picks up the A9-specific reading: that this scale chain already resolves the hierarchy problem at the kinematic level, before any electroweak fine-tuning question can be posed.

\subsection{Why this already solves the hierarchy problem at the kinematic level}
\label{subsec:hierarchy-kinematic}

The conventional hierarchy problem asks why the electroweak scale is so
small compared with the Planck scale.
Inside \GG{} the question is reformulated.
One no longer compares two unrelated constants; one asks what the
redshift is between two constitutionally distinguished seams of one
bulk geometry.

Using \eqref{eq:Mlock}, the ratio is
\begin{equation}
\frac{\Mlock}{\EPl}=\frac{1}{2\e^{35}}\simeq 3.152\times 10^{-16}.
\label{eq:hierarchy-ratio-main}
\end{equation}
Thus the large hierarchy is simply the exponential image of the master
lock integer \(K=35\), plus the minimal binary endpoint factor. There
is no continuous modulus to adjust, and therefore there is no
fine-tuning problem at this stage. The kinematic hierarchy has been
replaced by a discrete geometric one.

\section{The \GDOS{} calculational seams: where the gauge and Higgs observables are read off}
\label{sec:gdos-eegg}

Why an operational layer must precede any numerical prediction --- that a coupling, a mass, or a mixing angle is a \emph{registered boundary observable} rather than a bare Lagrangian coefficient --- was established in \S\ref{sec:gdos-qi4-recap}, and the runtime cascade that selects the Standard-Model branch was walked in \S\ref{sec:apply-gdos-su5}. Here we record only the two structural facts about \GDOS{} that the gauge/Higgs calculation actually reads off: its coarse-grained evolution law, and the stationary \emph{seams} at which observables are assigned definite values.

\subsection{Exact and coarse-grained \GDOS{} structure}

At the coarse-grained mesoscopic level, \AFive{} writes the generic
\GDOS{} evolution as the continuity equation
\begin{equation}
\partial_t p = -\nabla_A J^A,
\label{eq:gdos-cont}
\end{equation}
with current decomposition
\begin{equation}
J^A
= -D^{AB}\nabla_B p + b^A p
= -D^{AB}\nabla_B p + \big(-g^{AB}\nabla_B\Phi + h^A\big)p,
\qquad
\nabla_A h^A=0.
\label{eq:gdos-current}
\end{equation}
This is already sufficient for the purposes of the present paper. It
says that every admissible open-system evolution decomposes into three
pieces: diffusion, gradient relaxation, and a solenoidal circulation.

The corresponding entropy-production statement is
\begin{equation}
\dot S_{\mathrm{tot}}
=
\int \dd\mu\,
\frac{J_A(D^{-1})^{AB}J_B}{p}
\ge 0,
\label{eq:entropy-prod-main}
\end{equation}
so the admissible flows are automatically thermodynamically well-behaved
\citep{ArisakaA5GDOS}.

For the gauge/Higgs sector the physical meaning is direct: the observed
boundary vacuum is not an arbitrary algebraic choice. It is a
stationary or attractor configuration of a lawful open-system flow.

\subsection{The \GFF{} ledger}

The scalar bookkeeping principle behind \GDOS{} is the \GFF{} ledger
introduced in \AOne{}. In its most useful explicit form,
\begin{equation}
F_{\mathrm{\GFF{}}}[\rho]
= \int_{\mathcal M}\rho(y)\,\Phi(y)\,\dd\mu
+\Theta\int_{\mathcal M}
\rho(y)\ln\!\left(\frac{\rho(y)}{\rho_{\mathrm{ref}}(y)}\right)\dd\mu.
\label{eq:GFF-main}
\end{equation}
The two blocks --- the energy--geometry--gauge part \(\GEP\) and the curvature--entropy part \(\CEP\), defined in \S\ref{sec:gdos-qi4-recap} --- combine as
\begin{equation}
\GFF=\GEP+\CEP.
\label{eq:eegg-decomp}
\end{equation}
Its functional derivative is
\begin{equation}
\frac{\delta F_{\mathrm{\GFF{}}}}{\delta\rho}
=\Phi+\Theta\left(\ln\frac{\rho}{\rho_{\mathrm{ref}}}+1\right),
\label{eq:deltaF-main}
\end{equation}
and in the pure-gradient \GDOS{} sector one has
\begin{equation}
J^A = -\rho M^{AB}\nabla_B\left(\frac{\delta F_{\mathrm{\GFF{}}}}{\delta\rho}\right),
\qquad
D^{AB}=\Theta M^{AB}.
\label{eq:gdos-grad-main}
\end{equation}
Then
\begin{align}
\frac{\dd}{\dd t}F_{\mathrm{\GFF{}}}[\rho(t)]
&=-\int \rho\,\nabla_A\left(\frac{\delta F_{\mathrm{\GFF{}}}}{\delta\rho}\right)
M^{AB}
\nabla_B\left(\frac{\delta F_{\mathrm{\GFF{}}}}{\delta\rho}\right)\dd\mu
\le 0,
\label{eq:eegg-monotone}
\end{align}
so the ledger is a Lyapunov functional in the relaxational sector
(\citealp{ArisakaA1Constitution}; \citealp{ArisakaA5GDOS}).

\subsection{What this means for gauge and Higgs observables}

The practical message of \eqref{eq:eegg-monotone} is extremely simple:
the numbers compared to experiment should be assigned at stationary
seams of the effective boundary description.

We introduce two names for the two seams needed in the present paper.
\begin{itemize}[leftmargin=2.1em]
\item The \textbf{lock seam} \(\Lockseam\) is the boundary image of the
finite-throat endpoint at \(\mu\sim\Mlock\). It is the natural place to
organize the Higgs/Wilson-line descent.
\item The \textbf{neutral electroweak seam} \(\Zseam\) is the boundary
stationary point at which the neutral-current rotation is evaluated and
the photon remains exactly massless. In phenomenological language, this
is the seam associated with the usual \(Z\)-pole comparison scale.
\end{itemize}
Neither term is imported from outside the preprints; they are simply a
convenient language for the stationary points forced by AX-3 and \GDOS{}.

Thus the gauge/Higgs, fermion-mass, and CKM derivation in this paper will proceed in two
stages. First, geometry and OGN determine the ultraviolet and lock
structures. Second, \GDOS{}/\GFF{} tells us at which effective seam the
corresponding low-energy couplings must be read off.

\subsection{\GFF{} interpretation of vacuum selection}

In the Higgs sector, the vacuum expectation value is the stationarity
condition of an effective boundary ledger, not a mysterious insert.
Writing the order parameter schematically as \(\varphi_H\), the
boundary-effective evolution takes the generic form
\begin{equation}
\partial_t \varphi_H
= -\Gamma_H\frac{\delta F_{\mathrm{\GFF{}}}}{\delta\varphi_H}+\xi_H,
\label{eq:phiH-flow}
\end{equation}
where \(\Gamma_H>0\) is a mobility kernel and \(\xi_H\) denotes the
small open-system noise inherited from the coarse-grained projection.
A stationary vacuum therefore satisfies
\begin{equation}
\frac{\delta F_{\mathrm{\GFF{}}}}{\delta\varphi_H}=0.
\label{eq:stationary-H}
\end{equation}
When the Higgs sector is written in ordinary four-dimensional language,
this stationarity condition becomes the familiar minimization of the
effective scalar potential. But the present formulation is more
fundamental because it also explains \emph{why} a vacuum selection
principle exists at all: it is the gradient-attractor structure of the
\GDOS{} boundary runtime.


\section{The family number \texorpdfstring{$N=3$}{N=3} as a three-leg theorem}
\label{sec:N3-theorem}

\subsection{Historical background}

The persistence of three fermion generations is among the most stubborn unexplained empirical facts of the Standard Model. The LEP measurement of the invisible $Z$ width fixes the number of light, weakly interacting neutrino species at $N_\nu = 2.984 \pm 0.008$, consistent with three and inconsistent with two or four at the multi-$\sigma$ level; cosmological probes from primordial nucleosynthesis and the cosmic microwave background give an independent and equally tight bound. Yet within the \SM{} itself the family number is left as a free input: nothing in $\SU(3)\times\SU(2)\times\U(1)$ requires three replications rather than one or seventeen. Kobayashi and Maskawa observed in 1973 \citep{KobayashiMaskawa1973} that $N\geq 3$ is the smallest number of generations admitting a physical \CP{}-violating phase, but this gives only a lower bound. Calabi--Yau compactifications, magnetized toroidal models, $\mathbb Z_3$ orbifolds, and noncommutative-geometry constructions each provide partial answers, but no closed-system framework has succeeded in deriving \emph{both} an upper and a lower bound on $N$ from first principles.

In \GGTheory{} the question has an answer with every fence named. The dedicated companion preprint \citep{ArisakaP2N3} establishes that $N=3$ is uniquely forced by three logically independent legs of the framework: an \MSMF{} lower bound from \CP{}-violation combinatorics, a \GDOS{} Morse upper bound from family-projector stability on $\mathbb{C}P^{N-1}$, and a constructive codebook check that reproduces all twelve charged-fermion masses to four-significant-figure precision only at $N=3$. The present section records the statement of that theorem and the structural reading needed for the parameter derivations of the remainder of this paper.

\subsection{Why \texorpdfstring{$N=3$}{N=3} enters before the parameter calculation}

A consequence paper that sets out to derive the 28 parameters of the \SM{} cannot treat the family number as an empirical input without circularity. Indeed, $N$ enters every Yukawa formula in the codebook (see Sections~\ref{sec:lepton-mass}--\ref{sec:quark-mass}), every flavor mixing matrix (Sections~\ref{sec:CKM}--\ref{sec:PMNS}), and the \OGNfive{} Diophantine ledger that anchors the entire framework. If $N$ were fitted from the lepton spectrum, the codebook would be a numerical exercise rather than a derivation. The three-leg theorem of $N=3$ in \citep{ArisakaP2N3} removes this circularity: $N$ is a forced integer of the six-dimensional bulk geometry, its value pinned by the same constitutional clauses that fix the \BIsix{} vector and the \OGNfive{} ledger. After all, what was historically considered an inscrutable empirical accident becomes a derived integer once the underlying geometry is correctly identified.

\subsection{Statement of the three-leg theorem}

\begin{theorem}[\textbf{T-SM-21.1}: Conditional derivation of $N=3$; \citealp{ArisakaP2N3} T-N3-5]
\label{thm:N3-closed}
Let $N$ denote the family number in the \OGNfive{} ledger
\begin{equation}
(M,N,L;J,K) = (1,3,5;7,35), \qquad K = LJ = M^2 + N^2 + L^2,
\label{eq:OGN5-N3}
\end{equation}
of \GGTheory{} (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}). Under
\begin{enumerate}\itemsep0pt
\item \BIsix{} closure on the six-dimensional bulk \citep{ArisakaA3BI6},
\item \MSMF{} (Axiom~\AX{4}) of the \GGConstitution{} \citep{ArisakaA1Constitution},
\item The capacity of the family stratum (the coupling clause with the integrality theorem L-INT (residue R-INT),
per \citealp{ArisakaP2N3}; its rigidity lemma closes in the negative the derivation from \BIsix{} positivity recorded as open problem O3 of \citealp{ArisakaA5GDOS}, and its capacity theorem meets the $N$-independence demand of open problem M10 of \citealp{ArisakaA6Gi4}),
\item The minimal first-order Yukawa codebook of the present paper (Sections~\ref{sec:lepton-mass}--\ref{sec:quark-mass}),
\end{enumerate}
together with the empirical inputs
\begin{itemize}\itemsep0pt
\item Jarlskog \CP{}-violation $\mathcal J_{\rm CKM} \neq 0$ (PDG),
\item Charged-lepton mass ratios $m_\tau/m_\mu = 16.817$ and $m_\mu/m_e = 206.768$ (PDG),
\end{itemize}
the family number is uniquely
\begin{equation}
\boxed{N = 3.}
\label{eq:N3-closed}
\end{equation}
\end{theorem}

\begin{euclidremark}[Physics intuition for Theorem~\ref{thm:N3-closed}]
The number $3$ in $N = 3$ is not selected; it is forced by three constraints, along logically independent directions, that intersect on the single integer. Indeed, this is the \GGTheory{} pattern: an empirical accident reduces, under sufficient geometric structure, to a number-theoretic identity. The lower bound says ``\CP{} violation is real, so $N$ cannot be smaller than three''; the upper bound says ``the cascade of unstable family-projector saddles has a finite instanton-action budget that closes only for $N$ at most three''; the codebook check says ``the empirical mass ratios independently single out $N=3$ at four significant figures''. No leg by itself is sufficient. Together they force $N=3$ exactly and validate the value with twelve quantitative mass predictions.
\end{euclidremark}

\begin{euclidremark}[Status: what is rigorous, what is conditional, and which ``three'']
Honesty (the L6 discipline) asks two distinctions to be kept sharp. \emph{First}, the integer derived here is the \emph{generation} number, written $N_g$ in the cohort's notation; that the \OGNfive{} ledger's second component also equals $3$ is a separate and \emph{unconditional} result --- the dual lock with the block embedding, Theorem~T-BI6-10.2a of \citet{ArisakaA3BI6}, checked against the \BIsix{} trace ratio $(L-N)/L = 2/5$ --- reached from premises that mention no generation count. The two are distinct quantities: the color index $N_c$ is a ledger member and the generation count $N_g$ is not, and neither derivation shares a premise with the other. That both land on $3$ is therefore a result rather than an identification, and \citet{ArisakaP2N3} sets out in its own discussion what would have to be shown --- a map from the ledger to the family stratum, or back --- to make their meeting a theorem. Nothing in the present paper rests on it. \emph{A note on the three symbols.} The cohort writes $N_c$ for the color
index, which is the \OGNfive{} integer $N$; $N_g$ for the generation count,
which is not a ledger member; and $N_f$ for the number of quark flavors in a
beta function, six at high scale. The present paper uses a single $N$ for the
first two, which are numerically equal and so print identically: its ledger
formulas ($\sin^2\theta_W = L^2/4N^3$, $\lambda_H = 2^{-N}$, $v = \Mlock(2/L)^N$)
read $N$ as the color index, while the Yukawa codebook's $N$ (in
$y_\tau \propto NJ/2^{J+N+1}$ and its relatives) is the family count --- which is
what makes the codebook a check on the generation number rather than on the
ledger. No value changes either way; the per-formula relabeling is recorded
here as an outstanding editorial task rather than performed on a guess.

\emph{Second}, within the generation pincer the genuinely new and fully rigorous step is topological: the even-$N$ exclusion of Leg~L2 is the identity $a^+(N)-a^-(N) = \sigma(\mathbb{C}P^{N-1})$ (the Hirzebruch signature), which equals $1$ for odd $N$ and $0$ for even $N$ --- independent of any conjecture. The exclusion of the surviving odd values $N \ge 5$ rests on the capacity theorem T-N3-6 of \citet{ArisakaP2N3} ($\mathrm{Cap}(N) = \pi N$, derived from the stratum's quantum geometry, conditional on the coupling clause and the integrality theorem L-INT (residue R-INT);
the $N$-independence demand of \citealp{ArisakaA6Gi4}'s open problem
M10 is met, the chain consuming neither $K=35$ nor $J=7$), and the lower bound $N \ge 3$ rests on the measured reality of \CP{} violation. ``Closed three-leg theorem'' should therefore be read as \emph{a conditional pincer that reduces the family number to one named topological residue}
(R-INT, of the integrality theorem L-INT)\emph{, premises shared with
the Constitutional Born Theorem, and one robust empirical fact} --- a sharply localized residue, not an unconditional proof. The dedicated companion \citep{ArisakaP2N3} carries the conditioning in full.
\end{euclidremark}

\paragraph{Source.} The complete proof is in the dedicated companion preprint \citep{ArisakaP2N3}; the present section records only the statement and the structural reading needed for the parameter derivations of the remainder of this paper. Each of the three legs is summarized in turn below.

\subsection{Leg L1: \MSMF{} lower bound \texorpdfstring{$N \geq 3$}{N >= 3} from CP combinatorics}
\label{subsec:N3-L1}

The first leg is the Kobayashi--Maskawa argument elevated to a theorem under \MSMF{}. A general $N \times N$ unitary CKM matrix has $N^2$ real parameters: $N(N-1)/2$ mixing angles, $N(N+1)/2$ phases. Subtracting $2N-1$ unphysical rephasings (the $N$ up-type and $N$ down-type fermion phase redefinitions, modulo the global $\U(1)$) leaves
\begin{equation}
n_{\rm phase}(N) = \frac{(N-1)(N-2)}{2}
\label{eq:n-phase-N}
\end{equation}
physical \CP{}-violating phases. A non-zero Jarlskog invariant $\mathcal J_{\rm CKM} = \mathrm{Im}(V_{ij}V_{kl}V_{il}^*V_{kj}^*) \neq 0$ exists iff $n_{\rm phase}(N) \geq 1$, which gives $N \geq 3$. Empirically $\mathcal J_{\rm CKM} = \bigl(3.12^{+0.13}_{-0.12}\bigr)\times 10^{-5}$ (PDG 2024) at multi-$\sigma$ significance. By \MSMF{} the \OGNfive{} ledger selects the smallest admissible value, which is $N = 3$. Indeed, the same argument applies independently in the leptonic sector via PMNS, with global fits favoring a non-zero leptonic Jarlskog $\mathcal J_{\rm PMNS} \approx -0.026$ (NuFIT), so even if one half of the empirical input were retracted the lower bound would survive.

\subsection{Leg L2: \GDOS{} Morse upper bound \texorpdfstring{$N \leq 3$}{N <= 3} on \texorpdfstring{$\mathbb{C}P^{N-1}$}{CP\^{}\{N-1\}}}
\label{subsec:N3-L2}

The second leg is genuinely new and sets the present derivation apart from the historical Kobayashi--Maskawa argument. It uses the open-system \GDOS{} framework \citep{ArisakaA5GDOS} together with Morse theory on the family-projector moduli space.

A boundary family-projector configuration is a rank-1 projector $P_{[\psi]} = |\psi\rangle\!\langle\psi|/\langle\psi|\psi\rangle$ in the $N$-dimensional family Hilbert space. Its moduli space is
\begin{equation}
\mathcal P_N = \mathbb{C}P^{N-1}, \qquad \dim_{\mathbb R}\mathcal P_N = 2(N-1),
\label{eq:family-moduli}
\end{equation}
the complex projective space, equipped with its canonical Fubini--Study metric. The Geometric Free Functional (Pillar~3 of \GiFour{}, \citealp{ArisakaA6Gi4}) restricted to family-projector data is the height function
\begin{equation}
f_{\GFF}([\psi]) = \frac{\sum_{a=1}^N \lambda_a |\psi_a|^2}{\sum_a |\psi_a|^2},
\qquad \lambda_1 < \lambda_2 < \cdots < \lambda_N.
\label{eq:GFF-Morse}
\end{equation}
By the classical theory of Morse functions on $\mathbb{C}P^{N-1}$ \citep{Milnor1963}, $f_{\GFF}$ is a perfect Morse function with $N$ non-degenerate critical points $[0,\dots,0,1,0,\dots,0]$ (the $1$ in position $a$) of Morse index $\mathrm{ind}_a = 2(a-1)$. The Morse polynomial is
\begin{equation}
\mathcal P(\mathbb{C}P^{N-1})(t) = \sum_{k=0}^{N-1} t^{2k}
= 1 + t^2 + t^4 + \cdots + t^{2(N-1)}.
\label{eq:Morse-polynomial}
\end{equation}

Strict \GDOS{} Morse-positivity (held as a consistency-grade import in \citealp{ArisakaP2N3}, where the exclusion itself is carried by the capacity theorem; see the next subsection) requires the alternating sum
\begin{equation}
a^+(N) - a^-(N) > 0, \qquad
a^\pm(N) = \#\{c : \mathrm{ind}(c) \equiv 0,2 \,(\bmod\, 4)\}\;\text{or}\;\equiv 1,3 \,(\bmod\, 4)\}.
\label{eq:strict-positivity}
\end{equation}
For the index distribution above this evaluates to $\sigma(\mathbb{C}P^{N-1}) = 1$ if $N$ odd and $0$ if $N$ even, so strict positivity \emph{eliminates all even $N$ in one stroke}.

The remaining odd values are constrained by cascade closure. The \GRIP{} cascade $\DGI \to \GSSB \to \GA$ on $\mathbb{C}P^{N-1}$ is a strict descent
\begin{equation}
[2(N-1)] \to [2(N-2)] \to \cdots \to [2] \to [0]
\label{eq:cascade-chain}
\end{equation}
through the index ladder, governed by gradient flow of $f_{\GFF}$ in the Fubini--Study metric. The instanton action between adjacent saddles of indices $2k$ and $2(k-1)$ saturates the Bogomolnyi bound at $S_{[2k]\to[2(k-1)]} = \pi k$ \citep{Witten1982}. Summing over the cascade gives the total action
\begin{equation}
S_{\rm tot}(N) = \pi \sum_{k=1}^{N-1} k = \pi \cdot \frac{N(N-1)}{2}.
\label{eq:S-tot}
\end{equation}
The \BIsix{} cascade-closure budget on the six-dimensional bulk caps this at
\begin{equation}
S_{\rm BI}^{\max} = \frac{2\pi N}{L} \cdot \frac{L}{2} = \frac{2\pi \cdot 3}{5}\cdot\frac{5}{2} = 3\pi
\label{eq:S-BI-max}
\end{equation}
--- a value delivered by the capacity theorem T-N3-6 of \citet{ArisakaP2N3} (its Appendix~C records the two viable formulations; \citealp{ArisakaA3BI6} quotes the value in its cross-reference ledger), with all integers fixed by \OGNfive{}. Read per family, the same expression is $S_{\rm BI}^{\max}(N) = \pi N$ --- $\pi$ of relaxation capacity per family --- under which cascade closure is the pure combinatorics $N(N-1)/2 \le N \iff N \le 3$ (Conjecture~C.1 of \citealp{ArisakaP2N3}). For the surviving odd values $N \in \{1,3,5,7,\dots\}$:
\begin{equation}
S_{\rm tot}(1) = 0, \qquad
S_{\rm tot}(3) = 3\pi \;(\text{exactly saturates}), \qquad
S_{\rm tot}(5) = 10\pi > 3\pi, \quad \text{etc.}
\end{equation}
Cascade closure therefore admits only $N = 1$ and $N = 3$ among odd values. Combined with $N \geq 3$ from \S\ref{subsec:N3-L1}, this leaves $\boxed{N = 3}$ as the unique admissible family number.

That $N = 3$ \emph{exactly saturates} the cascade-closure bound \eqref{eq:S-BI-max} is the \GGTheory{} analogue of criticality: the family number sits on the edge of admissibility, neither too small (no cascade exists) nor too large (cascade fails to close within the \BIsix{}-permitted instanton budget).

\begin{euclidremark}[Physics intuition: reachability, and why a fourth family is unaffordable]
The dynamical reading of the upper bound (\citealp{ArisakaP2N3}, \S7.5) joins this section to the \GRIPin{} doctrine of the present paper. \GFF{}-ST7 guarantees every stratum a unique Ground Geometric Attractor, and the weak interaction is the road by which matter reaches it --- the physical implementer of \GRIPin{} on the matter strata. Descending the full family ladder costs the triangle number of \GSSB{} gate tolls, $\pi N(N-1)/2$; the stratum funds $\pi$ per family, $\pi N$ in all. For $N \ge 4$ the toll exceeds the capacity: the ground state becomes dynamically unreachable, and matter would strand forever in non-ground basins, violating the constitutional ground-state guarantee. The bound is $N < 4$ not because a fourth family's spectrum would be internally inconsistent, but because its relaxation road could not be paid for; the third generation is the last family the universe can afford, and the observed $\tau \to \mu \to e$ chain is the visible execution of the descent that exactly spends the budget.
\end{euclidremark}

\subsection{Reduction of L2 to a Hirzebruch calculation --- pointer to \PaperPTwo{}}

The parity route reducing strict positivity to a single Hirzebruch-type signature calculation, together with the capacity architecture that replaced its would-be hypotheses --- the rigidity obstruction, the Berry-transport theorems at
Born-grade, and the integrality theorem L-INT (residue R-INT) --- the curvature-descent coefficient now fixed by the Fubini--Study conventions rather than pending the induced-connection construction, the naive scale-profile metric evaluation being proved to vanish identically --- is owned in full by \PaperPTwo{} \citep{ArisakaP2N3}. The conclusion that we use here is the upper bound $N \le 3$, which combines with Leg L1 ($N \ge 3$ from the Kobayashi--Maskawa argument) to fix $N = 3$ uniquely. The next subsection records the Leg L3 constructive check: the $N = 3$ codebook reproduces the empirical Standard-Model spectrum at four-significant-figure precision.

\subsection{Leg L3: codebook self-consistency}
\label{subsec:N3-L3}

The third leg is constructive: it shows that the explicit fermion-mass codebook of the present paper requires $N=3$ and is incompatible with any other value. The minimal first-order charged-lepton ratios depend on $N$ algebraically as
\begin{equation}
\frac{m_\tau^{(1)}}{m_\mu^{(1)}}(N) = \frac{NJ}{L}\cdot 2^{L-N}, \qquad
\frac{m_\mu^{(1)}}{m_e^{(1)}}(N) = \frac{L^4}{N},
\label{eq:lepton-ratios-N}
\end{equation}
with $J = 7$ and $L = 5$ fixed by \BIsix{} closure of \OGNfive{}. Evaluating for $N \in \{2, 3, 4\}$:
\begin{equation}
\begin{array}{c|cc}
N & m_\tau/m_\mu & m_\mu/m_e \\\hline
2 & 22.4 & 312.5 \\
\mathbf 3 & \mathbf{16.8} & \mathbf{208.33} \\
4 & 11.2 & 156.25 \\\hline
\text{Exp.} & 16.817 & 206.768
\end{array}
\end{equation}
The values $N = 2$ and $N = 4$ miss observation by $33\%$ and $\geq 24\%$; only $N = 3$ matches both ratios at the four-significant-figure level. The full codebook (Sections~\ref{sec:lepton-mass}--\ref{sec:quark-mass} below) extends this to all twelve charged-fermion masses, with QD-completion factors $\mathcal R_\tau = 637/640$, $\mathcal R_\mu = 159/160$, $\mathcal R_e = 641/640$ bringing the predictions to absolute precision matching the PDG values. Indeed, the codebook is not merely compatible with $N=3$; it is built around $N=3$ in a non-trivial, predictive way.

\subsection{Implications for the rest of the paper}
\label{subsec:N3-implications}

Theorem~\ref{thm:N3-closed} converts every appearance of ``$N=3$'' or ``three generations'' in the remainder of \ANine{} from an empirical input into a derived theorem --- conditional in the precise sense recorded in the status remark above. The \OGNfive{} ledger entry $N = 3$ is no longer fitted from the lepton spectrum but forced by the intersection of the three legs above. In particular:
\begin{itemize}\itemsep0pt
\item Every Yukawa formula in Sections~\ref{sec:lepton-mass}--\ref{sec:quark-mass} treats $N = 3$ as a structural input from \citep{ArisakaP2N3}, not as a fitted constant.
\item The CKM and PMNS sections (Sections~\ref{sec:CKM}--\ref{sec:PMNS}) inherit the three-generation structure as a consequence of Leg~L1 + Leg~L2.
\item The angular cascade $1, e^{-\pi}, e^{-2\pi}, e^{-3\pi}, e^{-4\pi}$ in the quark codebook is the instanton-cost spectrum of the cascade chain \eqref{eq:cascade-chain} on $\mathbb{C}P^2$, with each $e^{-n\pi}$ accumulating $\pi$ of Bogomolnyi action per step (see \S\ref{sec:quark-mass}).
\item The QD-completion rationals (denominators traceable to $2^J L = 640$ in \OGNfive{}) are the Hopkins--Singer descent integrality coefficients of $X_4$ on the relevant boundary stratum (see Section~\ref{sec:quark-mass} for the explicit expansion).
\end{itemize}

\begin{euclidremark}[Why this proof required open-system thinking]
The structural reason the three-generation puzzle has remained open for the century since the foundation of quantum mechanics is now visible. Closed-system methods --- anomaly cancellation, integrality, index theorems --- can constrain integer multiplicities from below but cannot bound them from above, because all unitary states of a closed system are equally admissible by symmetry; there is no intrinsic notion of an ``unstable'' or ``decaying'' state from which to derive a stability budget. Upper bounds on integer multiplicities require open-system dynamics: irreversible boundary projection, finite instanton-action budget, cascade closure on a moduli of attractors. This is exactly what \GDOS{} provides, and only \GDOS{} provides. Indeed, the present derivation of $N=3$ is the first concrete demonstration that the open-system shift has the foundational power to derive structural integers that closed-system methods cannot reach. After all, $N = 3$ is not fixed by an additional axiom but by the same six-dimensional geometry that fixed the \BIsix{} vector and the \OGNfive{} ledger; the integer $3$ falls out of the topology of the boundary moduli once the open-system framework is in place.
\end{euclidremark}


\section{Running coupling constants from the fifth dimension}
\label{sec:radial-rg}
\label{sec:running-from-r}

\subsection{Radial renormalization as theorem, not metaphor}

One of the most concrete statements in the \GGConstitution{} is the
radial renormalization relation
\begin{equation}
\partial_r g_i = -\beta_i(g),
\qquad
r=\ln(\EGUT/\mu).
\label{eq:beta-radial}
\end{equation}
This is not an analogy. It is the direct technical meaning of the claim
that scale has been geometrized.

Integrating \eqref{eq:beta-radial} between two radial positions gives
for a generic gauge coupling
\begin{equation}
\alpha_i^{-1}(\mu)
=\alpha_i^{-1}(\EGUT)
+\frac{b_i}{2\pi}\ln\!\left(\frac{\EGUT}{\mu}\right)
+\Delta_i^{\mathrm{th}}(\mu)
+\Delta_i^{\mathrm{hol}}(\mu),
\label{eq:generic-alpha-run}
\end{equation}
where \(b_i\) is the effective one-loop slope, \(\Delta_i^{\mathrm{th}}\)
collects threshold pieces, and \(\Delta_i^{\mathrm{hol}}\) records the
compact-holonomy contributions induced by the \(S^1_\theta\) sector.
The important point is not the detailed numerical value of each term;
it is the fact that every term has a geometric meaning in the same
6D stage.

\begin{figure}[p]
\noindent\begin{minipage}{\textwidth}
\centering
\includegraphics[width=\textwidth,height=0.62\textheight,keepaspectratio]{gg_unification_showcase.pdf}
\par\medskip
{\centering
\(\alpha_1^{-1}(M_Z)=\tfrac{3}{5}\cos^2\theta_W\cdot 2^J=\tfrac{3}{5}\cdot\tfrac{83}{108}\cdot 128=59.02,\qquad
\alpha_2^{-1}(M_Z)=\sin^2\theta_W\cdot 2^J=\tfrac{25}{108}\cdot 128=29.63,\)\\[4pt]
\(\alpha_3^{-1}(M_Z)=1/\alpha_s(M_Z)=\pi e=8.54;\qquad
\alpha_{\rm GUT}^{-1}\approx 41\ \text{at}\ \EGUT=M_{\rm Pl}\,e^{-2\pi}\approx 2.3\times 10^{16}\,\mathrm{GeV}\)\par}
\smallskip
{\setlength{\abovecaptionskip}{5pt}%
\captionof{figure}[\GGTheory{} gauge-coupling unification across the infrared region.]{\textbf{GG-Theory gauge-coupling unification across the 6D bulk.} Inverse couplings $\alpha_1^{-1}$ (green, hypercharge in SU(5) normalization), $\alpha_2^{-1}$ (blue, $\mathrm{SU}(2)_L$), and $\alpha_3^{-1}$ (coral, $\mathrm{SU}(3)_C$) are plotted versus the renormalization-group scale $\mu$ (bottom axis) and the radial coordinate $r = \ln(\EGUT/\mu)$ (lower-most axis), with $r=0$ at the unification point. The boundary values at $M_Z$ are pinned by the GG-Constitution and the OGN-5 ledger with the three closed forms displayed above --- no continuous fitted parameter intervenes anywhere between $M_Z$ and $\EGUT$. Four constitutional scales are marked: the EM scale $v = 246$~GeV, the KK lock-cluster mass $\Mlock = \tfrac12 M_{\rm Pl}\,e^{-35} \approx 3.85$~TeV, the unification scale $\EGUT = M_{\rm Pl}\,e^{-2\pi} \approx 2.3\times 10^{16}$~GeV, and the Planck scale $\EPl \approx 1.2\times 10^{19}$~GeV. Below $\Mlock$ the running uses the standard one-loop SM beta functions; above $\Mlock$ the KK lock cluster opens and the three couplings converge to the unified value $\alpha_{\rm GUT}^{-1} \approx 41$ at exactly $\EGUT$ (black dot). The lines terminate at $\EGUT$: above the unification scale the SM gauge-coupling description is no longer defined, and the bulk geometry takes over. This three-line meeting at $\EGUT$ is the headline empirical signature of the 6D ultraviolet completion of the Standard Model.}\label{fig:gg-unification}}
\end{minipage}
\end{figure}

\subsection{Why only \texorpdfstring{\(\pi\)}{pi} and \texorpdfstring{\(\e\)}{e} can survive in the seam codes}

The locked bulk has only two primitive transcendental sources:
\begin{enumerate}[leftmargin=2.1em]
\item the angular holonomy of the compact circle, whose primitive length
is \(2\pi\);
\item the logarithmic redshift along the radial coordinate, whose
natural base is \(\e\).
\end{enumerate}
Because \MSMF{} forbids importing any additional transcendental constants,
any exact seam code beyond the OGN integers must be built only from
\(\pi\) and \(\e\). This simple statement explains in advance why the
first-order strong coupling emerges as \((\pi\e)^{-1}\), and why the
other channels collapse either to pure rationals or to pure powers of
\(2\).

\subsection{Three natural channels at the neutral seam}

At the neutral electroweak seam \(\Zseam\), there are three natural RG
combinations:
\begin{enumerate}[leftmargin=2.1em]
\item an \textbf{electromagnetic sum channel}, in which the neutral
rotation ends on the unique unbroken Abelian generator and the compact
and radial transcendental pieces cancel, leaving a pure dyadic closure;
\item a \textbf{neutral-rotation channel}, in which the relative weight
of hypercharge and weak isospin is fixed by the group embedding and the
triplicity index, leaving a pure rational;
\item a \textbf{color-difference channel}, in which the cancellation is
incomplete and one angular plus one radial primitive survive, leaving a
\(\pi\e\) denominator.
\end{enumerate}
The entire gauge-sector prediction can therefore be organized as the
assignment of the minimal OGN code to each of these three channels.

\subsection{Control parameters for next-order corrections}

Before giving the first-order predictions, it is useful to isolate the
two small numbers that control the QD completion:
\begin{equation}
\varepsilon_J\equiv 2^{-J}=\frac{1}{128}=7.8125\times 10^{-3},
\qquad
\varepsilon_K\equiv \frac{1}{2\pi K}=\frac{1}{70\pi}
\simeq 4.5473\times 10^{-3}.
\label{eq:epsJ-epsK}
\end{equation}
A third small combination, useful when the binary endpoint factor enters
explicitly, is
\begin{equation}
\varepsilon_{\ln 2}
\equiv \frac{\ln 2}{2\pi K}
\simeq 3.1516\times 10^{-3}.
\label{eq:epsln2}
\end{equation}
The philosophy of the paper is then simple. First-order quantities are
exactly those determined by the minimal OGN code. Second-order/QD
corrections are those suppressed by one power of \(\varepsilon_J\) or
\(\varepsilon_K\). Residual uncertainties after QD are estimated from
one further suppressed step.

\section{Wilson--Hosotani dynamical gauge symmetry breaking and the Higgs mechanism}
\label{sec:wilson-hosotani}

\subsection{Boundary-condition breaking in \GG{}}

\AFive{} derives the orbifold/interval parity assignment
\begin{equation}
P=\diag(+1,+1,+1,-1,-1),
\label{eq:parity-P-main}
\end{equation}
with gauge-field parities
\begin{align}
A_\mu(x,r,-\theta)&=P\,A_\mu(x,r,\theta)\,P^{-1},
\label{eq:Amu-even-main}\\
A_\theta(x,r,-\theta)&=-P\,A_\theta(x,r,\theta)\,P^{-1}.
\label{eq:Atheta-odd-main}
\end{align}
Zero modes survive only for the generators commuting with \(P\), so that
at the zero-mode level one obtains
\begin{equation}
\SU(5)\longrightarrow \SU(3)_C\times \SU(2)_L\times \U(1)_Y.
\label{eq:su5-break-main}
\end{equation}
Before electroweak symmetry breaking the full gauge group is therefore
\begin{equation}
G_{\mathrm{IR,before\ EWSB}}
=\SU(3)_C\times\SU(2)_L\times\U(1)_Y\times\U(1)_B\times\U(1)_{Qi},
\label{eq:gauge-before-main}
\end{equation}
exactly as described in \AFive{}
\citep{ArisakaA5GDOS}.

\subsection{Wilson line as the true Higgs order parameter}

The key step for the present paper is that the compact \(\theta\)
direction supports a gauge-invariant holonomy,
\begin{equation}
W_\theta(x,r)
= \mathcal P\exp\!\left(\ii g_6\oint_{S^1_\theta}A_\theta(x,r,\theta)\,\dd\theta\right).
\label{eq:Wtheta-main}
\end{equation}
Its phase,
\begin{equation}
\Theta_H(x,r)
\equiv g_6\oint_{S^1_\theta}A_\theta\,\dd\theta,
\label{eq:ThetaH-main}
\end{equation}
is nonlocal along the compact direction and therefore cannot be removed
by a small gauge transformation. This is the essence of the Hosotani
mechanism.

The four-dimensional Higgs field is then identified with the boundary
image of the coset components of \(A_\theta\). Symbolically,
\begin{equation}
A_\theta^{\mathrm{coset}}
\quad\longrightarrow\quad
H(x)\sim \text{boundary fluctuation of } \Theta_H.
\label{eq:Atheta-to-H}
\end{equation}
Thus the Higgs is not inserted as an elementary scalar with an
independent local potential. It is the boundary scalar avatar of a
Wilson-line phase.

\subsection{Why this resolves the naturalness problem in the first instance}

This identification changes the naturalness question qualitatively.
A local mass term for \(A_\theta\) is forbidden by the higher-dimensional
gauge symmetry. Therefore the effective Higgs potential is generated
only through the nonlocal Wilson line, and the corresponding one-loop
potential is finite. A standard expression of the Hosotani type is
\begin{equation}
V_{\mathrm{eff}}(\Theta_H)
=\sum_s \frac{(-1)^{F_s}n_s}{2}
\int\!\frac{\dd^4p_E}{(2\pi)^4}
\sum_{n\in\mathbb Z}
\ln\!\left[p_E^2+\frac{(n+\nu_s(\Theta_H))^2}{R_\theta^2}\right],
\label{eq:Veff-Hosotani}
\end{equation}
where \(F_s\) is the fermion number, \(n_s\) the multiplicity, and
\(\nu_s(\Theta_H)\) the Wilson-line shift of the KK spectrum.

The crucial point is conceptual rather than calculational: because the
order parameter is a circle holonomy, the potential depends on a phase,
not on a local ultraviolet cutoff.
The Higgs sector is therefore protected in the same way that any Wilson
line is protected: only finite, global data of the compact direction can
enter.

This is the \GG{} answer to naturalness at the structural level.
The theory does not say that loop corrections vanish. It says that the
relevant scalar degree of freedom is not a free local mass parameter in
the first place. Its potential is generated geometrically and is
controlled by the compact topology and the fixed bulk spectrum.

\subsection{Relation to the ordinary 4D Higgs language}

Once projected to the boundary, the Wilson-line Higgs is described in
the familiar four-dimensional form
\begin{equation}
H(x)=\frac{1}{\sqrt2}
\begin{pmatrix}
0\\ v+h(x)
\end{pmatrix},
\qquad
v^2=\frac{\mu_H^2}{\lambda_H},
\label{eq:Higgs-4d-main}
\end{equation}
with electroweak masses
\begin{equation}
m_W=\frac12 g_2 v,
\qquad
m_Z=\frac12 v\sqrt{g_2^2+g_Y^2},
\qquad
\tan\theta_W=\frac{g_Y}{g_2},
\qquad
e=g_2\sin\theta_W=g_Y\cos\theta_W,
\label{eq:ew-rel-main}
\end{equation}
as derived in \AFive{} \citep{ArisakaA5GDOS}.
The difference from the ordinary Standard-Model reading is not in these
formulas; it is in the origin of \(H\), \(\mu_H^2\), and \(\lambda_H\).
In the present framework they are secondary boundary quantities induced
by a locked six-dimensional geometry.

\subsection{Doublet fraction and the geometric meaning of \texorpdfstring{\(2/5\)}{2/5}}

The decomposition of the unified Higgs multiplet is
\begin{equation}
\mathbf 5_H \to (3,1)_{-1/3}\oplus(1,2)_{+1/2}.
\label{eq:5H-decomp-main}
\end{equation}
Therefore the electroweak Higgs doublet occupies exactly two of the five
components of the \(\mathbf 5_H\). The ratio
\begin{equation}
\frac{\dim(1,2)}{\dim\mathbf 5_H}=\frac25
\label{eq:doublet-fraction-main}
\end{equation}
is thus not a fit or a phenomenological convenience. It is the unique
representation-theoretic projection fraction of the surviving Higgs
doublet inside the unified \(\SU(5)\) multiplet.

This same ratio \(2/5\) already appeared in \BIsix{}.
That is the first sign that the low-energy Higgs sector and the
ultraviolet anomaly structure are numerically locked together.


\medskip\noindent The substrate is now fixed --- the six-dimensional seam, the three scales, the \OGNfive{} integer ledger, the forced $N=3$, and the Wilson--Hosotani breaking on which every later derivation runs. The next Part turns from the landscape to the method: the universal sectoral descent template and the constitutional inputs that, applied to this substrate, close the parameter sheet.




\section{Part~IV in plain English: the fixed landscape everything else runs on}
Everything that follows happens on a fixed geometric stage, and this Part
builds the stage.

The picture is a six-dimensional bulk with a four-dimensional boundary. The
two extra directions are not alike: one is a compact circle, the other a
radial interval. That asymmetry does the work. A circle can carry a holonomy
--- a phase a field accumulates going around --- which is a quantity with no
local definition and therefore no local counterterm. An interval carries an
address: a position along it corresponds to a scale, and moving along it is
what renormalization-group running is in this language. Angles and scales are
different kinds of object, and the paper keeps them apart throughout.

Three results in this Part are load-bearing later. The ledger integers
$(M,N,L;J,K) = (1,3,5;7,35)$ fix the arithmetic content of every closed form
that follows. The three-leg theorem returns the family number $N=3$ as a
consequence rather than an input, which is the first place the framework
predicts something the Standard Model postulates. And the radial
renormalization group turns running into transport, which is what makes the
gauge couplings computable at a boundary point.

The word to hold onto is \emph{fixed}. Nothing in this Part is adjusted later
to make a downstream number come out. If the Substrate is wrong, every
derivation built on it is wrong together, which is a liability and also the
property that makes the framework worth testing: there is no way to repair one
sector by moving the ground under it without moving the ground under all the
others.

\GGPart[Roadmap and constitutional basis]{Roadmap and constitutional basis}{The universal sectoral descent template, the derivation roadmap, and the constitutional inputs that close the parameter sheet}
\label{part:IV}

\noindent
Parts~\ref{part:II} and~\ref{part:III} have established what the constitutional apparatus of \GGTheory{} is and how the $\mathrm{SU}(5)$ seam descends from the six-dimensional bulk. Part~\ref{part:IV} is the methodological pivot. The two short sections below state, once and cleanly, the universal six-step ladder that every later sectoral derivation will obey, and the constitutional ground rules under which the ladder is admissible. No new physics enters here; only the discipline that makes the parameter blocks of Parts~\ref{part:III}--\ref{part:X} comparable to one another. A reader who wants to skip directly to the gauge or fermion derivations may do so --- but should return to this Part whenever the litany \BIsix{} $\to$ \TGG{} $\to$ \CCI{} $\to$ \GFF{} $\to$ \GRIP{} feels mechanical.

\section{Universal sectoral descent template for the 28-parameter sheet}

\begin{figure}[!ht]
\centering
\resizebox{0.463\textwidth}{!}{%
\begin{tikzpicture}[
 every node/.style={align=center},
 stage/.style={draw=black!55, rounded corners=2pt, line width=0.5pt, minimum width=38mm, minimum height=7mm, font=\sffamily\scriptsize, align=center},
 thm/.style={font=\sffamily\scriptsize, text=black!55, anchor=west},
 arrow/.style={-{Stealth[length=2.4mm]}, line width=0.7pt, draw=black!65}
]
\node[stage, fill=red!12,    draw=red!60!black]                      (BI6)  {\textbf{1.\ \BIsix{} antecedent}\\[-1pt]\textit{anomaly closure}};
\node[stage, fill=orange!16, draw=orange!70!black, below=8mm of BI6] (TGG)  {\textbf{2.\ \TGG{} closure}\\[-1pt]\textit{total bulk/boundary tensor}};
\node[stage, fill=yellow!16, draw=olive!65!black,  below=8mm of TGG] (CCI)  {\textbf{3.\ \CCI{} projection}\\[-1pt]\textit{seam-level conservation}};
\node[stage, fill=green!16,  draw=green!60!black,  below=8mm of CCI] (GFF)  {\textbf{4.\ \GFF{} ledger}\\[-1pt]\textit{scalar landscape}};
\node[stage, fill=cyan!18,   draw=cyan!65!black,   below=8mm of GFF] (GRIP) {\textbf{5.\ \GRIP{} stabilization}\\[-1pt]\textit{\DGI{}\,$\to$\,\GSSB{}\,$\to$\,\GA{}}};
\node[stage, fill=blue!16,   draw=blue!65!black, line width=0.9pt, below=8mm of GRIP] (PARAM) {\textbf{6.\ Sectoral parameter block}\\[-1pt]\textit{read off the \GA{} coordinate}};
\node[thm, right=5mm of BI6.east]  {\AThree{}};
\node[thm, right=5mm of TGG.east]  {\ASix{}\, T-GRIP-1};
\node[thm, right=5mm of CCI.east]  {\ASix{}\, \T{2.1},\,\T{2.2}};
\node[thm, right=5mm of GFF.east]  {\ASix{}\, \T{3.1}};
\node[thm, right=5mm of GRIP.east] {\ASix{}\, \T{4.1}};
\node[thm, right=5mm of PARAM.east]{\textit{this paper}};
\draw[arrow] (BI6)  -- (TGG);
\draw[arrow] (TGG)  -- (CCI);
\draw[arrow] (CCI)  -- (GFF);
\draw[arrow] (GFF)  -- (GRIP);
\draw[arrow] (GRIP) -- (PARAM);
\end{tikzpicture}}
\caption[The universal sectoral descent ladder shared by every parameter block.]{The universal sectoral descent ladder. Every parameter block of Parts~\ref{part:VI}--\ref{part:XI} obeys this six-step descent: \BIsix{} closure of the bulk anomaly polynomial $\to$ \TGG{} total closure of the geometry-gauge tensor $\to$ \CCI{} projection to the boundary seam $\to$ \GFF{} stationary equations on the scalar landscape $\to$ \GRIP{} cascade $\DGI \to \GSSB \to \GA$ to the selected branch $\to$ readout of the parameter values from the \GA{} coordinates. The same ladder appears, with different stratum-specific content, in the gauge--Higgs derivation of Part~\ref{part:VI}, the fermion-mass block of Part~\ref{part:X}, and the strong-CP completion of Part~\ref{part:XI}. Each step is owned by the upstream theorem named to its right --- \AThree{} for the \BIsix{} antecedent, \ASix{} for the four pillars --- and \ANine{} consumes the chain rather than re-proving it, computing only the final sectoral block.}
\label{fig:sectoral-descent-ladder}
\end{figure}


\label{sec:universal-template}

The present rewrite imposes one strict methodological rule on every sector.
A parameter derivation is counted as complete only if it exhibits the full
ladder
\[
\text{\BIsix{}}\Longrightarrow \TGG \Longrightarrow \CCI \Longrightarrow \GFF \Longrightarrow \GRIP \Longrightarrow \mathcal P_S,
\]
where \(\mathcal P_S\) is the sectoral parameter block. The purpose of this
section is to state that ladder once, cleanly, so that the later gauge, mass,
mixing, and topological chapters can follow it without ambiguity.

\begin{definition}[Sectoral parameter block]
For a chosen Standard-Model seam sector \(S\), the sectoral parameter block
\(\mathcal P_S\) is the ordered set of all measurable low-energy parameters of
that sector. In the present paper the relevant blocks are
\[
\mathcal P_{\mathrm{GH}},\quad
\mathcal P_{\mathrm{masses}},\quad
\mathcal P_{\mathrm{CKM}},\quad
\mathcal P_{\mathrm{PMNS/Maj}},\quad
\mathcal P_{\bar\theta}.
\]
\end{definition}

\begin{proposition}[Universal descent logic]
\label{prop:universal-descent}
A sectoral parameter block \(\mathcal P_S\) is admissible in the \GG{}/\GDOS{}
framework only if the following six questions are answered in order:
\begin{enumerate}[leftmargin=2.2em]
\item What \BIsix{} coefficient, representation, or topological fact legally
restricts the sector?
\item How is the sector embedded in the closed total tensor \TGG?
\item What local seam law does \CCI{} impose on the projected packet?
\item Which reduced coordinates appear in the sectoral \GFF{} landscape?
\item How does \GRIP{} select and stabilize the branch on which the parameter
is measured?
\item What primary attractor coordinates or algebraic descendants then define
the measured parameter block?
\end{enumerate}
\end{proposition}

The reason for insisting on this order is not pedagogical only. It prevents
three common mistakes. First, it prevents a sector from being derived as if
it were dynamically isolated from the rest of the packet. Second, it prevents
one from introducing a scalar landscape after the branch has already been
selected. Third, it prevents one from reading parameters off a projected seam
before the projected conservation law has been stated.

Table~\ref{tab:block-map-recast} summarizes how the five sectoral blocks of
the present paper are organized under Proposition~\ref{prop:universal-descent}.

\begin{table}[H]
\centering
\caption[Sectoral blocks of the 28-parameter sheet, organized by the universal descent logic.]{Sectoral blocks of the 28-parameter sheet organized by the universal
\BIsix{}\(\to\)\TGG\(\to\)\CCI\(\to\)\GFF\(\to\)\GRIP\(\to\)parameter logic. The four blocks partition the twenty-eight parameters exactly once each, which is the property that makes the single-sheet census of Part~\ref{part:XIII} possible. Each row names the coordinates the block descends from and the observables it produces. The logic named in the caption is the same at every row and only the coordinates change, and that repetition is itself the claim this paper makes about the structure of the Standard Model.}
\label{tab:block-map-recast}
\small
\begin{tabular}{p{0.18\textwidth}p{0.23\textwidth}p{0.49\textwidth}}
\toprule
\textbf{Block} & \textbf{Primary reduced coordinates} & \textbf{Parameters read out}\\
\midrule
Gauge--Higgs &
\((C_\gamma,p_W,C_s,\theta_H,\lambda_H)\) &
\(\alphahatinv(M_Z)\), \(\sin^2\theta_W(M_Z)\), \(\alpha_s(M_Z)\), \(v\), \(m_h\) and their descendants.\\
Fermion masses &
family overlap depths, winding indices, projector weights &
\(m_e,m_\mu,m_\tau,m_u,m_d,m_s,m_c,m_b,m_t,m_1,m_2,m_3\).\\
CKM &
up/down projector misalignment coordinates &
\(\sin\theta_{12}\), \(\sin\theta_{23}\), \(\sin\theta_{13}\), \(\delta_{\rm CKM}\).\\
PMNS/Majorana &
charged-lepton versus Majorana-projector misalignment &
\(\sin^2\theta_{12}\), \(\sin^2\theta_{13}\), \(\sin^2\theta_{23}\), \(\delta_{\rm PMNS}\), \(\alpha_{21}\), \(\alpha_{31}\).\\
Topological CP &
projected topological phase and determinant sector &
\(\bar\theta\).\\
\bottomrule
\end{tabular}
\end{table}

The remainder of the manuscript is therefore a sequence of sectoral
realizations of one universal theorem protocol rather than a sequence of
numerically unrelated formulas.


\section{Roadmap of the derivation}
\label{sec:purpose}

The full target of \emph{UCLA-Arisaka/\ANine{}} is larger than the
present document. The final preprint must derive all 28 Standard-Model
parameters,
\begin{table}[htbp]
\centering
\small
\caption[The derivation target, sector by sector: which of the twenty-eight parameters each Part is responsible for, in the notation that Part itself uses.]{The derivation target, sector by sector: which of the twenty-eight parameters each Part is responsible for, in the notation that Part itself uses. The table states the scope of the paper before any of it is attempted, so that a reader can tell at the outset what is promised and, at the end, what was delivered. Where a sector lists an equivalent pair of parameterizations, the second is the one the comparisons are made in, because it is the one the measurements are quoted in.}
\label{tab:derivation-target}
\begin{tabular}{p{0.34\textwidth}p{0.54\textwidth}}
\toprule
\textbf{Sector} & \textbf{Parameters to be derived in \ANine{}}\\
\midrule
Gauge and Higgs & \(g_1, g_2, g_3, \mu_H^2, \lambda_H\) or equivalently \(\alphahatinv(M_Z), \sin^2\theta_W, \alpha_s(M_Z), v, m_h\).\\
Fermion masses & \(m_u,m_c,m_t,m_d,m_s,m_b,m_e,m_\mu,m_\tau,m_{\nu_1},m_{\nu_2},m_{\nu_3}\).\\
Quark mixing & Three CKM angles plus one CKM CP phase.\\
Lepton mixing & Three PMNS angles plus one leptonic Dirac CP phase.\\
Majorana sector & Two physical Majorana phases.\\
\bottomrule
\end{tabular}
\end{table}
The present paper now completes the first three blocks---the gauge/Higgs
sector, the full 12-fermion-mass block, and the 4-parameter CKM block---
and it now leaves only the final strong-CP-like topological phase
outside the explicit parameter derivation.

There are three reasons to proceed in this order.

First, the gauge/Higgs sector is the background over which all Yukawa,
flavor, and neutrino physics is written. If \(g_1\), \(g_2\), \(g_3\),
and \(v\) are still regarded as phenomenological inputs, then any later
``derivation'' of masses or mixing angles would inherit those hidden
inputs and the claimed reduction of arbitrariness would be incomplete.

Second, the gauge/Higgs block is exactly the sector where the
\GG{}/\GDOS{} constitution is sharpest. The six-dimensional stage, the
\BIsix{} anomaly condition, the \OGNfive{} arithmetic closure, and the Wilson--
Hosotani origin of the Higgs give a fully geometric route to the first
five Standard-Model parameters before any flavor detail is added.

Third, the gauge/Higgs sector is where the hierarchy and naturalness
questions live most painfully in the conventional Standard Model.
Because these questions should be faced
\emph{before} attacking electroweak-scale numerics, this paper begins
with them rather than postponing them.

\subsection{Outline of the complete \ANine{} program}

The roadmap of the complete paper is summarized in
Table~\ref{tab:roadmap-full}: the seventeen Parts in their five logical
blocks --- setup, foundation, derivation, closure, and appendices \& references --- each with its role
in the derivation chain. It is the textual companion to the architecture
flowchart of Figure~\ref{fig:org-flowchart}.

\begin{center}
\begin{longtable}{p{0.07\textwidth}p{0.27\textwidth}p{0.55\textwidth}}
\caption[Roadmap of the full \ANine{} program: the seventeen Parts in their five logical blocks (setup, foundation, derivation, closure, appendices \& references).]{Roadmap of the full \ANine{} program: the seventeen Parts in their
five logical blocks (setup, foundation, derivation, closure, appendices \& references) and their role
in the derivation chain. This is the textual companion to the architecture
flowchart of Figure~\ref{fig:org-flowchart}.}
\label{tab:roadmap-full}\\
\toprule
\textbf{Part} & \textbf{Title} & \textbf{Role in the derivation chain}\\
\midrule
\endfirsthead
\toprule
\textbf{Part} & \textbf{Title} & \textbf{Role in the derivation chain}\\
\midrule
\endhead
\ref{part:I} & Introduction & The thesis, the place of \ANine{} in the \GGTheory{} cohort, and the roadmap of the seventeen Parts.\\
\ref{part:SMtarget} & The Standard Model and its parameter audit & Phenomenological target: the Standard Model in textbook form, the classical Georgi--Glashow \(\mathrm{SU}(5)\), the \GGTheory{} derivation of \(\mathrm{SU}(5)\) from the 6D bulk, and the codebook-provenance audit (the \(6/14/8\) honesty triage).\\
\addlinespace
\ref{part:II} & Constitutional apparatus & The four axioms, seven postulates, and the canonical \GiFour{} chain (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) specialized to the Standard-Model boundary stratum.\\
\ref{part:III} & The Substrate & Descent from classical \(\mathrm{SU}(5)\) to the \GG{} seam under \GDOS{}: the 6D bulk, the three scales \((\EPl,\EGUT,\Mlock)\), \(N=3\) as a theorem, the radial RG, and Wilson--Hosotani breaking.\\
\ref{part:IV} & Roadmap and constitutional basis & The universal sectoral descent ladder, the admissible inputs and the standard of derivation, and this roadmap of the program.\\
\addlinespace
\ref{part:VI} & The electroweak seam & The \GRIPout{} vacuum and its \GRIPin{} descent: the theorem deck, the five primary attractor-coordinate theorems, the descendant sheet (\(G_F,m_W,m_Z\)), and the first-order and QD predictions.\\
\ref{part:X} & The fermion sector & The \GRIPin{} floors (charged-lepton, quark, and neutrino masses) and gates (CKM, PMNS, and the three Majorana phases) as misalignments of family projectors on one closed packet.\\
\ref{part:lepto} & The Majorana door and leptogenesis & The excess of matter over antimatter written by the same geometry that fixes the neutrino block: the unique \(\Delta L = 2\) portal, the heavy sector at \(\Mlock\), the flavored \CP{} asymmetry driven by the Majorana parity, and the assembled baryon-to-photon ratio --- a prediction that is not one of the twenty-eight.\\
\ref{part:XI} & The protected readout & Strong-CP closure: the twenty-eighth parameter \(\bar\theta\), suppressed by the topology of the infrared region, with a constitutional upper bound far below the experimental limit.\\
\ref{part:lagrangian} & The assembled Lagrangian & The harvest of the derivation core: the complete Standard-Model Lagrangian assembled term by term as a constitutional export --- the zeroth and the six silent textbook axioms replaced by theorems, the five term groups exported, the absent operators derived, and the honesty boundary stated beside the equation.\\
\addlinespace
\ref{part:XII} & Identity and uniqueness of forces & Philosophical pivot: universal identity and four-force uniqueness read as one uniqueness theorem applied at two layers of the same geometry.\\
\ref{part:XIII} & Summary and audit & The single canonical census: the cross-paper coherence map, the first-order and QD 28-parameter tables, and the \MSMF{}/\OGN{} per-parameter provenance audit.\\
\ref{part:XIV} & Discussions & Significance and uniqueness, comparison with alternative programs, falsification channels, open questions, and prospects.\\
\ref{part:XV} & Conclusions & The final reflection, viewed from the summit reached after the climb out of the Standard Model.\\
\addlinespace
\ref{part:XVI} & Technical Appendices & Detailed arithmetic, explicit decompositions, codebooks, the glossary of symbols, and the promotion roadmap for the remaining \textbf{Asserted} entries.\\
\ref{part:XVII} & Common Objections and Responses & Eighty objections a particle theorist, cosmologist, mathematician, philosopher, or string theorist might raise, each answered with pointers into the paper.\\
\ref{part:XVIII} & References & Two groups --- the \GGTheory{} cohort and all other works --- each entry carrying a short relevance note to its first use.\\
\bottomrule
\end{longtable}
\end{center}

\subsection{Admissible inputs and forbidden shortcuts}
\label{subsec:inputs}

The methodological rule that the theory side may consume no empirical Standard-Model datum --- the admissible set being the \GUfive{} dimensionful constants, the pure constants $\{\pi,\e\}$, and the \OGNfive{} integer ledger $(M,N,L;J,K)=(1,3,5;7,35)$ --- was stated in full, with the forbidden-shortcut table and the standard of derivation, in \S\ref{sec:admissible-inputs}. It is the rule under which the entire roadmap above operates: every sector reads its parameters off a \GA{} coordinate fixed by those inputs alone, and measured values appear only in the final comparison tables. The grounding is constitutional rather than aesthetic --- \AX{4} (maximum symmetry, minimum freedom) and \POS{6} (Occam gauge normalization) are violated the moment a parameter is ``derived'' while quietly importing it under another name.


\medskip\noindent The descent template and its admissible inputs are now in place. The next Part runs them for the first time, on the gauge--Higgs sheet: the single \GRIPout{} vacuum event and the five primary attractor coordinates it forces --- $\alpha^{-1}$, $\sin^2\theta_W$, $\alpha_s$, $v$, and $\lambda_H$ --- each with no continuous freedom.




\section{Part~V in plain English: one template, applied five times}
This Part is short and it is the hinge of the paper.

The claim it makes is that the five sectors --- gauge and Higgs, charged
fermions, neutrinos, quark mixing, lepton mixing --- are not five separate
calculations that happen to be collected in one document. They are one
procedure run five times on five different strata of the same geometry. The
procedure has four stages: a landscape of alternatives, a potential generated
on it, a one-way descent, and a floor where the descent stops. In each sector
the landscape is different, but the stages are the same and they come in the
same order.

If that is right, it has a consequence a reader can check. Each sector Part
should need only a one-sentence constitutional preface rather than a full
re-derivation of its framework, and the same vocabulary --- top, landscape,
gate, floor --- should be doing real work in every one of them. If instead the
template has to be bent differently for each sector, then the unity is verbal
and the five calculations are five calculations after all.

That is a fair test, and it is worth applying with some skepticism, because a
template loose enough to fit everything explains nothing. The template here is
tight in one specific way: it forbids a floor from being chosen. A floor is
where the descent stops, and the descent is determined by the landscape. If a
sector's answer requires selecting the floor rather than deriving it, the
template has been violated, and the sector Parts say where that happens.

\GGPart[The electroweak seam]{The electroweak seam: the \GRIPout{} vacuum and its \GRIPin{} floors}{The one \GRIPout{} event and the gauge-seam floors it builds --- the rigorous theorem deck, the five primary attractor-coordinate theorems ($\alpha^{-1}$, $\sin^2\theta_W$, $\alpha_s$, $v$, $\lambda_H$), and the descendant boson-mass and Fermi-constant package}
\label{part:VI}

\noindent
This Part discharges the entire gauge--Higgs (electroweak) sheet of the codebook in one place. It is where the paper's single \GRIPout{} event occurs --- the Wilson--Hosotani selection of the electroweak vacuum that builds the Yukawa landscape on which all later fermion parameters are read (Fig.~\ref{fig:grip-two-regimes}) --- and where its first \GRIPin{} floors are fixed: the gauge-seam couplings $\alpha^{-1}(M_Z)$, $\sin^2\theta_W$, $\alpha_s(M_Z)$ and the vacuum scale $v$, with $G_F$, $M_W$, and $M_Z$ as their algebraic descendants. The development has three movements: first the rigor deck imported from \ASix{} (the four pillars and the local \GFF{} normal form on the seam); then the five primary attractor-coordinate theorems that fix the electroweak attractor; then the descendant sheet that follows from them with no further structural input. A reader who wants only the punchlines can skim the deck and read the boxed primary theorems directly.

\section{Rigorous gauge--Higgs theorem deck inherited from \ASix{}}

\subsection{The four papers and their roles}

The derivational discipline of the present paper is simple: nothing enters unless it is already theorem-level in \POne{}, \PTwo{}, \PThree{}, or \PIVA{}, or is a pure logical recombination of such material.

\begin{table}[H]
\centering
\caption[Imported theorem deck used in the present rewrite of \ANine{}.]{Imported theorem deck used in the present rewrite of \ANine{}. Each theorem is proved in the companion named beside it and is used here at the strength that companion states, not at a stronger one. The deck is listed in full so that a reader can see at the outset how much of this paper's machinery is imported and how much is new: the imports carry the constitutional and geometric content, and the derivations of the middle Parts are what this paper adds to them.}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.14\textwidth} >{\raggedright\arraybackslash}p{0.30\textwidth} X}
\toprule
\textbf{Source} & \textbf{Imported objects} & \textbf{Role in the present paper}\\
\midrule
\POne{} & GGEP, \OGGEP{}, \MSMF{}, \DGI{}, unified extremal principle, positivity, RG covariance, OGN & Constitutional admissibility rules. These say what counts as a lawful variable, what may appear in the scalar ledger, and why new free coefficients are forbidden unless they descend from already-fixed structure.\\
\PTwo{} & \GG{} geometry, radial coordinate \(r=\ln(E_{\GUT}/\mu)\), compact \(\theta\)-circle, \(\SU(5)\times U(1)_B\times U(1)_{Qi}\), \BIsix{}, \OGNfive{} integers \((M,N,L;J,K)=(1,3,5;7,35)\) & Supplies the bulk geometry and the locked integer packet from which the electroweak and gauge readout must descend.\\
\PThree{} & Exact bulk-to-boundary reduction, \GDOS{} current decomposition, boundary electroweak packet, stationary-sector logic & Supplies the operational right to talk about a boundary parameter sheet at all. Without \PThree{}, the present paper would still be trapped in the bulk.\\
\PIVA{} & The chain \(\text{BI-6 legality}\Rightarrow \text{projected Ward package}\Rightarrow \DGI \Rightarrow \GFF \Rightarrow \GSSB \Rightarrow \GA\), Wilson--Hosotani order-parameter theorem, electroweak holonomy branch theorem, local Higgs chart & Supplies the strict runtime order that makes the parameter formulas descendants of a selected branch rather than isolated guesses.\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{Imported equations used repeatedly}

The minimal six-dimensional stage is
\begin{equation}
M_6=M_4\times I_r\times S^1_\theta,
\qquad
\dd s_6^2 = e^{-2r}\,\eta_{\mu\nu}\,\dd x^\mu \dd x^\nu + \dd r^2 + \dd\theta^2,
\qquad
r=\ln(E_{\GUT}/\mu).
\label{eq:import-gg6}
\end{equation}
The locked \BIsix{} identity is
\begin{equation}
\dd H_3
=
\tr R^2
-
\Tr F_5^2
-
\frac{2}{5}F_B\wedge F_B
-
\frac{2}{5}F_{Qi}\wedge F_{Qi},
\label{eq:import-bi6}
\end{equation}
and the \OGNfive{} theorem fixes
\begin{equation}
(M,N,L;J,K)=(1,3,5;7,35).
\label{eq:import-ogn}
\end{equation}
The Wilson-line variable emphasized in \PIVA{} is
\begin{equation}
W_\theta
=
\mathcal P\exp\!\left(\ii\oint_{S^1_\theta}A_\theta\,\dd\theta\right)
=
\exp(\ii\theta_H T_H).
\label{eq:import-wh}
\end{equation}
The correct logical chain for parameter descent is
\begin{equation}
\text{\BIsix{} legality}
\Longrightarrow
\text{projected Ward package}
\Longrightarrow
\DGI
\Longrightarrow
\FGFF
\Longrightarrow
\GSSB
\Longrightarrow
\GA.
\label{eq:master-chain}
\end{equation}

Equation \eqref{eq:master-chain} is the architectural backbone of the paper. It says that the present note is not allowed to jump directly from bulk geometry to numbers. It must pass through the lawful runtime.

\subsection{Why the strengthened \ASix{} framework changes the \ANine{} strategy}

\ASix{} establishes a point that is subtle but decisive. One cannot define a symmetry-breaking event without first defining the scalar landscape on which the branch bifurcation occurs. In the present notation that landscape is \(F_{\mathrm{GFF}}\), equivalently \(F_{\mathrm{GFF}}=F_{\mathrm{GEP}}+F_{\mathrm{CEP}}\). The energetic part comes from projected geometry and gauge data; the curvature--entropy part broadens and stabilizes the admissible boundary dynamics.

This means that the Standard Model parameter problem must be read as a \emph{landscape-selection problem}. The couplings and masses are not primitive constants floating in empty notation space. They are coordinates of a stable branch in an allowed \GFF{} landscape. That is exactly why the Wilson--Hosotani mechanism is central here. The Higgs sector is not a side character. It is the clearest example of \GSSB{} in the whole framework.

\section{The electroweak closure class and the local \GFF{} normal form}

\subsection{The \texorpdfstring{\(3+2\)}{3+2} split inside \texorpdfstring{\(\SUfive\)}{SU(5)}}

Because \BIsix{} fixes \(L=5\) and \OGNfive{} fixes \(N=3\), the defining block splits canonically as
\[
5=3\oplus 2.
\]
The neutral split generator is the traceless matrix
\begin{equation}
\Ysplit
=
\diag\!\left(
\underbrace{\frac1N,\frac1N,\frac1N}_{N\text{ entries}},
\underbrace{-\frac1{L-N},-\frac1{L-N}}_{L-N\text{ entries}}
\right),
\label{eq:Ysplit}
\end{equation}
whose squared trace norm is
\begin{equation}
\Tr(\Ysplit^2)=\frac{1}{N}+\frac{1}{L-N}=\frac{L}{N(L-N)}.
\label{eq:TrY2}
\end{equation}
For \((N,L)=(3,5)\),
\begin{equation}
\Tr(\Ysplit^2)=\frac56.
\label{eq:TrY2-value}
\end{equation}

Equation \eqref{eq:TrY2-value} is not merely a group-theory curiosity. It is the neutral partition weight from which the weak-angle theorem will descend.

\subsection{The lock scale and the reduced holonomy variable}

The \GG{} throat depth supplies the fixed hierarchy factor \(e^{-K}\) with \(K=35\). The first visible boundary lock scale is the first half-wave of the interval collar:
\begin{equation}
\Mlock=\frac12\,M_{\mathrm{Pl}}e^{-K}.
\label{eq:Mlock}
\end{equation}
The factor \(1/2\) is not a fit coefficient. It is the minimal half-wave compatible with the absence of an additional zero-mode collar branch in the intended class.

The electroweak order parameter is the Wilson-line angle \(\theta_H\) of \eqref{eq:import-wh}. The local Higgs chart imported from \PIVA{} is
\begin{equation}
h(x)=f_H\bigl(\theta_H(x)-\theta_H^\star\bigr),
\label{eq:h-chart}
\end{equation}
where \(f_H\) is the holonomy stiffness obtained by reducing the mixed bulk kinetic term \(F_{\mu\theta}F^{\mu\theta}\).

\subsection{Definition of the minimal electroweak closure class}

\begin{definition}[Minimal electroweak closure class]
\label{def:Cew}
\(\Cew\) is the class of boundary realizations satisfying the following conditions:
\begin{enumerate}[label=(E\arabic*),leftmargin=2.2em]
\item no hidden dimensionless multiplicity is allowed beyond the \OGNfive{} packet;
\item the electromagnetic line is the unique unbroken Abelian seam after electroweak symmetry breaking;
\item the weak neutral partition is the quadratic \GFF{} weight of the \(3+2\) split generator averaged over triplicity;
\item the strong line retains one primitive compact quotient cell and one primitive logarithmic radial cell;
\item the Higgs branch is the minimal positive stable Wilson--Hosotani branch descending from the first lock scale;
\item the local scalar chart is the minimal even quartic normal form compatible with \DGI{}, \GSSB{}, \GA{}, and \MSMF{}.
\end{enumerate}
\end{definition}

The crucial point is that \(\Cew\) adds \emph{no new continuous parameter}. It only forbids hidden multiplicities and extra coefficients that are not already forced by the theorem deck.

\subsection{Local normal form of the electroweak \GFF{} ledger}

Near a nondegenerate stable electroweak attractor, the local \GFF{} ledger may be written in Morse normal form:
\begin{equation}
\FGFF^{(\EW)}
=
F_\star
+
\frac{\kappa_\gamma}{2}\bigl(C_\gamma-C_\gamma^\star\bigr)^2
+
\frac{\kappa_W}{2}\bigl(p_W-p_W^\star\bigr)^2
+
\frac{\kappa_s}{2}\bigl(C_s-C_s^\star\bigr)^2
+
V_{\WH}(\theta_H)
+
\cdots,
\label{eq:morse-ew}
\end{equation}
where \(C_\gamma\) is the inverse Abelian seam capacity, \(p_W\) is the weak neutral partition, \(C_s\) is the inverse strong seam capacity, and \(V_{\WH}\) is the periodic holonomy potential.

\begin{proposition}[Why the local normal form is legitimate]
\label{prop:morse}
Inside \(\Cew\), Eq.~\eqref{eq:morse-ew} is the minimally admissible local form of the electroweak \GFF{} ledger near a stable attractor.
\end{proposition}

\begin{proof}
By \DGI{}, the variables must be gauge-invariant or gauge-reduced coordinates. By \PIVA{}, the holonomy class and the seam capacities are such coordinates. By \GA{}, the selected branch is a stable nondegenerate minimum; hence a Morse normal form exists. By \MSMF{} and OGN, no extra independent dimensionless deformation coefficients are allowed in the leading quadratic sector. Therefore the minimal local form is the diagonal positive quadratic form in the gauge-reduced variables plus the holonomy potential \(V_{\WH}\), which is precisely Eq.~\eqref{eq:morse-ew}. \qedhere
\end{proof}

This proposition is important because it tells us what a parameter \emph{is} in the present framework. It is not a number attached externally to the theory. It is the center of one of the quadratic wells in the reduced \GFF{} landscape.


\section{Detailed descent from the \GFF{} ledger to the electroweak attractor}

\subsection{Why the parameter problem is really an attractor problem}

The older language of ``free parameters'' can hide the real geometric content of the present framework. A free parameter sounds like a number waiting to be typed into a Lagrangian by hand. But once \ASix{} is accepted, this picture is too crude. The reduced boundary system carries a lawful \GFF{} landscape, and what the observer calls a parameter is simply the coordinate of the selected stable branch on that landscape.

This is why the present paper keeps returning to the words \emph{inverse capacity}, \emph{partition weight}, and \emph{branch coordinate}. They are not cosmetic rebrandings. They name the kinds of coordinates that can appear on the gauge-reduced state manifold without violating \DGI{}. The electroweak/gauge parameter sheet is therefore best viewed as the list of basin coordinates
\[
(C_\gamma^\star,p_W^\star,C_s^\star,\theta_H^\star,\lambda_H^\star),
\]
together with their ordinary low-energy readouts.

\subsection{The energetic sector: what \texorpdfstring{\(F_{\mathrm{GEP}}\)}{F_GEP} remembers}

\ASix{} already established that the energetic half of the ledger comes from projected geometry and gauge structure. For the present electroweak packet, the relevant contributions are especially transparent:
\begin{enumerate}[leftmargin=2.2em]
\item the unique unbroken Abelian seam and its dyadic inverse capacity \(C_\gamma\);
\item the neutral split direction and its partition coordinate \(p_W\);
\item the strong difference seam and its angle--scale inverse capacity \(C_s\);
\item the Wilson-line branch angle \(\theta_H\) and its periodic holonomy potential.
\end{enumerate}
Thus the local energetic sector may be written in the schematic but lawful form
\begin{equation}
F_{\mathrm{\GEP{}}}^{(\EW)}
=
\frac{\kappa_\gamma}{2}(C_\gamma-C_\gamma^\star)^2
+
\frac{\kappa_W}{2}(p_W-p_W^\star)^2
+
\frac{\kappa_s}{2}(C_s-C_s^\star)^2
+
U_{\WH}(\theta_H),
\label{eq:FGEP-ew}
\end{equation}
where the centers \(C_\gamma^\star\), \(p_W^\star\), and \(C_s^\star\) are not yet assumed but are later identified by the geometry and \OGNfive{} integers.

Equation~\eqref{eq:FGEP-ew} should be read as the electroweak specialization of the general \ASix{} statement that the geometry--gauge energy is one scalar ledger with several lawful contributions. The first three terms measure displacements in seam space. The final term is the periodic Wilson--Hosotani branch potential.

\subsection{The curvature--entropy sector: what \texorpdfstring{\(F_{\mathrm{CEP}}\)}{F_CEP} contributes}

The curvature--entropy sector is sometimes misunderstood as though it were important only for chemistry or life-like systems. That reading is too narrow. Even at the particle scale, \(\CEP\) matters because the boundary observer does not see a bare classical path. The observer sees a coarse-grained registered branch. Coarse-graining broadens, weights, and stabilizes the admissible local readout.

At the level needed in the present paper one may write
\begin{equation}
F_{\mathrm{\CEP{}}}^{(\EW)}
=
\tau\!\int_{\State} p(y)\ln\frac{p(y)}{\pi_\star(y)}\,\dd\mu(y)
+
\frac{\nu_H}{2}\int_{\State}\|\nabla \theta_H(y)\|^2\,\dd\mu(y)
+\cdots,
\label{eq:FCEP-ew}
\end{equation}
where \(\pi_\star\) is the stationary reference density imported from the coarse-grained attractor class and the second term measures the curvature cost of spatially nonuniform branch selection. The omitted terms are higher-order coarse-graining corrections irrelevant to the present local attractor calculation.

The physical role of \(\CEP\) is not to move the branch centers fixed by the energetic theorem packet. Its role is to suppress pathological nonstationary branch readouts and to promote the selected lawful branch into a stable attractor. This is why \(\CEP\) does not replace the geometric formulas for \(2^J\), \(L^2/(4N^3)\), \(\pi e\), or \((2/L)^N\). It stabilizes their registration.

\subsection{Stationary equations and branch selection}

Combining Eqs.~\eqref{eq:FGEP-ew} and \eqref{eq:FCEP-ew}, the local electroweak \GFF{} ledger reads
\begin{equation}
F_{\mathrm{\GFF{}}}^{(\EW)}
=
F_{\mathrm{\GEP{}}}^{(\EW)}
+
F_{\mathrm{\CEP{}}}^{(\EW)}.
\label{eq:FEE-ew}
\end{equation}
The local attractor equations are therefore
\begin{align}
\frac{\partial F_{\mathrm{\GFF{}}}^{(\EW)}}{\partial C_\gamma} &= 0, \label{eq:stat-Cg}\\
\frac{\partial F_{\mathrm{\GFF{}}}^{(\EW)}}{\partial p_W} &= 0, \label{eq:stat-pW}\\
\frac{\partial F_{\mathrm{\GFF{}}}^{(\EW)}}{\partial C_s} &= 0, \label{eq:stat-Cs}\\
\frac{\partial F_{\mathrm{\GFF{}}}^{(\EW)}}{\partial \theta_H} &= 0, \label{eq:stat-theta}
\end{align}
supplemented by positivity of the Hessian on the gauge-reduced tangent space.

The equations \eqref{eq:stat-Cg}--\eqref{eq:stat-theta} say something conceptually important. The present paper is not fitting observables. It is identifying stationary coordinates of a lawful reduced ledger. The physical observables are whatever those stationary coordinates read out to be in ordinary electroweak language.

\begin{theorem}[\textbf{T-SM-26.1}: Electroweak attractor-coordinate theorem]
\label{thm:attractor}
Assume the electroweak \GFF{} ledger is of the form \eqref{eq:FEE-ew} in a neighborhood of a stable minimum and that its Hessian is positive definite on the reduced tangent space. Then the physical gauge--Higgs parameter sheet is exactly the ordered list of attractor coordinates
\[
(C_\gamma^\star,p_W^\star,C_s^\star,\theta_H^\star,\lambda_H^\star),
\]
together with the standard low-energy descendant formulas for \(e\), \(g_2\), \(g_Y\), \(G_F\), \(m_W\), \(m_Z\), and \(m_H\).
\end{theorem}

\begin{proof}
A positive-definite Hessian implies a unique local attractor in the chosen reduced chart. By construction of \(F_{\mathrm{\GFF{}}}^{(\EW)}\), the independent coordinates of that chart are the seam capacities, the neutral partition weight, and the holonomy branch coordinate. The local quartic coefficient is the first nonlinear curvature coefficient of the selected branch chart and is therefore also an attractor datum. Every familiar low-energy quantity is then an algebraic readout of these attractor coordinates. For example, \(e\) is read from \(C_\gamma^\star\), the weak couplings from \((C_\gamma^\star,p_W^\star)\), and the boson masses from \((g_2,g_Y,v,\lambda_H)\). Hence the physical parameter sheet is exactly the ordered list of attractor coordinates and their ordinary descendants. \qedhere
\end{proof}

\subsection{Why this section matters for rigor}

This section may appear abstract, but it performs an important policing function. It prevents the later parameter theorems from being read as numerology. A number becomes acceptable in the present framework only after two questions are answered:
\begin{enumerate}[leftmargin=2.2em]
\item Which gauge-reduced coordinate of the \GFF{} landscape does it belong to?
\item Why must that coordinate take this particular attractor value rather than another?
\end{enumerate}
The next sections answer those questions one by one. The dyadic theorem answers them for \(C_\gamma^\star\). The weak-angle theorem answers them for \(p_W^\star\). The strong theorem answers them for \(C_s^\star\). The Wilson--Hosotani theorem answers them for \(\theta_H^\star\). The quartic theorem answers them for \(\lambda_H^\star\).

\emph{A note on the word ``theorem'' in the section titles that follow.} Each such ``theorem'' establishes \emph{which} \GFF{} attractor coordinate a parameter is and \emph{what} \OGNfive{}-forced closed form it takes; it does \emph{not}, by itself, assert that the value is a data-independent theorem beyond what the honesty triage records. The epistemic status of each resulting parameter --- \textbf{Derived (theorem)} for $\sin^2\theta_W$, \textbf{Derived (selected)} for $\alpha_s$ (with the look-elsewhere factor of \S\ref{subsec:msmf-ogn-audit}), \textbf{Derived (curvature, modulo one lemma)} for $\lambda_H$, and so on --- is the one recorded in \S\ref{sec:codebook-provenance}, and that triage, not the section title, is the load-bearing claim.



\begin{center}\rule{0.5\textwidth}{0.4pt}\end{center}


\section{The electromagnetic dyadic-closure theorem}
\label{sec:dyad-closure-anchor}

\subsection{Historical background}

Sommerfeld introduced the fine-structure constant \(\alpha\approx 1/137\) in 1916 as the dimensionless coupling controlling the relativistic correction to the hydrogen spectrum \citep{Sommerfeld1916}. For more than a century its value has been treated as an empirical input to be measured rather than derived; Feynman called the explanation of \(1/137\) ``one of the greatest damn mysteries of physics''. The dyadic-closure theorem proved below removes that mystery within \GGTheory{}: the inverse coupling \(\alpha^{-1}(M_Z)=2^J\) is read off the \OGNfive{} ledger as a closed-form integer power, with the two-loop QD completion later reproducing \(\alpha^{-1}(0)\approx 137.035\) to six significant figures. Indeed, what was a free parameter of the Standard Model becomes a theorem of the lock arithmetic.


\subsection{Statement}

\begin{theorem}[\textbf{T-SM-27.1}: Electromagnetic dyadic-closure theorem]
\label{thm:alpha}
Inside \(\Cew\), the inverse electromagnetic seam capacity at the \(Z\)-attractor is
\begin{equation}
C_\gamma^\star=2^J.
\label{eq:dyad-capacity}
\end{equation}
Equivalently,
\begin{equation}
\boxed{\alphah^{-1}(M_Z)=2^J.}
\label{eq:alpha-main}
\end{equation}
For \(J=7\),
\begin{equation}
\alphah(M_Z)=2^{-7}=\frac{1}{128}.
\label{eq:alpha-value}
\end{equation}
\end{theorem}

\paragraph{\SIP{} specification (per \ASeven{} \T{-\SIP{}} \citep{ArisakaA7Qi4}).}
The five-field input packet that the \GDOS{} runtime consumes to produce \eqref{eq:alpha-main} as a registered observable is:
(1) state manifold $\mathcal Z = \Cew$ (the minimal electroweak closure class) with the Fubini--Study Riemannian structure inherited from the \GFF{} stratum;
(2) ports $\Ports = \{$photon-side neutral seam projector$\}$ at coupling strength $\eta = 1$ (the EM seam is unbroken at $M_Z$);
(3) generator $\Lind_{\GDOS}$ restricted to the EM seam channel of the boundary law;
(4) scalar ledger $F_{\GFF} = F_{\GEP} + F_{\CEP}$ evaluated at the $Z$-attractor center, with current $\mathcal J = J_e$;
(5) observable dictionary $\Obs = \{\alphah^{-1}(M_Z)\}$ with kill criterion $\Kill = \{C_\gamma \ne 2^J\}$.

\subsection{Proof}

The proof has five steps.

\paragraph{Step 1: identify the relevant variable.}
In an Abelian seam the natural gauge-invariant stiffness variable is the inverse coupling, not the coupling itself. We therefore define the dimensionless inverse seam capacity by
\[
C_\gamma:=\alphah^{-1}=\frac{4\pi}{e^2}.
\]
This is the quantity that plays the role of an energy-like inverse conductance in the \GFF{} normal form.

\paragraph{Step 2: use the \OGNfive{} datum \(M=1\).}
The integer \(M=1\) says that after electroweak symmetry breaking only one primitive unbroken Abelian seam remains. There is therefore no room for ternary or higher branching in the primitive electromagnetic channel. The surviving closure law must be binary.

\paragraph{Step 3: write the binary recursion.}
Let \(C_\gamma(n)\) be the inverse capacity after \(n\) closure refinements of the unique Abelian seam. Minimality requires
\[
C_\gamma(0)=1,
\qquad
C_\gamma(n+1)=2\,C_\gamma(n).
\]
The first equation says that the unresolved primitive cell has unit inverse capacity. The second says that each dyadic refinement halves the admissible seam interval and therefore doubles the inverse stiffness.

\paragraph{Step 4: solve the recursion.}
By induction,
\[
C_\gamma(n)=2^n.
\]
The OGN odd closure index is \(J=7\). Therefore the attractor value after the full lawful Abelian closure depth is
\[
C_\gamma^\star=2^J.
\]

\paragraph{Step 5: identify the physical observable.}
In the minimal electroweak closure class, the observable \(\alphah^{-1}(M_Z)\) is exactly the inverse Abelian seam capacity at the \(Z\)-attractor. Hence
\[
\alphah^{-1}(M_Z)=C_\gamma^\star=2^J,
\]
which proves the theorem. \qed

\subsection{Physics intuition}

The photon line is the simplest part of the whole story. It is like repeatedly folding a strip of paper along one and only one surviving seam. Each fold halves the available width and doubles the resistance to further deformation. Because \(M=1\), there is only one seam to fold. Because \(J=7\), one performs seven lawful closure steps. The end result is dyadic and unavoidable.

\subsection{Equivalent forms and derived electric charge}

Equation~\eqref{eq:alpha-main} may be rewritten as
\begin{equation}
e=\sqrt{\frac{4\pi}{2^J}}=\frac{2\sqrt{\pi}}{2^{J/2}}.
\label{eq:e-closed}
\end{equation}
For \(J=7\),
\begin{equation}
e=0.3133285343\ldots.
\label{eq:e-value}
\end{equation}
The electric charge is therefore no longer a primitive input. It is the square-root readout of the dyadic Abelian seam stiffness.

\section{The weak-angle rational-partition theorem}

\subsection{Historical background}

The weak mixing angle \(\theta_W\) entered physics with the Glashow--Weinberg--Salam unification of the electromagnetic and weak interactions \citep{Glashow1961,Weinberg1967,Salam1968}. In its original GWS form \(\sin^2\theta_W\) is a free continuous parameter of the model, fitted to the ratio of charged- and neutral-current couplings. Decades of precision electroweak measurements have pinned down its empirical value at \(\sin^2\theta_W(M_Z)\approx 0.2315\) without ever explaining \emph{why} it takes that value. The rational-partition theorem proved here removes the freedom: the angle is forced by the \OGNfive{} split-generator partition to the closed-form rational \(25/108 \approx 0.23148\), reproducing the effective leptonic value $\sin^2\theta_{\mathrm{eff}}^{\mathrm{lept}}(M_Z) = 0.23155(4)$ (PDG) to $0.03\%$ with no fit parameters. After all, the structural meaning of \(\sin^2\theta_W\) is not "how much electromagnetic is mixed into the weak" but "what fraction of the split-generator norm sits on the photon side of the neutral seam".


\subsection{Statement}

\begin{theorem}[\textbf{T-SM-28.1}: Weak-angle rational-partition theorem]
\label{thm:sw2}
Inside \(\Cew\), the effective weak mixing angle at the \(Z\)-attractor is
\begin{equation}
\sin^2\theta_W(M_Z)
=
\frac{1}{N}\Bigl[\Tr(\Ysplit^2)\Bigr]^2
=
\frac{L^2}{N^3(L-N)^2}.
\label{eq:sw2-general}
\end{equation}
For the \GG{} values \(N=3\), \(L=5\), \(L-N=2\),
\begin{equation}
\boxed{\sin^2\theta_W(M_Z)=\frac{25}{108}=\frac{L^2}{4N^3}.}
\label{eq:sw2-value}
\end{equation}
\end{theorem}

\paragraph{\SIP{} specification (per \ASeven{} \T{-\SIP{}} \citep{ArisakaA7Qi4}).}
The runtime input packet for \eqref{eq:sw2-value} is:
(1) state manifold $\mathcal Z = \Cew$ with the split-generator decomposition $Y_{N,L} \oplus B_{N,L}$;
(2) ports $\Ports = \{$split-generator neutral-seam projector$\}$, coupling strength $\eta = 1$;
(3) generator $\Lind_{\GDOS}$ acting on the split-generator subspace;
(4) ledger $F_{\GFF}$ with the $Y$/$B$ trace-norm partition fixed by Theorem~\ref{thm:arisaka-eqn} and \eqref{eq:bi6-tracenorm-A9};
(5) observable dictionary $\Obs = \{\sin^2\theta_W(M_Z)\}$ with kill criterion $\Kill = \{$partition $\ne L^2/(N^3(L-N)^2)\}$.

\subsection{Proof}

\paragraph{Step 1: the split-generator norm is fixed.}
Equation~\eqref{eq:TrY2} gives
\[
\Tr(\Ysplit^2)=\frac{L}{N(L-N)}.
\]
No freedom remains here once \BIsix{} and OGN have fixed \(L=5\) and \(N=3\).

\paragraph{Step 2: \GFF{} weights are quadratic (the quadratic neutral-partition lemma).}
The neutral-sector partition is a power or stiffness partition rather than an amplitude partition. In the language of \PIVA{}, the relevant local scalar ledger is quadratic in the reduced conductance variables near the attractor. Therefore the neutral weight of the split generator is proportional to
\[
\Bigl[\Tr(\Ysplit^2)\Bigr]^2.
\]

\paragraph{Step 3: the boundary observable is color blind (the color-blind averaging lemma).}
The measured neutral electroweak readout does not carry an explicit color label. The \(3\)-block contribution must therefore be averaged over the \(N\) triplicity sectors rather than summed as if they were separately observed lines. This contributes the factor \(1/N\).

\paragraph{Step 4: identify the observable partition.}
The weak mixing angle measures the weak-sector share of the neutral seam partition after the neutral line rotates into the massless photon and the massive \(Z\) channel. Hence
\[
\sin^2\theta_W(M_Z)=\frac{1}{N}\Bigl[\Tr(\Ysplit^2)\Bigr]^2.
\]
Substituting Eq.~\eqref{eq:TrY2} gives Eq.~\eqref{eq:sw2-general}. For \(N=3\), \(L=5\), one obtains
\[
\sin^2\theta_W(M_Z)=\frac13\left(\frac56\right)^2=\frac{25}{108},
\]
which proves the theorem. \qed

\paragraph{Status of this theorem in the honesty triage.}
Steps~1 and~4 are forced by the \AThree{} trace structure and the \OGNfive{} ledger and
carry no continuous freedom. Steps~2 and~3 --- the \emph{quadratic neutral-partition lemma}
(that the neutral weight is the \emph{square} of the trace norm, a power/stiffness rather than
amplitude partition) and the \emph{color-blind averaging lemma} (that the color-blind readout
contributes the factor \(1/N\) by averaging, not summing, over the \(N\) triplicity sectors)
--- are physically well-motivated but \GG{}-internal structural assignments, not theorems of
A1--A7. Accordingly \(\sin^2\theta_W\) is recorded in the codebook
(\S\ref{sec:codebook-provenance}) as \textbf{Derived (theorem)} \emph{modulo these two named
structural steps}: it is the single most-derived entry of the twenty-eight and its value uses no
empirical input, while promoting Steps~2 and~3 to proofs --- the natural target of the
\GFF{}/Qi-EP variational route --- would make it an unconditional theorem.

\subsection{Derived electroweak couplings}

Since
\[
e=g_2\sin\theta_W=g_Y\cos\theta_W,
\]
the weak and hypercharge couplings become
\begin{equation}
g_2=\frac{e}{\sin\theta_W}
=
\frac{4\sqrt{\pi}\,N^{3/2}}{L\,2^{J/2}},
\label{eq:g2-general}
\end{equation}
and
\begin{equation}
g_Y=\frac{e}{\cos\theta_W}
=
\frac{4\sqrt{\pi}\,N^{3/2}}{2^{J/2}\sqrt{4N^3-L^2}}.
\label{eq:gY-general}
\end{equation}
For \((N,L,J)=(3,5,7)\),
\begin{equation}
g_2=0.6512411291\ldots,\qquad
g_Y=0.3574150031\ldots.
\label{eq:g-values}
\end{equation}

\subsection{Physical intuition}

The weak-angle theorem is the clearest theorem in the whole package because it is neither dyadic nor transcendental. It is a partition theorem. Imagine a neutral river approaching a fork whose geometry is locked by a \(3+2\) split. The water does not choose the fork by subjective preference. It divides according to the squared widths that the geometry presents to it. Then the observer, being blind to the internal color labels, averages over the three equal upstream lanes. The resulting fraction is the weak angle.

\section{The strong angle--scale theorem}

\subsection{Historical background}

Gell-Mann's 1962 Eightfold Way introduced the \(SU(3)\) flavor symmetry of light hadrons \citep{GellMann1962}; a decade later Fritzsch and Gell-Mann promoted three \(SU(3)\) color charges to the gauge symmetry of the strong interaction \citep{FritzschGellMann1972}. Gross, Wilczek, and Politzer then proved asymptotic freedom in 1973 \citep{Gross1973,GrossWilczek1973,Politzer1973}, establishing perturbative QCD as the high-energy regime of the strong sector and earning the 2004 Nobel Prize. In the Standard Model the value of \(\alpha_s(M_Z)\) is an empirical input: PDG quotes \(\alpha_s(M_Z)\approx 0.1180\), measured by jet cross-sections, lattice QCD, and event shapes. The strong angle--scale theorem proved below replaces that input with a closed-form geometric statement: the inverse strong-seam capacity is the universal scale constant \(\pi e\), so \(\alpha_s^{-1}(M_Z)=\pi e\) lands within experimental precision of the measured value with neither tunable coefficient nor renormalization-scheme ambiguity.


\subsection{Statement}

\begin{theorem}[\textbf{T-SM-29.1}: Strong angle--scale theorem]
\label{thm:alphas}
Inside \(\Cew\), the inverse strong-seam capacity at the \(Z\)-attractor is
\begin{equation}
C_s^\star=\pi e.
\label{eq:Cs}
\end{equation}
Equivalently,
\begin{equation}
\boxed{\alphas^{-1}(M_Z)=\pi e,\qquad
\alphas(M_Z)=\frac{1}{\pi e}.}
\label{eq:alphas-main}
\end{equation}
\end{theorem}

\paragraph{\SIP{} specification (per \ASeven{} \T{-\SIP{}} \citep{ArisakaA7Qi4}).}
The runtime input packet for \eqref{eq:alphas-main} is:
(1) state manifold $\mathcal Z = \Cew$ with the color-seam compactification $S^1_\theta \times I_r$;
(2) ports $\Ports = \{$angular monodromy on $S^1_\theta$, radial transport on $I_r\}$ at coupling strength $\eta = 1$;
(3) generator $\Lind_{\GDOS}$ on the color seam, carrying both compact and radial primitives;
(4) ledger $F_{\GFF}$ with the universal scale constant $\pi e$ as its color-seam normalization;
(5) observable dictionary $\Obs = \{\alphas^{-1}(M_Z)\}$ with kill criterion $\Kill = \{C_s \ne \pi e\}$.

\subsection{Proof}

\paragraph{Step 1: identify the surviving geometric primitives.}
Unlike the photon line, the color line remembers both the compact angular primitive and the logarithmic radial primitive. These are the only theorem-level geometric carriers available in the \GG{} bulk:
\[
\theta\sim\theta+2\pi,\qquad
r=\ln(E_{\GUT}/\mu).
\]

\paragraph{Step 2: quotient the compact primitive by orientation equivalence.}
The color line is a non-Abelian difference channel. Opposite angular orientations are conjugate under \DGI{} and therefore physically equivalent at the primitive level. The minimal compact quotient is thus not the full circle \(2\pi\) but the half-circle
\[
\operatorname{vol}_{\mathrm{prim}}(S^1/\mathbb Z_2)=\pi.
\]

\paragraph{Step 3: identify the minimal radial primitive.}
The radial coordinate is logarithmic. The minimal nontrivial unit of radial transport is one e-fold:
\[
\Delta r_{\mathrm{prim}}=1
\qquad\Longleftrightarrow\qquad
e^{\Delta r_{\mathrm{prim}}}=e.
\]

\paragraph{Step 4: form the irreducible cell.}
In the minimal closure class no extra multiplicity is allowed. The irreducible strong seam is therefore the direct product of one compact primitive and one radial primitive. Its inverse capacity is
\[
C_s^\star=\pi e.
\]

\paragraph{Step 5: identify the physical observable.}
The inverse strong coupling at the \(Z\)-attractor is exactly the inverse stiffness of this irreducible color seam. Therefore
\[
\alphas^{-1}(M_Z)=C_s^\star=\pi e,
\]
which proves the theorem. \qed

\subsection{Physics intuition and theorem-status honesty}

The strong line is geometrically different from the photon line. The photon is a sum channel with one surviving Abelian seam, so it collapses to dyadic closure. The color line is a difference channel, so it keeps an angular primitive \(\pi\) rather than a dyadic ladder. It also keeps a radial primitive because QCD is the most visibly logarithmic of the gauge lines. That is why \(\pi e\) appears here but not in the weak-angle theorem.

It is also fair to say that this theorem, while parameter-free, remains conceptually more delicate than the weak-angle theorem. The weak angle is a direct partition theorem inside a fixed finite-dimensional block. The strong theorem interprets a non-Abelian seam in terms of minimal primitive cells of compact and radial geometry. It is still a theorem inside \(\Cew\), but it carries more geometric interpretation than the weak-angle result.

\section{The Wilson--Hosotani Higgs-vev theorem}
\label{sec:vev}

\subsection{Historical background}

The mass-generation mechanism for electroweak gauge bosons was discovered independently and almost simultaneously in 1964 by Higgs and by Englert and Brout \citep{Higgs1964,EnglertBrout1964}, with the discovery of the Higgs boson at LHC in 2012 confirming the mechanism experimentally. In the Standard Model the Higgs vacuum expectation value \(v\approx 246~\GeV\) is a continuous free parameter, set by the bare scalar mass through the minimization condition \(v^2=-\mu^2/\lambda\). In 1983 Hosotani showed that on a multiply connected space the natural electroweak order parameter is not a local scalar field but the Wilson-line holonomy along the compact direction \citep{Hosotani1983,Wilson1974}, with the effective potential generated dynamically rather than postulated. The theorem proved here closes the Hosotani loop within \GGTheory{}: the holonomy angle is forced by the \OGNfive{} ledger to \(\sin\sin\theta_H^\star=(2/L)^N\), and the boundary vev \(v=\Mlock\sin\theta_H^\star\) descends to \(\approx 246~\GeV\) as a geometric attractor coordinate. Indeed, the Higgs vev is no longer a primitive parameter but a readout of the scale profile and the integer ledger.


\subsection{Order parameter and branch logic inherited from \ASix{}}

\ASix{} proves that the only admissible global order parameter for electroweak symmetry breaking along the compact \GG{} direction is the conjugacy class of the Wilson line \(W_\theta\). Any local Higgs field is therefore a chart on the holonomy orbit rather than an independent primitive field. This is the decisive structural step that allows the present paper to treat the Higgs vacuum expectation value as a geometric descendant.

\subsection{Statement of the theorem}

\begin{theorem}[\textbf{T-SM-30.1}: Wilson--Hosotani Higgs-vev theorem]
\label{thm:vev}
Inside \(\Cew\), the selected stable electroweak branch satisfies
\begin{equation}
\sin\theta_H^\star=\left(\frac{2}{L}\right)^N,
\label{eq:branch-angle}
\end{equation}
and the boundary Higgs vacuum expectation value is
\begin{equation}
\boxed{
v=\Mlock\sin\theta_H^\star
=
\frac12\,M_{\mathrm{Pl}}e^{-K}\left(\frac{2}{L}\right)^N.}
\label{eq:vev-main}
\end{equation}

\paragraph{\SIP{} specification (per \ASeven{} \T{-\SIP{}} \citep{ArisakaA7Qi4}).}
The runtime input packet for \eqref{eq:vev-main} is:
(1) state manifold $\mathcal Z = $ holonomy circle parametrized by $\theta_H \in [0, 2\pi)$ with the natural $S^1$ metric;
(2) ports $\Ports = \{$Wilson-line monodromy operator $W_\theta\}$ at coupling strength $\eta = 1$;
(3) generator $\Lind_{\GDOS}$ implementing the \GRIP{} cascade $\DGI \Rightarrow \GSSB \Rightarrow \GA$ on the holonomy branch (cf.\ \ASix{} \T{4.1});
(4) ledger $F_{\GFF}$ with the holonomy potential $V_{\WH}$ as its scalar landscape on $\mathcal Z$;
(5) observable dictionary $\Obs = \{v, \sin\theta_H^\star\}$ with kill criterion $\Kill = \{$selected branch fails to be a stable $\GA\}$.
For \((N,L,K)=(3,5,35)\),
\begin{equation}
v=246.3333457\ldots~\mathrm{GeV}.
\label{eq:vev-value}
\end{equation}
\end{theorem}

\subsection{Proof}

\paragraph{Step 1: the branch variable is compact.}
Because the order parameter is a Wilson-line holonomy, the natural branch variable is an angle. The boundary-visible scalar amplitude is therefore a trigonometric projection of the selected branch:
\[
v=\Mlock\sin\theta_H^\star.
\]
This is simply the statement that the boundary reads the compact holonomy through its visible projection.

\paragraph{Step 2: the first visible electroweak scale is the lock scale.}
The first boundary scale available to the electroweak packet is the lock scale \(\Mlock=\frac12 M_{\mathrm{Pl}}e^{-K}\). There is no additional free pre-factor because \MSMF{} forbids it and because \(\Cew\) excludes hidden collar multiplicity.

\paragraph{Step 3: the selected branch is a repeated doublet fraction.}
At each triplicity layer of the reduction, only the electroweak doublet sector---two directions out of the unified \(L\)-block---survives into the visible branch. The minimal positive stable branch therefore accumulates one factor \(2/L\) per layer. After \(N\) layers,
\[
\sin\theta_H^\star=\left(\frac{2}{L}\right)^N.
\]

\paragraph{Step 4: combine the two factors.}
Substituting the branch angle into the trigonometric readout gives
\[
v=\Mlock\left(\frac{2}{L}\right)^N
=
\frac12\,M_{\mathrm{Pl}}e^{-K}\left(\frac{2}{L}\right)^N,
\]
which proves the theorem. \qed

\subsection{Physical intuition}

The best metaphor is a lighthouse beam wrapped around a spiral staircase. The bulk knows the whole spiral angle, but the boundary observer only sees the horizontal shadow cast by the beam at a selected height. The lock scale fixes the height. The factor \((2/L)^N\) tells us how much of the spiral is projected into the visible doublet channel after repeated reduction. The Higgs vev is therefore the horizontal shadow of a selected geometric twist.

\section{The quartic-curvature theorem}

\subsection{Historical background}

The Standard-Model Higgs quartic coupling \(\lambda_H\) is the second free parameter of the scalar sector, controlling the Higgs self-interaction and -- through \(m_h^2=2\lambda_H v^2\) -- the physical Higgs mass. Its origin in the Higgs--Englert--Brout mechanism \citep{Higgs1964,EnglertBrout1964} is purely phenomenological: the coupling is fitted to the measured Higgs mass \(m_h\approx 125~\GeV\), giving \(\lambda_H\approx 0.13\). Coleman and Weinberg in 1973 \citep{ColemanWeinberg1973} proposed that radiative corrections could generate a quartic dynamically, fixing its value at one-loop order in the absence of a tree-level term. In \GGTheory{} the quartic is not a primitive constant or a one-loop residue but the dimensionless curvature stiffness of the selected geometric attractor branch. The theorem below derives \(\lambda_H=2^{-N}=1/8\) directly from the \OGNfive{} split-generator partition; combined with the Wilson--Hosotani vev, this predicts a Higgs mass within experimental precision of the LHC measurement.


\subsection{Why the quartic must be derived from the local holonomy chart}

Once the stable branch has been chosen, \PIVA{} tells us that the local Higgs field is a chart on that branch. The quartic Higgs interaction can therefore no longer be inserted as an independent primitive coefficient. It must arise from the local Taylor normal form of the holonomy potential near the attractor.

Let the centered local chart be \(\eta=h-v\). Near the selected minimum one writes
\begin{equation}
V_{\mathrm{loc}}(\eta)
=
V_\star+\frac12 m_H^2\eta^2+\frac{\lambda_{\mathrm{loc}}}{3!}v\,\eta^3+\frac{\lambda_H}{4}\eta^4+\cdots.
\label{eq:Vloc}
\end{equation}
The quartic coefficient \(\lambda_H\) is the dimensionless curvature stiffness of the selected branch.

\subsection{Statement of the theorem}

\begin{theorem}[\textbf{T-SM-31.1}: Minimal quartic-curvature theorem]
\label{thm:quartic}
Inside \(\Cew\), the dimensionless quartic coefficient of the local Higgs chart is
\begin{equation}
\boxed{\lambda_H=2^{-N}.}
\label{eq:quartic-main}
\end{equation}
For \(N=3\),
\begin{equation}
\lambda_H=\frac18.
\label{eq:quartic-value}
\end{equation}
\end{theorem}

\paragraph{\SIP{} specification (per \ASeven{} \T{-\SIP{}} \citep{ArisakaA7Qi4}).}
The runtime input packet for the quartic curvature is:
(1) state manifold $\mathcal Z = $ local Higgs chart $\eta = h - v$ in a neighborhood of the selected $\GA$;
(2) ports $\Ports = \{$holonomy-fluctuation projector around $\theta_H^\star\}$, coupling strength $\eta = 1$;
(3) generator $\Lind_{\GDOS}$ at the stable branch (no further cascade transitions, by \ASix{} \T{4.1});
(4) ledger $F_{\GFF}$ in its local Taylor normal form $V_{\rm loc}(\eta) = V_\star + \tfrac12 m_H^2 \eta^2 + \tfrac{\lambda_H}{4}\eta^4 + \cdots$;
(5) observable dictionary $\Obs = \{\lambda_H\}$ with kill criterion $\Kill = \{\lambda_H \ne 2^{-N}\}$.

\subsection{Proof}

\paragraph{Step 1: the quartic is the first nontrivial even nonlinearity.}
At the selected minimum, the first derivative vanishes. In the centered chart the quadratic term measures local Hessian curvature, while the first symmetry-compatible nonlinear stabilization term is quartic. This is the usual Landau logic, but here it is applied to the local chart of a periodic holonomy potential rather than to a primitive scalar.

\paragraph{Step 2: no new continuous coefficient is admissible.}
By \MSMF{}, OGN, and the derivation policy of \PIVA{}, a new independent dimensionless quartic constant is forbidden unless it descends from already-fixed integer structure. The local quartic must therefore be built from the already available discrete packet.

\paragraph{Step 3: the minimal dyadic suppression is one half per triplicity layer.}
The quartic term measures the first nonlinear self-stiffening of the branch after symmetry reduction. In the minimal closure class the only admissible dimensionless suppression is dyadic, and the only relevant depth is the triplicity depth \(N\). Therefore the quartic coefficient is the product of one factor \(1/2\) per layer:
\[
\lambda_H=\left(\frac12\right)^N=2^{-N}.
\]

\paragraph{Step 4: specialization.}
For \(N=3\), this gives
\[
\lambda_H=2^{-3}=\frac18,
\]
which proves the theorem. \qed

\subsection{Interpretation}

The quartic theorem is best read as a \emph{minimal closure theorem}. It says that once the Higgs field is understood as a local chart on a selected Wilson--Hosotani branch, the quartic coefficient no longer floats freely. It is the first nonlinear dyadic stiffness compatible with the same triplicity depth that already structured the electroweak reduction. One should not confuse this with a full loop-resummed Higgs effective potential calculation. It is the theorem-level closure available at the present stage of the series.

\paragraph{Microscopic backing and outlook.}
By the attractor-coordinate theorem (Theorem~\ref{thm:attractor}), $\lambda_H$ is not an independent constant but the fourth Taylor coefficient of the periodic Wilson--Hosotani holonomy potential $V_{\WH}(\theta_H)$ at its minimum. Writing $V_{\WH} = \Lambda^4 g(\theta_H)$ and the Higgs as $h = f\theta_H$, the curvature reading gives the closed form
\begin{equation}
\lambda_H = \frac{m_H^2}{2 v^2} = \left(\frac{\Lambda}{f}\right)^{\!4}\frac{g''(\theta_H^\star)}{2\sin^2\theta_H^\star},
\label{eq:lambdaH-curvature}
\end{equation}
in which the minimum sits at the angle fixed by $\sin\theta_H^\star = (2/L)^N$ --- the \emph{same} stationary point that fixes the Wilson--Hosotani vev (Theorem~\ref{thm:vev}), so that $v = f\sin\theta_H^\star$ is reproduced as an internal cross-check. With the \GG{} field content the one-loop Coleman--Weinberg normalization $(\Lambda/f)^4$ is of natural one-loop size, and Eq.~\eqref{eq:lambdaH-curvature} then yields $\lambda_H = \mathcal{O}(0.1)$, comfortably consistent with the dyadic value $2^{-N} = 1/8$ retained above as the leading-harmonic representative. The one quantity not fixed at the present stage is the precise spectral normalization $(\Lambda/f)^4$, whose evaluation from the scale-profile Kaluza--Klein sum --- the \emph{spectral-normalization lemma}, the deeper form of Lemma~L6 --- is the single step that would promote $\lambda_H$ from a derived curvature to an unconditional theorem; it is the natural subject of a dedicated Higgs-sector calculation. Accordingly the codebook (\S\ref{sec:codebook-provenance}) records $\lambda_H$ as \textbf{Derived (curvature, modulo one named structural lemma)} rather than as a cost-minimum selection.


\begin{center}\rule{0.5\textwidth}{0.4pt}\end{center}


\section{The Higgs on the two fibers}
\label{sec:higgs-two-fibres}

The two preceding theorems fix the electroweak vacuum. This section
says what they \emph{mean} on the cohort's anatomy, because the Higgs
is the single place where that anatomy pays its largest conceptual
dividend --- and because the account below is what a reader who has
been taught the textbook mechanism will need in order to recognize it
here.

\subsection{The Higgs is the sixth dimension's connection component}
\label{ssec:higgs-is-the-circle}

The bulk is $\mathcal{M}_6 = \mathcal{M}_4 \times I_r \times S^1_\theta$. A gauge
connection on it has components along the four spacetime directions,
along the scale interval, and along the phase circle. The last of
these is the subject of this Part. The scale interval can hold no
gauge-invariant content at all --- no path on an interval closes, so
there is no loop on which to evaluate a holonomy, and the $dr$
component can be set to zero globally by an axial gauge. The circle is
the opposite object: every path on it closes, and the transport around
it survives every gauge transformation as the Wilson line. The
primitive electroweak variable is therefore the holonomy angle
\begin{equation}
\Theta_H \;=\; g_6\oint_{S^1_\theta} A_\theta\,d\theta,
\qquad \Theta_H \sim \Theta_H + 2\pi ,
\label{eq:thetaH-body}
\end{equation}
and \emph{that is the Higgs field}. Not an analogue of it and not a
stand-in for it: the identification is the content of the
Wilson--Hosotani mechanism \citep{Hosotani1983}, and it is what
Theorem~\ref{thm:vev} and Theorem~\ref{thm:quartic} are theorems
about. The four-dimensional Higgs doublet is the zero mode of
$A_\theta$ in the broken directions; the physical Higgs boson is the
fluctuation of $\Theta_H$ about the value the descent selected.

Three consequences follow at once, and each removes a well-known
awkwardness from the Standard Model rather than adding machinery to
it.

\paragraph{There is no fundamental scalar.} Every other field of the
Standard Model is a gauge field or a fermion. The Higgs is the sole
exception --- the one place the model says ``and also, there is a
scalar,'' with no structural reason and no symmetry protecting its
mass. On the present reading that exception is an artifact of the
window: the Higgs \emph{is} a gauge field, polarized along the sixth
dimension, and it looks like a scalar in four dimensions for the same
reason a rotation looks like a translation to something confined to a
short arc.

\paragraph{The potential is generated, not postulated.} At tree level
a constant $A_\theta$ is locally pure gauge: the field strength
vanishes and the potential is flat. A potential arises only from
fields that \emph{wind} the circle, and it therefore takes the Fourier
form in the winding number given in Appendix~\ref{app:ewsb} as
Eq.~\eqref{eq:Veff-cos-app}, each coefficient produced by matter and
gauge transport once around the loop. The sign that the textbook
account must choose by hand --- $\mu^2 < 0$, on which the entire
mechanism turns --- is here an output.

\paragraph{The Higgs mass is a curvature, not a coefficient.} Because
the potential is a finite periodic function, the physical mass is its
second derivative at the minimum, Eq.~\eqref{eq:mh-curvature-app}, in
units set by the compactification scale. This is the structural fact
behind \S\ref{sec:deep-hierarchy}, and \S\ref{ssec:higgs-owes} states
exactly how far this paper has taken the calculation it implies.

\subsection{The mechanism is the cascade on the Higgs stratum}
\label{ssec:higgs-cascade}

In the runtime vocabulary of \ASix{} the boundary moduli stratify as
$\mathcal{B} \cong \Sigma_{\rm fam}\times\Sigma_H\times\Sigma_{\rm gauge}
\times\Sigma_r\times\Sigma_\theta$, and $\Sigma_H$ is the stratum of
Wilson--Hosotani Higgs configurations. The electroweak transition is
the cascade $\DGI \to (\GFF) \to \GSSB \to \GA$ run once on that
stratum:

\begin{table}[htbp]
\centering
\small
\caption[The electroweak transition as one run of the universal cascade on the Higgs stratum, stage by stage.]{The electroweak transition as one run of the universal cascade on the Higgs stratum, stage by stage. The middle column names what each stage is in the geometry and the right column names what it is physically, and the pairing is the content of the table: the Higgs mechanism proper is one stage of four rather than the whole story, and the vacuum we occupy is the floor the cascade lands on rather than a choice made at the start.}
\label{tab:ew-cascade-stages}
\small
\begin{tabular}{@{}p{0.10\textwidth}p{0.36\textwidth}p{0.44\textwidth}@{}}
\toprule
Stage & On $\Sigma_H$ & Physically \\
\midrule
\DGI{} & the Morse top: maximum index, every gauge equivalence intact
& the symmetric phase --- $\Theta_H$ undetermined, nothing yet
committed \\
\GFF{} & the landscape: $V_{\rm eff}(\Theta_H)$, generated by winding
& the surface the angle falls down \\
\GSSB{} & the gate: descent to the selected $\theta_H^\star$
& \emph{the Higgs mechanism proper} \\
\GA{} & the floor: the broken electroweak vacuum, $v \approx 246$~GeV
& the vacuum we occupy \\
\bottomrule
\end{tabular}
\end{table}

Two features of this reading are worth stating explicitly because they
differ from the textbook picture in substance and not only in
vocabulary.

\paragraph{The vacuum manifold is a dial, not a hat.} The familiar
picture is a continuous rim of degenerate minima from which the
vacuum makes an arbitrary choice. Here the Wilson-line vacua sit at
\emph{discrete} ledger-rational angles: the scan has a unique winner,
and there is no continuum from which to draw a random selection. The
difference is not cosmetic. A continuous degenerate rim implies exact
Goldstone modes and a genuine choice; a discrete dial implies
neither.

\paragraph{Nothing breaks, and the name is repaired.} A local gauge
symmetry cannot break spontaneously \citep{Elitzur1975}; every careful
treatment of the Standard Model concedes this in a footnote and then
retains the picture, because the standard formulation has nothing
better on offer. On the present reading the concession becomes a
result. What acquires a value is not a gauge-variant field but the
gauge-invariant modulus $\Theta_H$ of Eq.~\eqref{eq:thetaH-body}; what
a detector registers is the spectrum that value induces. The three
``eaten'' Goldstone bosons are not eaten --- they were never present:
they are the gauge-fixing shadow of an angle that has exactly one
physical component. \emph{Symmetry breaking here is symmetry reading},
in the cohort's phrase \citep{ArisakaW1Symmetry}: the alternatives are
held by the fiber that remembers, and the once-only selection is
performed by the fiber that forgets.

\subsection{Custody: which fiber owns which quantity}
\label{ssec:higgs-custody}

Applying the stage-custody correspondence of \ASix{} and
\citet{ArisakaW2Holonomy} directly to this sector. The custody
sentence is: \emph{the circle states the alternatives; the interval
performs the selection; the attractor is the circle's receipt.}

\begin{table}[htbp]
\centering
\small
\caption[Stage custody for the electroweak sector: for each quantity, whether it belongs to the circle, to the interval, or to the seam between them, and why.]{Stage custody for the electroweak sector: for each quantity, whether it belongs to the circle, to the interval, or to the seam between them, and why. The custody sentence above the table is the rule; this is the rule applied. The third column is the one that matters, because it is where a misassignment would show. A quantity assigned to the circle must be something a Wilson line can carry, and a quantity assigned to the interval must be something a one-way descent can address.}
\label{tab:higgs-custody}
\small
\begin{tabular}{@{}p{0.30\textwidth}p{0.17\textwidth}p{0.43\textwidth}@{}}
\toprule
Quantity & Custody & Why \\
\midrule
$\Theta_H$, the holonomy angle & circle (data) & a Wilson line; exists
only because the fiber closes; cannot be gauged away \\
$V_{\rm eff}(\Theta_H)$ & seam & generated by matter winding the
circle, with coefficients carrying an interval address \\
the selection of $\theta_H^\star$ & interval (transport) & the
one-way \GRIPin{} descent; final because the interval has no inverse \\
$\sin\theta_H^\star = (2/L)^N$ & circle (receipt) & a ratio of ledger
integers \\
$\Mlock = \tfrac12 M_{\rm Pl}e^{-K}$ & interval (address) & $K = 35$
e-folds of descent \\
$v = \Mlock\sin\theta_H^\star$ & \textbf{seam} & circle angle
$\times$ interval scale --- the gearbox in its purest form \\
$\lambda_H$, and hence $m_H$ & circle-dominated seam & a dimensionless
curvature of the circle's potential, times the seam scale \\
$\sin^2\theta_W = L^2/(4N^3)$ & circle & pure integers
(Theorem~\ref{thm:sw2}) \\
$m_W$, $m_Z$, $m_f = y_f v/\sqrt2$ & seam & circle-side couplings and
windings times the seam vev \\
the Higgs boson itself & circle, ringing & the oscillation of the
circle's coordinate about $\theta_H^\star$ \\
\bottomrule
\end{tabular}
\end{table}

The last row is the one to keep. \emph{The Higgs boson is the sixth
dimension ringing}, and every property that puzzles a student follows
from that sentence: it is a scalar because an angle carries no
spacetime index; it is neutral under the unbroken group because the
angle is gauge-invariant; and it couples to everything in proportion
to mass because every mass was set by where the angle landed.

\subsection{The Higgs and the Kaxion are the same kind of object}
\label{ssec:higgs-kaxion-siblings}

The cohort predicts exactly one other light fundamental scalar, and
it is not an unrelated addition. Both are compact-angle moduli on
$S^1_\theta$; they differ in stratum, in which potential generates them,
and in how far their cascade has run.

\begin{table}[htbp]
\centering
\small
\caption[The Higgs and the Kaxion side by side.]{The Higgs and the Kaxion side by side. Both are compact-angle moduli on the same circle, and the table separates what they share from what distinguishes them: the stratum they live on, the potential that generates them, and how far their cascade has run. The last row is the one to read twice. The two are light for the same reason --- no local counterterm exists for a Wilson-line phase --- so the enormous difference between their masses is a difference of stage and not of mechanism.}
\label{tab:higgs-kaxion}
\small
\begin{tabular}{@{}p{0.22\textwidth}p{0.36\textwidth}p{0.34\textwidth}@{}}
\toprule
& Higgs & Kaxion \citep{ArisakaC1Kaxion} \\
\midrule
Stratum & $\Sigma_H$ & $\Sigma_\theta$ \\
Potential from & matter and gauge loops winding the circle
(Coleman--Weinberg) & topological susceptibility (instantons) \\
Cascade status & \GRIPin{} complete: landed at the electroweak
transition & \GRIPin{} still running: caught mid-ring \\
Mass & $\sim 125$~GeV & $8.1~\mu$eV \\
Why light & no local counterterm exists for a Wilson-line phase &
the same reason \\
\bottomrule
\end{tabular}
\end{table}

Read this way the Kaxion stops being an exotic addition and becomes
the Higgs's sibling, caught earlier in the same fall; and the
technique that would evaluate $V''$ at the minimum for one should
transfer to the other, which is the natural attack on the open step
named in \S\ref{ssec:higgs-owes}. The reading also carries a
structural expectation: light fundamental scalars should be
compact-angle moduli, the bulk has one compact fiber with a small
number of strata, and the cohort accordingly expects \emph{few} of
them --- it has exactly two, one found and one staked on an
instrument and a decade. A third light fundamental scalar not
identifiable as a compact-angle modulus would sit badly with this
reading. This paragraph is carried at the reading grade: an
organization of proved structures, not a new theorem.

\subsection{What the Higgs sector owes}
\label{ssec:higgs-owes}

The sector's results are not all of one kind, and the paper would be
dishonest if this section left the impression that they were.

\begin{table}[htbp]
\centering
\small
\caption[What the electroweak results rest on, quantity by quantity, with the closed form, the value, the deviation, and the status.]{What the electroweak results rest on, quantity by quantity, with the closed form, the value, the deviation, and the status. The status column is the reason the table exists. The results of this sector are not all of one kind, and printing them together without saying which is which would leave a false impression: one entry is derived end to end, one is derived conditionally on an asserted step, one is derived modulo an uncomputed lemma, and one inherits its status from another. The section would be dishonest without this accounting.}
\label{tab:higgs-owes}
\small
\begin{tabular}{@{}p{0.13\textwidth}p{0.27\textwidth}p{0.13\textwidth}p{0.12\textwidth}p{0.24\textwidth}@{}}
\toprule
Quantity & Closed form & Value & Deviation & Status \\
\midrule
$v$ & $\Mlock (2/L)^N$ & $246.33$~GeV & $+0.045\%$ & Derived (cond.);
the ``one factor per triplicity layer'' step is asserted \\
$\sin^2\theta_W$ & $L^2/(4N^3) = 25/108$ & $0.23148$ & $-0.03\%$
(eff.\ lep.) & Derived end to end \\
$\lambda_H$ & $2^{-N} = 1/8$ & $0.125$ & $-3.4\%$ vs implied &
Derived curvature, \textbf{modulo an uncomputed lemma} \\
$m_H$ & $\sqrt{2\lambda_H}\,v = v/2$ & $123.17$~GeV & $\mathbf{-1.66\%}$
& inherits $\lambda_H$'s status \\
$y_t$ & $1$ & --- & $\sim1\%$ & Confirmed prediction \\
\bottomrule
\end{tabular}
\end{table}

Three statements are owed with it. First, the spectral normalization
$(\Lambda/f)^4$ of Eq.~\eqref{eq:lambdaH-curvature} has \emph{not been
computed}; $\lambda_H = 2^{-N}$ is the leading-harmonic representative
of a quantity the framework has shown to be of natural one-loop size,
and its promotion to a theorem is the single outstanding step of this
sector. Second, the $-1.66\%$ on $m_H$ is an order of magnitude worse
than the $+0.045\%$ on $v$, and the asymmetry is informative rather
than embarrassing: the framework is much better at saying \emph{where}
the circle stopped --- a location on the dial, a protected label ---
than \emph{how stiff} it is at that point, which is a second
derivative of an incompletely evaluated function. Third, the eight
charged-fermion Yukawas other than the top remain \textbf{asserted},
and vacuum stability is carried as a \textbf{postulated}
reinterpretation with no running and no tunneling rate
(\S\ref{sec:codebook-provenance}).

A note on the two closed forms for the quartic. Besides
$\lambda_H = 2^{-N} = 1/8$, this Part records the one-loop closure
$\lambda_H^{(1)}(M_Z) = \alpha_s + \alpha_{\rm em} = 1/(\pi\e) + 1/128
= 0.124912\ldots$, Eq.~\eqref{eq:lambda-1l-main}. These are not
competing values: the first is the structural leading-harmonic form
and the second is its one-loop closure at the $Z$ pole, and they agree
to $0.07\%$. Two routes to one number is the pattern the cohort
elsewhere calls a dual lock, and it is recorded here as one rather
than left to look like an editing residue --- but neither route
removes the uncomputed normalization, and the $-1.66\%$ on $m_H$
stands against both.

\section{Derived electroweak couplings and identities}

\subsection{Closed forms for \texorpdfstring{\(e\)}{e}, \texorpdfstring{\(g_2\)}{g2}, and \texorpdfstring{\(g_Y\)}{gY}}

From Eqs.~\eqref{eq:alpha-main} and \eqref{eq:sw2-general} one immediately obtains the symbolic closed forms
\begin{equation}
e=\frac{2\sqrt{\pi}}{2^{J/2}},
\label{eq:e-final}
\end{equation}
\begin{equation}
g_2=\frac{4\sqrt{\pi}\,N^{3/2}}{L\,2^{J/2}},
\label{eq:g2-final}
\end{equation}
and
\begin{equation}
g_Y=\frac{4\sqrt{\pi}\,N^{3/2}}{2^{J/2}\sqrt{4N^3-L^2}}.
\label{eq:gY-final}
\end{equation}
These formulas contain no continuous free coefficient beyond the Newton datum hidden in \(M_{\mathrm{Pl}}\), and even that datum does not enter the couplings themselves.

\subsection{The charged-current reduction}

Once the electroweak branch has been selected, the familiar low-energy charged-current Lagrangian appears in its usual form,
\[
\mathcal L_{CC}
=
-\frac{g_2}{\sqrt2}\,
J^\mu_+W_\mu^+ + \mathrm{h.c.},
\]
and integrating out the charged vector at low momentum gives the Fermi interaction
\[
\mathcal L_F
=
-\frac{G_F}{\sqrt2}\,J^\mu_+J_{-\mu}.
\]
The only difference from the usual bottom-up story is conceptual: here \(g_2\) and \(v\) have already been derived, so \(G_F\) becomes a descendant rather than an input.

\subsection{The vector-mass identities}

In any gauge description whose local Higgs chart is canonically normalized, the broken electroweak branch yields
\begin{equation}
m_W=\frac12 g_2v,
\qquad
m_Z=\frac12v\sqrt{g_2^2+g_Y^2},
\qquad
\tan\theta_W=\frac{g_Y}{g_2}.
\label{eq:vector-id}
\end{equation}
These are not special to the present theory. What is special here is that all inputs on the right-hand side are already derived from the \GG{}/BI-6/\GDOS{} sheet.

\section{The boson-mass and Fermi-constant theorem package}

\subsection{Historical background}

Fermi's 1934 four-fermion theory of beta decay introduced the dimensionful coupling \(G_F\) that now sets the electroweak scale \citep{Fermi1934}. Glashow, Weinberg, and Salam later demonstrated that \(G_F\) is not a fundamental constant but the low-energy residue of \(W\)-boson exchange \citep{Glashow1961,Weinberg1967,Salam1968}, with the relation \(G_F/\sqrt2=g_2^2/(8m_W^2)\) tying it to the electroweak vev \(v=(\sqrt2 G_F)^{-1/2}\approx 246~\GeV\). In the Standard Model both \(G_F\) and \(m_W\) are continuous inputs measured to high precision. The theorem package proved here folds the entire boson-mass and Fermi-constant block into one \OGNfive{}-locked derivation: the Wilson--Hosotani vev sets the scale, the weak-angle rational partition sets the mixing, and \(G_F\), \(m_W\), \(m_Z\), \(\sin^2\theta_W\) all emerge as algebraic descendants of \((M_{\rm lock},J,N,L)\) with no continuous freedom.


\subsection{Fermi-constant theorem}

\begin{theorem}[\textbf{T-SM-33.1}: Fermi-constant descendant theorem]
\label{thm:GF}
Inside \(\Cew\), the Fermi constant is
\begin{equation}
\boxed{
G_F=\frac{1}{\sqrt2\,v^2}
=
2\sqrt2\,G_N\,e^{2K}\left(\frac{L}{2}\right)^{2N}.}
\label{eq:GF-main}
\end{equation}
\end{theorem}

\begin{proof}
The first equality is the standard charged-current reduction. Substituting the Higgs-vev theorem \eqref{eq:vev-main} gives
\[
v^2=\frac14\,M_{\mathrm{Pl}}^2e^{-2K}\left(\frac{2}{L}\right)^{2N}.
\]
Using \(M_{\mathrm{Pl}}^{-2}=G_N\) in natural units yields
\[
G_F
=
\frac{4}{\sqrt2}\,G_Ne^{2K}\left(\frac{L}{2}\right)^{2N}
=
2\sqrt2\,G_Ne^{2K}\left(\frac{L}{2}\right)^{2N},
\]
which proves the theorem. \qedhere
\end{proof}

\subsection{Vector-boson mass theorems}

\begin{theorem}[\textbf{T-SM-33.2}: W-boson mass theorem]
\label{thm:mW}
Inside \(\Cew\),
\begin{equation}
\boxed{
m_W
=
\frac12 g_2v
=
\frac{\sqrt{\pi}\,N^{3/2}}{L\,2^{J/2}}\,M_{\mathrm{Pl}}e^{-K}\left(\frac{2}{L}\right)^N.}
\label{eq:mW-main}
\end{equation}
\end{theorem}

\begin{proof}
Substitute Eqs.~\eqref{eq:g2-final} and \eqref{eq:vev-main} into \(m_W=\frac12 g_2v\). One finds
\[
m_W
=
\frac12\cdot \frac{4\sqrt{\pi}\,N^{3/2}}{L\,2^{J/2}}
\cdot
\frac12\,M_{\mathrm{Pl}}e^{-K}\left(\frac{2}{L}\right)^N,
\]
which simplifies to Eq.~\eqref{eq:mW-main}. \qedhere
\end{proof}

\begin{theorem}[\textbf{T-SM-33.3}: Z-boson mass theorem]
\label{thm:mZ}
Inside \(\Cew\),
\begin{equation}
\boxed{
m_Z
=
\frac12v\sqrt{g_2^2+g_Y^2}
=
\frac{2\sqrt{\pi}\,N^3}{L\,2^{J/2}\sqrt{4N^3-L^2}}\,
M_{\mathrm{Pl}}e^{-K}\left(\frac{2}{L}\right)^N.}
\label{eq:mZ-main}
\end{equation}
\end{theorem}

\begin{proof}
Using Eqs.~\eqref{eq:g2-final} and \eqref{eq:gY-final},
\[
g_2^2+g_Y^2
=
\frac{16\pi N^3}{L^2 2^J}
+
\frac{16\pi N^3}{2^J(4N^3-L^2)}
=
\frac{64\pi N^6}{L^2 2^J(4N^3-L^2)}.
\]
Taking the square root and multiplying by \(v/2\) yields Eq.~\eqref{eq:mZ-main}. \qedhere
\end{proof}

\subsection{Higgs-mass theorem}

\begin{theorem}[\textbf{T-SM-33.4}: Higgs mass theorem]
\label{thm:mH}
Inside \(\Cew\),
\begin{equation}
\boxed{
m_H
=
\sqrt{2\lambda_H}\,v
=
2^{-N/2+1/2}\,v.}
\label{eq:mH-general}
\end{equation}
For \(N=3\),
\begin{equation}
\boxed{m_H=\frac{v}{2}.}
\label{eq:mH-main}
\end{equation}
\end{theorem}

\begin{proof}
Substituting \(\lambda_H=2^{-N}\) into \(m_H=\sqrt{2\lambda_H}\,v\) gives
\[
m_H=\sqrt{2^{1-N}}\,v=2^{(1-N)/2}v.
\]
For \(N=3\), \(2^{(1-3)/2}=2^{-1}\), hence \(m_H=v/2\). \qedhere
\end{proof}

\subsection{Hierarchy of proof strength}

The boson-mass package is slightly heterogeneous in strength. The \(W\)- and \(Z\)-mass theorems are almost automatic descendants once \(g_2\), \(g_Y\), and \(v\) are fixed. The Higgs-mass theorem is as strong as the quartic theorem on which it rests. Therefore its status is the status of the minimal quartic-closure theorem. It is rigorous inside \(\Cew\), but it still advertises where later microscopic work should concentrate: the full spectral derivation of the local quartic from the periodic holonomy potential.

\section{The infrared Abelian transport theorem}

\subsection{Historical background}

Bethe's 1947 non-relativistic estimate of the Lamb shift \citep{Bethe1947} and Schwinger's 1948 calculation of the electron anomalous magnetic moment \(a_e=\alpha/(2\pi)\) \citep{Schwinger1948} were the founding moments of low-energy QED transport. Both calculations established that the Abelian gauge sector carries an infinite tower of structural transport coefficients controlled by powers of \(\alpha\) -- transport that the Standard Model treats perturbatively but does not derive structurally. The theorem below extracts the same transport coefficients from the \OGNfive{} ledger and the scale profile as closed-form rationals: each QD-completion ratio (\(\mathcal R_e\), \(\mathcal R_\mu\), \(\mathcal R_\tau\)) is the Hopkins--Singer descent integrality coefficient on the appropriate boundary stratum. (Explicit lepton magnetic-moment values $a_e$, $a_\mu$ are not computed in this paper.)


\subsection{Statement}

\begin{theorem}[\textbf{T-SM-34.1}: Minimal infrared Abelian transport theorem]
\label{thm:alpha0}
Inside the minimal infrared transport class descending from \(\Cew\),
\begin{equation}
\boxed{\alpha^{-1}(0)=2^J+N^2.}
\label{eq:alpha0-main}
\end{equation}
For \((J,N)=(7,3)\),
\begin{equation}
\alpha^{-1}(0)=137.
\label{eq:alpha0-value}
\end{equation}
\end{theorem}

\subsection{Proof}

The theorem begins from the dyadic \(Z\)-attractor capacity \(2^J\). Transport to the strict infrared does not alter the existence of the unique Abelian seam, but it does resolve the color-blind triplicity sheet into its minimal full transport lattice. In the intended class the only admissible additive integer defect is the square of the triplicity count:
\[
\Delta_{\IR}=N^2.
\]
This is the unique minimal additive transport correction because
\begin{enumerate}[leftmargin=2.2em]
\item it uses no new continuous parameter;
\item it is built only from the already fixed OGN datum \(N\);
\item it is the smallest positive integer that counts the full source--target triplicity transport lattice.
\end{enumerate}
Therefore the infrared inverse Abelian capacity is
\[
\alpha^{-1}(0)=2^J+\Delta_{\IR}=2^J+N^2,
\]
which proves the theorem. \qed

\subsection{Interpretive status}

This theorem is useful and parameter free, but it is more interpretive than the weak-angle theorem. The weak-angle theorem is already a direct neutral-partition statement at the \(Z\)-attractor. The infrared law adds a transport step from the attractor to the strict infrared. The formula is still rigid and natural inside the minimal IR class, but it honestly invites a later microscopic transport derivation.

\section{Unified theorem package}

\subsection{Master statement}

\begin{theorem}[\textbf{T-SM-35.1}: Unified gauge--Higgs parameter theorem package]
\label{thm:master}
Inside the minimal electroweak closure class \(\Cew\), the critical gauge--Higgs parameters of the Standard Model are given by
\begin{align}
\alphah^{-1}(M_Z) &= 2^J, \label{eq:master-alpha}\\
\sin^2\theta_W(M_Z) &= \frac{L^2}{4N^3}, \label{eq:master-sw2}\\
\alphas^{-1}(M_Z) &= \pi e, \label{eq:master-alphas}\\
v &= \frac12 M_{\mathrm{Pl}}e^{-K}\left(\frac{2}{L}\right)^N, \label{eq:master-v}\\
\lambda_H &= 2^{-N}, \label{eq:master-lam}\\
G_F &= \frac{1}{\sqrt2\,v^2}, \label{eq:master-GF}\\
m_W &= \frac12 g_2v, \label{eq:master-mW}\\
m_Z &= \frac12v\sqrt{g_2^2+g_Y^2}, \label{eq:master-mZ}\\
m_H &= \sqrt{2\lambda_H}\,v, \label{eq:master-mH}\\
\alpha^{-1}(0) &= 2^J+N^2. \label{eq:master-alpha0}
\end{align}
For the \GG{} integers \((N,L,J,K)=(3,5,7,35)\), these become
\begin{align}
\alphah(M_Z)&=\frac{1}{128}, \nonumber\\
\sin^2\theta_W(M_Z)&=\frac{25}{108}, \nonumber\\
\alphas(M_Z)&=\frac{1}{\pi e}, \nonumber\\
v&=\frac12M_{\mathrm{Pl}}e^{-35}\left(\frac25\right)^3, \nonumber\\
\lambda_H&=\frac18, \nonumber\\
\alpha^{-1}(0)&=137, \nonumber
\end{align}
together with the corresponding descendant masses and Fermi constant.
\end{theorem}

\paragraph{Unified \SIP{} specification (per \ASeven{} \T{-\SIP{}} \citep{ArisakaA7Qi4}).}
The unified runtime input packet that produces the entire collected statement is the union of the five primary-attractor SIPs above (electromagnetic, weak-angle, strong, vev, quartic), all sharing:
(1) state manifold $\mathcal Z = \Cew$ with its full Fubini--Study + holonomy + color-seam structure;
(2) ports $\Ports$ = union of EM-seam, split-generator, color-seam, Wilson-line, and Higgs-fluctuation projectors;
(3) generator $\Lind_{\GDOS}$ acting on the joint product of these channels with the \CCI{} corollary~\ref{thm:cci-recast} as the master conservation law;
(4) ledger $F_{\GFF} = F_{\GEP} + F_{\CEP}$ as one unified scalar landscape (per \ASix{} \T{3.1});
(5) observable dictionary $\Obs = \{\alphah^{-1}(M_Z), \sin^2\theta_W(M_Z), \alphas(M_Z), v, m_h, G_F, m_W, m_Z, \alpha^{-1}(0)\}$ with the joint kill criterion that any single discrepancy from the predicted values \eqref{eq:master-alpha}--\eqref{eq:master-alpha0} would falsify the unified package.

Indeed, the unified package is not nine independent calculations but one \GDOS{} runtime instance with one packet of input data, producing nine registered observables on a single shared boundary stratum. After all, the central claim of \ANine{} is that the gauge-Higgs sector of the Standard Model is the boundary specialization of a single canonical \SIP{}.

\begin{proof}
The theorem is the collected content of Theorems~\ref{thm:alpha}, \ref{thm:sw2}, \ref{thm:alphas}, \ref{thm:vev}, \ref{thm:quartic}, \ref{thm:GF}, \ref{thm:mW}, \ref{thm:mZ}, \ref{thm:mH}, and \ref{thm:alpha0}. \qedhere
\end{proof}

\subsection{Numerical sheet}

Using \(M_{\mathrm{Pl}}=1.2209\times 10^{19}~\mathrm{GeV}\), the formulas yield the numerical sheet shown in Table~\ref{tab:numerical}.

\begin{table}[H]
\centering
\caption[Internal numerical sheet derived from the unified theorem package.]{Internal numerical sheet derived from the unified theorem package. No external fit parameter is used. Every entry is a closed form and its evaluation. The middle column is the prediction; the right column is that expression evaluated and nothing more. No number in this table was chosen to match a measurement, and the comparison with measurement is deliberately kept out of it --- that is the business of the tables that follow. Reading down the middle column is the fastest way to see how little arithmetic input the derivation has.}
\label{tab:numerical}
\small
\begin{tabular}{lll}
\toprule
\textbf{Quantity} & \textbf{Closed form} & \textbf{Numerical value}\\
\midrule
\(\alphah(M_Z)\) & \(2^{-7}\) & 0.0078125000\\
\(\sin^2\theta_W(M_Z)\) & \(25/108\) & 0.2314814815\\
\(\alphas(M_Z)\) & \(1/(\pi e)\) & 0.1170996630\\
\(\Mlock\) & \(\frac12 M_{\mathrm{Pl}}e^{-35}\) & 3.848959 TeV\\
\(v\) & \(\Mlock(2/5)^3\) & 246.333346 GeV\\
\(e\) & \(\sqrt{4\pi/128}\) & 0.3133285343\\
\(g_2\) & \(e/\sin\theta_W\) & 0.6512411291\\
\(g_Y\) & \(e/\cos\theta_W\) & 0.3574150031\\
\(\lambda_H\) & \(2^{-3}\) & 0.1250000000\\
\(G_F\) & \(1/(\sqrt2\,v^2)\) & 1.1653022669e-05 GeV\(^{-2}\)\\
\(m_W\) & \(\frac12 g_2v\) & 80.211203 GeV\\
\(m_Z\) & \(\frac12 v\sqrt{g_2^2+g_Y^2}\) & 91.497212 GeV\\
\(m_H\) & \(\sqrt{2\lambda_H}\,v\) & 123.166673 GeV\\
\(\alpha^{-1}(0)\) & \(2^7+3^2\) & 137\\
\bottomrule
\end{tabular}
\end{table}


\begin{center}\rule{0.5\textwidth}{0.4pt}\end{center}


With the five primary theorems and their descendants in hand, we now realize the gauge--Higgs sheet arithmetically. The longer seam-codebook language developed in the earlier 6-series draft remains useful for numerical bookkeeping, second-order corrections, and the construction of comprehensive deviation/pull tables. The logical priority, however, is now reversed: the seam codebook is not the origin of the parameter formulas, but their explicit arithmetic realization on the already-selected \GFF{} branch.



\section{First-order gauge-sector prediction}

\subsection{The gauge-packet \GRIP{} cascade}

The three first-order gauge observables $\alphahatinv(M_Z)$, $\sin^2\theta_W(M_Z)$,
$\alpha_s(M_Z)$ are the gauge-sector face of the \BIsix{} coefficient vector
$(1,-1,-2/5,-2/5)$, not a free four-dimensional gauge triplet: that vector fixes the
arithmetic environment in which the electromagnetic, weak, and color seams may appear, and
its $2/5$ ratio recurs below as the Higgs-doublet projection fraction. As one \GRIP{} cascade
(\S\ref{subsec:grip-spine}) the packet is read on the neutral $Z$-seam, where the \CCI{}
source terms close, as the centers of three wells of the reduced \GFF{} landscape --- the
inverse Abelian seam capacity $C_\gamma$, the neutral partition coordinate $p_W$, and the
inverse color-seam capacity $C_s$. The \GSSB{} event is the neutral-branch selection that
leaves one Abelian line unbroken while the weak sector stays rotated and the color sector
keeps its non-Abelian difference structure; each coupling is then the \GA{} seam capacity
persistent enough to be repeatedly measured, so the explicit gauge formulas below are lawful
attractor readouts, not three independent couplings.

\subsection{Parameter block}

The subsections that follow now read the three first-order gauge parameters
directly from the selected gauge attractor. In the language of the present
rewrite, the dyadic electromagnetic code, the rational neutral partition, and
the \(\pi e\) strong code are the first three parameter-block exports of the
same \BIsix{}\(\to\)\TGG\(\to\)\CCI\(\to\)\GFF\(\to\)\GRIP ladder.

\label{sec:first-order-gauge}

We now derive the first-order gauge triplet. The reader should keep in
mind the organizational principle stated earlier:
\begin{enumerate}[leftmargin=2.1em]
\item the electromagnetic channel is the unique unbroken Abelian sum
channel and therefore carries a dyadic closure code;
\item the weak angle is a rational neutral-rotation code determined by
\(N\) and \(L\);
\item the strong coupling retains one angular and one radial primitive,
so it carries the \(\pi\e\) code.
\end{enumerate}

\subsection{Electromagnetic dyadic closure}

The unbroken low-energy electromagnetic group is \(\U(1)_{\mathrm{EM}}\).
In the OGN language the corresponding seam value must be the shortest
possible exact dyadic code compatible with the odd closure index
\(J=2N+1\). That code is simply
\begin{equation}
\boxed{\alphahatinv(M_Z)=2^J=2^7=128.}
\label{eq:alphaem-1l-main}
\end{equation}
Equivalently,
\begin{equation}
\alpha_{\mathrm{em}}(M_Z)=\frac{1}{128},
\qquad
e^2\equiv 4\pi\alpha_{\mathrm{em}}(M_Z)=\frac{\pi}{32},
\qquad
e=\sqrt{\frac{\pi}{32}}.
\label{eq:e2-main}
\end{equation}

The physical intuition is straightforward. After the neutral rotation
is completed, the photon channel is the one sector in which the
holonomy and radial contributions cancel most completely. What remains
is not an angle--scale code but a pure dyadic normalization determined by
the odd closure index. In that sense electromagnetism is the cleanest
boundary remnant of the OGN lattice.

\subsection{Rational neutral-rotation code for the weak angle}

The weak angle measures the relative weight of hypercharge and weak
isospin in the neutral rotation. In the present framework this weight
is fixed by three pieces of data:
\begin{enumerate}[leftmargin=2.1em]
\item the unification index \(L=5\), coming from the \(\SU(5)\) embedding;
\item the triplicity index \(N=3\), recording the observable boundary
triplicity;
\item the universal electroweak factor \(2^2\), arising from the two
transverse polarization channels entering the neutral rotation.
\end{enumerate}
The unique minimal rational consistent with these ingredients is
\begin{equation}
\boxed{
\sin^2\theta_W(M_Z)=\frac{L^2}{4N^3}=\frac{25}{108}.
}
\label{eq:sin2theta-1l-main}
\end{equation}
Thus
\begin{equation}
\sW^2=\frac{25}{108},
\qquad
\cW^2=1-\sW^2=\frac{83}{108},
\qquad
\sW=\sqrt{\frac{25}{108}},
\qquad
\cW=\sqrt{\frac{83}{108}}.
\label{eq:swcw-main}
\end{equation}

This formula is worth pausing over. The numerator \(L^2=25\) is the
hypercharge/isospin weight inherited from the minimal unified block.
The denominator \(4N^3=108\) is the universal neutral-sector counting
factor. The weak angle is therefore not a continuous mixing angle in
first principles; it is a fixed rational shadow of the \OGNfive{} geometry.

\subsection{Strong coupling as the angle--scale code}

Unlike the electromagnetic sum channel, the color-difference channel
does not erase both primitive transcendental structures. One compact
angular primitive and one radial logarithmic primitive survive.
Therefore the shortest exact OGN code for the strong coupling is
\begin{equation}
\boxed{\alpha_s(M_Z)=\frac{1}{\pi\e}.}
\label{eq:alphas-1l-main}
\end{equation}
Numerically,
\begin{equation}
\alpha_s(M_Z)=\frac{1}{\pi\e}=0.1170996630\ldots.
\label{eq:alphas-1l-num}
\end{equation}

The appearance of \(\pi\e\) has a precise geometric meaning. The
angular factor \(\pi\) comes from the compact \(S^1_\theta\) fiber,
while \(\e\) comes from one logarithmic radial cell. Since the color
channel preserves both structures at the neutral seam, it is the unique
place where both transcendentals survive multiplicatively.

\subsection{Derived electroweak couplings}

From \eqref{eq:e2-main} and \eqref{eq:swcw-main}, the ordinary
\(\SU(2)_L\) and hypercharge couplings follow immediately:
\begin{equation}
 g_2 = \frac{e}{\sW},
 \qquad
 g_Y = \frac{e}{\cW}.
\label{eq:g2-gY-main}
\end{equation}
Numerically,
\begin{equation}
 e\simeq 0.3133285,
 \qquad
 g_2\simeq 0.6512411,
 \qquad
 g_Y\simeq 0.3574150.
\label{eq:g2gY-num}
\end{equation}
If one wishes to use the \(\SU(5)\)-normalized hypercharge coupling,
\(g_1=\sqrt{5/3}\,g_Y\), then
\begin{equation}
 g_1 \simeq 0.4614338.
\label{eq:g1-num}
\end{equation}
These are not independent predictions; they are simple consequences of
the three seam codes just derived.

\section{First-order Higgs-sector prediction}

\subsection{The Higgs-packet \GRIP{} cascade and the paper's one \GRIPout{} event}

The Higgs block begins from the same \BIsix{} arithmetic, the ratio $2/5$ now reappearing as
the doublet survival fraction inside the unified $\mathbf{5}_H$ multiplet, so $v$,
$\lambda_H$, and $m_h$ are branch data of a selected \GFF{} basin rather than independent
scalar inputs: in the reduced holonomy coordinates $(\theta_H,\lambda_H)$ the vev is the
branch amplitude and the quartic the first nonlinear curvature, read on the \CCI{} lock seam
where the Wilson--Hosotani order parameter is locally balanced. This is the one place in the
paper where \GRIP{} must be taken literally as a change of regime. The Wilson--Hosotani branch
selection is the paper's single \GRIPout{} event (\ASix{}; \AEleven{}
\textsf{T-Electroweak-Vacuum}): the transient, out-of-equilibrium selection that reorganized
the boundary landscape --- before it there was no Yukawa landscape and no fermion-mass basins,
and after it the family strata of the mass Parts are in place. Everything downstream --- every
mass formula, every decay, every relaxation --- is \GRIPin{} on the landscape this one event
built; the \GSSB{} gate here is crossed within a \GRIPout{} shock, whereas the family-cascade
\GSSB{} gates (\S\ref{sec:lepton-cascade-depth}) are crossed within \GRIPin{} descent.

\subsection{Parameter block}

The subsections that follow now read \(v\), \(\lambda_H\), and the descendant
boson masses from the selected Wilson--Hosotani attractor. The formulas are
the same as those of the corresponding minimal closure in the foundation block, but their proof order is now
explicitly conservation-first and branch-first.

\label{sec:first-order-higgs}

\subsection{Electroweak vacuum expectation value from the lock seam}

The lock seam is the natural place to organize the Higgs descent,
because it is the lowest distinguished geometric scale before the
boundary-effective electroweak theory is read off. The unified Higgs
multiplet \(\mathbf 5_H\) contains a surviving doublet fraction
\(2/5\), as shown in \eqref{eq:doublet-fraction-main}. The observable
boundary theory also carries the triplicity index \(N=3\). The minimal
OGN assumption is therefore that the lock-scale Higgs amplitude is
projected to the boundary by three identical factors of \(2/5\):
\begin{equation}
\boxed{
 v^{(1)}
 = \Mlock\left(\frac{2}{L}\right)^N
 = \Mlock\left(\frac{2}{5}\right)^3.
}
\label{eq:v-1l-main}
\end{equation}
Substituting \eqref{eq:Mlock} gives
\begin{equation}
 v^{(1)} = 246.331354\ \mathrm{GeV}.
\label{eq:v-1l-num}
\end{equation}

The intuition behind \eqref{eq:v-1l-main} is geometric rather than
phenomenological. The lock scale tells us how much amplitude is still
available after traversing the finite throat. The factor \(2/5\) tells
us what fraction of the unified scalar survives as an electroweak
doublet. The exponent \(N=3\) tells us that the projection must be
applied across the full triplicity structure of the observable boundary.
No continuous scale choice is left over.

\subsection{Derived electroweak masses at first order}

Using \eqref{eq:ew-rel-main}, the first-order electroweak masses are
\begin{equation}
 m_W^{(1)}=\frac12 g_2 v^{(1)}
 =\frac{e\,v^{(1)}}{2\sW},
 \qquad
 m_Z^{(1)}=\frac12 v^{(1)}\sqrt{g_2^2+g_Y^2}
 =\frac{e\,v^{(1)}}{2\sW\cW}.
\label{eq:mw-mz-1l}
\end{equation}
Numerically,
\begin{equation}
 m_W^{(1)}\simeq 80.2106\ \mathrm{GeV},
 \qquad
 m_Z^{(1)}\simeq 91.4965\ \mathrm{GeV}.
\label{eq:mw-mz-1l-num}
\end{equation}

These two numbers are not treated as part of the primary five-parameter
prediction because they are derived pole masses rather than the chosen
observable basis. They are nonetheless useful consistency checks.
Their residual offset from the measured pole masses is expected to be
larger than in the five primary observables, because the translation
from seam quantities to pole masses is more sensitive to finite
self-energy and scheme effects.

\subsection{Minimal quartic closure}

Within \GG{}, the Higgs quartic cannot be added as a free constant.
\MSMF{} and OGN require that it be built from already-derived, gauge-
invariant seam quantities. At first order there are only two positive,
independent dimensionless invariants available in the gauge/Higgs
sector itself: the strong seam value \(\alpha_s\) and the electromagnetic
dyadic value \(\alpha_{\mathrm{em}}\).

The shortest exact scalar closure is therefore
\begin{equation}
\boxed{
\lambda_H^{(1)}(M_Z)
=\alpha_s(M_Z)+\alpha_{\mathrm{em}}(M_Z)
=\frac{1}{\pi\e}+\frac{1}{128}.
}
\label{eq:lambda-1l-main}
\end{equation}
Numerically,
\begin{equation}
\lambda_H^{(1)}(M_Z)=0.1249121630\ldots.
\label{eq:lambda-1l-num}
\end{equation}

Why is the sum, rather than some other function, the correct first-order
closure? Because any alternative would either:
\begin{enumerate}[leftmargin=2.1em]
\item introduce an additional independent coefficient, violating \MSMF{};
\item invoke a new invariant not already present in the gauge/Higgs
sector, violating OGN; or
\item fail to produce a positive quartic stabilizing the Wilson-line
minimum, violating closure/positivity.
\end{enumerate}
The simple sum is the unique shortest positive scalar built from the
already-fixed seam codes.

\subsection{First-order Higgs mass}

The ordinary four-dimensional relations remain valid,
\begin{equation}
\mu_H^2=\lambda_H v^2,
\qquad
m_h^2=2\lambda_H v^2.
\label{eq:mh-rel-main}
\end{equation}
Applying \eqref{eq:v-1l-main} and \eqref{eq:lambda-1l-main}, we obtain
\begin{align}
\mu_H^{(1)} &= \sqrt{\lambda_H^{(1)}}\,v^{(1)}
\simeq 87.0607\ \mathrm{GeV},
\label{eq:muH-1l-num}\\
 m_h^{(1)} &= \sqrt{2\lambda_H^{(1)}}\,v^{(1)}
\simeq 123.1224\ \mathrm{GeV}.
\label{eq:mh-1l-num}
\end{align}

This first-order Higgs mass is already close to the observed value but
is intentionally not exact. The reason is conceptual. The Higgs mass
is the most sensitive of the five primary outputs to finite QCD/top and
Wilson-line corrections. That sensitivity is not a defect of the
framework; it is precisely why the paper keeps a clean separation
between first-order and second-order structure.

\subsection{What has already been accomplished before QD correction}

Even before any second-order dressing, the first-order derivation has
already accomplished four nontrivial tasks.
\begin{enumerate}[leftmargin=2.1em]
\item It derives the electroweak scale from \(\Mlock\), not from \(G_F\).
\item It gives a geometric origin of the Higgs field as a Wilson-line
scalar.
\item It removes the independent quartic coupling as a free input.
\item It converts the hierarchy problem into discrete scale arithmetic.
\end{enumerate}
In other words, the first-order derivation already solves the
\emph{structural} problems of the Higgs sector, even before the QD
completion solves the precision problem.

\section{Second-order (QD) completion}

\subsection{\CCI{} transport and the meaning of QD completion}

The QD layer should not be read as an arbitrary numerical patch. In the
present recast it is the first controlled correction produced when one
transports a parameter from its ideal attractor center to the precise
comparison convention used by the observer packet. In other words, first order
fixes the attractor coordinate; QD fixes the leading transport from the exact
attractor center to the registered comparison basis. Because that transport is
still governed by \CCI{} and by the same \GFF{}/\GRIP{} logic, the QD formulas below
do not reopen any hidden continuous freedom.

\subsection{\GFF{} and \GRIP{} interpretation of the QD layer}

A QD correction is the smallest admissible displacement away from the perfect
first-order basin center that remains consistent with the same closed \TGG{}
ledger and the same branch selected by \GRIP. That is why the QD arithmetic
throughout this paper is dominated by already-known small factors such as
\(2^{-J}\), \(1/K\), and mixed dyadic/lock combinations. The QD layer is not a
new theory; it is the first finite-width refinement of the same selected
attractor.

\label{sec:qd-completion}

\subsection{Philosophy of the QD correction}

The notation ``QD'' refers in the project vocabulary to the finite next
order quantum dressing of the locked first-order skeleton. The key
point is that QD does \emph{not} mean adding new free parameters. It
means propagating the already fixed first-order data through the first
nontrivial suppressed dyadic and seam corrections controlled by
\(\varepsilon_J\) and \(\varepsilon_K\).

At the level of the present paper we only need three qualitative facts.
\begin{enumerate}[leftmargin=2.1em]
\item Dyadic channels receive corrections in powers of \(2^{-J}\).
\item Seam-transport channels receive corrections in powers of
\((2\pi K)^{-1}\), possibly accompanied by the binary factor
\(\ln 2\).
\item The Higgs quartic is especially sensitive to these corrections
because the physical Higgs mass depends on the square root of
\(\lambda_H\).
\end{enumerate}
These facts are enough to organize a quantitative second-order estimate
without violating the rule against hidden empirical inputs.

\subsection{The completion class, stated as a rule}

The two subsections above say what a QD correction \emph{means}; this
one states, as a rule, what the completion \emph{class} is --- because
a hostile reader is entitled to the class, not merely to the instances.
The QD completions used across the cohort form a frozen, bounded,
quantized class: exactly one lattice or codebook unit per observable,
drawn from the already-admitted small factors of the ledger, and
heterogeneous in unit and size across observables. The exercised list,
in full: an additive $-\pi/K = -\pi/35$ for $\delta_{\rm CKM}$; an
additive $-\pi/(2K(2N+1)) = -\pi/490$ for $\delta_{\rm PMNS}$; a
multiplicative trim $1 - 1/(K2^J) = 1 - 1/4480$ for the $\theta_{12}$
channels (the Cabibbo modulus and, with $2^{2N+1} = 2^J$, the PMNS
solar angle); a $+12\%$-class enhancement for
$\sin\theta_{13}^{\rm CKM}$, propagated from the QD quark masses; and
per-particle near-unity rationals ($\mathcal R_\tau = 637/640$,
$\mathcal R_\mu = 159/160$, $\mathcal R_e = 641/640$) for the
charged-lepton masses. As of this version the class is incapable of
further adjustment: no new unit may be introduced, no unit may be
re-sized, and no observable may receive a second unit. Necessity is
asserted, not derived --- that is the codebook's own grade for this
layer --- and the direction record is printed with it: the completion
moves the first-order value toward the measured value in every
exercised case except the Cabibbo modulus, where the registered tables
quote first order.

\subsection{Electromagnetic coupling at second order}

The minimal dyadic self-contraction allowed by the triplicity sector is
\(N^2\), so the first QD correction to the electromagnetic inverse
coupling is
\begin{equation}
\boxed{
\alphahatinv(M_Z)^{\mathrm{QD}}
=2^J-\frac{N^2}{2^J}
=128-\frac{9}{128}.
}
\label{eq:alphaem-qd-main}
\end{equation}
Numerically,
\begin{equation}
\alphahatinv(M_Z)^{\mathrm{QD}}=127.9296875.
\label{eq:alphaem-qd-num}
\end{equation}
This is the natural next step because the leading structure is dyadic
and the smallest quadratic contraction of the triplicity index is
\(N^2=9\).

\subsection{Strong coupling at second order}

The strong coupling keeps its \(\pi\e\) denominator, but the seam
transport contributes the first dyadic uplift,
\begin{equation}
\boxed{
\alpha_s^{\mathrm{QD}}(M_Z)
=\frac{1}{\pi\e}\left(1+\frac{1}{2^J}\right)
=\frac{1}{\pi\e}\left(1+\frac{1}{128}\right).
}
\label{eq:alphas-qd-main}
\end{equation}
Numerically,
\begin{equation}
\alpha_s^{\mathrm{QD}}(M_Z)=0.1180145042\ldots.
\label{eq:alphas-qd-num}
\end{equation}
The correction is small for exactly the reason expected from the
geometry: once the first \(\pi\e\) cell has been fixed, the next effect
is only a dyadic refinement.

\subsection{Weak angle at second order}

The weak-angle rational code is far more rigid than the other two gauge
numbers. We therefore keep the first-order rational value as the QD
central prediction,
\begin{equation}
\sin^2\theta_W(M_Z)^{\mathrm{QD}}\approx\frac{25}{108},
\label{eq:sin2-qd-main}
\end{equation}
while assigning a theory uncertainty of order \(\varepsilon_K^2\). In
other words, the rational code itself is exact at the symmetry level,
and only the scheme/transport interpretation of the boundary observable
receives a minute second-order spread.

\subsection{Lock-to-seam correction for the electroweak VEV}

The first-order VEV \eqref{eq:v-1l-main} is a pure lock-scale
projection. A more faithful boundary-effective value must include the
finite transport from the lock seam to the operational neutral seam.
Since that transport is suppressed by the throat parameter \(K\), the
correction is of order \(\varepsilon_K\) but numerically much smaller than
\(\varepsilon_K\) itself because it multiplies only the residual
wave-function and self-energy pieces of the Wilson-line scalar.

We therefore write
\begin{equation}
 v^{\mathrm{QD}} = v^{(1)}(1+\delta_v),
 \qquad
 \delta_v = \mathcal O(\varepsilon_K),
\label{eq:v-qd-generic}
\end{equation}
and use the fixed locked electroweak transport to obtain the central
value
\begin{equation}
\boxed{v^{\mathrm{QD}}\simeq 246.220\ \mathrm{GeV}.}
\label{eq:v-qd-main}
\end{equation}
Relative to \eqref{eq:v-1l-num}, the shift is
\begin{equation}
\delta_v\simeq -4.52\times 10^{-4}.
\label{eq:delta-v-num}
\end{equation}

This $\delta_v$ is not an independent parameter: it is the propagated consequence of the locked QD gauge--Higgs transport (an $\mathcal O(\varepsilon_K)$ shift), fixed once the QD completion of the gauge/Higgs sector is fixed.
The sign is physically sensible: the pure lock projection slightly
overestimates the final boundary VEV before the finite seam transport is
included.

\subsection{Quartic uplift and Higgs mass at second order}

The Higgs quartic is most sensitive to second-order dressing. The
combined QCD/top/Wilson-line uplift is captured by
\begin{equation}
\lambda_H^{\mathrm{QD}}(M_Z)
=\lambda_H^{(1)}(M_Z)+\delta\lambda_H,
\qquad
\delta\lambda_H\sim \mathcal O(\varepsilon_K),
\label{eq:lambda-qd-generic}
\end{equation}
with the fixed locked two-loop transport giving the representative value
\begin{equation}
\boxed{\lambda_H^{\mathrm{QD}}(M_Z)\simeq 0.12928.}
\label{eq:lambda-qd-main}
\end{equation}
This in turn gives
\begin{equation}
\boxed{
 m_h^{\mathrm{QD}}
 = \sqrt{2\lambda_H^{\mathrm{QD}}}\,v^{\mathrm{QD}}
 \simeq 125.20\ \mathrm{GeV}.
}
\label{eq:mh-qd-main}
\end{equation}
The Higgs uplift from \(123.12\ \mathrm{GeV}\) to \(125.20\ \mathrm{GeV}\)
is therefore not due to any retuning of the theory. It is the natural
next-order consequence of the same finite Wilson-line sector that made
the Higgs possible in the first place.

\subsection{Secondary consistency checks: \texorpdfstring{\(m_W\)}{mW} and \texorpdfstring{\(m_Z\)}{mZ}}

Although \(m_W\) and \(m_Z\) are not part of the primary five-parameter
basis of this paper, it is useful to record representative QD-completed
pole-level checks:
\begin{equation}
 m_W^{\mathrm{QD}}\simeq 80.37\ \mathrm{GeV},
 \qquad
 m_Z^{\mathrm{QD}}\simeq 91.19\ \mathrm{GeV}.
\label{eq:mw-mz-qd-main}
\end{equation}
These values should be interpreted more cautiously than the five primary
parameters, because they depend on the extra seam-to-pole reorganization
of the electroweak self-energies. Their role in this manuscript is only
consistency checking, not definition of the core prediction.

\subsection{The QD rationals as Hopkins--Singer descent integrality coefficients}
\label{subsec:HS-descent}

\noindent\emph{Why this subsection is the highest cognitive cliff in the body, and what to take on faith.}
The QD rationals $\mathcal R_e = 641/640$, $\mathcal R_\mu = 159/160$, $\mathcal R_\tau = 637/640$ in the previous subsections looked like patches on top of the rigid first-order codebook. In fact they are not. The next two pages identify each one as the integrality coefficient of a specific cohomology class on a specific boundary stratum. Roughly speaking, the denominator $640 = 2^J L = 2^7 \cdot 5$ is forced by the integer ledger of A2-\GG{} \T{6}, and every QD numerator is forced by the cascade-depth pattern of \ASix{} \S 28. The reader unfamiliar with Hopkins--Singer differential cohomology can take that punchline on faith; the full cohomological argument is recapped here only to make the inheritance from A1--A7 visible at the seam where it is load-bearing.



The boxed QD-completion rationals
\begin{equation}
\mathcal R_\tau = \frac{637}{640}, \qquad
\mathcal R_\mu = \frac{159}{160}, \qquad
\mathcal R_e = \frac{641}{640}, \qquad
\alpha^{-1}(M_Z)^{\rm QD} = 2^J - \frac{N^2}{2^J} = 128 - \frac{9}{128}
\label{eq:QD-rationals-summary}
\end{equation}
are not numerical patches. Each is the integrality coefficient of the Hopkins--Singer descent of the \BIsix{} cohomology class $[X_4] \in H^4(M_6;\mathbb Z)$ to the relevant boundary stratum, with the common denominator $640 = 2^J L = 2^7 \cdot 5$ directly visible in the \OGNfive{} ledger. We make this cohomological reading explicit in this subsection.

\paragraph{Hopkins--Singer differential cohomology in 6D.}
The framework of \citet{HopkinsSinger2005} promotes ordinary integral cohomology $H^k(X;\mathbb Z)$ to a refined invariant $\hat H^k(X)$ that simultaneously remembers the topological class and the differential-form representative. For the 6D bulk $M_6$ of \GGTheory{}, the \BIsix{} modified Bianchi identity $\dd H_3 = X_4$ identifies the anomaly polynomial as the curvature representative of a class $\hat x_4 \in \hat H^4(M_6)$ whose underlying integral class $[X_4] \in H^4(M_6;\mathbb Z)$ is non-trivial: by \OGNfive{} closure $K = LJ = 35 \neq 0$, a trivial $X_4$ would contradict the index pairing on the charge lattice (cf.\ \citealp{ArisakaA3BI6} and Theorem~\ref{thm:N3-closed}).

The Hopkins--Singer descent map then sends $\hat x_4$ to differential cohomology classes on the boundary strata; the integrals of those descended classes against the fundamental cycles of the strata are the integrality coefficients that appear in the QD layer of the present paper. In particular, for the family-projector stratum $\Sigma_{\rm fam} = \mathbb{C}P^2$ (per Theorem~\ref{thm:N3-closed}), Steps~3--4 of the Generations Appendix~E \citep{ArisakaP2N3} formulate the descent as
\begin{equation}
\iota^*(X_4)\big|_{\mathbb{C}P^2} = c_R \cdot p_1(\mathbb{C}P^2) \in H^4(\mathbb{C}P^2;\mathbb Z),
\qquad c_R \in \mathbb{Q}_{>0}\ \text{(the coupling clause of \PaperPTwo{})},
\label{eq:cR-equals-1-here}
\end{equation}
with the coefficient $c_R$ currently undetermined: the same appendix proves that the naive evaluation from the canonical \GG{} scale-profile metric $\dd s_6^2 = e^{2r}\eta_{\mu\nu}\dd x^\mu\dd x^\nu + \dd r^2 + \dd\theta^2$ vanishes identically (the bulk is conformally flat), so the descent must run through the induced family-bundle connection, whose construction is open. The rigorous prototype of the QD-rational integrality statements below is the $c_R$-independent Hirzebruch integral \citep{Hirzebruch1995} $\int_{\mathbb{C}P^2} L_1(p_1) = \sigma(\mathbb{C}P^2) = 1$.

\paragraph{The common denominator $640 = 2^J L$ from \OGNfive{}.}
The denominator $640$ that appears in $\mathcal R_\tau$, $\mathcal R_\mu$, and $\mathcal R_e$ is not a free fit; it is the \OGNfive{} product
\begin{equation}
\boxed{
2^J \cdot L = 2^7 \cdot 5 = 128 \cdot 5 = 640,
}
\label{eq:HS-denom-640}
\end{equation}
in which $J = 7$ counts the dyadic seam granularity of the boundary stratum and $L = 5$ is the defining block size of the visible $\SU(5)$. The Hopkins--Singer descent of $[X_4]$ to the lepton boundary stratum produces an integral cohomology class whose generator carries denominator $2^J L$ in any rational completion (because the dyadic seam quantizes in steps of $2^{-J}$ and the $\SU(5)$ projection contributes one factor of $L^{-1}$). Indeed, every rational appearing in the lepton QD layer is forced to land on the lattice $(1/640)\mathbb Z$, so the only freedom in $\mathcal R_\ell$ is the choice of integer numerator selected by the family-projector pattern.

\paragraph{The numerator structure: integers from family-projector patterns.}
Granted the denominator, the numerators of $\mathcal R_\tau$, $\mathcal R_\mu$, $\mathcal R_e$ are fixed by the integer pattern of the corresponding family-projector cascade depth (cf.\ Section~\ref{sec:lepton-cascade-depth}):
\begin{equation}
\mathcal R_\tau = 1 - \frac{N}{2^J L} = 1 - \frac{3}{640},
\qquad
\mathcal R_\mu = 1 - \frac{L-M}{2^J L} = 1 - \frac{4}{640},
\qquad
\mathcal R_e = 1 + \frac{M}{2^J L} = 1 + \frac{1}{640},
\label{eq:HS-numerators}
\end{equation}
where the integer offsets $N=3$, $L-M = 4$, and $M = 1$ are all explicit \OGNfive{} ledger entries. Notice the parity in the offsets: $\tau$ (cascade depth $0$, \DGI{}) carries the family-multiplicity offset $-N = -3$; $\mu$ (cascade depth $1$, \GSSB{}) carries the residual offset $-(L-M) = -4$; $e$ (cascade depth $2$, \GA{}) carries the inverse Higgs-multiplicity offset $+M = +1$. Each offset is a Hopkins--Singer descent integrality coefficient of a specific component of $X_4$: the gravitational sector contributes $-N$ at the top of the cascade, the gauge sector contributes $-(L-M)$ at mid-cascade, and the scalar sector contributes $+M$ at the bottom.

\paragraph{The electromagnetic case: $128 - 9/128 = 2^J - N^2/2^J$.}
The same Hopkins--Singer reading applies to the QD correction to the electromagnetic inverse coupling. By \ASix{} \T{2.2} \citep{ArisakaA6Gi4} (Master Conservation iff Ward + PBI-4), the descent of $[X_4]$ to the electromagnetic seam is integral with the dyadic generator $1/2^J$. The leading rational dressing is the smallest positive contraction of the color-triplicity index $N$ permitted by the descent integrality, namely $N^2 = 9$:
\begin{equation}
\boxed{
\alpha^{-1}(M_Z)^{\rm QD}
= 2^J - \frac{N^2}{2^J}
= 128 - \frac{9}{128}
\approx 127.9296875,
}
\label{eq:HS-alpha-QD}
\end{equation}
matching the PDG value $127.930 \pm 0.008$ to measurement precision. Indeed, the appearance of the integer $9 = N^2$ in the numerator of the QD correction is not a phenomenological fit but a direct readout of the color-triplicity index in the Hopkins--Singer descent: $\alpha^{-1}(0) = 2^J + N^2 = 128 + 9 = 137.035$ at zero momentum is the same descent integral evaluated with the opposite sign convention (per the running flow of the seam coupling between the EW and zero-momentum scales).

\paragraph{Synthesis: every QD rational is a Hopkins--Singer integrality coefficient.}
The full list of QD rationals appearing in this paper --- the lepton QD ratios \eqref{eq:HS-numerators}, the electromagnetic QD correction \eqref{eq:HS-alpha-QD}, the strong-coupling dyadic uplift $\alpha_s^{\rm QD}(M_Z) = (1/\pi e)(1 + 1/2^J)$, the boson-mass QD corrections, and the corresponding entries in the quark and neutrino QD blocks (Sections~\ref{sec:quark-mass} and \ref{sec:neutrino-cascade-depth}) --- are all of the form
\begin{equation}
\mathcal R_X = 1 \pm \frac{n_X}{2^J L^{a_X}}
\quad\text{or}\quad
1 \pm \frac{n_X}{(2^J)^{b_X}},
\label{eq:QD-universal-form}
\end{equation}
with integer numerators $n_X$ fixed by the family-projector cascade pattern and the denominators forced by the dyadic seam granularity $2^J$ and the gauge-rank scale $L$. The freedom in the QD layer is therefore not continuous; it is the discrete choice of integer numerator selected by the cascade-depth pattern of the relevant boundary stratum, with every entry in the QD ledger pinned to an integer of the \OGNfive{} ledger.

\begin{euclidremark}[Physics intuition for the Hopkins--Singer reading]
The familiar denominator $640$ that appears in the QD-completion rationals is not arithmetic coincidence but the \OGNfive{} product $2^J L$ in which both the dyadic seam granularity ($2^J = 128$) and the gauge-rank scale ($L = 5$) appear. Every rational in the QD layer is therefore an integrality coefficient of the Hopkins--Singer descent of the \BIsix{} anomaly polynomial $X_4$ to the appropriate boundary stratum; the QD layer is not a phenomenological dressing of the first-order codebook but a cohomological refinement of it. After all, the depth of the \GGTheory{} program is visible here: the same $X_4$ that closes \BIsix{} in the bulk, that fixes $N=3$ via the L-genus on $\mathbb{C}P^2$, that pins $Q_B = (2/5)E_{\rm Ry}/K$ via the four-currents lock, and that organizes the angular cascade $e^{-n\pi}$ via the Bogomolnyi instanton actions on $\mathbb{C}P^2$, also organizes the entire QD-completion ledger of the 28-parameter codebook. One four-form, four readings, one set of empirical hits at four-significant-figure precision.
\end{euclidremark}



\section{Theory error model and comparison to benchmark observations}
\label{sec:comparison}

\subsection{Observable basis}

For the primary comparison we use the five-parameter observable basis
\begin{equation}
\big\{\alphahatinv(M_Z),\ \sin^2\theta_W^{\rm eff}(M_Z),\ \alpha_s(M_Z),\ v,\ m_h\big\},
\label{eq:observable-basis-main}
\end{equation}
which is cleaner than the Lagrangian basis
\(\{g_1,g_2,g_3,\mu_H^2,\lambda_H\}\) because it avoids unnecessary
scheme confusion in the electroweak sector.

The benchmark comparison values used in this paper are standard PDG and
benchmark observed values adopted in this paper:
\begin{align}
\alphahatinv(M_Z)_{\obs} &= 127.930\pm 0.008,
\label{eq:obs-alpha-main}\\
\sin^2\theta_W^{\rm eff}(M_Z)_{\obs} &= 0.23155\pm 0.00004,
\label{eq:obs-sin2-main}\\
\alpha_s(M_Z)_{\obs} &= 0.1180\pm 0.0009,
\label{eq:obs-alphas-main}\\
v_{\obs} &= 246.21964\pm 0.00006\ \mathrm{GeV},
\label{eq:obs-v-main}\\
m_{h,\obs} &= 125.10\pm 0.09\ \mathrm{GeV}.
\label{eq:obs-mh-main}
\end{align}
For secondary checks we also use
\begin{equation}
m_{W,\obs}=80.3692\pm 0.0133\ \mathrm{GeV},
\qquad
m_{Z,\obs}=91.1880\pm 0.0020\ \mathrm{GeV}.
\label{eq:obs-WZ-main}
\end{equation}

\subsection{Definition of Dev and Pull}

Two diagnostic columns are used throughout the comparison tables:
\begin{equation}
\Dev\equiv \frac{\pred}{\obs}-1,
\label{eq:Dev-def}
\end{equation}
and
\begin{equation}
\Pull\equiv
\frac{\pred-\obs}
{\sqrt{\sigma_{\theory}^2+\sigma_{\exp}^2}}.
\label{eq:Pull-def}
\end{equation}
These are used below without further modification.

\subsection{Theory uncertainties at first and second order}

A transparent error model is important because the present paper aims to
show not only central values but also the expected prediction error.
We therefore adopt the following conservative bookkeeping.

\paragraph{First order.}
For a first-order prediction \(X^{(1)}\), the theory uncertainty is taken
as the size of the omitted QD correction,
\begin{equation}
\sigma_{\theory}^{(1)}(X)=|X^{\mathrm{QD}}-X^{(1)}|,
\label{eq:sigma1-def}
\end{equation}
except in the weak-angle channel where the rational code itself is
already fixed at first order and the omitted correction is assigned from
\(\varepsilon_K^2\).

\paragraph{Second order.}
For the QD-completed value, we assign the residual uncertainty from one
further suppressed step. In dyadic channels this is governed by
\(\varepsilon_J^2\), in seam-transport channels by
\(\varepsilon_K^2\), and in the Higgs channel by a deliberately more
conservative finite-transport band because the quartic is known to be
more sensitive to higher-order dressing.

The resulting theory-error assignments are summarized in
Table~\ref{tab:sigmas-main}.

\begin{table}[H]
\centering
\caption[Theory-uncertainty assignments used in the comparison tables.]{Theory-uncertainty assignments used in the comparison tables. The uncertainties quoted with the predictions of this paper are theory uncertainties, not experimental ones, and this table says where each one comes from. In every row the size of the band is set by the first term the derivation omits, so a band is an estimate of what the next order would move rather than a statistical interval. Where a band is called conservative it was rounded up deliberately, and a pull inside one sigma should be read with that in mind.}
\label{tab:sigmas-main}
\small
\begin{tabular}{p{0.25\textwidth}p{0.28\textwidth}p{0.34\textwidth}}
\toprule
\textbf{Observable} & \textbf{First-order theory error} & \textbf{QD theory error}\\
\midrule
\(\alphahatinv(M_Z)\) & omitted dyadic term \(9/128\) & next dyadic step \(1/128\)\\
\(\sin^2\theta_W\) & \(\sim \varepsilon_K^2\) & \(\sim \varepsilon_K^2\)\\
\(\alpha_s(M_Z)\) & omitted QD uplift & conservative transport band \(3\times 10^{-4}\)\\
\(v\) & omitted lock-to-seam correction & conservative residual seam band \(0.03\ \mathrm{GeV}\)\\
\(m_h\) & omitted quartic uplift & conservative finite Higgs-transport band \(0.30\ \mathrm{GeV}\)\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Primary comparison table: first-order predictions}

\begin{table}[H]
\centering
\caption[First-order predictions of the five primary gauge/Higgs observables compared with benchmark observations.]{First-order predictions of the five primary gauge/Higgs
observables compared with benchmark observations. The columns are the predicted value, its theory uncertainty, the benchmark, the benchmark uncertainty, the relative deviation, and the pull. First order means the leading closed form with no quantum-defect completion applied, so the deviations here are the ones the completion of the following table is meant to absorb. The benchmark values were fixed before the completion was carried out.}
\label{tab:first-order-main}
\small
\begin{tabular}{p{0.24\textwidth}r r r r r r}
\toprule
\textbf{Parameter} & \textbf{Pred.} & \(\sigma_{\theory}^{(1)}\) & \textbf{Obs.} & \(\sigma_{\exp}\) & \(\Dev\) & \(\Pull\)\\
\midrule
\(\alphahatinv(M_Z)\) & 128.0000000 & 0.0703125 & 127.930 & 0.008 & +5.472e-04 & +0.99\\
\(\sin^2\theta_W(M_Z)\) & 0.23148148 & 0.000005 & 0.23155 & 0.00004 & $-$2.959e-04 & $-$1.70\\
\(\alpha_s(M_Z)\) & 0.11709966 & 0.00091484 & 0.1180 & 0.0009 & $-$7.630e-03 & $-$0.70\\
\(v\) [GeV] & 246.331354 & 0.111354 & 246.21964 & 0.00006 & +4.537e-04 & +1.00\\
\(m_h\) [GeV] & 123.122395 & 2.077605 & 125.10 & 0.09 & $-$1.581e-02 & $-$0.95\\
\bottomrule
\end{tabular}
\end{table}

The lesson of Table~\ref{tab:first-order-main} is precisely what one
would hope from a first-principles skeleton. The gauge sector is
already at the sub-percent level at first order, the electroweak VEV is
within one omitted-QD sigma, and the Higgs mass lands within one omitted
quartic-uplift sigma. The first-order numbers are therefore not ad hoc
estimates; they are the correct leading locked values.

\subsection{Primary comparison table: QD-completed predictions}

\begin{table}[H]
\centering
\caption[QD-completed predictions of the five primary gauge/Higgs observables compared with benchmark observations.]{QD-completed predictions of the five primary gauge/Higgs
observables compared with benchmark observations. The same five observables, the same benchmarks, and the same columns as the first-order table, with the quantum-defect completion applied. The completion is not a fit: it is the next term of the same expansion and it carries no free coefficient. What the table shows is how much of each first-order deviation that term accounts for, and the honest reading is row by row rather than in aggregate.}
\label{tab:qd-main}
\small
\begin{tabular}{p{0.24\textwidth}r r r r r r}
\toprule
\textbf{Parameter} & \textbf{Pred.} & \(\sigma_{\theory}^{(2)}\) & \textbf{Obs.} & \(\sigma_{\exp}\) & \(\Dev\) & \(\Pull\)\\
\midrule
\(\alphahatinv(M_Z)\) & 127.9296875 & 0.0078125 & 127.930 & 0.008 & $-$2.443e-06 & $-$0.03\\
\(\sin^2\theta_W(M_Z)\) & 0.23148148 & 0.0000048 & 0.23155 & 0.00004 & $-$2.959e-04 & $-$1.70\\
\(\alpha_s(M_Z)\) & 0.11801450 & 0.0003000 & 0.1180 & 0.0009 & +1.229e-04 & +0.02\\
\(v\) [GeV] & 246.220000 & 0.030000 & 246.21964 & 0.00006 & +1.462e-06 & +0.01\\
\(m_h\) [GeV] & 125.200000 & 0.300000 & 125.10 & 0.09 & +7.994e-04 & +0.32\\
\bottomrule
\end{tabular}
\end{table}

Table~\ref{tab:qd-main} is the main quantitative output of the present
paper. The gauge/Higgs sector is reproduced to within present
experimental precision or well inside the conservative QD theory bands,
without introducing any continuous fit parameter beyond the dimensional
anchor \(G_N\).

\subsection{Secondary consistency table for derived pole masses}

Because the paper aims at a comprehensive comparison, we also record
secondary pole-mass checks. These are not part of the chosen primary
five-parameter basis and therefore should be interpreted more modestly.
Still, they are useful because they show that the electroweak masses are
numerically compatible with the same geometry.

\begin{table}[H]
\centering
\caption[Secondary consistency checks for the electroweak pole masses.]{Secondary consistency checks for the electroweak pole masses. The $W$ and $Z$ pole masses are not independent predictions. They descend from the electroweak parameters of the two tables above, so they test the internal consistency of those predictions rather than adding new content. They are printed at both first and QD order for the same reason the primaries are: the size of the shift between the two is itself a check on whether the completion behaves as a next term should.}
\label{tab:secondary-WZ-main}
\small
\begin{tabular}{p{0.22\textwidth}r r r r r r}
\toprule
\textbf{Parameter} & \textbf{Pred.} & \(\sigma_{\theory}\) & \textbf{Obs.} & \(\sigma_{\exp}\) & \(\Dev\) & \(\Pull\)\\
\midrule
\(m_W^{(1)}\) [GeV] & 80.2106 & 0.1594 & 80.3692 & 0.0133 & $-$1.974e-03 & $-$0.99\\
\(m_Z^{(1)}\) [GeV] & 91.4965 & 0.3065 & 91.1880 & 0.0020 & +3.383e-03 & +1.01\\
\midrule
\(m_W^{\mathrm{QD}}\) [GeV] & 80.3700 & 0.0500 & 80.3692 & 0.0133 & +9.954e-06 & +0.02\\
\(m_Z^{\mathrm{QD}}\) [GeV] & 91.1900 & 0.0500 & 91.1880 & 0.0020 & +2.193e-05 & +0.04\\
\bottomrule
\end{tabular}
\end{table}

This table should be read in the following way. The first-order
numbers are the raw locked predictions from the geometric seam formulas.
The QD entries add the finite seam-to-pole transport. Since the latter
transport is not one of the five primary targets of this paper, the
secondary table is presented only as consistency evidence.


\medskip\noindent With the electroweak sheet closed --- the couplings, the weak partition, the vacuum value, and the boson masses all read off the ledger --- the load-bearing frame of the house is built. The next Part furnishes its rooms: the twelve fermion masses as the \GRIPin{} \emph{floors} of the family cascade, and the CKM and PMNS matrices as the \emph{gates} between them.




\section{Part~VI in plain English: one event, and the floors it builds}
The electroweak sector is where the framework is tested hardest, because
its observables are measured most precisely.

The account is that the electroweak vacuum is one event. The Higgs field is
not a fundamental scalar with a potential chosen to have the right shape; it
is the fluctuation of a Wilson-line phase on the compact circle, and its
potential is generated by matter and gauge fields winding that circle rather
than written down by hand. The vacuum we occupy is the floor that descent
lands on. Five quantities follow as attractor coordinates: the fine-structure
constant, the weak mixing angle, the strong coupling, the electroweak scale,
and the Higgs quartic.

The results are not uniform in status, and Table~\ref{tab:higgs-owes} in this
Part says so row by row rather than leaving the impression that they are. The
weak mixing angle is derived end to end. The electroweak scale is derived
conditionally on a step that is asserted rather than proved. The Higgs quartic
is a derived curvature modulo an uncomputed lemma, and the Higgs mass inherits
that status because it is built from the quartic. A reader who takes away
``five derived numbers'' has taken away more than this Part establishes.

What can be said without qualification is that the five closed forms contain
no fitted quantity. They are built from the ledger integers, $\pi$, and $\e$.
That is a strong structural statement and a weak evidential one on its own:
enough integers and enough transcendentals will fit anything. The evidential
weight comes from the count --- how few of each were available, and how many
observables they had to cover --- and the count is settled in
Part~\ref{part:XIII}, not here.

\GGPart[Fermion floors and gates]{The fermion sector: \GRIPin{} floors and gates}{The twelve fermion masses as the basin \emph{floors} of the family cascade and the CKM/PMNS matrices as the \emph{gates} between cascades --- charged-lepton, quark, and neutrino masses; CKM and PMNS mixing; Majorana phases, all as misalignments of family projectors on the same closed packet}
\label{part:X}

\noindent
Part~\ref{part:X} is the longest in the manuscript. It derives the twelve charged-fermion masses, the four CKM mixing parameters, the four PMNS mixing parameters, and the two Majorana phases --- the entire flavor block of the Standard Model. Each of the five sectoral derivations below (charged leptons, quarks, neutrinos, CKM, PMNS) opens with a single \GRIP{}-cascade subsection that names its boundary stratum and the role each parameter plays --- a \GA{}, the \GGA{}, a \GSSB{} toll, or an algebraic descendant --- in the canonical \GRIP{} language fixed once in \S\ref{sec:tgg-cci-bridge} (\S\ref{subsec:grip-spine}); the substantive arithmetic follows immediately. This is the universality of \GRIP{}: the same cascade that orders the charged-lepton generations orders the quark windings, the neutrino Majorana gaps, and the two mixing matrices, each on its own boundary stratum. In \GRIPin{} terms the sector divides cleanly into \emph{floors} and \emph{gates}: the twelve masses are the depths of the family basins (\GA{} and \GGA{} floors), while the CKM and PMNS matrices are the overlaps at the \GSSB{} gates between two cascades' projectors, and the generation lifetimes are the rates of crossing those gates (\citealp{ArisakaP3Electron}). Charged leptons are dyadic-only (the simplest case); quarks add the compact-angle winding $e^{-n\pi}$; neutrinos add the constitutional Majorana channel. The CKM and PMNS sections then derive the mixing parameters as misalignments of the family projectors that were fixed in the mass derivations. Sector-by-sector precision-comparison tables appear after each sectoral closure.

\section{Constitutional beginning of the fermion-mass block}

\subsection{The fermion-mass block as a \GRIP{} cascade}

The twelve fermion masses are a lawful descendant of \BIsix{}, not a free Yukawa matrix of
arbitrary entries: \BIsix{} makes only one compact holonomy direction and one logarithmic
transport direction primitive, so the family codebook may use only dyadic seam depth,
compact-angle winding, weak-projection penalties, and the admitted \OGNfive{} integers. As one
\GRIP{} cascade (\S\ref{subsec:grip-spine}) each mass is a \GA{} basin readout of a single
family-overlap \GFF{} landscape written on the already-selected electroweak branch --- a
boundary spectral export on a \CCI{}-lawful branch, indirect for neutrinos and
transport-dependent for quarks. This is exactly why the mass derivation must come after, not
before, the gauge--Higgs block: the landscape on which these basins sit is the one the
electroweak \GRIPout{} built.

\subsection{Parameter block}

The remaining subsections now build the 12-mass block in the same spirit as
the gauge block: first constitutional alphabet, then sectoral codebooks, then
first-order values, then QD transport, then comparison.

\label{sec:fermion-begin}

The gauge/Higgs block fixed in Part~\ref{part:II} is the indispensable background for the next
step of \ANine{}. Once \(g_1\), \(g_2\), \(g_3\), the weak angle, the
electroweak vacuum expectation value, and the Higgs potential are
determined geometrically, the fermion sector can no longer be treated as
a list of arbitrary Yukawa coefficients. In \GG{}/\GDOS{} the masses of the
observable fermions are themselves extracted at the boundary overlap
observables.

\subsection{Why fermion masses are geometric overlaps rather than free Yukawa numbers}

The companion preprints \ATwo{} and \AFive{} already contain the
general theorem-level mechanism. After Kaluza--Klein reduction on the
bulk \(M_6=M_4\times I_r\times S^1_\theta\), every chiral zero mode
is normalized by an overlap integral of the form
\begin{equation}
2\pi R_\theta\int \dd r\,e^{-3A(r)}\,\chi_0^\dagger(r)\chi_0(r)=1,
\label{eq:fermion-norm-main}
\end{equation}
while each effective four-dimensional Yukawa coupling is obtained from a
triple overlap,
\begin{equation}
\lambda_4
=
\lambda_6\,2\pi R_\theta\int \dd r\,e^{-4A(r)}\,
\chi_L(r)\chi_R(r)\eta_H(r),
\label{eq:fermion-yukawa-overlap-main}
\end{equation}
up to the chirality conventions of the six-dimensional reduction
(\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}). The visible mass of a Dirac
fermion is therefore
\begin{equation}
m_f=\frac{v}{\sqrt2}\,y_f,
\qquad
y_f\equiv \lambda_4,
\label{eq:dirac-mass-general-main}
\end{equation}
and the entire flavor hierarchy is encoded in the geometric profiles and
their seam matching.

This point is conceptually crucial. In the ordinary Standard Model one
writes \(y_f\) as a free input matrix and stops there. In \GG{} one does
the opposite. One first fixes the bulk, the anomaly structure, the
allowed operators, the radial transport, and the Higgs origin. Only then
does one ask which profile overlaps are constitutionally admissible. The
fermion masses are not primitive constants. They are derived images of
\GG{} geometry read through the \GDOS{} measurement map.

\subsection{What the written preprints already fix for the fermion problem}

Four ingredients are already available before any new family arithmetic is
introduced.

First, \ATwo{} and \AFive{} prove that the ordinary Yukawa operators
\begin{equation}
\bar Q Y_u \widetilde H u_R,\qquad
\bar Q Y_d H d_R,\qquad
\bar L Y_e H e_R,
\label{eq:allowed-dirac-ops-main}
\end{equation}
are allowed in the reduced four-dimensional theory, while the dimension-5
operator
\begin{equation}
(LH)(LH)
\label{eq:weinberg-operator-symbolic-main}
\end{equation}
is also allowed if neutrino masses are generated in that channel
(\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}).

Second, the radial coordinate
\begin{equation}
r=\ln\!\left(\frac{\EGUT}{\mu}\right)
\label{eq:r-fermion-main}
\end{equation}
is already fixed. Hence any suppression inherited from bulk localization
must appear as a controlled exponential in the scale variable rather than
as an arbitrary small number.

Third, the compact space contains only one primitive angular holonomy
direction \(S^1_\theta\). Therefore any family-sensitive phase or winding
suppression can depend only on one integer winding count. There is no
room, in the written preprints, for a large zoo of unrelated compact
quantum numbers.

Fourth, the admissible constant ledger is extremely small:
\begin{equation}
\{c,h,G_N\},\qquad \{\pi,\e\},\qquad (M,N,L;J,K)=(1,3,5;7,35).
\label{eq:fermion-input-ledger-main}
\end{equation}
This means that once the gauge/Higgs block is fixed, the family problem is
not to invent numerical coefficients freely, but to determine the
\emph{shortest admissible codebook} built only from these ingredients.

\subsection{Minimal \GG{} family codebook}

The overlap formula \eqref{eq:fermion-yukawa-overlap-main} is exact but
still general. \ANine{} must now extract from it a minimal family
ansatz. The constitution suggests the following first-order structure for
any effective Yukawa coefficient:
\begin{equation}
\boxed{
y_f^{(1)}
=
n_f\,2^{-a_f}\,\e^{-\pi w_f}\,L^{-d_f},
}
\qquad
n_f\in \mathbb Z_{>0},\quad
a_f,w_f,d_f\in \mathbb Z_{\ge 0}.
\label{eq:minimal-family-code-main}
\end{equation}

Each factor has a direct geometric meaning.

\begin{enumerate}[leftmargin=2.1em]
\item
\(n_f\) is a small integer multiplicity coming from the allowed \OGNfive{}
ledger, family triplicity, or color/electroweak branch counting.

\item
\(2^{-a_f}\) is the dyadic seam depth. The appearance of dyadic factors is
not optional once the neutral electroweak code closes through \(2^J\).
Every local branch splitting on the boundary inherits this binary ledger.

\item
\(\e^{-\pi w_f}\) is the compact-angle suppression. Because there is only
one primitive circle \(S^1_\theta\), the only admissible monodromy
suppression is an integer multiple of a single angular unit. The
first-order unit is \(\pi\), corresponding to the minimal nontrivial seam
mismatch between opposite boundary images.

\item
\(L^{-d_f}=5^{-d_f}\) counts extra doublet-projection penalties. The
factor \(L=5\) is already distinguished in the Higgs descent
\(\mathbf 5_H\to \mathbf 2_H\), so repeated weak-doublet reprojection
naturally generates additional powers of \(1/5\) for the lightest modes.
\end{enumerate}

Equation~\eqref{eq:minimal-family-code-main} is not yet the family
solution. It is the \emph{constitutional alphabet}. The actual family
codebook is obtained by imposing four further requirements.

\begin{enumerate}[leftmargin=2.1em]
\item
\textbf{Minimality:} use the lexicographically shortest code compatible
with a three-generation hierarchy.

\item
\textbf{Sector coherence:} quarks, charged leptons, and neutrinos should
each have one dominant organizing principle rather than a separate
unrelated rule for every particle.

\item
\textbf{No hidden fit parameters:} every numerator and denominator must be
expressible directly in terms of \(M,N,L,J,K\), small integers, or a
single primitive angular factor \(\e^{-\pi}\).

\item
\textbf{Monotone family order:} the third family should be least
suppressed, the second family intermediate, and the first family most
suppressed, unless a sector-specific null direction is protected at first
order.
\end{enumerate}

The remainder of this section shows that these simple principles already
lead to a surprisingly rigid 12-mass block.

\subsection{How the already-fixed gauge/Higgs block enters}

For the nine charged fermions the gauge/Higgs block enters through only one
normalization factor,
\begin{equation}
\frac{v^{(1)}}{\sqrt2},
\label{eq:v-over-root2-main}
\end{equation}
multiplied by the appropriate geometric code \(y_f^{(1)}\). Hence once
\(v^{(1)}\) is fixed, all relative hierarchy information is carried by the
dimensionless overlaps. This is exactly why it was necessary to solve the
hierarchy problem and the Higgs origin first.

For the neutrinos the role of the Higgs block is even sharper. The
allowed operator \((LH)(LH)\) is dimension-5, so after the Higgs acquires
its VEV the neutrino masses scale as \(v^2/\Lambda\). In other words, the
neutrino sector is quadratic in the electroweak scale. Any ambiguity in
the Higgs sector would therefore have been amplified in the neutrino
sector. The present ordering of the derivation avoids that problem.

\section{The charged-lepton floor: masses from dyadic boundary overlaps}
\label{sec:lepton-mass}

\subsection{Historical background}

Yukawa's 1935 paper introduced the dimensionless coupling between fermions and a scalar field that bears his name \citep{Yukawa1935}. In the Standard Model the three charged-lepton masses (\(m_e,m_\mu,m_\tau\)) are encoded as three independent Yukawa couplings \(y_\ell=m_\ell\sqrt2/v\) with no structural explanation for their hierarchy \(m_e:m_\mu:m_\tau\approx 1:207:3477\). The theorems below derive the same hierarchy as a cascade of dyadic seam-overlap depths on the \OGNfive{} ledger: each lepton sits at a specific dyadic stratum, and its mass is the readout of the overlap between its boundary projector and the Higgs branch. Indeed, the Yukawa hierarchy is no longer three free numbers but three rational depths.


\subsection{The charged-lepton \GRIP{} cascade}

The charged leptons are the cleanest mass block: color-singlet, so they pay no color-seam
price, and dyadic before angular at first order. On the family stratum
$\Sigma_{\rm fam} = \mathbb{C}P^2$ they form a single \GRIP{} cascade
(\S\ref{subsec:grip-spine}): the three generations are its Geometric Attractors (\GA{}'s), the
index-$0$ basins whose floors are the spectral gaps, normalized by the shared closed
electroweak packet (the universal $v/\sqrt2$ of Eq.~\eqref{eq:v-over-root2-main}), with
\BIsix{} forbidding any second independent compact flavor angle or suppression ladder so that
only the dyadic seam depth and the weak-doublet projection factor $L^{-1}$ survive; the
first-order masses are the ideal-seam \GA{} exports and the QD corrections are the \CCI{}
transport to the practical comparison convention. The cascade terminates not on an ordinary
\GA{} but on the \GGA{}: the electron is the unique deepest basin, the one attractor from which
no admissible exit channel remains, which is why it never decays. That ground-state distinction
is invisible in the masses themselves but is exactly what the $\tau \to \mu \to e$ lifetime
ladder measures --- a single \GRIPin{} descent paying the two Bogomolnyi tolls $S = 2\pi$ and
$S = \pi$, worked in full in \PaperPThree{} \citep{ArisakaP3Electron}.

The same cascade fixes more than the basin floors. Because each non-ground \GA{}
must leave its basin across a \GSSB{} gate to reach the \GGA{}, the charged-lepton
stratum carries a second, dynamical readout alongside the masses --- the generation
lifetimes, which \ASix{}'s \GRIP{}-ST7 identifies with the boundary projection of
the cumulative cascade action. The muon is the clean case: its decay
$\mu^-\!\to e^-\bar\nu_e\nu_\mu$ is a single \GRIPin{} descent across the gate
$S_{[2]\to[0]}=\pi$, implemented by the weak charged current whose coupling is the
geometric weak scale $G_F = 1/(\sqrt2\,v^2)$, so that the width is
$\Gamma_\mu = G_F^2 m_\mu^5/192\pi^3 = m_\mu^5/(384\pi^3 v^4)$ and the lifetime is
$\tau_\mu = \Gamma_\mu^{-1} = 2.197~\mu\text{s}$ (leading order $2.187~\mu$s; the
exact phase space and the leading radiative correction recover the measured value
to the part-per-million). The electron, having no exit gate, has $\Gamma_e=0$ and
$\tau_e=\infty$, so its absolute stability and the muon's finite lifetime are one
statement evaluated at zero versus one gate. Mass and lifetime are thus the static
and dynamical faces of the same geometric object --- the depth of a basin and the
rate of crossing its gate. As with the masses the reading is unconditional but the
number is conditioned: $G_F$ (equivalently $v$) is Derived (cond.) while $m_\mu$ is
Asserted (Table~\ref{tab:codebook-provenance}), so $\tau_\mu$ is derived modulo the
Asserted muon-mass depth. The full \GRIP{}-language derivation --- the GKLS form of
the descent, the $\DGI\!\to\!\GSSB\!\to\!\GA$ staging, and the worked numbers ---
is given as a proposition in \PaperPThree{} \citep{ArisakaP3Electron}.

\label{sec:charged-leptons}

The charged leptons are the cleanest entry point into the 12-mass block.
They do not feel QCD, they carry the same electric charge, and at first
order they need not traverse the compact angle nontrivially. Their
hierarchy can therefore be organized almost entirely by the dyadic seam
depth and by extra weak-doublet projection factors.

\subsection{\GiFour{} cascade-depth reading for the charged leptons}

\label{sec:lepton-cascade-depth}

Before presenting the dyadic codebook, we cast the three charged-lepton masses in the canonical \GiFour{} cascade-depth language of \ASix{} (specifically A6 §25 ``\DGI{}--\GSSB{}--GA cascade in detail'' and A6 §28 Theorem~\T{4.1} ``\GRIP{} cascade closure'') \citep{ArisakaA6Gi4} --- and we do so in the ground-state vocabulary of \ASix{} (cf.\ \S\ref{subsec:gga-refinement}): the cascade terminates not merely on a \GA{} but on the \GGA{}, and that distinction, invisible in the mass formulas, is precisely what the generation \emph{lifetime} pattern measures. The family-projector stratum carrying the three generations is, by Theorem~\ref{thm:N3-closed} of \S\ref{sec:N3-theorem},
\begin{equation}
\Sigma_{\rm fam} = \mathbb{C}P^{N-1} = \mathbb{C}P^2,
\qquad \dim_{\mathbb R} \Sigma_{\rm fam} = 4.
\label{eq:family-stratum-leptons}
\end{equation}
By \ASix{} \T{3.1} (\GFF{} Morse-positivity from \CCI{}), the projected free-energy functional restricted to $\Sigma_{\rm fam}$ is a perfect Morse function with three non-degenerate critical points of indices $0$, $2$, $4$ (Lemma~4.2 of \citealp{ArisakaP2N3}). By \ASix{} \T{4.1} (\GRIP{} cascade closure), the three critical points form the canonical \GRIP{} chain
\begin{equation}
\underbrace{[\,\mathrm{ind}=4\,]}_{\DGI{}\;=\;\text{depth }0}
\;\xrightarrow{\;S_{[4]\to[2]}=2\pi\;}\;
\underbrace{[\,\mathrm{ind}=2\,]}_{\GSSB{}\;=\;\text{depth }1}
\;\xrightarrow{\;S_{[2]\to[0]}=\pi\;}\;
\underbrace{[\,\mathrm{ind}=0\,]}_{\GGA{}\;=\;\text{depth }2,}
\label{eq:lepton-grip-chain}
\end{equation}
with cumulative cascade actions $0$, $2\pi$, $3\pi$ and total $S_{\rm tot}(3) = 3\pi$ exactly saturating the \BIsix{} budget $S_{\rm BI}^{\max} = 3\pi$ (cf.\ Theorem~\ref{thm:N3-closed}). Indeed, this is the same cascade chain that appeared in Leg~L2 of \S\ref{subsec:N3-L2}. One precision, supplied by the ground-state refinement, governs how the chain is to be read: \eqref{eq:lepton-grip-chain} is the Morse \emph{skeleton} of the stratum --- its critical-point count is what forces $N=3$, and nothing below touches that --- while the physical generations are the index-$0$ basins of the depth-$n$ effective landscapes hung on this skeleton, \emph{not} the saddle points themselves. The index-4 and index-2 critical points are the \GSSB{} gates the descent crosses, with their Bogomolnyi tolls $2\pi$ and $\pi$.

\paragraph{Generation-to-cascade assignment.} The three charged leptons sit on the three critical points of \eqref{eq:lepton-grip-chain} as
\begin{equation}
\boxed{
\begin{array}{lcl}
\tau & \longleftrightarrow & \text{depth } 0:\ \GA{} \;\text{(non-ground $\Rightarrow$ metastable; $m_\tau$ sharply defined)}, \\
\mu & \longleftrightarrow & \text{depth } 1:\ \GA{} \;\text{(non-ground $\Rightarrow$ metastable; $m_\mu$ sharply defined)}, \\
e & \longleftrightarrow & \text{depth } 2:\ \GGA{} \;\text{(ground state $\Rightarrow$ absolutely stable)}.
\end{array}
}
\label{eq:lepton-cascade-assignment}
\end{equation}
All three generations are genuine Geometric Attractors: each is an index-$0$ minimum of the effective free energy on its own flavor stratum, and this is not pedantry but spectroscopy --- a saddle supports no sharply defined mass, whereas $m_\tau$ and $m_\mu$ are measured at the $10^{-4}$ relative level or better. Precisely measurable masses are the experimental certificate that $\tau$ and $\mu$ are \emph{perfect \GA{}'s}. What they are not is the ground state: only the depth-2 basin is the global minimum --- the \GGA{} --- and the non-ground \GA{}'s at depths 0 and 1 therefore relax toward it through the weak channels, $\tau\to\mu\to e$, until the population reaches the ground state, which never decays. In the regime vocabulary of \ASix{}, the whole ladder is one \GRIPin{} trajectory on the fixed family landscape, pausing at two way-station basins and halting at the third; the landscape itself was built earlier, by the electroweak \GRIPout{} of Part~\ref{part:VI}.

\paragraph{Why the lepton sector is dyadic rather than angular.}
The charged leptons are color singlets and electroweak doublets, so the only seam structure that distinguishes them at first order is the dyadic seam depth $2^J = 2^7 = 128$ together with the weak-projection factor $L^{-3}$. The three generations therefore inherit the cascade depth $0, 1, 2$ as a multiplicative dyadic suppression rather than as an angular winding (the angular cascade is reserved for the quarks; see \S\ref{sec:quark-cascade-depth}). Concretely the first-order Yukawas, derived in detail below, satisfy the proportionality
\begin{equation}
y_\tau^{(1)}:y_\mu^{(1)}:y_e^{(1)}
\;\propto\;
\frac{NJ}{2^{J+N+1}}:\frac{L}{2^{J+L+1}}:\frac{N}{L^3 \cdot 2^{J+L+1}},
\label{eq:lepton-yukawa-ratios}
\end{equation}
which makes the cascade-depth pattern visible as an increasing dyadic depth $J{+}N{+}1 = 11$, $J{+}L{+}1 = 13$, $J{+}L{+}1 + 3\log_2 L \approx 13 + 6.97$ for the three generations.

\begin{euclidremark}[Physics intuition for the lepton cascade]
The three charged-lepton generations are not three independent species coexisting at separate energy scales, nor are they saddle points of the landscape; they are three \emph{basins} of one cascade landscape on $\mathbb{C}P^2$ --- two metastable \GA{}'s and one \GGA{}. After all, the lifetime hierarchy $\tau_\tau \approx 2.9 \times 10^{-13}$~s versus $\tau_\mu \approx 2.2 \times 10^{-6}$~s versus $\tau_e = \infty$ is exactly what the ground-state refinement predicts: the shallower non-ground basin relaxes fastest, the deeper non-ground basin six orders of magnitude more slowly, and the \GGA{} never relaxes at all. Identity and stability are again one fact: every electron is the same electron \emph{and} immortal for the same reason --- it is the ground state of its stratum (\S\ref{subsec:gga-refinement}).
\end{euclidremark}

\begin{figure}[H]
\centering
\resizebox{0.676\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=22mm and 32mm,
 morsenode/.style={circle, draw=black!70, line width=0.6pt, minimum size=11mm, font=\small\bfseries\sffamily, align=center},
 topnode/.style={morsenode, fill=red!18},
 midnode/.style={morsenode, fill=orange!16},
 botnode/.style={morsenode, fill=green!16},
 arrowstyle/.style={-{Latex[length=2.6mm]}, line width=0.8pt},
 actionlabel/.style={font=\footnotesize\sffamily, fill=white, inner sep=1.5pt}
]
 \node[topnode] (top) at (0,8) {$\tau$ \\[-0.3em] {\scriptsize \GA{}, depth 0}};
 \node[midnode] (mid) at (0,4) {$\mu$ \\[-0.3em] {\scriptsize \GA{}, depth 1}};
 \node[botnode] (bot) at (0,0) {$e$ \\[-0.3em] {\scriptsize \GGA{}, depth 2}};

 \draw[arrowstyle] (top) -- node[actionlabel, right=2pt]{\GSSB{} gate (ind $4\to2$): $S = 2\pi$} (mid);
 \draw[arrowstyle] (mid) -- node[actionlabel, right=2pt]{\GSSB{} gate (ind $2\to0$): $S = \pi$} (bot);

 \node[font=\small\sffamily, right=42mm of top] {3rd generation: $\tau$, $t$, $b$};
 \node[font=\small\sffamily, right=42mm of mid] {2nd generation: $\mu$, $c$, $s$};
 \node[font=\small\sffamily, right=42mm of bot] {1st generation: $e$, $u$, $d$};

 \node[font=\scriptsize\sffamily, left=4pt of top, text=red!70!black] {non-ground \GA{}: metastable};
 \node[font=\scriptsize\sffamily, left=4pt of mid, text=orange!80!black] {non-ground \GA{}: metastable};
 \node[font=\scriptsize\sffamily, left=4pt of bot, text=green!50!black] {ground state (\GGA{}): stable};

 \node[font=\scriptsize\sffamily\itshape, below=8mm of bot]
 {Total cascade action $S_{\rm tot}(3) = 3\pi$ \emph{exactly saturates} the BI-6 budget $S_{\rm BI}^{\max} = 3\pi$.};

\end{tikzpicture}%
}
\caption[The \GRIP{} cascade on the family-projector moduli.]{The \GRIP{} cascade on the family-projector moduli $\Sigma_{\rm fam} = \mathbb{C}P^{N-1} = \mathbb{C}P^2$ at $N=3$, in the ground-state (\GGA{}) reading of \ASix{}. The Morse skeleton of the \GFF{} height function keeps its three critical points (indices 4, 2, 0) and Bogomolnyi actions $S_{[2k]\to[2(k-1)]} = \pi k$ \citep{Witten1982}, accumulating to $S_{\rm tot}(3) = 3\pi$ and exactly saturating the BI-6 budget --- the cascade-closure count that forces $N=3$ (Theorem~\ref{thm:N3-closed}) is untouched. What the figure states precisely is the particle identification: all three generations are genuine Geometric Attractors --- index-$0$ basins of their depth-$n$ effective landscapes, which is why $m_\tau$ and $m_\mu$ are sharply measurable --- but only the depth-2 basin is the \emph{Ground} Geometric Attractor. The non-ground \GA{}'s relax toward it ($\tau\to\mu\to e$); the \GGA{} never decays. The index-4 and index-2 critical points are the \GSSB{} gates the descent crosses, not the particles. In regime language (\ASix{}): the two arrows are steps of a single \GRIPin{} descent on a fixed landscape; the \GGA{} is where \GRIPin{} halts.}
\label{fig:grip-cascade-cp2}
\end{figure}

The same cascade has an equivalent, and more intuitive, reading as a \emph{landscape}, which the companion electron paper develops in full (\PaperPThree{}, Part~IV). Figure~\ref{fig:lepton-basin-landscape} draws a one-dimensional cut through the \GFF{} Morse landscape on the family stratum $\mathbb{C}P^2$: the three generations are three basins, their floors are the masses, and the two \GSSB{} gates between them carry the Bogomolnyi action tolls $S = 2\pi$ and $S = \pi$. The observed $\tau \to \mu \to e$ weak-decay chain is then a single \GRIPin{} descent that pays both tolls and halts in the deepest basin --- the \GGA{}, which has no exit gate. The grammar is exactly Gamow's: as an alpha-decay rate carries an action in the exponent, $e^{-S}$, each gate crossing here is suppressed by its toll, and the lifetimes order themselves by what stands between a state and its destination (the quantitative $\tau$--$\mu$ accounting is \PaperPTwo{}~\S7.4). Four facts that the cascade-depth codebook records separately --- the identity of every electron (one deepest basin), the absolute stability of the electron (no gate out of it), the existence of exactly two excited copies (the two way-stations), and the decay hierarchy (tolls in exponents) --- are all visible in this single drawing.

\begin{figure}[!htb]
\centering
\resizebox{0.733\textwidth}{!}{%
\begin{tikzpicture}[
 yscale=1.40,
 font=\small,
 arr/.style={->, line width=0.80pt, draw=black!70},
 floorline/.style={dashed, draw=black!45, line width=0.50pt},
 gatemark/.style={draw=red!70!black, line width=0.90pt},
 desc/.style={-{Stealth[length=2.6mm]}, line width=1.10pt, draw=blue!60!black},
 lab/.style={font=\footnotesize, align=center}
]
%% axes
\draw[arr] (-0.3, -0.4) -- (-0.3, 6.0) node[anchor=south] {$\FGFF$};
\draw[arr] (-0.4, -0.3) -- (12.8, -0.3);
\node[font=\footnotesize\itshape, anchor=north east] at (12.6, -0.50)
 {configuration coordinate on the family stratum $\mathbb{C}P^{2}$ (schematic 1D cut)};
%% the landscape curve
\draw[line width=1.10pt, draw=black!80] plot[smooth, tension=0.75] coordinates {
 (0.0, 5.4) (1.0, 4.3) (2.3, 3.45) (3.0, 3.55) (3.9, 4.30)
 (4.9, 3.1) (5.9, 2.55) (6.7, 2.7) (7.5, 3.30)
 (8.6, 1.6) (10.0, 0.85) (11.2, 1.4) (12.2, 4.9) };
%% basin floors (dashed levels)
\draw[floorline] (2.3, 3.45) -- (13.0, 3.45) node[anchor=west, font=\footnotesize] {$m_\tau$};
\draw[floorline] (5.9, 2.55) -- (13.0, 2.55) node[anchor=west, font=\footnotesize] {$m_\mu$};
\draw[floorline] (10.0, 0.85) -- (13.0, 0.85) node[anchor=west, font=\footnotesize] {$m_e$};
\node[font=\scriptsize\itshape, anchor=west, text=black!60] at (11.55, 0.45) {(levels not to scale)};
%% particles in basins
\fill[black!75] (2.55, 3.46) circle (1.8pt) node[lab, below=2pt, xshift=-5mm] {$\tau$ --- depth 0\\ \GA{} (metastable)};
\fill[black!75] (6.2, 2.56) circle (1.8pt) node[lab, below=2pt, xshift=4mm] {$\mu$ --- depth 1\\ \GA{} (metastable)};
\fill[blue!60!black] (10.0, 0.86) circle (2.2pt) node[lab, below=3pt, text=blue!60!black]
 {$e$\\ depth 2\\ \textbf{\GGA{}: no exit gate}};
%% gates (saddles)
\draw[gatemark] (3.9, 4.42) -- (3.9, 4.18);
\node[lab, text=red!70!black, anchor=south] at (3.9, 4.50) {\GSSB{} gate\\ toll $S = 2\pi$};
\draw[gatemark] (7.5, 3.42) -- (7.5, 3.18);
\node[lab, text=red!70!black, anchor=south] at (7.5, 3.50) {\GSSB{} gate\\ toll $S = \pi$};
%% GRIP-In descent arrows along the curve
\draw[desc] (2.9, 3.75) to[bend left=28] (4.85, 3.45);
\draw[desc] (6.6, 3.0) to[bend left=28] (8.5, 2.0);
\node[lab, text=blue!60!black, anchor=west] at (0.0, 1.05)
 {\textbf{\GRIPin{} descent $\tau \to \mu \to e$}\\
 implemented by the weak interaction;\\
 each hop pays its gate toll $e^{-S}$};
%% no lower basin annotation
\node[lab, text=black!65, anchor=west] at (8.7, 4.9)
 {nothing below depth 2:\\ the landscape's global minimum\\ (\GFF{}-ST7 uniqueness)};
\draw[-{Stealth[length=2mm]}, draw=black!55] (9.5, 4.45) to[bend right=20] (9.85, 1.2);
\end{tikzpicture}%
}
\caption[The charged-lepton landscape in one picture (from \PaperPThree{}).]{\textbf{The charged-lepton landscape in one picture} (reproduced from \PaperPThree{}, Part~IV). A schematic one-dimensional cut through the \GFF{} Morse landscape on the family stratum $\mathbb{C}P^2$. The three generations are the three basins; their floors are the masses (levels schematic, not to scale); the two \GSSB{} gates carry the Bogomolnyi tolls $S = 2\pi$ and $S = \pi$; and the observed $\tau \to \mu \to e$ weak-decay chain is a single \GRIPin{} descent that pays both tolls and halts in the unique deepest basin --- the \GGA{}, which has no exit gate. Identity (one deepest basin), stability (no gate out of it), the excited copies (the two way stations), and the decay hierarchy (tolls in exponents) are all visible in this one drawing.}
\label{fig:lepton-basin-landscape}
\end{figure}





\subsection{Why the charged-lepton sector is purely dyadic at first order}

The compact circle \(S^1_\theta\) is needed to support the Abelian
holonomy and the Wilson-line Higgs. However, a field is not forced to pay
a nontrivial angular suppression at first order unless its chirality or
charge assignment distinguishes different angular branches. The
color-singlet charged leptons are the most economical sector in this
respect. Their first-order family hierarchy can therefore be taken to
have
\begin{equation}
w_e=w_\mu=w_\tau=0.
\label{eq:lepton-w-zero-main}
\end{equation}

This statement has an intuitive reading. The charged leptons live closest
to the boundary measurement layer. At leading order they are separated
not by genuine compact monodromy, but by how deeply the boundary
projector must resolve their profiles. Their hierarchy is therefore
dyadic before it is angular.

\subsection{Minimal first-order charged-lepton codebook}

The shortest charged-lepton codebook compatible with the OGN ledger and
with the required family order is
\begin{equation}
\boxed{
y_\tau^{(1)}=\frac{NJ}{2^{J+N+1}}
=\frac{21}{2^{11}}
=\frac{21}{2048},
}
\label{eq:y-tau-main}
\end{equation}
\begin{equation}
\boxed{
y_\mu^{(1)}=\frac{L}{2^{J+L+1}}
=\frac{5}{2^{13}}
=\frac{5}{8192},
}
\label{eq:y-mu-main}
\end{equation}
\begin{equation}
\boxed{
y_e^{(1)}=\frac{N}{L^3\,2^{J+L+1}}
=\frac{3}{5^3\,2^{13}}
=\frac{3}{1024000}.
}
\label{eq:y-e-main}
\end{equation}

Let us explain each code rather than merely display it.

\paragraph{The tau.}
The third family charged lepton should be the least suppressed. The
largest admissible numerator available without introducing a new free
integer is \(NJ=3\times 7=21\), namely the product of family triplicity
and the fundamental dyadic closure index. The mildest dyadic depth that
still reflects both the neutral closure \(J\) and the triplicity index
\(N\) is \(J+N+1=11\). This gives
\[
y_\tau^{(1)}=\frac{NJ}{2^{J+N+1}}.
\]
It is the minimal code in which both the family structure and the dyadic
closure visibly appear.

\paragraph{The muon.}
The second family must lie below the tau but above the electron. The
smallest available numerator that naturally encodes the electroweak
doublet ancestry is \(L=5\). The corresponding dyadic depth shifts from
\(J+N+1\) to \(J+L+1\), i.e.\ from 11 to 13, reflecting that the muon lies
one electroweak layer deeper than the tau in the boundary-resolution
ordering. Thus
\[
y_\mu^{(1)}=\frac{L}{2^{J+L+1}}.
\]

\paragraph{The electron.}
The lightest charged lepton must be suppressed relative to the muon
without introducing a new transcendental. The only remaining geometric
penalty already distinguished in the Higgs descent is the doublet
projection factor \(1/L=1/5\). Applying it three times gives the triple
boundary re-projection \(L^{-3}\), so that
\[
y_e^{(1)}=\frac{N}{L^3 2^{J+L+1}}.
\]
The numerator returns to \(N\), because by the time the lightest mode is
reached the dominant structure is no longer the dyadic closure itself but
the persistence of family triplicity through three successive weak
projections.

\subsection{First-order charged-lepton masses}

Using \eqref{eq:dirac-mass-general-main} and the already derived Higgs
vacuum expectation value \(v^{(1)}=246.331354~\mathrm{GeV}\), the charged
lepton masses become
\begin{equation}
m_\tau^{(1)}=\frac{v^{(1)}}{\sqrt2}\,\frac{21}{2048}
=1.78605175~\mathrm{GeV},
\label{eq:m-tau-1-main}
\end{equation}
\begin{equation}
m_\mu^{(1)}=\frac{v^{(1)}}{\sqrt2}\,\frac{5}{8192}
=106.312604~\mathrm{MeV},
\label{eq:m-mu-1-main}
\end{equation}
\begin{equation}
m_e^{(1)}=\frac{v^{(1)}}{\sqrt2}\,\frac{3}{1024000}
=0.510300500~\mathrm{MeV}.
\label{eq:m-e-1-main}
\end{equation}

These numbers are already striking because no charged-lepton mass has been
used as an input. Only the electroweak vacuum scale, itself previously
derived from the lock geometry, has been used together with the dyadic
codebook.

\subsection{First-order ratio structure and intuition}

The codebook immediately yields two exact mass ratios:
\begin{equation}
\frac{m_\tau^{(1)}}{m_\mu^{(1)}}
=
\frac{y_\tau^{(1)}}{y_\mu^{(1)}}
=
\frac{NJ}{L}\,2^{L-N}
=
\frac{84}{5}
=16.8,
\label{eq:tau-mu-ratio-main}
\end{equation}
and
\begin{equation}
\frac{m_\mu^{(1)}}{m_e^{(1)}}
=
\frac{y_\mu^{(1)}}{y_e^{(1)}}
=
\frac{L^4}{N}
=
\frac{625}{3}
=208.333\ldots.
\label{eq:mu-e-ratio-main}
\end{equation}

These relations are useful for two reasons. First, they show that the
charged-lepton hierarchy is not a random list of unrelated numbers.
Second, they make the role of the OGN integers transparent:
\(N=3\) organizes family triplicity, \(J=7\) sets the dyadic seam
granularity, and \(L=5\) governs electroweak projection depth.

\subsection{QD completion of the charged-lepton block}

The first-order codebook is the rigid skeleton, not yet the final
low-energy seam value. The finite QD transport from the idealized seam to
the comparison convention is encoded by a common denominator
\begin{equation}
2^J L = 2^7\cdot 5 = 640.
\label{eq:lepton-common-den-main}
\end{equation}
The shortest allowed rational closures are then
\begin{equation}
\boxed{
\mathcal R_\tau = 1-\frac{N}{2^J L}
=1-\frac{3}{640}
=\frac{637}{640},
}
\label{eq:R-tau-main}
\end{equation}
\begin{equation}
\boxed{
\mathcal R_\mu = 1-\frac{L-M}{2^J L}
=1-\frac{4}{640}
=\frac{159}{160},
}
\label{eq:R-mu-main}
\end{equation}
\begin{equation}
\boxed{
\mathcal R_e = 1+\frac{M}{2^J L}
=1+\frac{1}{640}
=\frac{641}{640}.
}
\label{eq:R-e-main}
\end{equation}

The corresponding QD masses are defined as transport-corrected seam masses,
\begin{equation}
m_\ell^{\mathrm{QD}}=\mathcal R_\ell\,m_\ell^{(1)},
\label{eq:lepton-qd-general-main}
\end{equation}
so that
\begin{equation}
m_\tau^{\mathrm{QD}}=1.777679634~\mathrm{GeV},
\qquad
m_\mu^{\mathrm{QD}}=105.6481505~\mathrm{MeV},
\qquad
m_e^{\mathrm{QD}}=0.511097845~\mathrm{MeV}.
\label{eq:lepton-qd-values-main}
\end{equation}

The physical meaning of these corrections is simple. The tau is shifted
slightly downward because the least-suppressed third-family mode still
experiences a finite leakage into neighboring dyadic cells. The muon is
shifted downward by a slightly larger relative amount because it sits one
resolution layer deeper. The electron is shifted upward because, once the
triple \(L^{-3}\) suppression is applied, the leading correction is a
minimal recovery of one boundary cell rather than an additional loss.

\section{Why is the weak interaction special?}
\label{sec:weak-special}

The cascade of \S\ref{sec:lepton-cascade-depth} (Figures~\ref{fig:grip-cascade-cp2} and~\ref{fig:lepton-basin-landscape}) raises a question the Standard Model never answers: of the four fundamental forces, why is it always the \emph{weak} interaction that carries one generation into the next? The textbook gives two replies --- that lepton and baryon number are conserved, and that heavier states decay to lighter ones --- but neither is an explanation. The first is a constraint the weak interaction happens to respect; the second is a statement about energetics that says nothing about \emph{which} force does the carrying. Gravity, electromagnetism, and the strong force respect every conservation law just as faithfully, and all three ``know'' that the electron is lighter than the muon; yet not one of them can turn a muon into an electron. Something singles out the weak interaction, and the Standard Model takes that something as a brute fact.

\GGTheory{} does not. In the landscape reading the answer is structural and, we will argue, close to inevitable: \emph{the weak interaction is the only force that acts on the gates of the family landscape rather than within its basins}. The other three forces conserve the charges that label the generations and can therefore move a state only inside its own generation; the weak interaction is the unique force built to carry a state across the saddle between generations. The remainder of this section makes that precise, then draws two consequences we have not seen stated elsewhere: that the weak interaction and the three-generation structure are two faces of a single fact, and that the neutrino sector is the ledger the cascade requires. We close with the honest status of the argument and the lemma that would make it a theorem.

\subsection{The question, sharpened}
\label{subsec:weak-question}

On the family stratum $\mathbb{C}P^{2} = \mathbb{C}P^{N-1}$ the three generations are not arbitrary copies but the three critical points of the \GFF{} Morse height function (Figure~\ref{fig:grip-cascade-cp2}): index-$4$, index-$2$, and index-$0$ basins, with Bogomolnyi saddles of action $S = 2\pi$ and $S = \pi$ between them. A particle \emph{is} a Geometric Attractor at the floor of its basin; a weak decay \emph{is} the crossing of a \GSSB{} gate from one basin to the adjacent deeper one. The vague question ``why the weak force?'' thus becomes a precise geometric one:
\begin{quote}
of the four interactions, which can transport a configuration across a Morse saddle on the family stratum --- and why is the answer unique?
\end{quote}
We measure the four forces against exactly this requirement.

\subsection{Basin forces and the gate force}
\label{subsec:basin-gate}

Three of the four forces turn out to be structurally incapable of crossing a gate; the fourth is structurally built to. Table~\ref{tab:basin-gate} summarizes the verdict; the five paragraphs below are its proof sketch.

\begin{table}[H]
\centering
\caption[The four fundamental interactions measured against a single requirement: can the force carry a state across a gate of the family Morse landscape?.]{The four fundamental interactions measured against a single requirement: can the force carry a state across a gate of the family Morse landscape? Three are \emph{basin forces} (unbroken or universal, parity-even, family-diagonal); only the weak interaction is the \emph{gate force} (broken, chiral, family-off-diagonal, short-range).}
\label{tab:basin-gate}
{\small
\begin{tabular}{@{}p{0.135\textwidth}p{0.205\textwidth}p{0.10\textwidth}p{0.115\textwidth}p{0.11\textwidth}p{0.16\textwidth}@{}}
\toprule
\textbf{Force} & \textbf{Gauge status} & \textbf{Range} & \textbf{Parity} & \textbf{Family} & \textbf{Role on the landscape}\\
\midrule
Gravity & Bulk geometry ($\mathrm{Spin}(1,5)$) & Long & Even & Diagonal & Universal --- all basins alike\\
\addlinespace
Strong & Unbroken $\SU(3)_C$ & Confining & Even & Diagonal & Binds within a basin\\
\addlinespace
Electro\-magnetism & Unbroken $\mathrm{U}(1)_{\rm EM}$ & Long & Even & Diagonal & Intra-basin transitions\\
\addlinespace
\textbf{Weak} & \textbf{Broken $\SU(2)_L\!\times\!\mathrm{U}(1)_Y$} & \textbf{Short} & \textbf{Maximally violating} & \textbf{Off-diagonal} & \textbf{Crosses the gates}\\
\bottomrule
\end{tabular}}
\end{table}

\paragraph{1.~The basin labels are the unbroken charges.} Each generation sits at a definite value of every \emph{exactly conserved} quantum number --- color, electric charge, baryon and lepton number. A force whose gauge symmetry is \emph{unbroken} holds its charge fixed and can therefore move a state only within a fixed-charge sheet. That is exactly what gravity, the strong force, and electromagnetism do. Gravity is the bulk $\mathrm{Spin}(1,5)$ geometry: flavor-blind, diagonal, universal --- it bends trajectories but never changes what a particle is. The strong force ($\SU(3)_C$, unbroken) conserves color and quark flavor exactly; it binds the basin without transmuting it. Electromagnetism ($\mathrm{U}(1)_{\rm EM}$, unbroken) conserves charge exactly; an electromagnetic transition changes a particle's \emph{state} --- an atomic level, an emitted photon --- but never its \emph{generation}. These three are \emph{basin forces}: an exact symmetry forbids precisely the family-changing transition a cascade needs.

\paragraph{2.~The weak force is the broken one.} $\SU(2)_L\times\mathrm{U}(1)_Y$ is the one Standard-Model gauge structure that is spontaneously broken; in \GGTheory{} it \emph{is} the electroweak \GRIPout{} event, the landscape-reorganizing symmetry breaking itself. Its charged current is intrinsically \emph{off-diagonal}: it rotates up $\leftrightarrow$ down and, through the CKM/PMNS projector misalignment, connects adjacent generations. A force that is a manifestation of symmetry breaking is, by construction, a force that lives on the symmetry-breaking gates. So the weak interaction is the unique candidate gate operator --- not by stipulation, but because ``being the broken force'' and ``being the gate force'' are the same statement.

\paragraph{3.~Chirality is the orientation of the descent.} A Morse gradient flow has a \emph{direction}: strictly downhill, heavier $\to$ lighter. A direction is an orientation, and an orientation can be carried only by a \emph{parity-violating} coupling. The weak force is maximally chiral --- it couples only to left-handed fermions --- 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.'' The same fact has a second face: the weak sector is the unique home of CP violation, because the CKM and PMNS phases are the \emph{holonomies} acquired in transporting the family projector across the gates, and a holonomy is an intrinsically oriented, complex quantity. Chirality and CP violation are not two separate weak peculiarities; they are the single statement ``the descent is oriented,'' seen twice.

\paragraph{4.~Short range is the locality of the gate.} A saddle is a \emph{localized} feature of the landscape, not a long-range potential, so the mediator of a gate crossing must be short-range --- that is, massive. The $W$ and $Z$ acquire their mass from the \GSSB{} vacuum (the Wilson--Hosotani holonomy): they are literally the massive modes of the broken symmetry, the quanta of the gates. The long-range forces (massless photon, graviton) are long-range precisely because they are basin or universal forces, attached to no gate. ``The weak force is short-range'' and ``the gates are localized saddles'' are the same sentence.

\paragraph{5.~The open system is why the descent happens, and where it stops.} The boundary sector is an open system (\GDOS{}). A non-ground Geometric Attractor is \emph{metastable}: it has an exit gate, but it needs dissipation to cross it, and the coupling to the boundary bath --- carried by the weak current --- supplies exactly that dissipation. So ``heavier decays to lighter'' is not a thermodynamic platitude but the precise statement that every non-ground \GA{} has a downhill gate and the weak interaction is the transport across it. The Ground Geometric Attractor --- the electron, the lightest neutrino, the proton's ground content --- has \emph{no} exit gate; there is no channel for the open system to relax it, and it is absolutely stable. The weak force still acts on it, but there is nowhere downhill to go.

Read together, the five points say that of the four forces exactly one is broken, chiral, off-diagonal, short-range, and dissipative on the boundary --- and those are exactly the properties a Morse-saddle transport operator must have. The weak interaction is special because it is the only force that is a property of the landscape's \emph{gates} rather than its \emph{basins}, and a cascade is, by definition, a sequence of gate crossings.

\subsection{Why the weak interaction and three families are one fact}
\label{subsec:weak-three-families}

The argument has a consequence that reaches past the weak interaction itself. A gate force is meaningless without gates; gates require at least two basins; and at least two basins require at least two generations. Conversely, a landscape with several basins but no gate force could never drain: its excited basins would be eternally stable, and the very distinction between ground and excited would be physically inert, betrayed by no transition. \emph{A universe therefore has both a weak interaction and a replicated family structure, or it has neither.} One does not explain the other; they are mutually entailed --- the gate force needs the gates, and the gates need the gate force to mean anything.

What fixes the \emph{number} of basins is closure. The gate tolls accumulate as $S_{\rm tot}(N) = \sum_{k=1}^{N-1}\pi k = \tfrac{1}{2}\pi N(N-1)$, and the cascade closes only when this exactly exhausts the \BIsix{} anomaly-inflow budget $S_{\rm BI}^{\max} = 3\pi$ (Theorem~\ref{thm:N3-closed}). The closure condition $\tfrac12 N(N-1) = 3$, i.e.\ $N(N-1) = 6$, has the unique positive solution $N = 3$: with two families the budget is under-used ($S_{\rm tot} = \pi$) and the cascade does not close; with four it is over-run ($S_{\rm tot} = 6\pi$) and the geometry is inconsistent; only three families spend the budget exactly. The same closure that forces $N=3$ forces the existence of the two gates, and the two gates are what the weak interaction exists to cross.

This is, we believe, the deepest available answer to a question usually treated as unanswerable --- ``why are there three generations?'' --- and it arrives bound to its twin, ``why is there a weak force?'' The family landscape must possess gates for the geometry to settle into its ground state; the closure of those gates against the anomaly budget admits exactly three basins; and crossing them demands a broken, chiral, short-range current. Nature contains three families and a weak interaction for one reason: a closed, drainable family landscape requires exactly that, and nothing less.

\subsection{The neutrino ledger}
\label{subsec:weak-neutrino-ledger}

The gate force must conserve the exactly-conserved charges even as it changes generation --- it is off-diagonal in the family index but strictly diagonal in $(B,L)$. That double requirement cannot be met by a charged lepton alone: descending a basin while holding each family lepton number fixed demands a neutral partner to carry the family label away. The neutrinos are precisely those partners. The chain $\tau \to \mu\,\nu_\tau\,\bar\nu_\mu$, then $\mu \to e\,\nu_\mu\,\bar\nu_e$, conserves $L_e$, $L_\mu$, and $L_\tau$ individually because each emitted neutrino balances the family ledger entry that the charged lepton vacates. On this reading the neutrino sector is not incidental decoration: a gate force that conserves $(B,L)$ but changes generation \emph{requires} a per-family neutral bookkeeping partner, and the three neutrinos are the three ledger columns the three-basin cascade demands. The question ``why does each charged lepton come with a neutrino?'' is answered together with ``why a weak force?'': the cascade cannot conserve its exact charges without them.

\subsection{The cosmic office of the weak force}
\label{subsec:weak-cosmic}

The structural argument has a physical payoff that explains why a universe \emph{needs} the weak interaction at all. Were the gate force absent, the excited basins --- the muon and tau, and the heavier quarks --- would be absolutely stable: the landscape would have basins but no drain. There would be no $\beta$ decay and hence no nucleosynthesis beyond hydrogen, no proton--proton chain and hence no stellar burning, and no route by which matter could relax to its ground configuration. The weak interaction is the channel through which a multi-basin geometry settles into its Ground Geometric Attractor everywhere; it is the force of cosmic descent to the minimum. That the Sun shines on the ten-billion-year timescale of one slow \GRIP{} rung --- the weak proton--proton bottleneck --- is the same fact at stellar scale: the pace of the universe's drainage to its ground state is set by the difficulty of the gate. The weak interaction is not an optional fourth force bolted onto the other three; it is what makes a closed family landscape physically settle, and therefore what makes a universe of stable hydrogen, stars, and chemistry possible. The full development of this relaxation channel \emph{as realized in Nature} --- the lepton and quark ladders, the valley of $\beta$ stability, the Sun, primordial nucleosynthesis, and the neutron star, together with the argument for why the channel must be \emph{weak} in magnitude --- is given in \S\ref{subsec:weak-grip}.

\subsection{Status, and the gate-operator lemma}
\label{subsec:weak-status}

The machinery this argument stands on is established in the cohort: the \GRIP{} cascade and the \GRIPin{}/\GRIPout{} distinction, the \GDOS{} open-system law, the Morse basin--gate--toll picture on $\mathbb{C}P^{N-1}$, the $N=3$ closure (Theorem~\ref{thm:N3-closed}), gauged $\UoneB{}$ with its exact baryon number, and the four-force-uniqueness theorem (Theorem~\ref{thm:four-force-uniqueness}). What is new here is the \emph{synthesis}: the identification of the weak interaction as the unique gate force, of chirality with the orientation of the Morse gradient and of CP violation with the gate holonomy, and the co-entailment of the weak interaction with the three-family structure. We state plainly that this synthesis is a structural argument, not yet a theorem.

The single step that would convert it into one we name the \emph{Gate-Operator Lemma}: \emph{on the family stratum $\mathbb{C}P^{N-1}$, a gauge connection that parallel-transports the family projector between adjacent critical points of the \GFF{} height function is necessarily} (i) \emph{off-diagonal in the critical-point basis}, (ii) \emph{chiral --- it carries the orientation of the downhill gradient flow}, and (iii) \emph{Higgs/Stueckelberg-massive --- its range is the saddle width; and within the boundary gauge group the unique factor admitting all three is $\SU(2)_L$.} A proof would upgrade ``the weak interaction is the gate force'' from a reading to a theorem and would, through the co-entailment above, give a first-principles derivation of the joint existence of the weak interaction and the three generations. We record it as the natural successor lemma to the $N=3$ closure, and as the flavor-side complement of the four-force-uniqueness theorem of \AEleven{}: the weak interaction's entire qualitative profile --- short-ranged, chiral, CP-violating, generation-changing --- is exactly, and only, the profile a gate force must have.

\section{The quark floors: masses from compact-angle winding and radial overlap}
\label{sec:quark-mass}

\subsection{Historical background}

The quark model was introduced independently by Gell-Mann and by Zweig in 1964 as the explanation of the Eightfold Way \citep{GellMann1962}, with the six-flavor spectrum completed by the discovery of the top quark at Tevatron in 1995. Like the charged-lepton sector, the six quark masses are encoded in the Standard Model as six independent Yukawa couplings \citep{Yukawa1935} with no structural explanation. The hierarchy is even more extreme than the lepton case (\(m_u:m_t\approx 1:75000\)) and the across-sector pattern (down-type vs up-type) is left as an empirical mystery. The theorems below derive the quark spectrum from compact-angle winding numbers around the \GG{} \(\theta\)-circle combined with radial transport overlaps; the up-type and down-type packets share the same compact alphabet but settle on different winding strata, giving the asymmetric mass hierarchy as a structural rather than empirical consequence.


\subsection{The quark \GRIP{} cascade}

The quark packet differs from the charged-lepton one by retaining both color and compact-angle
memory, so \BIsix{} admits a wider first-order alphabet: dyadic seam depth \emph{and} primitive
compact-angle winding, which is why $\e^{-\pi}$, $\e^{-2\pi}$, $\e^{-3\pi}$, and $\e^{-4\pi}$
enter the quark codebook while remaining absent from the lepton one. As one \GRIP{} cascade the
six quark masses are \GA{} basin readouts of one baryonic packet --- the same closed packet
that also constrains the later CKM misalignment and the topological completion --- with the
\GSSB{} winding tolls $\e^{-n\pi}$ supplying the hierarchy; the first-order values are
ideal-seam \GA{} exports and the rational QD factors are the shortest admissible \CCI{}
transport to the benchmark conventions, not free QCD knobs.

\label{sec:quark-masses}

The quark sector is more structured than the charged-lepton sector because
colored fields feel the compact holonomy already at first order. Their
hierarchy is therefore organized by two ingredients at once: angular
winding on \(S^1_\theta\) and small dyadic/color branch factors.

\subsection{\GiFour{} cascade-depth reading and the angular cascade \texorpdfstring{$e^{-n\pi}$}{e\^{}\{-n pi\}}}
\label{sec:quark-cascade-depth}

The quark sector exhibits the most explicit constructive realization of the \GiFour{} cascade-depth language in the entire codebook. The angular suppression $e^{-n\pi}$ for $n = 0, 1, 2, 3, 4$ that organizes the quark Yukawas below is, exactly and term by term, the instanton-cost spectrum of the \GFF{} Morse cascade on the family-projector moduli $\Sigma_{\rm fam} = \mathbb{C}P^2$, doubled by the up-type/down-type isospin branch structure of $\SU(2)_L$.

\paragraph{Cascade chain on $\mathbb{C}P^2$ doubled by isospin.} The three Morse critical points of \GFF{} on $\mathbb{C}P^2$ \eqref{eq:lepton-grip-chain} are doubled by the up-down isospin grading, producing a chain of $5$ effective cascade levels (one ``reference'' top plus four cascade legs):
\begin{equation}
\boxed{
\begin{array}{c|c|c|c|c}
\text{Level} & \text{Quark} & \text{Suppression} & \text{Cascade step} & \text{Bogomolnyi action} \\\hline
\text{top} & t & 1 & \text{reference (shallowest \GA{})} & 0 \\
\text{depth } 1 & b & e^{-\pi} & \text{first descent (down-type)} & \pi \\
\text{depth } 2 & c & e^{-2\pi} & \text{second descent (up-type)} & 2\pi \\
\text{depth } 3 & s & e^{-3\pi} & \text{third descent (down-type)} & 3\pi \\
\text{depth } 4 & u, d & e^{-4\pi} & \text{fourth descent (ground level)} & 4\pi \\
\end{array}
}
\label{eq:quark-cascade-table}
\end{equation}

\paragraph{Identification of $e^{-n\pi}$ as a cascade-step exponential.} The minimal first-order quark Yukawas of the codebook below are
\begin{equation}
y_t^{(1)} = 1, \quad
y_b^{(1)} = \tfrac{N}{2^N}\,e^{-\pi}, \quad
y_c^{(1)} = N\,e^{-2\pi}, \quad
y_s^{(1)} = 2N\,e^{-3\pi}, \quad
y_{u,d}^{(1)} = (1, 2)\,N\,e^{-4\pi},
\label{eq:quark-yukawa-cascade}
\end{equation}
with $N = 3$ entering as a family-multiplicity prefactor in every line except the reference $y_t = 1$. The exponential factor $e^{-n\pi}$ in the $n$-th line is exactly the WKB amplitude for the cascade step across the \GSSB{} gates between the adjacent Morse critical points of indices $2k$ and $2(k-1)$ on $\mathbb{C}P^2$ as proved by \citet{Witten1982}: the Bogomolnyi-saturating gradient flow has action $S_{[2k]\to[2(k-1)]} = \pi k$ (cf.\ Lemma~4.6 of \citealp{ArisakaP2N3}), so each cascade step contributes one factor of $e^{-\pi k}$ to the $k$-th Yukawa. In other words the angular cascade is not a fitting parameter; it is the explicit instanton suppression spectrum of the \GFF{} Morse function on the family-projector stratum.

\paragraph{Generation-to-cascade assignment for quarks.} The standard up-type and down-type triplets fold onto the cascade table \eqref{eq:quark-cascade-table} as
\begin{equation}
\begin{array}{lcl}
(t, c, u) & \longleftrightarrow & (\text{depth } 0, \text{depth } 2, \text{depth } 4)~\text{up-type sequence}, \\
(b, s, d) & \longleftrightarrow & (\text{depth } 1, \text{depth } 3, \text{depth } 4)~\text{down-type sequence},
\end{array}
\label{eq:quark-up-down-cascade}
\end{equation}
with the characteristic $(t, c, u)$ even-step / $(b, s, d)$ odd-step interleaving accounting for the asymmetry between up- and down-type mass ratios that is otherwise mysterious. Indeed, the asymmetry is no longer an empirical fit; it is a structural consequence of the SU$(2)_L$ doubling of the Morse cascade.

\paragraph{The ground-state (\GGA{}) reading.} As in the lepton sector (\S\ref{sec:lepton-cascade-depth}), the five levels are \emph{basins}, not saddles: each flavor is a genuine \GA{} of its depth-$n$ effective landscape --- which is why all six quark masses are sharply defined codebook parameters --- while only the deepest level is the ground state. The depth-4 basin, jointly occupied by $u$ and $d$ through the isospin doubling, is the \GGA{} of the quark sector, and the weak-decay chains $t\to b\to c\to s\to u/d$ are the runtime's relaxation of non-ground \GA{}'s toward it. Confinement then adds one further ground-state clause downstream: the lightest baryon built on the depth-4 \GGA{} --- the proton --- is the \GGA{} of the gauged-$\mathrm{U}(1)_B$ baryonic stratum (\S\ref{subsec:identity-cascade}), which is why quark-sector relaxation terminates in stable hydrogen rather than continuing without end.

\begin{euclidremark}[Physics intuition for the quark cascade]
The five-level structure $(t, b, c, s, u/d)$ arose historically from particle physics as five empirical Yukawa couplings spanning five orders of magnitude. The cascade-depth reading collapses these five orders of magnitude into one structural pattern: each quark sits one Bogomolnyi step further down the \GFF{} Morse cascade than the preceding one, with a uniform per-step penalty $e^{-\pi}$. The mass hierarchy $m_t : m_b : m_c : m_s : m_{u,d}$ is therefore not five independent measurements but one cascade-closure pattern read at five depths. After all, this is the same cascade that closes at $S_{\rm tot}(3) = 3\pi$ to fix $N=3$; here we are reading off, depth by depth, what each cascade step contributes to the boundary mass spectrum. In the ground-state vocabulary: five non-ground \GA{}'s relaxing, step by step, toward the depth-4 \GGA{} --- the up and down quarks of which all stable matter is made.
\end{euclidremark}


\subsection{Why quarks feel the angular monodromy at first order}

\ATwo{} identifies the compact direction \(S^1_\theta\) as the minimal
holonomy channel needed for Abelian phase information and for the
Wilson-line Higgs. Colored matter is necessarily more sensitive to that
holonomy than the color-singlet charged leptons because it lives in a
nontrivial gauge bundle before boundary descent. The natural first-order
quark hierarchy is therefore organized by nonzero winding integers,
\begin{equation}
w_q\neq 0\qquad \text{for light and intermediate quarks at first order.}
\label{eq:wq-nonzero-main}
\end{equation}

Intuitively, each additional family step in the quark sector costs angular
coherence. Because the only compact primitive is the single circle
\(S^1_\theta\), the only admissible suppression factor is again
\(\e^{-\pi}\) per minimal mismatch unit.

\subsection{Minimal first-order quark codebook}

The shortest quark codebook consistent with one unsuppressed top state,
one down-type branch shift, and monotone family descent is
\begin{equation}
\boxed{y_t^{(1)}=1,}
\label{eq:y-t-main}
\end{equation}
\begin{equation}
\boxed{
y_b^{(1)}=\frac{N}{2^N}\,\e^{-\pi}
=\frac{3}{8}\,\e^{-\pi},
}
\label{eq:y-b-main}
\end{equation}
\begin{equation}
\boxed{
y_c^{(1)}=N\,\e^{-2\pi}
=3\,\e^{-2\pi},
}
\label{eq:y-c-main}
\end{equation}
\begin{equation}
\boxed{
y_s^{(1)}=2N\,\e^{-3\pi}
=6\,\e^{-3\pi},
}
\label{eq:y-s-main}
\end{equation}
\begin{equation}
\boxed{
y_u^{(1)}=N\,\e^{-4\pi}
=3\,\e^{-4\pi},
\qquad
y_d^{(1)}=2N\,\e^{-4\pi}
=6\,\e^{-4\pi}.
}
\label{eq:y-u-y-d-main}
\end{equation}

Again, each line has a clear geometrical interpretation.

\paragraph{The top quark.}
The top is the unique quark with essentially maximal overlap at the lock
seam. The shortest admissible code is therefore simply
\[
y_t^{(1)}=1.
\]
No suppression is imposed because the top mode defines the dominant
up-type overlap.

\paragraph{The bottom quark.}
The heaviest down-type mode must differ from the top by the smallest
nontrivial branch change. The minimal angular price is one unit
\(\e^{-\pi}\). The down-type branch also carries the smallest family-coded
dyadic split \(N/2^N=3/8\). Hence
\[
y_b^{(1)}=\frac{N}{2^N}\e^{-\pi}.
\]

\paragraph{The charm quark.}
The second-family up-type mode is the first genuinely wound quark state.
The smallest even-winding suppression consistent with retaining up-type
sector coherence is \(\e^{-2\pi}\), accompanied by the triplicity factor
\(N\). Thus
\[
y_c^{(1)}=N\e^{-2\pi}.
\]

\paragraph{The strange quark.}
The strange quark sits on the down-type branch one angular unit deeper
than the charm. The extra factor of 2 relative to \(N\) represents the
two-sheeted down-type branch multiplicity, so
\[
y_s^{(1)}=2N\e^{-3\pi}.
\]

\paragraph{The up and down quarks.}
The first-family quarks live at the deepest first-order winding level,
\(w=4\). The up quark preserves the triplicity numerator \(N\), while the
down quark again carries the two-sheeted branch factor, giving
\[
y_u^{(1)}=N\e^{-4\pi},
\qquad
y_d^{(1)}=2N\e^{-4\pi}.
\]
This directly predicts the simple exact relation \(m_d/m_u=2\) at first
order.

\subsection{First-order quark masses}

Multiplying by the same electroweak normalization factor \(v^{(1)}/\sqrt2\)
gives
\begin{equation}
m_t^{(1)}=\frac{v^{(1)}}{\sqrt2}=174.18257~\mathrm{GeV},
\label{eq:m-t-1-main}
\end{equation}
\begin{equation}
m_b^{(1)}=\frac{v^{(1)}}{\sqrt2}\,\frac{3}{8}\e^{-\pi}
=2.82266677~\mathrm{GeV},
\label{eq:m-b-1-main}
\end{equation}
\begin{equation}
m_c^{(1)}=\frac{v^{(1)}}{\sqrt2}\,3\e^{-2\pi}
=0.975827928~\mathrm{GeV},
\label{eq:m-c-1-main}
\end{equation}
\begin{equation}
m_s^{(1)}=\frac{v^{(1)}}{\sqrt2}\,6\e^{-3\pi}
=84.3386966~\mathrm{MeV},
\label{eq:m-s-1-main}
\end{equation}
\begin{equation}
m_d^{(1)}=\frac{v^{(1)}}{\sqrt2}\,6\e^{-4\pi}
=3.64460554~\mathrm{MeV},
\qquad
m_u^{(1)}=\frac{v^{(1)}}{\sqrt2}\,3\e^{-4\pi}
=1.82230277~\mathrm{MeV}.
\label{eq:m-u-d-1-main}
\end{equation}

These formulas show why it was worth deriving \(v^{(1)}\) first. Once the
electroweak normalization is known, the entire quark hierarchy reduces to
a small list of winding and branch integers.

\subsection{Exact first-order quark ratios}

Several exact ratios follow immediately:
\begin{equation}
\frac{m_b^{(1)}}{m_c^{(1)}}=\frac{\e^\pi}{2^N}=\frac{\e^\pi}{8},
\label{eq:mbmc-main}
\end{equation}
\begin{equation}
\frac{m_c^{(1)}}{m_s^{(1)}}=\frac{\e^\pi}{2},
\label{eq:mcm_s-main}
\end{equation}
\begin{equation}
\frac{m_s^{(1)}}{m_d^{(1)}}=\e^\pi,
\label{eq:msmd-main}
\end{equation}
\begin{equation}
\frac{m_d^{(1)}}{m_u^{(1)}}=2.
\label{eq:mdmu-main}
\end{equation}

These are the quark analogues of the exact charged-lepton ratios in
\eqref{eq:tau-mu-ratio-main} and \eqref{eq:mu-e-ratio-main}. Their
importance is structural. The quark hierarchy is not being fitted one
particle at a time. It is being generated by a single compact-angle code.

\subsection{QD completion of the quark block}

The QD transport for the quark sector is naturally sector-dependent
because QCD dressing depends on the family depth and on whether the state
lies on the up-type or down-type branch. The shortest rational closures
compatible with the OGN ledger are
\begin{equation}
\boxed{
\mathcal R_t = 1-\frac{M}{2^J}
=1-\frac{1}{128}
=\frac{127}{128},
}
\label{eq:R-t-main}
\end{equation}
\begin{equation}
\boxed{
\mathcal R_b = 1+\frac{M}{2}
=\frac32,
}
\label{eq:R-b-main}
\end{equation}
\begin{equation}
\boxed{
\mathcal R_c = 1+\frac{M}{N}
=\frac43,
}
\label{eq:R-c-main}
\end{equation}
\begin{equation}
\boxed{
\mathcal R_s = 1+\frac{M}{2^N}
=\frac98,
}
\label{eq:R-s-main}
\end{equation}
\begin{equation}
\boxed{
\mathcal R_u=\mathcal R_d
=1+\frac{M}{L-M}
=1+\frac14
=\frac54.
}
\label{eq:R-u-d-main}
\end{equation}

The transport-corrected seam masses are then
\begin{equation}
m_q^{\mathrm{QD}}=\mathcal R_q\,m_q^{(1)},
\label{eq:q-qd-general-main}
\end{equation}
which gives
\begin{equation}
m_t^{\mathrm{QD}}=172.821770~\mathrm{GeV},
\qquad
m_b^{\mathrm{QD}}=4.23400015~\mathrm{GeV},
\label{eq:t-b-qd-main}
\end{equation}
\begin{equation}
m_c^{\mathrm{QD}}=1.30110390~\mathrm{GeV},
\qquad
m_s^{\mathrm{QD}}=94.8810337~\mathrm{MeV},
\label{eq:c-s-qd-main}
\end{equation}
\begin{equation}
m_d^{\mathrm{QD}}=4.55575693~\mathrm{MeV},
\qquad
m_u^{\mathrm{QD}}=2.27787846~\mathrm{MeV}.
\label{eq:u-d-qd-main}
\end{equation}

The interpretation of these closures is as follows. The top quark is so
dominant at first order that the leading transport correction is a minimal
dyadic subtraction \(2^{-J}\). The bottom and charm quarks require
positive branch dressing because their first-order wound overlaps
underestimate the full boundary mass after QCD transport. The strange,
up, and down quarks sit on deeper boundary seams and therefore inherit the
next-shortest rational uplifts compatible with their family depth.

\section{The neutrino floors: masses from the minimal constitutional Majorana channel}

\subsection{Historical background}

Pauli postulated the neutrino in his 1930 letter to the T\"ubingen conference to rescue energy conservation in beta decay \citep{Pauli1930}. Majorana in 1937 introduced the self-conjugate fermion that bears his name \citep{Majorana1937}, and Pontecorvo in 1957 introduced neutrino oscillations \citep{Pontecorvo1957}. After Super-Kamiokande and SNO confirmed atmospheric and solar neutrino oscillations near the turn of the millennium, the Standard Model was extended to accommodate three nonzero neutrino masses with two mass-squared splittings \(\Delta m_{21}^2\approx 7.4\times 10^{-5}~\mathrm{eV}^2\) and \(|\Delta m_{31}^2|\approx 2.5\times 10^{-3}~\mathrm{eV}^2\) -- but the absolute mass scale, the mass ordering, and the Dirac-vs-Majorana nature remain free. The theorems below derive the neutrino-mass block from a single constitutional Majorana channel: the smallness of \(m_\nu\sim 10^{-2}\) eV is the boundary-projection image of the same \OGNfive{} lock that fixes the lock scale, with no independent see-saw or right-handed-neutrino sector required.

\subsection{The constitutional door: why the channel is Majorana}
\label{subsec:constitutional-door}

The word ``Majorana'' in this section's title is not a model choice,
and the following statement --- co-stated with its cosmological twin
(T-Cosmos-18 of \citealp{ArisakaA10Cosmos}) and with entry R10 of the
real-bulk series of \citealp{ArisakaA8QM}, the series whose standalone,
full-length development is the Foundations Branch member paper \PaperPFour{}
(\citealp{ArisakaP4Real}) --- is why.

\begin{theorem}[\textbf{T-SM-44.1}: The dial-free door --- a Majorana mass exists iff the fiber weight vanishes (R10)]
\label{thm:dial-free-door-sm}
Let a fermion multiplet carry weight $n$ under the compact phase
direction $S^1_\theta$ (spinor and family indices commute with the
fiber action). 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 $\psi\bar\psi$;
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 weight-$n$ sector is the pair $(a, b)$
rotated by $R(n\alpha)$. Its symmetric square is spanned by
$a^2 + b^2$ (rotation-invariant for every $n$: the Dirac pairing) and
the doublet $(a^2 - b^2,\, 2ab) = (\mathrm{Re}\,\psi^2,
\mathrm{Im}\,\psi^2)$, which rotates by $2n\alpha$ under $R(n\alpha)$:
for $n \neq 0$ invariance forces it to vanish; for $n = 0$ the action
is trivial and the reality condition of the sector is precisely the
Majorana condition. The spinor factor's charge-conjugation pairing is
fiber-inert and cannot compensate a non-zero weight. A trilinear is
fiber-neutral iff its weights sum to zero; with $L$ at $-\tfrac12$ and
$H$ at $+\tfrac12$, the singlet $N$ at weight $0$ closes the unique
such term linear in $N$.
\end{proof}

Three consequences frame everything below. Lepton number is winding
number: exact on every dial-carrying species, so the charged-lepton
ledger conserves $L$ identically. The sole legal $\Delta L = 2$
structure in the matter sector is the weight-zero self-pairing --- the
Majorana channel of this section's title, which is thereby \emph{the
minimal and the unique} constitutional option, not the minimal of
several. And the ultraviolet completion, when it is opened, has no
freedom either: the portal is fixed, and the heavy-sector realization
and its leptogenesis are carried, in completed form, by
\citealp{ArisakaA10Cosmos}.

\subsection{The neutrino \GRIP{} cascade}

The neutrino block is the most indirect part of the mass sheet. \BIsix{} forbids any new
intermediate mass scale and any second independent compact loop, so the Majorana/Weinberg
channel must reuse the already-derived ultraviolet scale $\EGUT$, and the coefficient
$\pi/\EGUT$ below is the shortest allowed Majorana seam coefficient compatible with the same
compact-angle topology used elsewhere in the paper. As one \GRIP{} cascade the neutrino masses
are projected Majorana spectral gaps (\CCI{} exports) of a rank-two \GFF{} landscape carrying
the $v^2/\EGUT$ suppression: the same Higgs branch that fixed $v$ enters the dimension-five
operator twice, and the same family packet later carries the PMNS angles, so \GRIP{} selects
the normal-ordering branch and stabilizes it as the lepton-attractor background on which the
PMNS block is then read. One downstream bookkeeping convention is recorded here, because the cosmology companion's $\eta_B$ chain consumes this coefficient in two distinct roles: the heavy-sector resonance width is evaluated at the seam-matched one-generation Yukawa --- a property of the seam-localized texture, whose effective washout mass is the seam scale $\pi v^2/\EGUT$ itself, $\widetilde m\simeq8.4$~meV --- while the washout strength is evaluated at the light-neutrino mass scale the thermal bath actually sees, $\widetilde m\sim m_3\simeq50.5$~meV; the two conventions, and the factor-$\simeq13$ spread in $Y_\nu^2$ they induce, are declared, accounted for, and priced within the order-of-magnitude register in the companion's leptogenesis appendix (\citealp{ArisakaA10Cosmos}), with this paper's coefficient $\pi/\EGUT$ the upstream input of both.

\label{sec:neutrino-masses}

The neutrino sector is the most delicate part of the 12-mass block because
its masses are tiny, its flavor structure is nontrivial, and no direct
neutrino Yukawa operator appears at dimension four in the minimal
Standard-Model field content. Fortunately, the written Arisaka-GG preprints
already tell us which route is constitutionally preferred.

\subsection{\GiFour{} cascade-depth reading for the Majorana neutrinos}
\label{sec:neutrino-cascade-depth}

The neutrino sector occupies the same family-projector stratum $\Sigma_{\rm fam} = \mathbb{C}P^2$ as the charged leptons, but with two important structural differences: (i) the absolute mass scale is set by the Weinberg dimension-five operator $v^2/\EGUT$ rather than by the Higgs vev directly, and (ii) the Majorana nature of the channel produces a rank-two algebraic structure (with two independent mass-squared splittings rather than three independent masses) before any flavor-projection misalignment is imposed.

\paragraph{Cascade chain in the Majorana sector.} By \ASix{} \T{3.1} and \T{4.1} \citep{ArisakaA6Gi4} the same three-step \GRIP{} cascade \eqref{eq:lepton-grip-chain} applies, with the three neutrino mass eigenstates $\nu_3, \nu_2, \nu_1$ assigned to the three cascade depths --- basins of the depth-$n$ effective landscapes, as in \S\ref{sec:lepton-cascade-depth}; the critical points of \eqref{eq:lepton-grip-chain} are the gates between them:
\begin{equation}
\boxed{
\begin{array}{lcl}
\nu_3 & \longleftrightarrow & \text{depth } 0:\ \GA{} \;\text{(non-ground; relaxation portal-suppressed)}, \\
\nu_2 & \longleftrightarrow & \text{depth } 1:\ \GA{} \;\text{(non-ground; relaxation portal-suppressed)}, \\
\nu_1 & \longleftrightarrow & \text{depth } 2:\ \GGA{} \;\text{(ground state of the Majorana stratum)}.
\end{array}
}
\label{eq:neutrino-cascade-assignment}
\end{equation}

\paragraph{The ground-state (\GGA{}) reading, and why neutrinos nonetheless persist.}
As in the charged-lepton sector, all three mass eigenstates are genuine \GA{}'s --- sharply defined mass-squared splittings are the certificate --- and only the depth-2 basin is the ground state: $\nu_1$ is the \GGA{} of the Majorana stratum. Two consequences follow. First, an \emph{ordering prediction}: the depth assignment \eqref{eq:neutrino-cascade-assignment} requires the normal mass ordering ($m_1<m_2<m_3$, the \GGA{} lightest); an experimentally established inverted ordering would falsify the assignment outright --- a clean, near-term kill criterion, since the ordering is within reach of the current long-baseline and reactor program. Second, an instructive contrast with the charged leptons: the \GGA{} fixes the \emph{direction} of relaxation, while the portals fix its \emph{rate}. The muon and tau relax fast because the weak charged current offers a broad channel; the non-ground neutrinos $\nu_3,\nu_2$ can relax only through radiative and neutral channels so feeble that their lifetimes exceed the age of the universe by tens of orders of magnitude. Metastability with a cosmologically long clock is still metastability: the same theorem, read at a different coupling strength.

\paragraph{Why the Majorana channel produces only two mass-squared splittings.} The rank-two Majorana algebraic structure means that the $\nu_3$ and $\nu_2$ mass-squared splittings $\Delta m_{31}^2$ and $\Delta m_{21}^2$ are independent observables, while the absolute $\nu_1$ mass scale enters only through the dimension-five operator coefficient and is not a third independent splitting. This is exactly the empirical pattern observed by Super-Kamiokande, SNO, KamLAND, and the long-baseline accelerator experiments: $\Delta m_{21}^2 \approx 7.4 \times 10^{-5}~\text{eV}^2$ and $|\Delta m_{31}^2| \approx 2.5 \times 10^{-3}~\text{eV}^2$, with the absolute $m_1$ remaining undetermined down to the current cosmological bound $\sum m_\nu < 0.064$~eV ($95\%$, $\Lambda$CDM) \citep{DESI2025}. Indeed, the \GiFour{} cascade reading explains \emph{why} only two splittings are independent: the rank-two structure of the Majorana ledger pins the third degree of freedom to the absolute scale, not to a third splitting.

\paragraph{Why neutrino masses are tiny.} In the \GiFour{} cascade reading, the smallness of the neutrino mass scale $m_\nu \sim 10^{-2}$~eV (compared to the electroweak vev $v \sim 246$~GeV) is not a fine-tuned see-saw mass but a structural consequence of the dimension-five suppression
\begin{equation}
m_\nu \sim \frac{v^2}{\EGUT}
\sim \frac{(246~\text{GeV})^2}{2.28 \times 10^{16}~\text{GeV}}
\sim 2.7~\text{meV},
\label{eq:nu-mass-scale}
\end{equation}
which the 6D scale chain delivers automatically. The cascade-depth reading then assigns the three observed Majorana masses to the three \GFF{} Morse levels of \eqref{eq:neutrino-cascade-assignment}, with no additional right-handed neutrino sector or independent see-saw scale required.

\begin{euclidremark}[Physics intuition for the neutrino cascade]
The three Majorana neutrinos populate the same \GiFour{} cascade as the charged leptons but read it through a different observability window: only mass-squared splittings (not absolute masses) are accessible to oscillation experiments, so the rank-two algebraic structure is what observation actually probes. After all, the absence of a third independent splitting in the empirical neutrino spectrum is not a lacuna but a prediction of the Majorana cascade structure. So, in its way, is the mass ordering: the depth assignment makes normal ordering structural (the \GGA{} sits at the bottom), handing the oscillation program a second, sharper falsifier.
\end{euclidremark}


\subsection{Why the Weinberg/Majorana channel is the minimal route}

\ATwo{} and \AFive{} explicitly list the dimension-5 operator
\((LH)(LH)\) as allowed if neutrino masses are generated that way
(\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}). No additional low-energy scale is
admitted in the constitutional ledger, and no right-handed neutrino sector
has yet been made obligatory in the written preprints. Therefore the
minimal neutrino mass mechanism of \ANine{} is the Majorana/Weinberg
channel.

The corresponding effective term is taken to be
\begin{equation}
\mathcal L_{\nu}^{(5)}
=
\frac{2\pi}{\EGUT}\,
\big(LH\big)\,\Xi_\nu\,\big(LH\big)
+\mathrm{h.c.},
\label{eq:Lnu5-main}
\end{equation}
where \(\Xi_\nu\) is a dimensionless family matrix. The coefficient
\(2\pi/\EGUT\) is the minimal constitutional choice for two reasons:
\begin{enumerate}[leftmargin=2.1em]
\item the suppression scale must be the already derived ultraviolet seam
\(\EGUT\), because no new arbitrary intermediate scale is allowed;
\item the prefactor \(2\pi\) represents one full compact traversal on
\(S^1_\theta\), the natural closed Majorana loop.
\end{enumerate}

After electroweak symmetry breaking, each Higgs field contributes
\(v/\sqrt2\), so the common neutrino scale becomes
\begin{equation}
\boxed{
m_\star
=
\pi\,\frac{v^2}{\EGUT}.
}
\label{eq:mstar-main}
\end{equation}
This is one of the cleanest formulas in the entire paper: the neutrino
scale is not imported from observation but generated by the electroweak
scale squared divided by the GUT scale, multiplied by the unique compact
angle constant \(\pi\).

\subsection{First-order neutrino scale}

Using the first-order electroweak scale \(v^{(1)}\) derived earlier, we
obtain
\begin{equation}
m_\star^{(1)}
=
\pi\,\frac{[v^{(1)}]^2}{\EGUT}
=
8.36113789~\mathrm{meV}.
\label{eq:mstar1-main}
\end{equation}

The size is immediately reasonable on physical grounds. The neutrino
sector is doubly suppressed: once by the electroweak scale relative to the
GUT scale, and once more because the mass is generated only after two
Higgs insertions. In \GG{} language, neutrinos are the most infrared of
the fermion observables precisely because they are the most indirect.

\subsection{Minimal family projector and normal ordering}

The family matrix \(\Xi_\nu\) must now be fixed. The minimal first-order
choice consistent with three families, one protected null direction, and
monotone normal ordering is
\begin{equation}
\Xi_\nu^{(1)} = P_2 + 2N\,P_3,
\label{eq:Xinu1-main}
\end{equation}
where \(P_i\) are orthogonal family projectors satisfying
\(
P_iP_j=\delta_{ij}P_i
\)
and
\(
P_1+P_2+P_3=\mathbf 1.
\)

Why is a null first-order direction natural? In the \GDOS{} language the
boundary-visible neutrino sector is the most weakly projected part of the
fermion spectrum. A rank-two first-order projector is therefore the
minimal way to encode the fact that one family direction remains
unresolved until higher order. The coefficient of the third projector is
\(2N=6\), the smallest OGN-coded enhancement that cleanly separates the
third neutrino mass from the second without introducing new integers.

The resulting first-order eigenvalues are
\begin{equation}
\boxed{
(m_1,m_2,m_3)^{(1)}
=
(0,\,1,\,2N)\,m_\star^{(1)}
=
\big(0,\ 8.36113789,\ 50.1668273\big)\ \mathrm{meV}.
}
\label{eq:nu-masses-1-main}
\end{equation}

\subsection{First-order oscillation observables}

The mass-squared splittings are therefore
\begin{equation}
\Delta m_{21}^{2\,(1)}
=
m_2^{2\,(1)}-m_1^{2\,(1)}
=
6.990864\times10^{-5}\ \mathrm{eV}^2,
\label{eq:dm21-1-main}
\end{equation}
\begin{equation}
\Delta m_{31}^{2\,(1)}
=
m_3^{2\,(1)}-m_1^{2\,(1)}
=
2.516711\times10^{-3}\ \mathrm{eV}^2.
\label{eq:dm31-1-main}
\end{equation}
The first-order sum of neutrino masses is
\begin{equation}
\Sigma m_\nu^{(1)}=58.5280~\mathrm{meV}.
\label{eq:sum-nu-1-main}
\end{equation}

Even before any QD correction is applied, the neutrino sector already
lands in the phenomenologically correct regime. This is a nontrivial
result: it means that the same scale architecture that solved the
hierarchy problem also generates neutrino masses of the right order of
magnitude.

\subsection{QD lift of the neutrino spectrum}

At QD order the first-order null direction is lifted by the smallest
dyadic amount, the second eigenvalue receives the shortest throat-depth
uplift, and the third eigenvalue inherits the same dyadic closure that
already appeared in the gauge sector:
\begin{equation}
\boxed{
m_1^{\mathrm{QD}}=\frac{1}{2^J}\,m_\star,
\qquad
m_2^{\mathrm{QD}}=\left(1+\frac1K\right)m_\star,
\qquad
m_3^{\mathrm{QD}}=2N\left(1+\frac{1}{2^J}\right)m_\star,
}
\label{eq:nu-qd-ansatz-main}
\end{equation}
with
\begin{equation}
m_\star=\pi\,\frac{v_{\mathrm{QD}}^2}{\EGUT},
\qquad
v_{\mathrm{QD}}=246.220~\mathrm{GeV}.
\label{eq:mstarqd-main}
\end{equation}
Numerically,
\begin{equation}
m_\star = 8.35358030~\mathrm{meV},
\label{eq:mstarqd-num-main}
\end{equation}
so that
\begin{equation}
\boxed{
(m_1,m_2,m_3)^{\mathrm{QD}}
=
\big(0.0652623,\ 8.59225345,\ 50.5130555\big)\ \mathrm{meV}.
}
\label{eq:nu-masses-qd-main}
\end{equation}

The corresponding oscillation observables are
\begin{equation}
\Delta m_{21}^{2\,\mathrm{QD}}
=
7.382256\times10^{-5}\ \mathrm{eV}^2,
\label{eq:dm21-qd-main}
\end{equation}
\begin{equation}
\Delta m_{31}^{2\,\mathrm{QD}}
=
2.551565\times10^{-3}\ \mathrm{eV}^2,
\label{eq:dm31-qd-main}
\end{equation}
and
\begin{equation}
\Sigma m_\nu^{\mathrm{QD}}=59.1706~\mathrm{meV}.
\label{eq:sum-nu-qd-main}
\end{equation}

The conceptual picture is attractive. The first-order theory gives a
rank-two Majorana block with normal ordering. The QD theory then lifts
the null mode and slightly stretches the heavy eigenvalue by the same
dyadic transport that already sharpened the gauge couplings. In this
sense the neutrino sector is not alien to the rest of the paper; it is
the natural low-energy echo of the same locked arithmetic.

\section{Precision comparison for the 12-fermion-mass block}
\label{sec:fermion-comparison}

We now compare the derived fermion masses with standard benchmark values.
As before, these comparison numbers are \emph{not} inputs. They enter
only after the derivation is complete.

\subsection{Benchmark conventions and theory uncertainties}

For the charged leptons and quarks we use the standard benchmark masses in
the conventional comparison schemes summarized by CODATA and the Particle
benchmark data adopted in this paper. For the neutrino sector we compare
not the individual masses, which are not directly known with precision,
but the oscillation observables \(\Delta m_{21}^2\) and \(\Delta m_{31}^2\)
as summarized by the benchmark oscillation targets adopted in this paper.

The theory errors in the tables are defined constitutionally rather than
statistically.

\begin{itemize}[leftmargin=2.1em]
\item
For each first-order mass, the quoted theory uncertainty
\(\sigma_{\theory}^{(1)}\) is the magnitude of the omitted QD correction:
\[
\sigma_{\theory}^{(1)} = \left|m_f^{\mathrm{QD}}-m_f^{(1)}\right|.
\]

\item
For each QD mass, the quoted residual theory uncertainty is taken to be
one further dyadic suppression of the leading correction:
\[
\sigma_{\theory}^{(2)} =
\frac18\left|m_f^{\mathrm{QD}}-m_f^{(1)}\right|.
\]
The factor \(1/8\) reflects three additional binary boundary steps beyond
the explicitly kept QD layer.
\end{itemize}

This prescription is simple, transparent, and consistent with the locked
dyadic logic used throughout the paper.

\subsection{Direct-mass comparison: first-order seam prediction}

\begin{table}[H]
\centering
\small
\caption[First-order predictions for the nine directly benchmarked charged fermions.]{First-order predictions for the nine directly benchmarked charged
fermions. Experimental comparison values are standard CODATA/PDG
benchmarks used only after the derivation is complete. The nine are the charged fermions whose masses are measured sharply enough for a comparison at this precision to carry information.}
\label{tab:fermion-first-order-main}
\begin{tabular}{p{0.22\textwidth}r r r r r r}
\toprule
\textbf{Parameter} & \textbf{Pred.} & \(\sigma_{\theory}^{(1)}\) & \textbf{Obs.} & \(\sigma_{\exp}\) & \(\Dev\) & \(\Pull\)\\
\midrule
\(m_e\) [MeV] & 0.510300500 & 0.000797345 & 0.510998951 & 0.000000000 & $-$1.367e-03 & $-$0.88\\
\(m_\mu\) [MeV] & 106.312604 & 0.6644535 & 105.658376 & 0.0000023 & +6.192e-03 & +0.98\\
\(m_\tau\) [GeV]& 1.78605175 & 0.00837212 & 1.77693 & 0.00009 & +5.133e-03 & +1.09\\
\midrule
\(m_u\) [MeV] & 1.82230277 & 0.45557569 & 2.20 & 0.07 & $-$1.717e-01 & $-$0.82\\
\(m_d\) [MeV] & 3.64460554 & 0.91115139 & 4.69 & 0.05 & $-$2.229e-01 & $-$1.15\\
\(m_s\) [MeV] & 84.3386966 & 10.5423371 & 92.74 & 0.54 & $-$9.059e-02 & $-$0.80\\
\(m_c\) [GeV] & 0.975827928 & 0.325275976 & 1.275 & 0.009 & $-$2.346e-01 & $-$0.92\\
\(m_b\) [GeV] & 2.82266677 & 1.41133338 & 4.196 & 0.012 & $-$3.273e-01 & $-$0.97\\
\(m_t\) [GeV] & 174.182571 & 1.36080108 & 172.56 & 0.31 & +9.403e-03 & +1.16\\
\bottomrule
\end{tabular}
\end{table}

Two features should be emphasized. First, the first-order masses already
come out in the correct numerical neighborhoods across more than five
orders of magnitude. Second, the residual discrepancies are organized,
not chaotic. They point in exactly the direction one would expect if the
main missing ingredient were finite seam transport rather than a failure
of the first-order codebook itself.

\subsection{Direct-mass comparison: QD-completed prediction}

\begin{table}[H]
\centering
\small
\caption[QD-completed predictions for the nine directly benchmarked charged fermions.]{QD-completed predictions for the nine directly benchmarked
charged fermions. The nine charged fermion masses with the quantum-defect completion applied, against the same CODATA and PDG benchmarks as the first-order table and in the same column order. The light-quark rows carry the largest benchmark uncertainties in the set, so their pulls are the least informative entries here even where the central agreement looks close; the lepton rows are where the completion is actually being tested.}
\label{tab:fermion-qd-main}
\begin{tabular}{p{0.22\textwidth}r r r r r r}
\toprule
\textbf{Parameter} & \textbf{Pred.} & \(\sigma_{\theory}^{(2)}\) & \textbf{Obs.} & \(\sigma_{\exp}\) & \(\Dev\) & \(\Pull\)\\
\midrule
\(m_e\) [MeV] & 0.511097845 & 0.0000996681 & 0.510998951 & 0.000000000 & +1.935e-04 & +0.99\\
\(m_\mu\) [MeV] & 105.648150 & 0.0830567 & 105.658376 & 0.0000023 & $-$9.677e-05 & $-$0.12\\
\(m_\tau\) [GeV]& 1.77767963 & 0.00104651 & 1.77693 & 0.00009 & +4.219e-04 & +0.71\\
\midrule
\(m_u\) [MeV] & 2.27787846 & 0.05694696 & 2.20 & 0.07 & +3.540e-02 & +0.86\\
\(m_d\) [MeV] & 4.55575693 & 0.11389392 & 4.69 & 0.05 & $-$2.862e-02 & $-$1.08\\
\(m_s\) [MeV] & 94.8810337 & 1.31779214 & 92.74 & 0.54 & +2.309e-02 & +1.50\\
\(m_c\) [GeV] & 1.30110390 & 0.04065950 & 1.275 & 0.009 & +2.047e-02 & +0.63\\
\(m_b\) [GeV] & 4.23400015 & 0.17641667 & 4.196 & 0.012 & +9.056e-03 & +0.21\\
\(m_t\) [GeV] & 172.821770 & 0.17010014 & 172.56 & 0.31 & +1.517e-03 & +0.74\\
\bottomrule
\end{tabular}
\end{table}

Table~\ref{tab:fermion-qd-main} is the main quantitative result of the
fermion-mass extension. The QD layer improves every charged-lepton mass
and every heavy-quark mass substantially, while keeping the light-quark
sector within plausible comparison bands despite the well-known
scheme-dependence of light-quark masses.

\subsection{Predicted neutrino masses}

Because individual neutrino masses are not yet directly known with the
same precision as the charged fermion masses, the most honest presentation
is to display the predicted eigenvalues and then separately compare the
oscillation observables.

\begin{table}[H]
\centering
\caption[Predicted neutrino mass eigenvalues and mass sums.]{Predicted neutrino mass eigenvalues and mass sums. The two columns are first order and QD-completed. The lightest eigenvalue vanishes at first order and acquires a small value only through the completion, which is a structural statement about the mass matrix rather than a fitted outcome. The sum is printed because it is the quantity cosmology bounds; the individual eigenvalues are not separately measured, and the comparison with the oscillation splittings is made in the table that follows.}
\label{tab:nu-masses-main}
\small
\begin{tabular}{p{0.30\textwidth}r r}
\toprule
\textbf{Observable} & \textbf{First order} & \textbf{QD}\\
\midrule
\(m_1\) [meV] & 0 & 0.0652623\\
\(m_2\) [meV] & 8.36113789 & 8.59225345\\
\(m_3\) [meV] & 50.1668273 & 50.5130555\\
\(\Sigma m_\nu\) [meV] & 58.5280 & 59.1706\\
\bottomrule
\end{tabular}
\end{table}

The normal ordering is an output rather than an assumption. It follows
from the minimal rank-two first-order projector together with the QD lift
in \eqref{eq:nu-qd-ansatz-main}.

\subsection{Oscillation comparison}

\begin{table}[H]
\centering
\small
\caption[Comparison of the derived neutrino oscillation observables with the named benchmarks of \S\ref{sec:pmns-benchmark}.]{Comparison of the derived neutrino oscillation observables with
the named benchmarks of \S\ref{sec:pmns-benchmark}. JUNO's first-data
determination of the solar splitting is printed beside the global fit. JUNO's determination and the global fit are printed side by side because they are independent, and the paper is scored against both rather than against whichever is closer.}
\label{tab:nu-osc-main}
\begin{tabular}{p{0.22\textwidth}r r r r r r}
\toprule
\textbf{Observable} & \textbf{Pred.} & \(\sigma_{\theory}\) & \textbf{Obs.} & \(\sigma_{\exp}\) & \(\Dev\) & \(\Pull\)\\
\midrule
\(\Delta m_{21}^{2\,(1)}\) {\footnotesize (NuFIT 6.1)} & 6.990864e-05 & 3.913917e-06 & 7.530000e-05 & 1.000000e-06 & $-$7.160e-02 & $-$1.33\\
\(\Delta m_{21}^{2\,(1)}\) {\footnotesize (JUNO)} & 6.990864e-05 & 3.913917e-06 & 7.500000e-05 & 1.200000e-06 & $-$6.788e-02 & $-$1.24\\
\(\Delta m_{31}^{2\,(1)}\) {\footnotesize (NuFIT 6.1)} & 2.516711e-03 & 3.485390e-05 & 2.511000e-03 & 2.100000e-05 & +2.274e-03 & +0.14\\
\midrule
\(\Delta m_{21}^{2\,\mathrm{QD}}\) {\footnotesize (NuFIT 6.1)} & 7.382256e-05 & 4.892396e-07 & 7.530000e-05 & 1.000000e-06 & $-$1.962e-02 & $-$1.33\\
\(\Delta m_{21}^{2\,\mathrm{QD}}\) {\footnotesize (JUNO)} & 7.382256e-05 & 4.892396e-07 & 7.500000e-05 & 1.200000e-06 & $-$1.570e-02 & $-$0.91\\
\(\Delta m_{31}^{2\,\mathrm{QD}}\) {\footnotesize (NuFIT 6.1)} & 2.551565e-03 & 4.356738e-06 & 2.511000e-03 & 2.100000e-05 & +1.615e-02 & +1.89\\
\bottomrule
\end{tabular}
\end{table}

The neutrino comparison is already encouraging at first order and remains
within the order-one-sigma to order-two-sigma regime after QD completion
against both the global fit and JUNO's direct measurement --- the QD solar
splitting sits \(0.91\sigma\) from the JUNO value. More importantly, the entire neutrino scale has been obtained
without inserting any measured neutrino mass or oscillation parameter into
the derivation.

\subsection{Summary table for the full 12-mass block}

For clarity we collect the full set of fermion-mass predictions in one
place.

\begin{center}
\footnotesize
\begin{longtable}{p{0.12\textwidth}p{0.13\textwidth}r r p{0.24\textwidth} r}
\caption[Summary of the full 12-fermion-mass block derived in the present paper.]{Summary of the full 12-fermion-mass block derived in the present
paper. Observed values (PDG/CODATA) and Dev (the QD
fractional deviation) reproduce the benchmark column of
Table~\ref{tab:second28-summary-v08}; absolute neutrino masses are not
directly measured (only \(\Delta m^2\) constrained).}
\label{tab:full-12mass-summary}\\
\toprule
\textbf{Sector} & \textbf{Parameter} & \textbf{First order} & \textbf{QD} & \textbf{Observed} & \textbf{Dev}\\
\midrule
\endfirsthead
\toprule
\textbf{Sector} & \textbf{Parameter} & \textbf{First order} & \textbf{QD} & \textbf{Observed} & \textbf{Dev}\\
\midrule
\endhead
Charged leptons & \(m_e\) [MeV] & 0.510300500 & 0.511097845 & \(0.510998951\) & +1.935e-04\\
 & \(m_\mu\) [MeV] & 106.312604 & 105.648150 & \(105.658376\pm0.0000023\) & -9.677e-05\\
 & \(m_\tau\) [GeV] & 1.78605175 & 1.77767963 & \(1.77693\pm0.00009\) & +4.219e-04\\
\midrule
Quarks & \(m_u\) [MeV] & 1.82230277 & 2.27787846 & \(2.20\pm0.07\) & +3.540e-02\\
 & \(m_d\) [MeV] & 3.64460554 & 4.55575693 & \(4.69\pm0.05\) & -2.862e-02\\
 & \(m_s\) [MeV] & 84.3386966 & 94.8810337 & \(92.74\pm0.54\) & +2.309e-02\\
 & \(m_c\) [GeV] & 0.975827928 & 1.30110390 & \(1.275\pm0.009\) & +2.047e-02\\
 & \(m_b\) [GeV] & 2.82266677 & 4.23400015 & \(4.196\pm0.012\) & +9.056e-03\\
 & \(m_t\) [GeV] & 174.182571 & 172.821770 & \(172.56\pm0.31\) & +1.517e-03\\
\midrule
Neutrinos & \(m_1\) [meV] & 0 & 0.0652623 & not directly measured & ---\\
 & \(m_2\) [meV] & 8.36113789 & 8.59225345 & not directly measured & ---\\
 & \(m_3\) [meV] & 50.1668273 & 50.5130555 & not directly measured & ---\\
\bottomrule
\end{longtable}
\end{center}


The diagonal mass sector by itself accounts for 17 of the 28
Standard-Model parameters. The next step is the quark-mixing problem.
Section~\ref{sec:ckm} derives the 4 CKM parameters from the already-fixed
quark codebooks, raising the total of the present paper to 21 parameters.

\section{The quark gates: CKM mixing from the misalignment of quark family projectors}
\label{sec:CKM}

\subsection{Historical background}

Cabibbo's 1963 paper introduced the rotation angle that mixes strange and non-strange semileptonic decays \citep{Cabibbo1963}, the original quark-mixing matrix. A decade later Kobayashi and Maskawa generalized Cabibbo to three quark generations \citep{KobayashiMaskawa1973}, predicting a third family before its discovery and showing that three generations admit one irreducible \(CP\)-violating phase. The resulting \(3\times 3\) CKM matrix carries four physical parameters (three angles and one phase) measured to high precision but lacking any structural explanation. The theorems below derive the CKM block as the misalignment between the up-type and down-type quark family projectors on the same closed \OGNfive{} packet: the four free parameters become four compact-angle ratios fixed by the integer ledger, with the CP phase emerging from a topological-projection remainder rather than from an arbitrary insertion.


\subsection{The CKM \GRIP{} cascade}

The CKM block is not an arbitrary unitary matrix inserted after the masses: \BIsix{} has
already fixed the compact-angle alphabet --- the integer $L=5$ (the $\SU(5)$ block size) and
the single compact holonomy circle $S^1_\theta$ surviving in the flavor sector --- from which
the pentagonal phase structure (the interference phase $3\pi/5$ and the rephasing step $\pi/5$)
is assembled here in \ANine{}, with no second flavor torus on which a separate quark CP angle
could float. As one \GRIP{} cascade the CKM observables are the misalignment face of the same
stabilized quark packet whose diagonal masses were just derived: \CCI{} compares the up-type
and down-type family projectors only on a shared admissible branch, so the common left-handed
piece is unphysical and cancels and the $2\!\leftrightarrow\!3$ angle becomes multiplicative,
and the Cabibbo angle, the smaller angles, and the Dirac phase are the \GA{} basin coordinates
of that projector-comparison \GFF{} landscape --- which is why CKM must come after the diagonal
mass block, never before it.

\subsection{Two structural theorems: the discrete-symmetry table, and the flavor connection}

Before the parameter block, two statements deserve theorem status,
because everything the block asserts about phases rests on them. Both
are proved in the real presentation of the compact fiber --- the
presentation in which a charge-$n$ field is the real pair $(a,b)$
rotated by $R(n\alpha)$, a coupling phase is a rigid fiber rotation, and
complex conjugation is orientation reversal of the circle
\citep{ArisakaA8QM}. The real presentation itself --- its two real
sources, the derived complex structure, and the holomorphy bridge they
support --- receives its standalone, full-length development in the
Foundations Branch member paper \PaperPFour{} (\citealp{ArisakaP4Real}).

\begin{theorem}[\textbf{T-SM-46.1}: Discrete symmetries as orientation bookkeeping (R8)]
\label{thm:discrete-orientation}
On the quasi-free real bulk with any coupling texture built from real
geometric data: (i) charge conjugation --- orientation reversal of the
fiber, $\theta \mapsto -\theta$ --- commutes with the complex structure
$J = \partial_\theta / Q$, because it reverses the generator and the
signed charge together: $C$ is $J$-linear. Time reversal --- orientation
reversal of the constitutional clock --- anticommutes with the polar
phase of the time generator: $T$ is $J$-antilinear. Parity is
$J$-linear. Hence a discrete symmetry is antiunitary if and only if it
contains $T$: the textbook unitary/antiunitary table is derived, not
assigned. (ii) The total orientation reversal $CPT$ maps each
Feynman--St\"uckelberg-paired mode set to itself and is a symmetry of
the construction for every real texture. (iii) $CP$ --- fiber plus
space reversal, $J$-linear --- is a symmetry if and only if the coupling
texture is achiral on the circle: every closed-loop angle of the
texture lies in $\{0, \pi\}$.
\end{theorem}

\begin{proof}
(i) Under $\theta \mapsto -\theta$ the weight-$n$ sector maps to the
weight-$(-n)$ sector and $\partial_\theta \mapsto -\partial_\theta$;
since $J$ is normalized by the signed charge,
$J \mapsto (-\partial_\theta)/(-Q) = J$. Under $t \mapsto -t$ the
frequency orientation flips, so the quarter-period advance becomes a
quarter-period delay: $T J T^{-1} = -J$. Parity touches neither fiber
nor clock. The composition rule follows since $J$-(anti)linearity
multiplies. (ii) On the paired real modes $\cos(n\theta - Et)$,
$\cos(-n\theta - Et)$ the total reversal acts as the identity map of
the pair; the metric, the symplectic form, moduli of couplings, and
winding sums are even under total reversal, and reversing the sign of
every texture angle simultaneously is a relabeling of the same real
texture. (iii) A fiber reversal alone negates every texture angle; the
theory is invariant iff the negated texture is a per-family re-zeroing
of the original, which on rephasing-invariant coordinates (Theorem
T-SM-46.2) says every loop angle equals its own negative mod $2\pi$,
i.e.\ lies in $\{0, \pi\}$.
\end{proof}

\begin{theorem}[\textbf{T-SM-46.2}: Flavor holonomy --- rephasing invariants are loop angles (R9)]
\label{thm:flavour-holonomy}
Regard a Yukawa texture as a $\U(1)$-valued connection on the flavor
graph: vertices are fields, each carrying a dial; edges are couplings,
each block $|y_{ij}|\,R(\varphi_{ij})$ carrying a fiber twist. Then:
(i) the rephasing group is the group of per-family dial re-zeroings,
acting as $\varphi_{ij} \mapsto \varphi_{ij} + \alpha_i - \beta_j$;
(ii) in a spanning-tree gauge every tree edge has angle zero and the
residual physical data are exactly the loop holonomies; for the
$N \times N$ bipartite quark graph their number is
$N^2 - (2N - 1) = (N-1)^2$; (iii) the quartet invariants
$\mathrm{Im}(V_{ij}V_{kl}V_{il}^{*}V_{kj}^{*})$ are the plaquette
sines, and unitarity ties all of them to a single Jarlskog invariant
for $N = 3$; CP violation holds iff some loop holonomy lies outside
$\{0, \pi\}$; (iv) with the \BIsix{} compact-angle alphabet --- one
circle and the integer $L = 5$, with no second flavor torus ---
first-order holonomies are valued in the pentagonal lattice
$(\pi/5)\,\mathbb{Z}$.
\end{theorem}

\begin{proof}
(i) is the fiber dictionary applied at vertices. (ii) Fix a spanning
tree of the connected flavor graph and re-zero dials along it to null
its edge angles; each remaining edge closes one independent loop, and
the gauge action on a closed loop telescopes to zero, so the loop
angles are invariant and coordinatize the quotient. Counting: $N^2$
edge angles minus the $2N - 1$ effective re-zeroings (the global common
shift acts trivially). (iii) A quartet is the plaquette product of four
blocks with the conjugated legs transposed; its antisymmetric part is
the sine of the loop angle weighted by the moduli. For unitary $V$ with
$N = 3$ the nine quartets are equal up to sign --- the standard
Jarlskog uniqueness --- so one real number carries all of CP violation.
The achirality criterion is (iii) of T-SM-46.1. (iv) is the pentagonal
assembly of this Part: every physical angle is built from steps of
$\pi/L$ on the single circle. 
\end{proof}

The parameter block below is the realization of this frame: the
Cabibbo interference phase ($3\pi/5$), the unphysical common step
($\pi/5$), and the residual physical Dirac phase
$\delta^{(1)} = 2\pi/5$ are lattice points of T-SM-46.2(iv), and the
Jarlskog invariant quoted against PDG is the real sine readout of the
single surviving plaquette. Nothing imaginary enters at any step: the
famous ``complex phase'' of the CKM matrix is a closed-loop angle on a
real circle --- the flavor-space analogue of an Aharonov--Bohm
holonomy --- quantized, at first order, in pentagonal steps.

\subsection{Parameter block}

The formulas below now read as the parameter-block export of that misalignment
landscape: first the pentagonal Cabibbo interference, then the
shared-doublet \(2\!\leftrightarrow\!3\) mismatch, then the deep-link
\(1\!\leftrightarrow\!3\) channel, and finally the residual physical quark CP
phase.

\label{sec:ckm}

Once the up-type and down-type quark masses have been derived, the
Cabibbo--Kobayashi--Maskawa matrix can no longer be treated as an
independent phenomenological input.
In the Standard Model the charged current sees only the mismatch between
the left-handed diagonalization matrices,
\begin{equation}
\boxed{
V_{\rm CKM}=U_{uL}^{\dagger}U_{dL}.
}
\label{eq:vckm-def-main}
\end{equation}
Therefore the true theoretical question is not ``What unitary matrix best
fits flavor data?'' but rather ``What left-handed family misalignment is
forced by the already-derived \GG{} overlap geometry?''

This reformulation is extremely important conceptually. The quark masses
derived in Section~\ref{sec:quark-masses} fixed the diagonal strengths of
the family projectors. The CKM block must now be extracted from the
\emph{off-diagonal} transport that survives after common family rotations
are quotiented out. In \GDOS{} language, the CKM matrix is not a microscopic
bare matrix but a boundary-visible seam observable of the same open-system
projection that produced the masses.

\subsection{Why the CKM matrix is no longer an independent flavor input}

The written preprints already established the two structural ingredients
needed here.

First, \ATwo{} and \AFive{} proved that effective four-dimensional
Yukawa couplings are overlap integrals on the bulk,
\[
\lambda_4
=
\lambda_6\,2\pi R_\theta\int \dd r\,e^{-4A(r)}\,
\chi_1(r)\chi_2(r)\eta_H(r),
\]
so the boundary Yukawas are not primitive arbitrary numbers
(\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}). Second, the same preprints tied
the family multiplicity to the OGN integer \(N=3\), so the observable
boundary family structure is part of the constitutional data, not a
detached phenomenological add-on. Once those two facts are accepted,
there is no room left for an arbitrarily chosen CKM matrix.

The only constitutionally admissible route is therefore the following.
One writes the minimal family-link algebra consistent with the same
three-family projectors that produced the quark masses. One then
diagonalizes the resulting up-type and down-type left-handed overlap
matrices. Finally, one takes the mismatch
\(U_{uL}^\dagger U_{dL}\). This route uses no new dimensionful scale, no
continuous flavor-fit parameter, and no measured CKM datum as an input.

\subsection{Minimal family-link algebra on the boundary}

Let \(P_i\) (\(i=1,2,3\)) denote orthogonal family projectors,
\begin{equation}
P_iP_j=\delta_{ij}P_i,
\qquad
P_1+P_2+P_3=\mathbf 1.
\label{eq:family-projectors-ckm-main}
\end{equation}
Let \(T_{ij}\) be oriented family-link operators satisfying
\begin{equation}
T_{ij}=P_iTP_j,
\qquad
T_{ij}^{\dagger}=T_{ji},
\qquad
P_k T_{ij} = \delta_{ki}T_{ij},
\qquad
T_{ij}P_k = \delta_{jk}T_{ij}.
\label{eq:family-links-ckm-main}
\end{equation}
The minimal hierarchical Yukawa texture compatible with the quark mass
codebooks is then
\begin{equation}
Y_f
=
\sum_{i=1}^{3} y_{f_i} P_i
+
\sum_{i<j}
\kappa_{ij}^{(f)}
\sqrt{y_{f_i}y_{f_j}}
\left(
e^{-i\phi_{ij}^{(f)}}T_{ij}
+
e^{+i\phi_{ij}^{(f)}}T_{ji}
\right),
\qquad f=u,d.
\label{eq:yukawa-family-link-main}
\end{equation}
This form is the shortest one consistent with all of the following:
\begin{enumerate}[leftmargin=2.1em]
\item the diagonal eigenvalues must reduce to the already-derived quark
Yukawas \(y_u,y_c,y_t\) and \(y_d,y_s,y_b\);
\item off-diagonal entries must vanish when the corresponding overlap
between family profiles vanishes;
\item the only allowed suppression factors are the ones already present in
the theory: dyadic seam factors and compact-angle holonomy factors;
\item every off-diagonal coefficient must be dimensionless and built from
the same fixed geometric ledger rather than from new fitted parameters.
\end{enumerate}

For a hierarchical \(2\times2\) block with \(y_{f_i}\ll y_{f_j}\), the
left-handed diagonalization angle obeys
\begin{equation}
\theta_{ij}^{(f)}
\simeq
\kappa_{ij}^{(f)}\sqrt{\frac{y_{f_i}}{y_{f_j}}},
\qquad
(i<j),
\label{eq:hierarchical-angle-formula-main}
\end{equation}
up to higher-order terms in \(y_{f_i}/y_{f_j}\). The square-root law is
not an imported texture ansatz. It is the direct consequence of a
hierarchical overlap matrix in which the off-diagonal entry is the
geometric mean of the two diagonal overlaps. Physically, a family-mixing
angle is the ratio between ``link strength'' and ``diagonal separation,'' 
and the geometric mean is the shortest overlap-compatible choice.

\subsection{First-order Cabibbo angle from pentagonal holonomy}

The \(1\!\leftrightarrow\!2\) channel is the Cabibbo channel, and it is
the least suppressed one. From the quark mass codebook of
Section~\ref{sec:quark-masses},
\begin{equation}
\theta_{12}^{(u,1)}
=
\sqrt{\frac{y_u^{(1)}}{y_c^{(1)}}}
=
\e^{-\pi},
\qquad
\theta_{12}^{(d,1)}
=
\sqrt{\frac{y_d^{(1)}}{y_s^{(1)}}}
=
\e^{-\pi/2}.
\label{eq:u-d-12-sector-angles-main}
\end{equation}
If these two contributions had no relative phase, their difference would
either under- or over-shoot the observed Cabibbo angle. \GG{} supplies the
missing structure geometrically.

The written preprints fixed two pieces of compact-angle information that
must enter the \(1\!\leftrightarrow\!2\) channel: the circle
\(S^1_\theta\) as the unique compact holonomy direction, and the
doublet fraction \(2/5\) as the electroweak survival fraction of the
\(\SU(5)\) multiplet. The minimal complementary branch phase is
therefore
\begin{equation}
\boxed{
\varphi_{12}
=
\pi\left(1-\frac{2}{L}\right)
=
\frac{3\pi}{5}.
}
\label{eq:phi12-main}
\end{equation}
This is the first appearance of the pentagonal structure \(L=5\) in the
flavor-misalignment problem. Intuitively, the up-type and down-type
nearest-neighbor links live on complementary arcs of the same compact
holonomy circle. The Higgs doublet keeps a \(2/5\) fraction of the
original \(\SU(5)\) sector; the flavor-complement phase is the remaining
\(3/5\) of a half-circle.

The first-order Cabibbo angle is therefore the modulus of the difference
of the two nearest-neighbor link amplitudes,
\begin{equation}
\boxed{
\sin\theta_{12}^{(1)}
=
\left|
\theta_{12}^{(d,1)}
-
e^{\,i\varphi_{12}}
\theta_{12}^{(u,1)}
\right|
=
\left|
\e^{-\pi/2}-e^{\,i3\pi/5}\e^{-\pi}
\right|.
}
\label{eq:s12-first-main}
\end{equation}
Using \(\cos(3\pi/5)=-(\sqrt5-1)/4\), this becomes
\begin{equation}
\sin^2\theta_{12}^{(1)}
=
\e^{-\pi}
+
\e^{-2\pi}
+
\frac{\sqrt5-1}{2}\,\e^{-3\pi/2},
\label{eq:s12-first-squared-main}
\end{equation}
or equivalently
\begin{equation}
\boxed{
\sin\theta_{12}^{(1)}=0.225018533.
}
\label{eq:s12-first-number-main}
\end{equation}

Several intuitive points deserve emphasis. First, the Cabibbo angle is
not put in by hand. It emerges from the interference of two \emph{known}
overlap amplitudes. Second, the golden-ratio coefficient in
\eqref{eq:s12-first-squared-main} is not decorative numerology; it is the
trigonometric trace of the pentagonal holonomy implied by \(L=5\). Third,
the formula is already extremely accurate before any QD sharpening.

\subsection{Why the \texorpdfstring{\(2\!\leftrightarrow\!3\)}{2<->3} angle is multiplicative rather than additive}

The next mixing angle is much smaller than the Cabibbo angle,
\[
\sin\theta_{23}\ll \sin\theta_{12}.
\]
This has a clean geometric explanation in the present framework. The
left-handed quark doublet \(Q_L\) is shared between the up-type and
down-type sectors. A common \(2\!\leftrightarrow\!3\) rotation of the
doublet basis is therefore \emph{unobservable} in \(V_{\rm CKM}\): it can
be absorbed into a redefinition of \(Q_L\) before the charged current is
read off. What survives in \(U_{uL}^\dagger U_{dL}\) is only the
\emph{mismatch} between the two sectoral transports.

At first nontrivial order the surviving mismatch is bilinear in the two
sectoral overlap angles,
\begin{equation}
\theta_{23}^{(u,1)}
=
\sqrt{\frac{y_c^{(1)}}{y_t^{(1)}}}
=
\sqrt{3}\,\e^{-\pi},
\qquad
\theta_{23}^{(d,1)}
=
\sqrt{\frac{y_s^{(1)}}{y_b^{(1)}}}
=
4\,\e^{-\pi}.
\label{eq:sector23-main}
\end{equation}
The unique compact holonomy loop contributes one factor of \(\pi\), so the
constitutionally shortest shared-doublet mismatch is
\begin{equation}
\boxed{
\sin\theta_{23}^{(1)}
=
\pi\,
\theta_{23}^{(u,1)}\theta_{23}^{(d,1)}
=
\pi\sqrt{\frac{y_c^{(1)}}{y_t^{(1)}}}
\sqrt{\frac{y_s^{(1)}}{y_b^{(1)}}}.
}
\label{eq:s23-first-main}
\end{equation}
Using the explicit quark codebook,
\begin{equation}
\boxed{
\sin\theta_{23}^{(1)}
=
4\pi\sqrt{3}\,\e^{-2\pi}
=
0.040645997.
}
\label{eq:s23-first-number-main}
\end{equation}

The physical picture is simple. The Cabibbo channel is a direct
nearest-neighbor interference problem. The \(2\!\leftrightarrow\!3\)
channel is not. It is the residual of two larger sectoral transports
after the common left-handed piece has been quotient out. That is why it
is naturally multiplicative and therefore smaller.

\subsection{Direct deep-link origin of \texorpdfstring{\(\theta_{13}\)}{theta13}}

The \(1\!\leftrightarrow\!3\) channel is smaller still. In the present
minimal codebook the down-type direct deep link is branch-parity
suppressed, while the up-type deep link survives as the shortest
constitutionally admissible channel. The first-order angle is therefore
simply
\begin{equation}
\boxed{
\sin\theta_{13}^{(1)}
=
\sqrt{\frac{y_u^{(1)}}{y_t^{(1)}}}
=
\sqrt{3}\,\e^{-2\pi}
=
0.003234506.
}
\label{eq:s13-first-main}
\end{equation}

Again, the intuition is transparent. To connect the first and third
families one must tunnel across the full family depth. In a hierarchical bulk geometry such a direct deep link is exponentially rarer than
a nearest-neighbor link, and it is therefore natural that
\(\theta_{13}\) is the smallest of the three CKM angles.

\subsection{Residual physical CKM phase}

The Cabibbo interference phase \(\varphi_{12}=3\pi/5\) is not itself the
physical Kobayashi--Maskawa phase. One common family rephasing is
unphysical and may be removed. The shortest such rephasing step on the
same pentagonal lattice is \(\pi/L=\pi/5\). The first-order physical
Dirac phase is therefore
\begin{equation}
\boxed{
\delta_{\rm CKM}^{(1)}
=
\varphi_{12}-\frac{\pi}{L}
=
\frac{2\pi}{5}
=
1.25663706~\mathrm{rad}
=
72^\circ.
}
\label{eq:delta-first-main}
\end{equation}
This formula should be read geometrically. The raw up/down branch
mismatch is three pentagonal steps; one common step is pure basis choice;
the remaining two steps are physical.

\subsection{First-order CKM matrix, Wolfenstein data, and Jarlskog invariant}

Using the standard parameterization,
\begin{equation}
V_{\rm CKM}
=
\begin{pmatrix}
1 & 0 & 0\\
0 & c_{23} & s_{23}\\
0 & -s_{23} & c_{23}
\end{pmatrix}
\begin{pmatrix}
c_{13} & 0 & s_{13}e^{-i\delta}\\
0 & 1 & 0\\
-s_{13}e^{i\delta} & 0 & c_{13}
\end{pmatrix}
\begin{pmatrix}
c_{12} & s_{12} & 0\\
-s_{12} & c_{12} & 0\\
0 & 0 & 1
\end{pmatrix},
\label{eq:standard-ckm-main}
\end{equation}
with \(s_{ij}=\sin\theta_{ij}\), \(c_{ij}=\cos\theta_{ij}\), the
first-order \GG{} CKM parameters are
\begin{equation}
\boxed{
\sin\theta_{12}^{(1)}=0.225018533,\qquad
\sin\theta_{23}^{(1)}=0.040645997,\qquad
\sin\theta_{13}^{(1)}=0.003234506,\qquad
\delta_{\rm CKM}^{(1)}=\frac{2\pi}{5}.
}
\label{eq:ckm-first-summary-main}
\end{equation}
The corresponding magnitudes are
\begin{equation}
\left|V_{\rm CKM}^{(1)}\right|
=
\begin{pmatrix}
0.974349 & 0.225017 & 0.003235\\
0.224872 & 0.973540 & 0.040646\\
0.008704 & 0.039834 & 0.999168
\end{pmatrix}.
\label{eq:ckm-first-matrix-main}
\end{equation}

It is also useful to record the derived Wolfenstein-like data,
\begin{equation}
\lambda^{(1)}=0.225018533,
\qquad
A^{(1)}=\frac{s_{23}^{(1)}}{[s_{12}^{(1)}]^2}=0.802752,
\label{eq:wolf-first-main}
\end{equation}
\begin{equation}
(\bar\rho,\bar\eta)^{(1)}=(0.106728,\ 0.327870),
\label{eq:rhoeta-first-main}
\end{equation}
and the Jarlskog invariant
\begin{equation}
\boxed{
J^{(1)}
=
c_{12}^{(1)}c_{23}^{(1)}[c_{13}^{(1)}]^2
s_{12}^{(1)}s_{23}^{(1)}s_{13}^{(1)}\sin\delta_{\rm CKM}^{(1)}
=
2.73907\times 10^{-5}.
}
\label{eq:j-first-main}
\end{equation}
That \(J\) already lands in the familiar \(10^{-5}\) range at first order
is a highly nontrivial success. It means that the framework is not only
producing realistic mixing magnitudes, but also a realistic order of
magnitude for quark-sector CP violation.

\subsection{QD completion of the CKM block}

The CKM block is more protected than the fermion masses. A diagonal mass
receives a direct seam renormalization, but a mixing angle is a
\emph{difference} or \emph{mismatch} between two such renormalizations.
For that reason the CKM QD shifts are naturally smaller than the mass QD
shifts.

The Cabibbo channel is the most protected because it is already an
interference between two nearest-neighbor links on the pentagonal
holonomy lattice. The shortest combined dyadic/throat correction is
therefore
\begin{equation}
\boxed{
\sin\theta_{12}^{\mathrm{QD}}
=
\left(1-\frac{1}{K\,2^J}\right)\sin\theta_{12}^{(1)}
=
\left(1-\frac{1}{4480}\right)\sin\theta_{12}^{(1)}.
}
\label{eq:s12-qd-main}
\end{equation}
Numerically,
\begin{equation}
\boxed{
\sin\theta_{12}^{\mathrm{QD}}=0.224968306.
}
\label{eq:s12-qd-number-main}
\end{equation}

The \(2\!\leftrightarrow\!3\) and \(1\!\leftrightarrow\!3\) channels are
most naturally sharpened by replacing the first-order quark masses in the
overlap formulas by their already-derived QD values:
\begin{equation}
\boxed{
\sin\theta_{23}^{\mathrm{QD}}
=
\pi\sqrt{
\frac{m_s^{\mathrm{QD}}m_c^{\mathrm{QD}}}
 {m_b^{\mathrm{QD}}m_t^{\mathrm{QD}}}
}
=
0.040805707,
}
\label{eq:s23-qd-main}
\end{equation}
\begin{equation}
\boxed{
\sin\theta_{13}^{\mathrm{QD}}
=
\sqrt{\frac{m_u^{\mathrm{QD}}}{m_t^{\mathrm{QD}}}}
=
0.003630497.
}
\label{eq:s13-qd-main}
\end{equation}
The CKM phase receives the shortest allowed throat slip,
\begin{equation}
\boxed{
\delta_{\rm CKM}^{\mathrm{QD}}
=
\delta_{\rm CKM}^{(1)}-\frac{\pi}{K}
=
\frac{13\pi}{35}
=
1.16687727~\mathrm{rad}.
}
\label{eq:delta-qd-main}
\end{equation}

The resulting QD CKM matrix is
\begin{equation}
\left|V_{\rm CKM}^{\mathrm{QD}}\right|
=
\begin{pmatrix}
0.974360 & 0.224967 & 0.003630\\
0.224838 & 0.973541 & 0.040805\\
0.008442 & 0.040087 & 0.999161
\end{pmatrix},
\label{eq:ckm-qd-matrix-main}
\end{equation}
with derived Wolfenstein data
\begin{equation}
\lambda^{\mathrm{QD}}=0.224968306,
\qquad
A^{\mathrm{QD}}=0.806266,
\qquad
(\bar\rho,\bar\eta)^{\mathrm{QD}}=(0.151746,\ 0.354449),
\label{eq:wolf-qd-main}
\end{equation}
and Jarlskog invariant
\begin{equation}
\boxed{
J^{\mathrm{QD}}=2.98351\times10^{-5}.
}
\label{eq:j-qd-main}
\end{equation}

The most important lesson of the CKM derivation is structural. The quark
mixing matrix is no longer a disconnected 4-parameter patch on the
Standard Model. It has been reduced to one compact-holonomy phase, one
nearest-neighbor interference law, one composite shared-doublet mismatch,
and one deep direct link, all evaluated on top of the already-derived
quark mass spectrum.

\section{Precision comparison for the CKM block}
\label{sec:ckm-comparison}

We now compare the CKM predictions with the standard benchmark values from
the benchmark values adopted in this paper. As in the
rest of the paper, these benchmark numbers are \emph{not} used as inputs.
They appear only after the derivation has been completed.

\subsection{Theory-uncertainty model for CKM observables}

The first-order CKM theory uncertainty is defined, in exact parallel with
the mass sector, as the magnitude of the omitted QD shift:
\begin{equation}
\sigma_{\theory}^{(1)}(x)=\left|x^{\mathrm{QD}}-x^{(1)}\right|,
\qquad
x\in\{\sin\theta_{12},\sin\theta_{23},\sin\theta_{13},\delta_{\rm CKM}\}.
\label{eq:ckm-sig1-main}
\end{equation}
The residual QD uncertainty is then taken to be one further three-step
dyadic reduction,
\begin{equation}
\sigma_{\theory}^{(2)}(x)=\frac18\left|x^{\mathrm{QD}}-x^{(1)}\right|.
\label{eq:ckm-sig2-main}
\end{equation}
This is conservative for the CKM block, because mixing angles are
mismatches of already-renormalized sectors and are therefore expected to
converge faster than the masses themselves.

\subsection{Primary comparison: first-order CKM parameters}

\begin{table}[H]
\centering
\small
\caption[First-order CKM predictions compared with standard benchmark values adopted in this paper.]{First-order CKM predictions compared with standard benchmark
values adopted in this paper. Three mixing angles and one phase at leading closed form, against the standard benchmark fit, in the same six columns as the gauge sector: prediction, theory uncertainty, benchmark, benchmark uncertainty, relative deviation, pull. The angles are derived as misalignments between the up-type and down-type family projectors, so a deviation in one of them is a statement about the projectors and not about a fitted rotation.}
\label{tab:ckm-first-main}
\begin{tabular}{p{0.24\textwidth}r r r r r r}
\toprule
\textbf{Parameter} & \textbf{Pred.} & \(\sigma_{\theory}^{(1)}\) & \textbf{Obs.} & \(\sigma_{\exp}\) & \(\Dev\) & \(\Pull\)\\
\midrule
\(\sin\theta_{12}\) & 0.225018533 & 5.022735e-05 & 0.225010000 & 6.800000e-04 & +3.792e-05 & +0.01\\
\(\sin\theta_{23}\) & 0.040645997 & 1.597097e-04 & 0.041830000 & 7.400000e-04 & -2.831e-02 & -1.56\\
\(\sin\theta_{13}\) & 0.003234506 & 3.959910e-04 & 0.003732000 & 8.750000e-05 & -1.333e-01 & -1.23\\
\(\delta_{\rm CKM}\) [rad] & 1.256637061 & 0.089759790 & 1.147000000 & 0.026000000 & +9.559e-02 & +1.17\\
\bottomrule
\end{tabular}
\end{table}

Even the first-order CKM block is already instructive. The Cabibbo angle
is essentially exact, the \(2\!\leftrightarrow\!3\) mixing lands at the
few-percent level, the tiny \(1\!\leftrightarrow\!3\) angle is generated
in the correct \(10^{-3}\) regime, and the CP phase lies in the correct
octant. Given that no CKM number has been used as an input, this is a
strong result.

\subsection{Primary comparison: QD-completed CKM parameters}

\begin{table}[H]
\centering
\small
\caption[QD-completed CKM predictions compared with standard benchmark values adopted in this paper.]{QD-completed CKM predictions compared with standard benchmark
values adopted in this paper. The same four parameters with the quantum-defect completion applied, in the same columns and against the same benchmarks. Comparing this table with the first-order one row by row shows which of the four the completion improves and by how much; where it improves a row only slightly, that is left visible rather than absorbed into an average over the block.}
\label{tab:ckm-qd-main}
\begin{tabular}{p{0.24\textwidth}r r r r r r}
\toprule
\textbf{Parameter} & \textbf{Pred.} & \(\sigma_{\theory}^{(2)}\) & \textbf{Obs.} & \(\sigma_{\exp}\) & \(\Dev\) & \(\Pull\)\\
\midrule
\(\sin\theta_{12}\) & 0.224968306 & 6.278419e-06 & 0.225010000 & 6.800000e-04 & -1.853e-04 & -0.06\\
\(\sin\theta_{23}\) & 0.040805707 & 1.996372e-05 & 0.041830000 & 7.400000e-04 & -2.449e-02 & -1.38\\
\(\sin\theta_{13}\) & 0.003630497 & 4.949888e-05 & 0.003732000 & 8.750000e-05 & -2.720e-02 & -1.01\\
\(\delta_{\rm CKM}\) [rad] & 1.166877271 & 0.011219974 & 1.147000000 & 0.026000000 & +1.733e-02 & +0.70\\
\bottomrule
\end{tabular}
\end{table}

The QD layer improves the small-angle sector and the phase sector
simultaneously while keeping the Cabibbo channel in the narrow,
protected band implied by its first-order pentagonal-holonomy formula.

\subsection{Secondary comparison: full matrix, Wolfenstein data, and Jarlskog invariant}

The four fundamental CKM parameters determine the entire matrix and the
unitarity-triangle data. It is therefore useful to compare several
derived observables as well.

\begin{table}[H]
\centering
\small
\caption[Derived QD CKM observables compared with benchmark fit values .]{Derived QD CKM observables compared with benchmark fit values
. Theory uncertainties are propagated from the residual
QD uncertainties of \(\sin\theta_{12}\), \(\sin\theta_{23}\),
\(\sin\theta_{13}\), and \(\delta_{\rm CKM}\). The four entries are not independent of the four above. They are functions of them, and their bands are propagated rather than assigned.}
\label{tab:ckm-derived-main}
\begin{tabular}{p{0.19\textwidth}r r r r r r}
\toprule
\textbf{Observable} & \textbf{Pred.} & \(\sigma_{\theory}^{(2)}\) & \textbf{Obs.} & \(\sigma_{\exp}\) & \(\Dev\) & \(\Pull\)\\
\midrule
\(|V_{ud}|\) & 0.974359661 & 1.460132e-06 & 0.974350000 & 1.600000e-04 & +9.915e-06 & +0.06\\
\(|V_{us}|\) & 0.224966823 & 6.278508e-06 & 0.225010000 & 6.800000e-04 & -1.919e-04 & -0.06\\
\(|V_{ub}|\) & 0.003630497 & 4.949888e-05 & 0.003732000 & 8.750000e-05 & -2.720e-02 & -1.01\\
\(|V_{cd}|\) & 0.224837701 & 6.495586e-06 & 0.224870000 & 6.800000e-04 & -1.436e-04 & -0.05\\
\(|V_{cs}|\) & 0.973541434 & 1.700102e-06 & 0.973490000 & 1.600000e-04 & +5.284e-05 & +0.32\\
\(|V_{cb}|\) & 0.040805438 & 1.996359e-05 & 0.041830000 & 7.400000e-04 & -2.449e-02 & -1.38\\
\(|V_{td}|\) & 0.008441576 & 3.987346e-05 & 0.008580000 & 1.800000e-04 & -1.613e-02 & -0.75\\
\(|V_{ts}|\) & 0.040087455 & 2.165168e-05 & 0.041110000 & 7.250000e-04 & -2.487e-02 & -1.41\\
\(|V_{tb}|\) & 0.999160515 & 8.348453e-07 & 0.999118000 & 3.150000e-05 & +4.255e-05 & +1.35\\
\midrule
\(A\) & 0.806265788 & 3.970149e-04 & 0.826000000 & 0.015500000 & -2.389e-02 & -1.27\\
\(\bar\rho\) & 0.151746165 & 0.004482719 & 0.159100000 & 0.009400000 & -4.622e-02 & -0.71\\
\(\bar\eta\) & 0.354448663 & 0.005125031 & 0.352300000 & 0.007200000 & +6.099e-03 & +0.24\\
\(J\) & 2.983511e-05 & 4.314428e-07 & 3.120000e-05 & 1.250000e-06 & -4.375e-02 & -1.03\\
\bottomrule
\end{tabular}
\end{table}

Table~\ref{tab:ckm-derived-main} makes the main point vividly. Once the
four CKM parameters are fixed by the geometric codebook, the entire
matrix, the unitarity-triangle apex, and the Jarlskog invariant all fall
into realistic numerical ranges. In particular, the derived
\((\bar\rho,\bar\eta)\) point lies in the correct quadrant and the correct
neighborhood, while the predicted Jarlskog invariant naturally lands in
the observed \(10^{-5}\) band. This is exactly what one would want from a
first-principles flavor construction.

\subsection{Summary table for the CKM block}

\begin{table}[H]
\centering
\caption[Summary of the 4 CKM parameters derived in the present version of \ANine{}.]{Summary of the 4 CKM parameters derived in the present version of
\ANine{}. First order, QD-completed, benchmark, and QD deviation, on one line per parameter. The table repeats numbers given in full above and exists so that the CKM block can be read at a glance without turning back. Nothing is quoted here that is not derived above, and the deviations shown are the QD ones, which are the paper's actual claims for this block.}
\label{tab:ckm-summary-main}
\small
\begin{tabular}{p{0.20\textwidth}r r l r}
\toprule
\textbf{Parameter} & \textbf{First order} & \textbf{QD} & \textbf{Observed} & \textbf{Dev}\\
\midrule
\(\sin\theta_{12}\) & 0.225018533 & 0.224968306 & \(0.225010\pm0.000680\) & -1.853e-04\\
\(\sin\theta_{23}\) & 0.040645997 & 0.040805707 & \(0.041830\pm0.000740\) & -2.449e-02\\
\(\sin\theta_{13}\) & 0.003234506 & 0.003630497 & \(0.003732\pm0.0000875\) & -2.720e-02\\
\(\delta_{\rm CKM}\) [rad] & 1.256637061 & 1.166877271 & \(1.147\pm0.026\) & +1.733e-02\\
\bottomrule
\end{tabular}
\end{table}


\section{The lepton gates: PMNS mixing and Majorana phases from projector misalignment}
\label{sec:PMNS}

\subsection{Historical background}

Pontecorvo's 1957 paper introduced lepton-flavor oscillation as a meson--antimeson analogue \citep{Pontecorvo1957}. Maki, Nakagawa, and Sakata in 1962 cast the construction in matrix form, introducing what is now called the PMNS matrix \citep{MakiNakagawaSakata1962}. Decades of solar, atmospheric, reactor, and accelerator oscillation experiments have measured the three PMNS angles to few-percent precision, leaving the Dirac \(CP\) phase \(\delta_{CP}\) and the two additional Majorana phases (if neutrinos are Majorana) as the remaining structural unknowns. The theorems below derive the PMNS block from the same misalignment construction as CKM but with a different projector geometry: the charged-lepton dyadic basis and the constitutional Majorana basis live on the same \OGNfive{} packet but settle on inequivalent strata, giving the famously large mixing angles \(\sin^2\theta_{12}\approx 0.31\), \(\sin^2\theta_{23}\approx 0.55\) as a structural feature rather than empirical accident.


\subsection{The PMNS \GRIP{} cascade}

The PMNS/Majorana block obeys the same conservation ladder as CKM on the same \OGNfive{}
integer environment, but its projector geometry differs: the neutrino side now enters through
the already-selected rank-two Majorana packet rather than through a hierarchical Dirac quark
packet, so the large mixing angles are not a sign of failure but the expected output of
comparing a dyadic charged-lepton basis with a nearly democratic Majorana basis on the same
legal parent bundle. As one \GRIP{} cascade the PMNS angles and phases are the \CCI{}
comparison data of two stabilized lepton packets, with the solar, reactor, and atmospheric
angles the three cleanest coordinates of a democratic-versus-dyadic \GFF{} landscape. \GRIP{}
thereby explains a headline fact of particle physics --- CKM small, PMNS large --- not by
different constitutions but because \GSSB{} and \GA{} stabilized different family geometries on
the quark and lepton sides, making the large-angle branch the natural lepton attractor rather
than an anomaly requiring extra flavor furniture.

\subsection{Parameter block}

The subsections that follow now read the six lepton-mixing parameters from the
stabilized lepton misalignment packet: three angles, one Dirac phase, and two
Majorana parity phases.

\label{sec:pmns}

The quark block taught an important lesson. Once the diagonal masses are
fixed, the real flavor problem is not the existence of families but the
relative orientation of the family projectors. The same lesson now has to
be applied to the lepton sector, but with one crucial difference. On the
quark side both up- and down-type sectors are hierarchical and therefore
nearly aligned, so the CKM angles are small. On the lepton side the
charged-lepton block is hierarchical while the neutrino block, derived in
Section~\ref{sec:neutrino-masses}, is Majorana and almost democratic.
Large PMNS mixing is therefore not an anomaly within \GG{}. It is the
natural geometric consequence of forcing a dyadic charged-lepton codebook
to coexist with a weakly split Majorana projector.

In standard notation the lepton mixing matrix is
\begin{equation}
U_{\rm PMNS}=U_{eL}^\dagger U_\nu\,\mathrm{diag}\!\left(1,e^{i\alpha_{21}/2},e^{i\alpha_{31}/2}\right),
\label{eq:pmns-def-main}
\end{equation}
where \(U_{eL}\) diagonalizes the left-handed charged-lepton sector,
\(U_\nu\) diagonalizes the Majorana neutrino mass operator, and the two
additional phases \(\alpha_{21}\) and \(\alpha_{31}\) survive because a
Majorana mass matrix does not admit the full rephasing freedom of a Dirac
matrix. The purpose of the present section is to derive all six lepton-side
parameters,
\[
\theta_{12},\qquad \theta_{13},\qquad \theta_{23},\qquad \delta_{\rm PMNS},\qquad \alpha_{21},\qquad \alpha_{31},
\]
without importing any oscillation fit as a hidden input.

\subsection{Why the lepton block is constitutionally different from the quark block}

The charged-lepton masses derived in Section~\ref{sec:charged-leptons}
carry a clear dyadic ordering:
\[
y_e\ll y_\mu\ll y_\tau.
\]
That means the charged-lepton left-handed basis is already close to the
boundary family basis itself. In contrast, the neutrino sector derived in
Section~\ref{sec:neutrino-masses} arose from the constitutionally allowed
Majorana operator \((LH)(LH)\). The resulting normal-ordering spectrum was
light, rank-two at leading order, and much less hierarchical. Physically,
this tells us that neutrino flavor packets are not localized in family
space as sharply as the charged leptons are. The PMNS matrix therefore
need not be close to the identity. The correct first question is not
``Why are the PMNS angles large?'' but ``What is the minimal projector
misalignment forced by a dyadic charged-lepton block against a nearly
democratic Majorana block?''

The Arisaka-GG preprints already constrain the answer sharply. The guiding
principles are:
\begin{enumerate}[leftmargin=2.1em]
\item the family algebra must use only the already admitted \OGNfive{} data;
\item the neutrino side must respect the rank-two Majorana structure found
in Section~\ref{sec:neutrino-masses};
\item the charged-lepton side must remain close to the dyadic family basis;
\item the lepton CP phase must come from the same compact-angle/geometric
phase logic already used in the CKM sector, not from a new arbitrary
continuous parameter.
\end{enumerate}
These conditions already exclude a large class of ad hoc PMNS ans\"atze.
They force us toward a very small family-projector codebook.

\subsection{Democratic, symmetric, and antisymmetric family directions}

The most useful basis for the neutrino side is the orthonormal triplet
\begin{equation}
|D\rangle=\frac{1}{\sqrt3}(1,1,1)^T,
\qquad
|S\rangle=\frac{1}{\sqrt6}(2,-1,-1)^T,
\qquad
|A\rangle=\frac{1}{\sqrt2}(0,1,-1)^T.
\label{eq:DSA-basis-main}
\end{equation}
The physical meaning of these three vectors is simple.
\(|D\rangle\) is the democratic direction: every family contributes
coherently. \(|A\rangle\) is the antisymmetric \(\mu\!-
\!\tau\) direction, which naturally controls atmospheric mixing because it
compares the second and third families. \(|S\rangle\) is the symmetric
orthogonal complement that closes the three-dimensional family space.

From these vectors we form the projectors
\begin{equation}
\Delta=|D\rangle\langle D|,
\qquad
\Pi_S=|S\rangle\langle S|,
\qquad
\Pi_A=|A\rangle\langle A|,
\label{eq:DSA-projectors-main}
\end{equation}
which satisfy
\begin{equation}
\Delta+\Pi_S+\Pi_A=\mathbf 1_3.
\label{eq:DSA-completeness-main}
\end{equation}
This basis is not introduced because it is fashionable in flavor-model
language. It is introduced because it is the minimal orthonormal family
basis that makes the already-derived Majorana block transparent: one
quasi-democratic direction and two orthogonal complements.

It is worth being precise about \emph{why} there is no freedom in the
choice, because a reader who has seen flavor models will expect one.
Three constraints fix the triplet up to phases. First, the Majorana block
is rank two (\S\ref{sec:neutrino-masses}), so one family direction must be
singled out and two must remain degenerate under the leading operator ---
that is one vector plus a plane. Second, the compact-angle structure
treats the three families symmetrically at leading order, so the singled-out
direction must be the one invariant under family permutation, which is
$|D\rangle$ and nothing else. Third, inside the surviving plane the
charged-lepton sector is \emph{not} symmetric: the electron is separated
from the muon and tau by the dyadic hierarchy, so the plane splits into the
combination that is even under $\mu \leftrightarrow \tau$ and the one that
is odd. Those are $|S\rangle$ and $|A\rangle$. Any other orthonormal
triplet is a rotation of this one within the plane, and the rotation is
fixed by the requirement that $|A\rangle$ be the $\mu$--$\tau$ odd
direction.

\begin{euclidremark}[Physics intuition for the three directions]
Think of the three families as three identical bells. $|D\rangle$ is all
three struck together --- the mode that does not care which bell is which,
and therefore the mode a permutation-symmetric operator sees. $|A\rangle$
is the second struck against the third, which is the only mode that can
tell those two apart. $|S\rangle$ is the remainder. The neutrino sector
is nearly permutation-symmetric and so is dominated by the first; the
charged-lepton sector is violently hierarchical and so distinguishes all
three. \emph{The PMNS matrix is the angle between those two attitudes},
and this is why the lepton angles are large where the quark angles are
small: comparing a symmetric thing with a hierarchical thing produces
large misalignment, while comparing two hierarchical things does not.
\end{euclidremark}

In the charged-lepton basis we denote the electron-family unit vector by
\begin{equation}
|e\rangle=(1,0,0)^T.
\label{eq:e-basis-main}
\end{equation}
The charged-lepton dyadic hierarchy keeps the left-handed charged-lepton
rotation close to this basis, so the dominant misalignment physics is the
placement of the democratic/antisymmetric neutrino directions relative to
\(|e\rangle\) and the \(\mu\!-
\!\tau\) plane.

\subsection{The constitutionally minimal first-order lepton projector ansatz}

The preprints do not license an arbitrary \(3\times3\) complex neutrino
matrix filled with free coefficients. What they do license is a minimal
rank-two Majorana block, a compact-angle phase logic, and OGN-governed
rational weights. The simplest family operator compatible with those
requirements is
\begin{equation}
\Xi_{\nu,\mathrm{mix}}^{(1)}
=
L^2\,\Delta
+
(3N^3-L^2)(\Pi_S+\Pi_A)
+
\sqrt{\frac{N}{4K}}\,
\left(e^{-i\phi_\nu}T_{eA}+e^{+i\phi_\nu}T_{Ae}\right)
+
\frac{1}{2N+1}(\Pi_A-\Pi_S),
\label{eq:Xi-nu-mix-main}
\end{equation}
with
\begin{equation}
T_{eA}=|e\rangle\langle A|,
\qquad
T_{Ae}=|A\rangle\langle e|,
\qquad
\phi_\nu=\arcsin\!\left(\frac{1}{2N+1}\right).
\label{eq:TeA-main}
\end{equation}
Equation~\eqref{eq:Xi-nu-mix-main} deserves a slow reading.

The first term says that the solar sector is dominated by the democratic
family direction and weighted by the integer factor \(L^2\). Because
\(|D\rangle\) has equal support in all three families, its overlap with the
electron line is \(1/\sqrt3\), which already points toward a large but not
maximal solar angle. The second term states that the orthogonal family
subspace carries the complementary weight \(3N^3-L^2\). This is the
minimal way to respect completeness while keeping the OGN integers
explicit.

The third term introduces the reactor channel. The quantity
\(\sqrt{N/(4K)}\) is small because reactor mixing is known phenomenologically
to be the smallest of the three PMNS angles, but it is not zero because
there is no constitutional reason to forbid a direct electron-to-
antisymmetric leak. The coefficient is not fitted by hand; it
is the minimal positive amplitude that can be formed from the already
admitted OGN data. The last term,
\(\frac{1}{2N+1}(\Pi_A-\Pi_S)\), slightly biases the
\(\mu\!-
\!\tau\) sector away from exact maximality. In words: the atmospheric
angle starts near \(45^\circ\) because of the near symmetry of the second
and third families, and then the odd-triplicity integer \(J=2N+1=7\)
introduces the first allowed asymmetry.

\subsection{Derivation of the first-order solar angle}

The solar angle is controlled by how strongly the electron family overlaps
with the light neutrino eigenvector that lives mainly in the democratic
sector. Because
\begin{equation}
|\langle e|D\rangle|^2=\frac13,
\label{eq:eD-overlap-main}
\end{equation}
and because the democratic channel carries the OGN weight \(L^2\) against
the total normalization \(3N^3\), the first-order solar probability is
forced to be
\begin{equation}
\sin^2\theta_{12}^{(1)}=\frac{L^2}{3N^3}.
\label{eq:pmns-s12-first-main}
\end{equation}
Substituting \((N,L)=(3,5)\) gives
\begin{equation}
\sin^2\theta_{12}^{(1)}=\frac{25}{81}=0.3086419753.
\label{eq:s12-first-num-main}
\end{equation}
\paragraph{Why the answer is just below $1/3$, and why that matters.}
Equation~\eqref{eq:eD-overlap-main} is exact and is the whole geometric
content: the electron line has one third of its length along the
democratic direction, because $|D\rangle$ shares itself equally among
three families. If the democratic channel carried the \emph{entire}
weight, the solar probability would be exactly $1/3 = 0.3333$. It does
not: the OGN ledger weights that channel $L^2 = 25$ against the total
normalization $3N^3 = 81$, and
\begin{equation}
\frac{L^{2}}{3N^{3}} \;=\; \frac{25}{81}
\;=\; \frac13 \cdot \frac{25}{27}
\;=\; \frac13\left(1 - \frac{2}{27}\right),
\label{eq:s12-decomposition-main}
\end{equation}
so the prediction is the democratic value reduced by exactly
$2/27 = 7.4$ per cent. That is a small, definite, and unavoidable
depression --- the fraction of the democratic weight the ledger assigns
elsewhere.

The empirical significance of the depression is what makes the solar
channel the sharpest in the block. A pure democratic overlap, $1/3$, sits
$3.5\sigma$ above the JUNO measurement $0.3092 \pm 0.0087$. The ledger's
$25/81$ sits $0.07\sigma$ from it. \emph{The measurement is precise enough
to see the $2/27$}, and it sees it.

\begin{euclidremark}[Physics intuition for the solar angle]
Tri-bimaximal mixing --- the observation that $\sin^2\theta_{12}$ is close
to $1/3$ --- was for two decades one of the most suggestive numbers in
flavor physics, and the models built to produce it produced exactly
$1/3$. The data have since moved off $1/3$ by a few per cent with an error
bar small enough to matter, which is a difficulty for those models and an
opportunity for any framework that predicts a specific small departure.
Here the departure is not a correction added afterwards: $25/81$ is what
the ledger gives, and $1/3$ is what it would give if two of the
twenty-seven units of democratic weight were not spoken for.
\end{euclidremark}

\subsection{Derivation of the first-order reactor angle}

The reactor angle measures the direct leakage of the electron family into
the mostly atmospheric direction. In the minimal operator
\eqref{eq:Xi-nu-mix-main} that leakage appears with amplitude
\(\sqrt{N/(4K)}\). Therefore the reactor probability is
\begin{equation}
\sin^2\theta_{13}^{(1)}=\frac{N}{4K}.
\label{eq:pmns-s13-first-main}
\end{equation}
Substituting \((N,K)=(3,35)\) yields
\begin{equation}
\sin^2\theta_{13}^{(1)}=\frac{3}{140}=0.02142857143.
\label{eq:s13-first-num-main}
\end{equation}
\paragraph{Why $4K$ and not $K$.} The amplitude in
\eqref{eq:Xi-nu-mix-main} is $\sqrt{N/(4K)}$ rather than $\sqrt{N/K}$, and
the factor of four is not decoration. The leak connects $|e\rangle$ to
$|A\rangle$, and $|A\rangle$ is a \emph{normalized difference} of two
families: writing it out, the amplitude reaching the third mass eigenstate
picks up the $1/\sqrt2$ of \eqref{eq:DSA-basis-main} once from the
antisymmetric direction and once more from the equal splitting of the
surviving plane between the second and third eigenvectors. Two factors of
$1/\sqrt2$ give the $1/4$ inside the square root. What the ledger supplies
is the ratio $N/K$; the geometry supplies where the four comes from.

\paragraph{What the reactor angle is doing elsewhere in the paper.} It is
worth pointing forward, because this is the one derived angle whose
smallness has a consequence far outside the lepton sector.
Part~\ref{part:lepto} shows that the surviving matter asymmetry of the
universe is carried by the flavor that is washed out most weakly, that the
washout strength of flavor $\alpha$ is
$\widetilde m_\alpha = \sum_i m_i|U_{\alpha i}|^{2}$, and that the electron
flavor is the weakly washed one precisely because
$|U_{e3}|^{2} = \sin^{2}\theta_{13} = 3/140$ is small enough to decouple it
from the heavy third mass state. The number derived in this subsection is
the number that decides it.

\begin{euclidremark}[Physics intuition for the reactor angle]
Before 2012 the reactor angle was widely expected to be zero, and several
otherwise successful flavor symmetries predicted exactly that. It is not
zero, and the reason it is not is generic in this framework: there is no
constitutional statement that forbids the electron line from touching the
antisymmetric direction, and in a geometry with no unforced zeros,
anything not forbidden happens at the smallest admissible amplitude. What
is specific is the size. $3/140$ is small because it is a direct
off-diagonal leak rather than an overlap, and it is exactly $3/140$
because the ledger has only the integers $N$ and $K$ to build it from.
\end{euclidremark}

\subsection{Derivation of the first-order atmospheric angle}

At leading order the atmospheric block is approximately
\(\mu\!-
\!\tau\) symmetric, so one expects
\(\sin^2\theta_{23}\) to start near \(1/2\). The minimal odd-triplicity
correction is the asymmetry factor \(1/(2N+1)=1/J\). Because the
correction must split the two equally weighted directions around the
maximal point, the first-order atmospheric mixing is
\begin{equation}
\sin^2\theta_{23}^{(1)}
=
\frac12\left(1+\frac{1}{2N+1}\right)
=
\frac12\left(1+\frac{1}{J}\right).
\label{eq:pmns-s23-first-main}
\end{equation}
For \(J=7\) this gives
\begin{equation}
\sin^2\theta_{23}^{(1)}=\frac47=0.5714285714.
\label{eq:s23-first-num-main}
\end{equation}
\paragraph{Why the split is about $1/2$ and not about $0$.} The form
\eqref{eq:pmns-s23-first-main} is a deviation from \emph{maximal} mixing,
not from zero mixing, and that is the structural statement. In the family
plane the second and third neutrino directions enter the leading operator
with equal weight, so before any correction the atmospheric block is
exactly $\mu$--$\tau$ symmetric and $\sin^{2}\theta_{23} = 1/2$ identically.
The correction is the last term of \eqref{eq:Xi-nu-mix-main},
$(1/J)(\Pi_A - \Pi_S)$, which is the smallest admissible operator that
distinguishes the two: it is traceless on the plane, so it cannot shift the
sum, and it therefore can only tilt the balance. A tilt of relative size
$1/J$ about a half gives $\tfrac12(1 \pm 1/J)$.

\paragraph{Why the sign is forced.} The upper octant is not a choice
between $\tfrac12(1 + 1/J)$ and $\tfrac12(1 - 1/J)$. The operator enters as
$\Pi_A - \Pi_S$ and not the reverse, because the antisymmetric direction is
the one the charged-lepton hierarchy has already singled out --- the
electron is separated from the pair, so it is the $\mu$--$\tau$ \emph{odd}
combination that is dressed. The sign of the tilt therefore follows from
which direction the charged-lepton sector distinguishes, and that is fixed
before the neutrino sector is consulted.

This is worth stating plainly because it is what makes the octant a real
prediction rather than a coin toss, and therefore what makes
\S\ref{sec:theta23-split} a real exposure. A framework whose atmospheric
sign were free could accommodate either octant and would learn nothing
from the measurement. This one cannot: if the octant resolves lower, it is
not the value of $J$ that fails but the ordering of $\Pi_A$ against
$\Pi_S$, and with it Lemma~L5.

\begin{euclidremark}[Physics intuition for the atmospheric angle]
Maximal atmospheric mixing means the muon and tau neutrinos are shared
equally between two mass states --- a perfect balance, and the kind of
exact statement a symmetry can make. The framework says the balance is
\emph{almost} exact and names what breaks it: the same odd integer
$J = 2N+1 = 7$ that counts the compact-angle structure, tilting the scale
by one part in seven. The prediction is not "near maximal", which any
number of models produce; it is maximal displaced by $1/7$, in a direction
fixed by the charged-lepton hierarchy.
\end{euclidremark}

\subsection{Derivation of the first-order leptonic Dirac phase}

The CKM derivation already taught us that CP phases in \GG{} should be read
as residual compact-angle holonomies after unphysical family rephasings are
removed. The same rule applies on the lepton side. The minimal neutrino
phase is
\begin{equation}
\phi_\nu=\arcsin\!\left(\frac{1}{2N+1}\right)=\arcsin\!\left(\frac17\right).
\label{eq:phi-nu-main}
\end{equation}
Because the PMNS phase is measured relative to a negative real reference
channel in the conventional parameterization, the physical Dirac phase is
shifted by \(\pi\):
\begin{equation}
\delta_{\rm PMNS}^{(1)}
=
\pi+2\phi_\nu
=
\pi+2\arcsin\!\left(\frac{1}{2N+1}\right).
\label{eq:delta-pmns-first-main}
\end{equation}
With \(N=3\),
\begin{equation}
\delta_{\rm PMNS}^{(1)}
=
\pi+2\arcsin\!\left(\frac17\right)
=
3.428287791~\mathrm{rad}
=
196.426421^\circ.
\label{eq:delta-pmns-first-num-main}
\end{equation}
\paragraph{The reference-channel argument, written out.} Two features of
\eqref{eq:delta-pmns-first-main} need justifying: why $\pi$ appears at all,
and why the residual is $2\phi_\nu$ rather than $\phi_\nu$.

The $\pi$ is a statement about the standard parameterization, not about
the geometry. In the conventional form the $e3$ element is written
$s_{13}e^{-i\delta}$, so the measured phase is defined against the channel
that carries the $1$--$3$ leak, and the sign convention makes that
reference channel enter with a \emph{negative} real coefficient. The
geometric phase in \eqref{eq:Xi-nu-mix-main} is carried by the same leak
$T_{eA}$, but referred to a positive real axis. Converting from one
convention to the other is a rotation of the reference by half a turn, and
that is the $\pi$. It is bookkeeping, and it is stated as such --- but it
is bookkeeping that must be done, because a prediction quoted in the wrong
convention is a wrong prediction.

The factor of two is physical. The leak appears in
\eqref{eq:Xi-nu-mix-main} as the Hermitian pair
$e^{-i\phi_\nu}T_{eA} + e^{+i\phi_\nu}T_{Ae}$: the operator carries
$e^{-i\phi_\nu}$ going one way and $e^{+i\phi_\nu}$ coming back. A
rephasing-invariant quantity must traverse the leak and return, so it
collects the phase twice, and the physical invariant is $2\phi_\nu$ rather
than $\phi_\nu$. This is the same rule the quark sector obeys, and it is
the reason the leptonic Jarlskog invariant below carries
$\sin\delta_{\rm PMNS}$ with $\delta$ already containing the doubling.

\paragraph{The number, and what it is tied to.} With
$\phi_\nu = \arcsin(1/7) = 8.213^\circ$, the residual is $16.43^\circ$ and
the prediction is $180^\circ + 16.43^\circ = 196.43^\circ$. The phase is
order one, unlike the quark sector, because the lepton projector geometry
is intrinsically more misaligned; but it is not an arbitrary number,
because it is tied to exactly the same odd integer $J = 7$ that fixes the
atmospheric tilt. \emph{The atmospheric angle and the Dirac phase are the
same prediction read twice}: if $J$ were not seven, both would move
together.

\begin{euclidremark}[Physics intuition for the leptonic phase]
A \CP{}-violating phase is an area, not a number: it is what survives when
every convention-dependent rephasing has been removed, and it can be
computed by walking a closed loop in family space and asking how much the
phase has turned on return. Here the loop is short --- out along the
electron-to-antisymmetric leak and back --- and the turn is $2\phi_\nu$
because the leak is traversed twice. That the loop is short is why the
answer is a simple inverse sine of a ledger integer rather than a
complicated function of several.
\end{euclidremark}

\subsection{Derivation of the first-order Majorana phases}

The Majorana phases answer a different question from the Dirac phase. They
measure the relative intrinsic parities of the neutrino eigenstates once
all allowed charged-lepton rephasings have been removed. The minimal
rank-two Majorana block derived earlier already picks one neutrino state of
positive parity and one of opposite parity. Therefore the constitutionally
minimal first-order choice is
\begin{equation}
(\alpha_{21},\alpha_{31})^{(1)}=(0,\pi).
\label{eq:majorana-first-main}
\end{equation}
\paragraph{Why rank two forces a parity \emph{pair}.} The argument is
short and it is the reason these two entries are Derived rather than
Asserted. A Majorana mass term is a symmetric bilinear in the field, so its
eigenvalues are real up to a phase that can be absorbed into the field ---
but only up to a \emph{sign}, because the absorbing rotation is squared by
the bilinear (this is the half-angle statement of
\S\ref{sec:lg-halfangle}). What cannot be rotated away is therefore not a
continuous phase but a discrete intrinsic parity, one bit per eigenstate.

Now impose the rank-two structure derived in \S\ref{sec:neutrino-masses}.
A rank-two block on a three-dimensional family space has exactly two
nonzero eigenvalues at leading order, and the leading operator
$\Xi_\nu^{(1)} = P_2 + 2N P_3$ of \eqref{eq:Xinu1-main} assigns them
opposite signs of the relevant invariant --- one from the plain projector
and one from the $2N$-enhanced one. Two states, opposite parities, no
third value available. Hence
\begin{equation}
(\alpha_{21},\alpha_{31}) = (0,\pi):
\qquad
0 \ \text{means same parity},
\quad
\pi \ \text{means opposite},
\label{eq:majorana-parity-reading-main}
\end{equation}
and there is no continuum to choose from.

This is stronger than a generic prediction in two distinct senses.
Oscillation experiments are insensitive to these phases, so no fit could
have determined them; and in this framework they are not extra continuous
parameters at all, but parity data of the Majorana projector. It is also
weaker than it might appear in one sense, and the next subsection says how:
the particular parity that is derived is close to the one that makes
$m_{\beta\beta}$ hardest to observe.

\subsection{First-order PMNS matrix, Jarlskog invariant, and neutrinoless-double-beta observable}

Once the four standard mixing parameters are known, the full PMNS matrix in
the conventional ordering is fixed. Using
\begin{equation}
\begin{gathered}
\sin^2\theta_{12}^{(1)}=0.3086419753,
\qquad
\sin^2\theta_{13}^{(1)}=0.02142857143,\\
\sin^2\theta_{23}^{(1)}=0.5714285714,
\qquad
\delta_{\rm PMNS}^{(1)}=196.426421^\circ,
\end{gathered}
\label{eq:first-pmns-inputs-main}
\end{equation}
we obtain
\begin{equation}
|U_{\rm PMNS}^{(1)}|
=
\begin{pmatrix}
0.822522 & 0.549571 & 0.146385\\
0.276669 & 0.603548 & 0.747786\\
0.496901 & 0.577669 & 0.647602
\end{pmatrix}.
\label{eq:Upmns-first-main}
\end{equation}
The corresponding leptonic Jarlskog invariant is
\begin{equation}
J_{\rm PMNS}^{(1)}
=
c_{12}c_{23}c_{13}^2s_{12}s_{23}s_{13}\sin\delta_{\rm PMNS}
=
-9.26009\times10^{-3}.
\label{eq:Jpmns-first-main}
\end{equation}
\paragraph{The two unitarity triangles, at true relative area.} The
Jarlskog invariant is twice the area of the unitarity triangle, so the two
sectors can be compared directly. This paper derives both:
\begin{equation}
\frac{\bigl|J_{\rm PMNS}\bigr|}{\bigl|J_{\rm CKM}\bigr|}
\;=\;
\frac{9.187\times10^{-3}}{2.984\times10^{-5}}
\;=\; 308
\qquad \text{(QD order; $338$ at first order).}
\label{eq:triangle-ratio-main}
\end{equation}
\emph{The leptonic unitarity triangle has about three hundred times the
area of the quark one.} Both numbers are outputs of the same ledger, and
the ratio is therefore a prediction rather than an observation --- the same
constitutional rules, read on two different projector geometries.

That is not a failure of universality; it is the sharpest single
expression of the framework's answer to the oldest question about flavor.
The quark side compares two hierarchical Dirac sectors against one another
and gets small angles and a thin triangle. The lepton side compares a
dyadic charged-lepton sector against a nearly democratic Majorana sector
and gets large angles and a fat one. \emph{Same constitution, different
projector geometry} --- and the factor of three hundred is how much
different.

The contrast also carries the cosmological weight of
Part~\ref{part:lepto}. It is often said that the quark sector's \CP{}
violation is too small to make the observed matter asymmetry by roughly ten
orders of magnitude, and that the lepton sector is therefore where one must
look. Equation~\eqref{eq:triangle-ratio-main} quantifies the first half of
that statement inside this framework, from derived quantities on both
sides. What it does not do is close the gap by itself: three hundred is not
ten orders of magnitude, and the reason the lepton sector can pay when the
quark sector cannot is not the size of $J_{\rm PMNS}$ at all but the
existence of the $\Delta L = 2$ door and the out-of-equilibrium lock
crossing. \S\ref{sec:lg-cp} makes that explicit, and indeed finds that the
\emph{unflavored} invariant --- the one $J_{\rm PMNS}$ belongs to ---
drops out of the matter asymmetry entirely.

The Majorana phases make it meaningful to quote the leading
neutrinoless-double-beta effective mass,
\begin{equation}
 m_{\beta\beta}
 =
 \left|
 m_1 c_{12}^2 c_{13}^2
 +m_2 s_{12}^2 c_{13}^2 e^{i\alpha_{21}}
 +m_3 s_{13}^2 e^{i(\alpha_{31}-2\delta_{\rm PMNS})}
 \right|.
 \label{eq:mbb-def-main}
\end{equation}
Rather than quote the result, it is worth working, because the way the
three terms combine is the content.

At first order $m_1 = 0$, so the first term vanishes identically and the
answer is a two-term interference. With
$s_{12}^{2} = 25/81$, $s_{13}^{2} = 3/140$ and hence
$c_{13}^{2} = 137/140$, and with $\alpha_{21} = 0$ so that the second term
is real and positive:
\begin{equation}
m_2 s_{12}^{2}c_{13}^{2}
= 8.36114 \times \tfrac{25}{81} \times \tfrac{137}{140}
= +2.52530~\mathrm{meV}.
\label{eq:mbb-term2-main}
\end{equation}
The third term carries the phase
$\alpha_{31} - 2\delta_{\rm PMNS} = \pi - 2(196.426^\circ)$, and
$m_3 s_{13}^{2} = 50.1668 \times 3/140 = 1.07500$~meV, so
\begin{equation}
m_3 s_{13}^{2}e^{i(\alpha_{31}-2\delta_{\rm PMNS})}
= -0.90307 + 0.58317\,i~\mathrm{meV}.
\label{eq:mbb-term3-main}
\end{equation}
The real parts \emph{oppose}: $+2.52530$ against $-0.90307$. Adding and
taking the modulus,
\begin{equation}
m_{\beta\beta}^{(1)}
= \bigl|\,2.52530 - 0.90307 + 0.58317\,i\,\bigr|
= 1.72386~\mathrm{meV}.
\label{eq:mbb-first-main}
\end{equation}
At QD order the first term is no longer exactly zero
($m_1^{\rm QD} = 0.0653$~meV contributes $+0.04414$~meV) and the other two
shift slightly, giving $m_{\beta\beta}^{\rm QD} = 1.79925$~meV.

\paragraph{A finding that must be stated: the derived parity is close to
the worst case.} The cancellation in \eqref{eq:mbb-first-main} is not
incidental --- it is most of the answer, and it is caused by the derived
parity. Two comparisons make the size of the effect concrete, on the same
derived spectrum and the same derived angles:

\begin{itemize}\itemsep2pt
\item With the \emph{other} admissible parity, $\alpha_{31} = 0$, the
third term's real part changes sign and adds instead of subtracting:
$m_{\beta\beta}$ would be $3.62$~meV, twice as large.
\item Scanning both Majorana phases freely over the full circle --- which
is what an experiment analyzing a generic normal-ordering spectrum must
allow --- gives the envelope
$m_{\beta\beta} \in [1.44,\ 3.74]$~meV.
\end{itemize}

The derived value $1.80$~meV therefore lands in the \emph{lowest quarter}
of the accessible range, near the maximally destructive end. That is an
uncomfortable result and the paper states it rather than resting on the
observation that $1.8$~meV is small: \textbf{this framework's Majorana
prediction is harder to confirm than a generic one}, and the difficulty is
a direct consequence of the parity the framework derives rather than of any
choice made to avoid a constraint.

Two things follow. The prediction remains one-sided in the sense of
F-SM-3b --- a confirmed $0\nu\beta\beta$ line anywhere in the tonne-scale
window of $9$--$21$~meV kills it, while a null result at that sensitivity
proves nothing --- and the margin by which the prediction escapes the
approved program is larger than the spectrum alone would require. The
cost is paid for the parity being \emph{derived}: a framework free to
choose $\alpha_{31}$ would place its prediction at the top of the envelope
and claim a better chance of confirmation, and would be claiming it about a
number it had fitted.

\subsection{QD completion of the PMNS/Majorana block}

The first-order formulas are the constitutional skeleton. The QD layer must
respect the same logic as everywhere else in the paper: finite corrections,
no new fitted constants, and suppression by already-admitted dyadic or
locked factors. The minimal QD completion is
\begin{equation}
\sin^2\theta_{12}^{\rm QD}
=
\left(1-\frac{1}{K2^{2N+1}}\right)
\sin^2\theta_{12}^{(1)},
\label{eq:pmns-s12-qd-main}
\end{equation}
\begin{equation}
\sin^2\theta_{13}^{\rm QD}
=
\left(1+\frac{1}{2K}+\frac{1}{2^{2N+1}}\right)
\sin^2\theta_{13}^{(1)},
\label{eq:pmns-s13-qd-main}
\end{equation}
\begin{equation}
\sin^2\theta_{23}^{\rm QD}
=
\frac12\left(1+\frac{1}{2N+1}-\frac{1}{K}\right),
\label{eq:pmns-s23-qd-main}
\end{equation}
\begin{equation}
\delta_{\rm PMNS}^{\rm QD}
=
\delta_{\rm PMNS}^{(1)}-\frac{\pi}{2K(2N+1)},
\label{eq:delta-pmns-qd-main}
\end{equation}
with the Majorana phases protected,
\begin{equation}
(\alpha_{21},\alpha_{31})^{\rm QD}=(0,\pi).
\label{eq:majorana-qd-main}
\end{equation}
Each correction has a direct physical meaning.

The solar channel is already dominated by the democratic projector, so its
QD correction is tiny and negative: it merely trims the idealized
first-order value by the smallest mixed dyadic/lock factor
\(1/(K2^{2N+1})=1/4480\). The reactor channel receives the largest relative
QD enhancement because it comes from a direct off-diagonal leak and is
therefore most sensitive to the first residual local dressing. The
atmospheric channel shifts from \(4/7\) to \(39/70\), showing beautifully
how a near-maximal angle can be nudged by the lock denominator \(K=35\)
without losing its first-order origin. The Dirac phase slides only very
slightly because a holonomy phase is already fixed at first order and later
corrections can only perturb it weakly.

Substituting the OGN values gives
\begin{equation}
\sin^2\theta_{12}^{\rm QD}=0.3085730820,
\qquad
\sin^2\theta_{13}^{\rm QD}=0.0219021046,
\qquad
\sin^2\theta_{23}^{\rm QD}=\frac{39}{70}=0.5571428571,
\label{eq:pmns-qd-angles-main}
\end{equation}
\begin{equation}
\delta_{\rm PMNS}^{\rm QD}=3.421876378~\mathrm{rad}=196.059074^\circ,
\qquad
(\alpha_{21},\alpha_{31})^{\rm QD}=(0,\pi).
\label{eq:pmns-qd-phase-main}
\end{equation}
The sharpened PMNS magnitude matrix is then
\begin{equation}
|U_{\rm PMNS}^{\rm QD}|
=
\begin{pmatrix}
0.822364 & 0.549377 & 0.147994\\
0.282542 & 0.612560 & 0.738201\\
0.493849 & 0.568292 & 0.658147
\end{pmatrix},
\label{eq:Upmns-qd-main}
\end{equation}
with
\begin{equation}
J_{\rm PMNS}^{\rm QD}=-9.18733\times10^{-3},
\qquad
m_{\beta\beta}^{\rm QD}=1.79925~\mathrm{meV}.
\label{eq:J-mbb-qd-main}
\end{equation}
The important point is that the QD layer sharpens the block without changing
its qualitative character. Large lepton mixing, a small reactor channel,
an order-one CP phase, and parity-like Majorana phases are already fixed at
first order.

\section{Precision comparison for the PMNS/Majorana block}
\label{sec:pmns-comparison}

\subsection{Benchmark convention}
\label{sec:pmns-benchmark}

Unlike the quark sector, the leptonic atmospheric angle and Dirac phase
remain analysis dependent, and at the time of writing that dependence is
not a technicality: the two leading determinations disagree with each
other about which octant the atmospheric angle lies in. The pull tables
below therefore quote \emph{named} benchmarks rather than a single
consensus point, and where the determinations disagree both are printed
with the pull against each. Three sources are used.

\paragraph{The global fit.} \textsc{NuFIT}~6.1, based on data available
in November 2025, supersedes the 6.0 release \citep{NuFIT61, NuFIT60}.
For normal ordering,
\begin{equation}
\sin^2\theta_{12}^{\obs}=0.308^{+0.0067}_{-0.0066},
\qquad
\sin^2\theta_{13}^{\obs}=0.02248^{+0.00055}_{-0.00059},
\qquad
\sin^2\theta_{23}^{\obs}=0.470^{+0.017}_{-0.014},
\label{eq:pmns-benchmark-angles-main}
\end{equation}
together with \(\Delta m_{21}^2 = (7.53^{+0.094}_{-0.10})\times10^{-5}\)~eV\(^2\)
and \(\Delta m_{31}^2 = (2.511^{+0.021}_{-0.020})\times10^{-3}\)~eV\(^2\).

\paragraph{The direct reactor measurement.} JUNO published first results
from 59.1 days of running in 2026 \citep{JUNO2026}, and they are already
the world's most precise determination of the solar pair,
\begin{equation}
\sin^2\theta_{12}^{\obs}=0.3092\pm0.0087,
\qquad
\Delta m_{21}^{2\,\obs}=(7.50\pm0.12)\times10^{-5}~\mathrm{eV}^2 .
\label{eq:pmns-benchmark-juno-main}
\end{equation}

\paragraph{The accelerator joint fit.} T2K and NOvA published a joint
analysis in 2025 \citep{T2KNOvA2025}. For normal ordering it gives the
Dirac phase and an atmospheric angle in the \emph{upper} octant,
\begin{equation}
\delta_{\rm PMNS}^{\obs}=203.4^{\circ\,+63.0}_{\phantom{\circ}-37.8},
\qquad
\sin^2\theta_{23}^{\obs}=0.56^{+0.03}_{-0.05}.
\label{eq:pmns-benchmark-delta-main}
\end{equation}
The Dirac phase is taken from this source rather than from the global fit
because it is the direct measurement, and because the two long-baseline
experiments together carry essentially all of the sensitivity to it.

\paragraph{Convention for asymmetric errors.} All three sources quote
asymmetric uncertainties, which the \(\Pull\) definition of
\S\ref{sec:theory-error-model} did not previously have to accommodate.
The convention adopted here, and used in every table below, is that the
observational uncertainty entering the pull is the one on the side
\emph{facing} the prediction: the upper uncertainty when the prediction
lies above the central value, the lower when it lies below. This is the
conservative choice in the sense that it never shrinks a disagreement.

The Majorana phases are presently unmeasured, so for those rows the tables
record only the theoretical prediction and the internal theory
uncertainty.

\subsection{Theory-error model for the PMNS block}
\label{sec:theory-error-model}

To keep the error assignment consistent with the rest of the manuscript, the
first-order theory error of each PMNS observable is defined as the size of
the omitted QD correction,
\begin{equation}
\sigma_{\theory}^{(1)}(X)=|X^{\rm QD}-X^{(1)}|,
\label{eq:pmns-sigma1-main}
\end{equation}
while the residual QD theory error is taken to be one eighth of that shift,
\begin{equation}
\sigma_{\theory}^{(2)}(X)=\frac18|X^{\rm QD}-X^{(1)}|.
\label{eq:pmns-sigma2-main}
\end{equation}
For the Majorana phases, which do not move between first order and QD, we
adopt a conservative protected-phase uncertainty based on the smallest
constitutional phase quantum,
\begin{equation}
\sigma_{\theory}^{(1)}(\alpha_{21})=\sigma_{\theory}^{(1)}(\alpha_{31})
=
\frac{\pi}{K2^{2N+1}}
=
0.0401786^\circ,
\label{eq:majorana-sigma1-main}
\end{equation}
\begin{equation}
\sigma_{\theory}^{(2)}(\alpha_{21})=\sigma_{\theory}^{(2)}(\alpha_{31})
=
\frac{\pi}{K2^{2N+4}}
=
0.00502232^\circ.
\label{eq:majorana-sigma2-main}
\end{equation}
These uncertainties are deliberately conservative. Their role is not to
inflate agreement but to acknowledge that the present manuscript has given a
precise constitutional codebook rather than a full multi-loop oscillation
fit.

\subsection{First-order comparison table}

\begin{table}[H]
\centering
\caption[First-order PMNS/Majorana predictions compared with the named oscillation benchmarks of \S\ref{sec:pmns-benchmark}.]{First-order PMNS/Majorana predictions compared with the named
oscillation benchmarks of \S\ref{sec:pmns-benchmark}. Where the global fit
and the accelerator joint fit disagree --- the atmospheric octant --- both
are printed. The Majorana phases are not yet experimentally measured.}
\label{tab:pmns-first-comparison-main}
\small\setlength{\tabcolsep}{3pt}%
\begin{tabular}{p{0.22\textwidth}r r r r r r}
\toprule
\textbf{Observable} & \textbf{Pred} & \(\sigma_{\theory}\) & \textbf{Obs} & \(\sigma_{\exp}\) & \textbf{Dev} & \textbf{Pull}\\
\midrule
\(\sin^2\theta_{12}\) {\footnotesize (NuFIT 6.1)} & 0.308641975 & 0.0000688933 & 0.308000000 & 0.006700000 & +0.0020843 & +0.10\\
\(\sin^2\theta_{12}\) {\footnotesize (JUNO)} & 0.308641975 & 0.0000688933 & 0.309200000 & 0.008700000 & -0.0018047 & -0.06\\
\(\sin^2\theta_{13}\) {\footnotesize (NuFIT 6.1)} & 0.021428571 & 0.000473533 & 0.022480000 & 0.000590000 & -0.0467718 & -1.39\\
\(\sin^2\theta_{23}\) {\footnotesize (NuFIT 6.1)} & 0.571428571 & 0.014285714 & 0.470000000 & 0.017000000 & +0.2158050 & +4.57\\
\(\sin^2\theta_{23}\) {\footnotesize (T2K$+$NOvA)} & 0.571428571 & 0.014285714 & 0.560000000 & 0.030000000 & +0.0204082 & +0.34\\
\(\delta_{\rm PMNS}\,[^\circ]\) {\footnotesize (T2K$+$NOvA)} & 196.426421 & 0.367347 & 203.400000 & 37.800000 & -0.0342850 & -0.18\\
\(\alpha_{21}\,[^\circ]\) & 0.000000 & 0.040179 & \multicolumn{4}{c}{not yet measured}\\
\(\alpha_{31}\,[^\circ]\) & 180.000000 & 0.040179 & \multicolumn{4}{c}{not yet measured}\\
\bottomrule
\end{tabular}
\end{table}

The first-order table already captures the dominant empirical structure:
a solar angle of order \(0.3\), a small reactor angle near
\(2\times10^{-2}\), an atmospheric angle in the upper octant, and a large
Dirac phase in the neighborhood favored by the direct measurement. The
solar and Dirac channels land well inside one sigma of every source. The
atmospheric row is where the sources part company, and it is treated on
its own below.

\subsection{QD comparison table}

\begin{table}[H]
\centering
\caption[QD-completed PMNS/Majorana predictions compared with the same named benchmark values.]{QD-completed PMNS/Majorana predictions compared with the same
named benchmark values. The lepton mixing angles and the Majorana phases at QD order. Two benchmarks are printed for the solar angle --- the global fit and JUNO's first-data determination --- because they are independent and this paper does not choose between them. The Majorana rows are included for completeness and are not testable by oscillation: no oscillation experiment can measure $\alpha_{31}$, which the appendix proves rather than asserts.}
\label{tab:pmns-qd-comparison-main}
\small\setlength{\tabcolsep}{3pt}%
\begin{tabular}{p{0.22\textwidth}r r r r r r}
\toprule
\textbf{Observable} & \textbf{Pred} & \(\sigma_{\theory}\) & \textbf{Obs} & \(\sigma_{\exp}\) & \textbf{Dev} & \textbf{Pull}\\
\midrule
\(\sin^2\theta_{12}\) {\footnotesize (NuFIT 6.1)} & 0.308573082 & 0.00000861166 & 0.308000000 & 0.006700000 & +0.0018607 & +0.09\\
\(\sin^2\theta_{12}\) {\footnotesize (JUNO)} & 0.308573082 & 0.00000861166 & 0.309200000 & 0.008700000 & -0.0020276 & -0.07\\
\(\sin^2\theta_{13}\) {\footnotesize (NuFIT 6.1)} & 0.021902105 & 0.0000591916 & 0.022480000 & 0.000590000 & -0.0257071 & -0.97\\
\(\sin^2\theta_{23}\) {\footnotesize (NuFIT 6.1)} & 0.557142857 & 0.001785714 & 0.470000000 & 0.017000000 & +0.1854100 & +5.10\\
\(\sin^2\theta_{23}\) {\footnotesize (T2K$+$NOvA)} & 0.557142857 & 0.001785714 & 0.560000000 & 0.050000000 & -0.0051020 & -0.06\\
\(\delta_{\rm PMNS}\,[^\circ]\) {\footnotesize (T2K$+$NOvA)} & 196.059074 & 0.0459184 & 203.400000 & 37.800000 & -0.0360911 & -0.19\\
\(\alpha_{21}\,[^\circ]\) & 0.000000 & 0.005022 & \multicolumn{4}{c}{not yet measured}\\
\(\alpha_{31}\,[^\circ]\) & 180.000000 & 0.005022 & \multicolumn{4}{c}{not yet measured}\\
\bottomrule
\end{tabular}
\end{table}

The QD layer moves the reactor channel toward the data --- the pull
against the global fit falls from \(-1.39\) to \(-0.97\) --- while leaving
the solar angle and the Dirac phase almost unchanged. This is what the
structure of the correction terms suggested in advance: the solar channel
was already a stable democratic overlap, whereas the reactor channel was
more sensitive to finite local dressing.

The atmospheric channel is the exception, and honesty requires stating it
plainly. Against the accelerator joint fit the QD layer is an improvement,
carrying the pull from \(+0.34\) to \(-0.06\). Against the global fit it is
not: the correction moves \(4/7 = 0.5714\) to \(39/70 = 0.5571\), which is
\emph{toward} maximal mixing and therefore away from a central value that
sits in the lower octant, and the pull grows from \(+4.57\) to \(+5.10\).
The QD dressing is a small downward shift of a prediction whose disagreement
with one of the two sources is not small; it can hardly repair it, and it
does not.

\subsection{The atmospheric octant: a disagreement between the sources}
\label{sec:theta23-split}

One row of the tables above deserves its own discussion, because the
disagreement it records is between two \emph{measurements}, not between
theory and measurement.

The global fit \textsc{NuFIT}~6.1 places the atmospheric angle in the
\emph{lower} octant, \(\sin^2\theta_{23} = 0.470^{+0.017}_{-0.014}\), with
a three-sigma range \([0.435, 0.584]\) that still admits the upper octant.
The T2K and NOvA joint analysis places it in the \emph{upper} octant,
\(\sin^2\theta_{23} = 0.56^{+0.03}_{-0.05}\), and reports a Bayes factor of
\(3.5\) favoring the upper octant once the reactor constraint on
\(\theta_{13}\) is imposed. The two determinations are not independent ---
the global fit contains the accelerator data --- but they weight the
atmospheric and reactor samples differently, and on this one parameter they
come out on opposite sides of maximal mixing.

The present framework derives
\(\sin^2\theta_{23} = \tfrac12(1+1/J) = 4/7\), QD-completed to
\(\tfrac12(1+1/J-1/K) = 39/70 = 0.5571\), and the upper octant is not an
accident of the arithmetic: it is forced by the sign of the odd-triplicity
splitting \(1/J\) about the \(\mu\)--\(\tau\) symmetric value \(1/2\). The
prediction therefore sits comfortably inside the accelerator result and
about five sigma from the global one. It cannot be reconciled with a
lower-octant central value by any dressing available in the codebook.

Three things follow, and the paper states all three rather than choosing
the source that suits it.

\emph{First}, this is now the sharpest near-term exposure in the lepton
block. A converged global-plus-accelerator determination placing
\(\sin^2\theta_{23}\) in the lower octant at high significance would
falsify the atmospheric assignment outright. That criterion is registered
in \S\ref{sec:predictions-falsification} and is expected to be settled by the
long-baseline program of the 2030s.

\emph{Second}, the exposure is structural rather than local. The
atmospheric value is Row~21 of the codebook, carried at
\textbf{Derived (cond.)} on the rank-2 atmospheric-deviation lemma L5 of
\S\ref{sec:appendix-promotion-roadmap}. L5's own statement selects
\(4/7\) over the equally cheap candidate \(5/9 = 0.5556\) by the
structural-naming criterion --- but \emph{both} candidates are in the upper
octant, so the tie-break is not what is at stake. What is at stake is the
sign of the deviation from \(\pi/4\). If the octant resolves lower, L5 does
not merely lose a tie-break; it falls, and Row~21 falls with it.

\emph{Third}, nothing else in the block moves. The solar angle, the reactor
angle, the Dirac phase, the Majorana parities and the mass splittings are
fixed by different structures and are unaffected by the octant, as the
tables above show: the solar prediction sits \(0.07\sigma\) from the JUNO
measurement and \(0.09\sigma\) from the global fit, and the Dirac phase
sits \(0.19\sigma\) from the joint-fit best value. A framework whose
lepton block agreed everywhere would be less informative than one that
agrees in five places and is exposed in the sixth.

\subsection{Comprehensive summary tables for the first-order and QD lepton block}

For a concise side-by-side overview, we collect the
entire PMNS/Majorana result in the two summary tables below. The first
summarizes the six directly parameter-counted observables. The second adds
two secondary but phenomenologically important derived quantities: the
leptonic Jarlskog invariant and the neutrinoless-double-beta effective
mass.

\begin{table}[H]
\centering
\caption[Comprehensive summary of the PMNS/Majorana block at first order and at QD order.]{Comprehensive summary of the PMNS/Majorana block at first order and
at QD order. First order, QD-completed, benchmark with its asymmetric interval, and the QD deviation, one line per parameter. As the note in the table says, the atmospheric row is scored against the global fit rather than against a single experiment, because the octant is not yet resolved and the global fit is the honest summary of what is currently known.}
\label{tab:pmns-summary-main}
\small
\begin{tabular}{p{0.19\textwidth}r r p{0.27\textwidth} r}
\toprule
\textbf{Parameter} & \textbf{First order} & \textbf{QD} & \textbf{Observed} & \textbf{Dev}\\
\midrule
\multicolumn{5}{l}{\footnotesize Dev is the QD relative deviation; the
atmospheric row is scored against the global fit (see
\S\ref{sec:theta23-split}).}\\
\(\sin^2\theta_{12}\) & 0.308641975 & 0.308573082 & \(0.308^{+0.0067}_{-0.0066}\) & +1.86066e-03\\
\(\sin^2\theta_{13}\) & 0.0214285714 & 0.0219021046 & \(0.02248^{+0.00055}_{-0.00059}\) & -2.57071e-02\\
\(\sin^2\theta_{23}\) & 0.571428571 & 0.557142857 & \(0.470^{+0.017}_{-0.014}\) (global); \(0.56^{+0.03}_{-0.05}\) (accel.) & +1.85410e-01\\
\(\delta_{\rm PMNS}\,[^\circ]\) & 196.426421 & 196.059074 & \(203.4^{+63.0}_{-37.8}\) & -3.60911e-02\\
\(\alpha_{21}\,[^\circ]\) & 0.000000 & 0.000000 & not yet measured & ---\\
\(\alpha_{31}\,[^\circ]\) & 180.000000 & 180.000000 & not yet measured & ---\\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[H]
\centering
\caption[Secondary derived lepton-sector observables associated with the first-order and QD PMNS/Majorana solutions.]{Secondary derived lepton-sector observables associated with the
first-order and QD PMNS/Majorana solutions. The mixing-matrix elements and the Jarlskog invariant are consequences of the angles and the phase rather than additional predictions, and are printed because they are what a reader of the oscillation literature will want to compare. The benchmark column carries the values implied by the same global fit used above, so a disagreement here restates a disagreement in the angles and is not new information.}
\label{tab:pmns-secondary-summary-main}
\small
\begin{tabular}{p{0.30\textwidth}r r p{0.22\textwidth}}
\toprule
\textbf{Observable} & \textbf{First order} & \textbf{QD} & \textbf{Representative benchmark/comment}\\
\midrule
\(|U_{e1}|\) & 0.822522 & 0.822364 & benchmark \(0.825038\)\\
\(|U_{e2}|\) & 0.549571 & 0.549377 & benchmark \(0.545263\)\\
\(|U_{e3}|\) & 0.146385 & 0.147994 & benchmark \(0.148324\)\\
\(|U_{\mu3}|\) & 0.747786 & 0.738201 & benchmark \(0.740054\)\\
\(|U_{\tau3}|\) & 0.647602 & 0.658147 & benchmark \(0.655988\)\\
\(J_{\rm PMNS}\) & $-9.26009\times10^{-3}$ & $-9.18733\times10^{-3}$ & benchmark \(-1.24076\times10^{-2}\)\\
\(m_{\beta\beta}\) [meV] & 1.72386 & 1.79925 & not yet measured; target for future \(0\nu\beta\beta\) searches\\
\bottomrule
\end{tabular}
\end{table}

The secondary table is useful for intuition. It shows explicitly that the
present construction does not merely reproduce the angle set abstractly. It
actually yields a realistic full matrix pattern: the electron row is close
to the canonical large-solar/small-reactor shape, while the third-column
entries land naturally in the atmospheric neighborhood. The derived
leptonic Jarlskog invariant is negative and of order \(10^{-2}\), which is
precisely what one expects when large mixing angles coexist with an order-one
CP phase.

\subsection{Significance and uniqueness of the PMNS/Majorana result}

The PMNS/Majorana block is significant for three distinct reasons.

\paragraph{First significance: one theory now explains why CKM is small and PMNS is large.}
A common criticism of unified flavor ideas is that quark and lepton mixing
look qualitatively different. In the present framework that difference is
not a problem to be patched over. It is a prediction. The quark side is
small-angle because two hierarchical Dirac sectors are compared against one
another. The lepton side is large-angle because a dyadic charged-lepton
sector is compared against a nearly democratic Majorana sector. Same
constitution, different projector geometry.

\paragraph{Second significance: the Majorana option becomes predictive rather than optional.}
The present derivation does not merely allow Majorana neutrinos. It uses
the constitutionally minimal Majorana channel to compute both the neutrino
mass scale and the two physical parity phases. That immediately produces a
meV-scale prediction for \(m_{\beta\beta}\). Even though current data do not
yet measure \(\alpha_{21}\), \(\alpha_{31}\), or \(m_{\beta\beta}\) directly,
the theory side is already specific enough to be falsifiable by future
neutrinoless-double-beta searches.

\paragraph{Third significance: the result is unusually short.}
Many flavor constructions require long chains of fitted textures,
flavon alignments, or continuous parameters. Here the visible final codes
are strikingly compact:
\[
\sin^2\theta_{12}=\frac{25}{81},
\qquad
\sin^2\theta_{13}=\frac{3}{140},
\qquad
\sin^2\theta_{23}=\frac47\to\frac{39}{70},
\]
\[
\delta_{\rm PMNS}=\pi+2\arcsin\!\left(\frac17\right)\to196.06^\circ,
\qquad
(\alpha_{21},\alpha_{31})=(0,\pi).
\]
That compactness is the uniqueness claim of the present section. The block
is not unique because no other algebra can be imagined in mathematics; it
is unique because once one insists on the written \GGConstitution{}, the
restricted input ledger, the dyadic charged-lepton hierarchy, the
Majorana-rank-two neutrino block, and the \OGNfive{} integers, the admissible
lepton projector arithmetic becomes very narrow.

The manuscript has already fixed the first 27 quantities by the end of the
PMNS/Majorana block. The next section then completes the last topological
parameter and closes the full 28-parameter census.




\section{Part~VII in plain English: masses as floors, mixings as gates}
Twelve masses and eight mixing parameters make up the largest block of
the twenty-eight, and this Part treats them as two different kinds of object.

A mass is a floor. Each family sits at a depth in the cascade, and the mass is
what the descent has left when it stops. That is why the mass ratios come out
as exponentials of $\pi$ rather than as arbitrary numbers: a depth is a
count, and the suppression per unit depth is fixed by the geometry. A mixing
parameter is not a floor at all. It is a gate --- the misalignment between the
family projectors of two sectors that have descended separately on the same
closed packet. The CKM and PMNS matrices are what that misalignment looks like
when it is written in the observable basis.

This is the Part where the paper's honesty boundary is most visible. The
depths are assigned, not derived: what is missing is a lemma that would fix
each depth as the dimension of a specific cohomology class, and until that
lemma exists the family structure is a working hypothesis with a good deal of
arithmetic behind it. The paper states this in the codebook grade and repeats
it here rather than letting the closeness of the numbers do the arguing.

So the fair summary is that ``flavor is unsolved'' and ``flavor is reduced to
one named calculation'' are both true statements about this Part. The second is
worth something only if the named calculation is genuinely a single one, and
the reader who wants to check that should look at how many separate
assumptions the twelve masses and eight mixings actually consume. That count
is in the audit, and it is the number that matters.

\GGPart[Leptogenesis at the lock]{The Majorana door and leptogenesis at the lock scale}{Why the excess of matter over antimatter is written by the same geometry that fixes the neutrino block --- the unique \texorpdfstring{$\Delta L = 2$}{Delta L = 2} portal, the heavy sector at \texorpdfstring{$\Mlock = 3.85$}{M\_lock = 3.85}~TeV, the flavored \texorpdfstring{\CP{}}{CP} asymmetry driven by the Majorana parity, the resonance question and its named hypothesis, the electron door, the Boltzmann integration, and the assembled baryon-to-photon ratio}
\label{part:lepto}

Part~\ref{part:X} closed the twelve fermion masses and the two mixing
matrices. In doing so it fixed, without adjustable freedom, five
quantities that between them decide whether the universe contains matter:
the scale $\Mlock$ at which the compact fiber locks, the unique
$\Delta L = 2$ portal through which lepton number can be violated at all,
the coefficient $\pi/\EGUT$ of the Majorana seam, the Majorana parity
$(\alpha_{21},\alpha_{31}) = (0,\pi)$, and the reactor angle
$\sin^2\theta_{13} = N/4K = 3/140$. Each was derived for a different
reason and none was derived with baryogenesis in view.

This Part asks what those five quantities imply when they are handed to
the early universe, and it does so in full rather than by reference. The
reason is not completeness for its own sake. It is that the answer is a
statement about \emph{this paper's own numbers}: the reactor angle A9
derives is what allows the asymmetry to survive; the Majorana parity A9
derives is what fixes its sign; the lock scale A9 derives is what places
the mechanism nine orders of magnitude below where the standard treatment
must put it. A reader who has followed the parameter derivation is
entitled to see that consequence worked out on the page, and not sent
elsewhere for it.

\paragraph{Custody, stated once and then honored.} The cosmological
transport calculation belongs to \ATen{}, and the hardest step in this
chain --- the finite-temperature identification of the quantity that
regulates the resonance --- is carried there and is \emph{open at
derivation level}. Nothing below quietly promotes it. Every step of the
chain carries its owner and its grade, the summary audit is
Table~\ref{tab:lg-audit}, and where this Part states a result whose home
is a companion it says so in the sentence that states it. What A9 owns
are the microscopic inputs, and it owns them because it derived them.

\paragraph{A note on the theorem identifiers.} The formal statements of
this Part are given as propositions and lemmas rather than as new
$\text{T-SM-}NN.M$ theorems. That scheme encodes a section number, and
inserting a Part in the middle of the manuscript moves every section
number after it; the identifiers are therefore not minted here.

\section{One survivor in a billion: Sakharov's conditions, located}
\label{sec:lg-sakharov}

\subsection{The size of the problem}

The observed baryon-to-photon ratio is
\begin{equation}
\etaB^{\obs} = (6.12 \pm 0.04)\times 10^{-10}
\label{eq:lg-etaB-obs}
\end{equation}
\citep{Planck2018}. The number is easy to write and hard to feel, so it
is worth converting once. In the early universe, before the electrons and
positrons annihilated, matter and antimatter were present in almost equal
numbers. Almost. For every billion antiquarks there were about a billion
and one quarks. The billion pairs annihilated into the photons that are
now the microwave background; the one left over is everything --- every
star, every planet, every reader of this sentence. The Standard Model,
taken alone, predicts that the leftover should be zero to a fantastic
precision, and the universe should be a bath of radiation with nothing in
it.

\subsection{The three conditions, and where this framework meets them}

\citet{Sakharov1967} gave the three conditions any mechanism must
satisfy. They are worth restating not as history but as a checklist,
because the interest here is \emph{which} of them the geometry supplies
and which it must borrow.

\paragraph{(i) Baryon-number violation.} Something must change $B$. In
the present framework baryon number is a \emph{gauged} quantity, residual
to a Stueckelberg-massive local $\U(1)_B$ (\S\ref{subsec:fsm4}),
and the selection rule is $\Delta B \in N\mathbb{Z}$ with $N = 3$: single
proton decay and neutron--antineutron oscillation are both forbidden,
while $\Delta B = 3$ is admissible and sphaleron-suppressed. That is a
sharper statement than the Standard Model's, and it cuts both ways --- it
is what makes $\tau_p = \infty$ a theorem of this framework, and it is
what forbids the framework from making baryons directly.

\paragraph{(ii) \CP{} violation.} Something must distinguish matter from
antimatter. The framework has \CP{} violation in abundance and has
derived it: the CKM phase $\delta_{\rm CKM} = 2\pi/5$, the leptonic Dirac
phase $\dPMNS = \pi + 2\arcsin(1/7)$, and the Majorana parity
$\alpha_{31} = \pi$. \S\ref{sec:lg-cp} establishes which of the three
actually pays, and the answer is not the one a reader would guess.

\paragraph{(iii) Departure from thermal equilibrium.} Something must
happen out of equilibrium, or \CPT{} guarantees that whatever is made is
unmade. The framework supplies this without being asked: the descent
crosses the lock at $\Mlock = 3.85$~TeV, and a heavy state decaying as
the bath cools through its own mass is the textbook out-of-equilibrium
condition.

\begin{euclidremark}[Why the quark sector cannot pay, in one paragraph]
The obvious candidate is the CKM phase, and it fails by a wide margin.
The Standard Model's \CP{} measure is the Jarlskog invariant multiplied
by the six quark-mass-squared differences, and made dimensionless it is
of order $10^{-20}$ at the electroweak scale --- some ten orders of
magnitude short of $10^{-10}$, and falling steeply with temperature
because the mass differences that carry it are Yukawa-suppressed. The
electroweak transition compounds the failure: in the Standard Model with
a $125$~GeV Higgs it is a smooth crossover, not a first-order transition,
so there are no bubble walls and condition (iii) is not met there either.
After all, the deficiency is not that the quark sector is a little too
small; it is that it is the wrong sector by twenty orders of magnitude,
which is why the search has always moved to the leptons.
\end{euclidremark}

So the framework must violate lepton number, not baryon number, and let
the electroweak sphalerons convert --- which is the mechanism
\citet{FukugitaYanagida1986} proposed, here with the heavy scale an output
rather than an input. The rest of this Part is the working
out of that sentence, and the first question is whether the geometry
permits a $\Delta L = 2$ interaction at all.

\section{The door: why the portal is Majorana, and why it is unique}
\label{sec:lg-door}

\subsection{Lepton number is winding number}

The constitutional statement was proved in Part~\ref{part:X} as
Theorem~T-SM-44.1 (\S\ref{subsec:constitutional-door}) and is recalled
here because the whole chain hangs from it: a Lorentz- and
gauge-invariant Majorana self-pairing exists \emph{if and only if} the
multiplet carries zero weight under the compact phase direction
$\Stheta$, and the trilinear $LHN$ is the unique fiber-neutral portal
touching the weight-zero sector linearly.

The physical content is worth saying in words, because it is the reason
this framework has no freedom here. In the real presentation a
weight-$n$ sector is a pair of fields rotated by an angle $n\alpha$ as
one goes around the compact fiber. The symmetric square of that pair
splits into a rotation-invariant piece --- the charge-conserving Dirac
pairing --- and a piece that rotates at \emph{twice} the rate. For any
$n \neq 0$, invariance kills the second piece outright. For $n = 0$ the
fiber does nothing at all, and the reality condition that survives is
precisely the Majorana condition.

\begin{euclidremark}[Physics intuition for the door]
Lepton number, in this framework, is not a bookkeeping label attached to
particles by hand. It is a winding number: how many times a field's phase
turns as one walks once around the compact sixth dimension. A quantity
that counts turns cannot change by a continuous process, which is why
every species that carries a nonzero winding conserves $L$ exactly, and
why the charged-lepton ledger is closed. The only place lepton number can
leak is where there is no winding to count --- the weight-zero sector.
There is exactly one such door, and it is Majorana by construction rather
than by assumption. That is the difference between this account and the
usual one: the usual one \emph{allows} Majorana neutrinos, and this one
\emph{permits nothing else}.
\end{euclidremark}

\paragraph{Relation to the historical seesaw.} The heavy-Majorana route
was written down three times independently in 1977--1980
(\citealp{Minkowski1977}; \citealp{GellMannRamondSlansky1979};
\citealp{Yanagida1979}; \citealp{MohapatraSenjanovic1980}), and the
effective operator that survives at low energy is
\citet{Weinberg1979}'s. The difference here is not the operator but its
standing: in the historical construction the Majorana route is one
admissible choice among several, and the heavy scale is free. Here the
route is the only legal one and the scale is an output.

\subsection{Three consequences that frame everything below}

\begin{enumerate}[leftmargin=2.1em]
\item \textbf{The $\Delta L = 2$ structure is unique, not merely
minimal.} It is common to describe the Weinberg operator as the minimal
choice among several. Here there is nothing else to choose: the
weight-zero self-pairing is the only legal $\Delta L = 2$ structure in
the matter sector.
\item \textbf{The ultraviolet completion has no freedom either.} Once the
portal is fixed, the heavy-sector realization is fixed with it. The
question is not which mechanism but at which scale, and
\S\ref{sec:lg-scale} answers that from quantities already derived.
\item \textbf{The prediction is one-sided and uncomfortable.} A Majorana
neutrino sector implies neutrinoless double beta decay, and the parity
$(0,\pi)$ this framework derives puts the effective mass at
$\mbb \simeq 1.8$~meV --- an order of magnitude below the reach of the
approved program. This is stated in F-SM-3 and repeated here because it
is the cost of the door being derived rather than chosen.
\end{enumerate}

\section{The heavy sector at the lock scale}
\label{sec:lg-scale}

\begin{figure}[htbp]
\centering
\resizebox{0.908\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize,
  tick/.style={line width=0.90pt},
  span/.style={|<->|, line width=0.70pt, draw=blue!55!black}]
\draw[line width=1.10pt,draw=black!80] (0,0) -- (13.6,0);
\node[anchor=west,font=\scriptsize\itshape] at (13.7,0) {$r=\ln(\EPl/\mu)$};
\draw[tick] (0.000,-0.18) -- (0.000,0.18);
\node[anchor=south,align=center] at (0.000,0.24) {$\EPl$};
\node[anchor=north,font=\scriptsize,text=black!65,align=center] at (0.000,-0.300) {$1.22\times10^{19}$~GeV};
\draw[tick] (2.125,-0.18) -- (2.125,0.18);
\node[anchor=south,align=center] at (2.125,0.24) {$\EGUT$};
\draw[line width=0.40pt,draw=black!45] (2.125,-0.24) -- (2.125,-0.700);
\node[anchor=north,font=\scriptsize,text=black!65,align=center] at (2.125,-0.860) {$2.28\times10^{16}$~GeV};
\draw[tick] (12.070,-0.18) -- (12.070,0.18);
\node[anchor=south,align=center] at (12.070,0.24) {$\Mlock$};
\node[anchor=north,font=\scriptsize,text=black!65,align=center] at (12.070,-0.300) {$3.85$~TeV};
\draw[tick] (13.000,-0.18) -- (13.000,0.18);
\node[anchor=south,align=center] at (13.000,0.24) {$v$};
\draw[line width=0.40pt,draw=black!45] (13.000,-0.24) -- (13.000,-0.700);
\node[anchor=north,font=\scriptsize,text=black!65,align=center] at (13.000,-0.860) {$246$~GeV};
\draw[span] (0.000,1.15) -- (2.125,1.15)
  node[midway,fill=white,inner sep=1.5pt] {$2\pi$ \emph{exactly}};
\draw[span] (2.125,1.15) -- (12.070,1.15)
  node[midway,fill=white,inner sep=1.5pt] {$29.41$};
\draw[span,draw=red!60!black] (0.000,1.95) -- (12.070,1.95)
  node[midway,fill=white,inner sep=1.5pt] {$K+\ln 2 = 35.693$};
\end{tikzpicture}}
\caption[The radial descent, and the one integer that fixes it.]{The radial descent, and the one integer that fixes it. The Planck scale, the grand-unification seam, the \OGNfive{} lock and the electroweak vev, placed on the coordinate $r=\ln(\EPl/\mu)$. Two spans are exact rather than fitted: $\ln(\EPl/\EGUT)=2\pi$, one traversal of the compact fiber, and $\ln(\EPl/\Mlock)=K+\ln2=35.693$, since $\Mlock=\tfrac12\EPl e^{-K}$. Everything Part~\ref{part:lepto} does downstream --- the splitting $\Mlock/K$, the seam Yukawa, the exponent $e^{-K/2}$ --- is read off the same $K=35$, which was fixed by an anomaly audit that never mentions neutrinos.}
\label{fig:lg-descent}
\end{figure}


\subsection{The scale is not chosen}

The lock scale is Part~\ref{part:III}'s, not this Part's. It is
\begin{equation}
\Mlock \;=\; \tfrac12\,M_{\mathrm{Pl}}\,e^{-K}
\;=\; \tfrac12\,M_{\mathrm{Pl}}\,e^{-35}
\;=\; 3.85~\mathrm{TeV},
\qquad
r \;=\; \ln\frac{\EGUT}{\Mlock} \;=\; 29.41,
\label{eq:lg-mlock}
\end{equation}
where $K = LJ = M^2 + N^2 + L^2 = 35$ is the ledger integer fixed by the
anomaly audit and the Green--Schwarz consistency of the six-dimensional
bulk (\AThree{}, \citealp{ArisakaA3BI6}). The same $K$ sets the Kaxion's
$1/35$ doublet spacing and the length of the radial descent.

It is worth being explicit about what that sentence does and does not
claim. \emph{No step in the derivation of \eqref{eq:lg-mlock} mentions
baryogenesis, neutrinos, or \CP{}.} The scale was fixed by an anomaly
count. That it should also be the scale at which a matter asymmetry can
be made is either a coincidence or a consequence, and the remainder of
this Part is the attempt to find out which.

\subsection{The splitting is the ledger's own level spacing}

The heavy states inherit the lock scale with splittings of relative order
$1/K$:
\begin{equation}
M_{N_i} \;=\; \Mlock\,(1 + \delta_i),
\qquad
\delta_i = \mathcal{O}(1/K),
\qquad
\Delta M \;\simeq\; \frac{\Mlock}{K} \;=\; \frac{3850}{35} \;=\; 110~\mathrm{GeV}.
\label{eq:lg-splitting}
\end{equation}
This is the single most consequential number in the Part after the lock
scale itself, and it is not fitted: $\Delta M$ is the ledger's own level
spacing, the same $K = 35$ divided into the same $\Mlock$.

\subsection{The launch condition}

Reheating after the inflationary phase deposits energy at
$\Trh \simeq 3\times10^{10}$~GeV (\ATen{}, \citealp{ArisakaA10Cosmos}) ---
comfortably above the heavy sector, so that it is populated thermally
without being asked to be, and comfortably below the GUT edge, so that
monopoles stay diluted. The bath then cools, and at
$t \sim 10^{-13}$~s it crosses the lock. The mechanism is therefore
\emph{selected by the launch} rather than chosen for it.

\begin{euclidremark}[Physics intuition for the scale]
Every treatment of thermal leptogenesis the authors are aware of
\emph{chooses} its scale: high, for the standard hierarchical mechanism,
or low, when the author wants collider signatures. Here the scale is an
output. The descent has one lock, the anomaly ledger fixes its depth at
$e^{-35}$, and the heavy sector sits there because that is where the
compact fiber closes. The reader is entitled to be suspicious of a
framework that finds what it needs where it needs it --- which is exactly
why \S\ref{sec:lg-resonance} states the one thing this chain assumes, in
sharp form, with a number attached and a way to be wrong.
\end{euclidremark}

\section{The seesaw at the lock, and the scale that cancels}
\label{sec:lg-seesaw}

The previous section placed a heavy sector at $\Mlock$. This one writes
down what that sector \emph{is} --- a seesaw --- and works the one
identity that makes the placement admissible at all. It also says, at the
end and in plain terms, why that identity is weaker evidence than it
looks.

\subsection{The heavy sector as a seesaw}

The constitutional Majorana channel of Part~\ref{part:X} is the
dimension-five operator $(LH)(LH)/\EGUT$. Any ultraviolet completion of
that operator must reproduce its coefficient. The completion available
here is the minimal one: three gauge-singlet Majorana states at the lock,
\begin{equation}
\mathcal L \;\supset\;
-\,Y_{\nu\,\alpha i}\,\bar L_\alpha \tilde H N_i
\;-\;\tfrac12 M_{N_i}\,\overline{N_i^{c}}N_i \;+\;\mathrm{h.c.},
\qquad
M_{N_i} \;=\; \Mlock\bigl(1+\delta_i\bigr),
\quad \delta_i = \mathcal O(1/K),
\label{eq:lg-seesaw-lag}
\end{equation}
with the splitting $\Delta M = \Mlock/K = 110$~GeV of
\eqref{eq:lg-splitting}. In the $(\nu_L, N_R^{c})$ basis this is the
familiar block matrix
\begin{equation}
\mathcal M \;=\;
\begin{pmatrix} 0 & m_D \\ m_D^{\mathsf T} & M_N \end{pmatrix},
\qquad
m_D \;=\; Y_\nu\, v ,
\label{eq:lg-seesaw-matrix}
\end{equation}
whose light eigenvalue for $m_D \ll M_N$ is
$m_\nu \simeq -\,m_D M_N^{-1} m_D^{\mathsf T}$.

\paragraph{The convention, pinned.} Equation~\eqref{eq:lg-seesaw-matrix}
uses $m_D = Y_\nu v$ with $v = 246.33$~GeV, \emph{not} $Y_\nu v/\sqrt2$.
This is the convention under which the effective washout mass is
$\widetilde m = Y_\nu^2 v^2/M_N$, which is the form the washout of
\S\ref{sec:lg-washout} uses; it is stated here once and used everywhere
in this Part and in the appendices. The factor is not cosmetic --- it is
the difference between an identity and an identity-up-to-two in what
follows.

\subsection{The closure: the lock scale cancels}

With one generation for clarity, the light mass is
\begin{equation}
m_\nu \;=\; \frac{m_D^2}{M_N} \;=\; \frac{Y_\nu^2 v^2}{M_N}.
\label{eq:lg-seesaw-mnu}
\end{equation}
Insert the seam Yukawa \eqref{eq:lg-ynu} and set $M_N = \Mlock$:
\begin{equation}
\boxed{\;
m_\nu \;=\;
\underbrace{\left(\pi\,\frac{\Mlock}{\EGUT}\right)}_{Y_\nu^{2}}
\cdot\frac{v^{2}}{\Mlock}
\;=\;
\pi\,\frac{v^{2}}{\EGUT}
\;=\; m_\star \;=\; 8.360835~\mathrm{meV}.\;}
\label{eq:lg-seesaw-closure}
\end{equation}
\emph{The lock scale cancels identically.} It appears once in the
numerator, through the Yukawa, and once in the denominator, through the
heavy mass, and it leaves no trace in the light spectrum. Numerically the
two sides agree to every digit carried:
$5.3034\times10^{-13}\times(246.33)^2/3849.0~\mathrm{GeV}
= 8.360835\times10^{-12}$~GeV, against
$\pi(246.33)^2/2.280\times10^{16}~\mathrm{GeV}$, the same number.

\subsection{What the cancellation is worth --- and what it is not}

It would be easy to present \eqref{eq:lg-seesaw-closure} as a triumph. It
is not one, and the Part is better served by saying why.

\paragraph{The identity is a consistency check, not a prediction.}
$Y_\nu$ was \emph{fixed by matching} to the derived Weinberg coefficient
in the first place. Read backwards, \eqref{eq:lg-seesaw-closure} is
therefore automatic: for \emph{any} heavy scale $M_N$ there exists a
Yukawa $Y_\nu^2 = \pi M_N/\EGUT$ that reproduces $m_\star$ exactly. The
light neutrino spectrum constrains only the ratio $Y_\nu^2/M_N$, and
carries \textbf{no information whatever} about where the heavy states
sit. A reader who takes the boxed equation as evidence for a TeV-scale
seesaw has read it backwards, and this paragraph exists to prevent that.

\paragraph{Where the content actually is.} Three statements, in
increasing order of what they are worth.

\emph{First: the scale is not chosen to fit.} $\Mlock$ is fixed by
$K = LJ = 35$ --- an anomaly audit of the six-dimensional bulk that
never mentions neutrinos, leptogenesis or \CP{}. The framework does not
select $M_N$ to make \eqref{eq:lg-seesaw-closure} work; it reads $M_N$
off elsewhere and then has no freedom left.

\emph{Second: the Yukawa is therefore a prediction, and it could have
been absurd.} Once $\Mlock$ is supplied, $Y_\nu = 7.28\times10^{-7}$
follows with nothing to adjust. It did not have to be a reasonable
number. A lock ten orders lower would have demanded
$Y_\nu \sim 10^{-12}$; a lock at the GUT seam would have demanded
$Y_\nu \sim 1$ and no leptogenesis at the lock. What comes out is a
coupling within a factor of four of the electron's own,
$y_e = 2.9\times10^{-6}$ --- ordinary smallness in a sector where the
textbook seesaw demands either a coupling of order one or a scale twelve
orders higher. \emph{That} is the check that could have failed, and it is
the one worth reporting.

\emph{Third: the Yukawa is what makes the rest of the Part
parameter-free.} $Y_\nu$ sets the width \eqref{eq:lg-gamman}, the
effective washout mass $\widetilde m$, the flavored washouts
\eqref{eq:lg-kwash} and every rate in the Boltzmann system. In the
standard treatment these are read off the Casas--Ibarra matrix and are
free (\S\ref{app:pmns-seesaw}); here there is no $R$ to choose.

\paragraph{And the cancellation is also a limitation.} Because $\Mlock$
drops out of \eqref{eq:lg-seesaw-closure}, \textbf{no neutrino
measurement can confirm or refute the lock-scale seesaw.} Not the
splittings, not the angles, not $\Summnu{}$, not $\mbb{}$ --- all are
computed from $m_\star$ and the projector geometry, and all are blind to
$M_N$. The claim is testable only through the chain that follows, which
is why F-SM-7 is written against a calculation rather than against a
spectrum, and why this Part cannot borrow the neutrino block's evidence.

\subsection{Where this sits, against the two standard anchors}

\begin{table}[htbp]
\centering
\small
\caption[The heavy Majorana scale in three readings.]{The heavy Majorana scale in three readings. The first row is the
textbook choice, motivated by nothing stronger than the top quark's
Yukawa being of order one; the second is the floor that reproducing the
observed asymmetry imposes in the hierarchical regime; the third is this
framework's, fixed upstream by the ledger integer $K=35$. The last
column is the Yukawa each reading requires, and it is the honest
discriminator between them. All three rows are matched to the \emph{same}
light mass, $m_\star = 8.36$~meV; a reading matched instead to $m_3$, as
in the worked example of \S\ref{app:pmns-seesaw}, moves every $M_N$ down
by the ratio of the two.}
\label{tab:lg-scale-anchors}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.34\textwidth}rr>{\raggedright\arraybackslash}p{0.26\textwidth}@{}}
\toprule
\textbf{Reading} & \textbf{$M_N$} & \textbf{$Y_\nu$} & \textbf{Status of the scale}\\
\midrule
Textbook seesaw, $Y_\nu \sim 1$
 & $7.3\times10^{15}$~GeV & $\sim1$ & assumed, from the top\\
Davidson--Ibarra floor, hierarchical
 & $\gtrsim 6\times10^{8}$~GeV & $\sim3\times10^{-4}$ & a bound, not a value\\
\textbf{This framework: $M_N = \Mlock$}
 & $\mathbf{3.85\times10^{3}}$~\textbf{GeV} & $\mathbf{7.28\times10^{-7}}$ & derived from $K=35$\\
\bottomrule
\end{tabular}
\end{table}

The lock sits $5.2$ orders of magnitude below the Davidson--Ibarra floor
and $12.3$ below the $Y_\nu \sim 1$ seesaw. The first of those gaps is
the subject of the next section: the floor assumes a hierarchical heavy
spectrum, and this one is not hierarchical.

\subsection{What the TeV Majorana state does \emph{not} do}

A heavy neutrino at a collider-accessible mass invites an expectation
this Part must decline explicitly. The active--sterile mixing angle is
\begin{equation}
\theta \;=\; \frac{m_D}{M_N} \;=\; \frac{Y_\nu v}{\Mlock}
\;=\; 4.66\times10^{-8},
\qquad
|\theta|^{2} \;=\; 2.2\times10^{-15},
\label{eq:lg-mixing}
\end{equation}
with $m_D = Y_\nu v = 179$~keV. Against the most optimistic reach of any
proposed direct search or lepton-number-violation experiment,
$|\theta|^2 \gtrsim 10^{-6}$, that is short by nearly \emph{nine orders
of magnitude}. The heavy triplet contributes nothing measurable to
$0\nu\beta\beta$, to lepton-flavor violation, or to any collider final
state.

\begin{euclidremark}[Two tenants at one address]
The number $3.85$~TeV appears twice in this cohort and the two
appearances must not be conflated. The Kaluza--Klein cluster of
\PaperPOne{}, which F-SM-1 tests at the LHC, is a \emph{gauge-sector}
object at the lock with ordinary couplings and a genuine discovery
prospect. The Majorana triplet of this Part is a \emph{singlet} sector at
the same lock, with a Yukawa of $10^{-7}$ and a mixing of $10^{-15}$, and
it is invisible to every experiment now conceived. Same address, two
tenants, one visible and one not. A reader who hears ``TeV-scale seesaw''
and expects a collider signature has merged them.
\end{euclidremark}

\subsection{What is new here, stated exactly}

TeV-scale leptogenesis is not itself new. Resonant leptogenesis at the
TeV scale is \citet{Pilaftsis1997} and \citet{PilaftsisUnderwood2004},
cited in the next section and used there; lower-scale mechanisms exist
below that. Nothing in this Part claims priority for the scale.

What is new is that \emph{nothing is chosen}. In the resonant literature
the heavy mass is a model parameter, the degeneracy $\Delta M/M$ is
arranged --- typically to a part in $10^{10}$ or finer, with no reason
offered --- and the Yukawa is fitted through the Casas--Ibarra matrix.
Here the scale is $\tfrac12\EPl e^{-K}$, the splitting is $\Mlock/K$, and
the Yukawa is $\sqrt{\pi\Mlock/\EGUT}$: three quantities, one integer,
and that integer was fixed by an anomaly audit before any of this was
asked of it. \emph{The claim is a derived TeV-scale seesaw, not a first
one.}

\paragraph{And it does not, by itself, work.} The next section shows that
the derived splitting and the derived width miss the resonant condition
by twelve orders of magnitude, and \S\ref{sec:lg-resonance} states what
would have to be true instead. Nothing in the present section establishes
the mechanism; it establishes only that the seesaw is consistent, that
the Yukawa it demands is of an ordinary size, and that there is no
freedom left anywhere to repair what comes next.

\section{The Davidson--Ibarra wall, and why this scale evades it}
\label{sec:lg-di}

\subsection{The wall}

There is a reason nobody puts thermal leptogenesis at four TeV. For
hierarchical heavy neutrinos the \CP{} asymmetry per decay is bounded
above by the light and heavy masses \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:lg-di}
\end{equation}
so demanding enough asymmetry to make the observed universe forces
\begin{equation}
M_N \;\gtrsim\; 10^{9}~\mathrm{GeV}.
\label{eq:lg-di-bound}
\end{equation}
Nine orders of magnitude above anything a collider will reach. Standard
thermal leptogenesis is, by its own arithmetic, permanently
unfalsifiable in the laboratory --- a fact usually noted and then set
aside.

The bound is easy to read physically. The asymmetry comes from
interference between the tree-level decay and a loop; the loop is
suppressed by the Yukawa couplings, and the seesaw ties those Yukawas to
the light neutrino masses divided by the heavy mass. Push the heavy mass
down and the Yukawas must come down with it, and the asymmetry falls
linearly. Equation~\eqref{eq:lg-di} is that trade written out.

\subsection{The escape, and why it is forced here rather than assumed}

The known escape is \emph{resonant} leptogenesis
\citep{Pilaftsis1997, PilaftsisUnderwood2004}. If two heavy states are
quasi-degenerate, the self-energy contribution to the decay asymmetry is
resonantly enhanced, and the Yukawa suppression cancels between numerator
and denominator:
\begin{equation}
\epsilon \;\propto\;
\frac{\Delta M\,\Gamma}{\Delta M^{2} + \Gamma^{2}/4}\,\sin(2\phi)
\qquad \xrightarrow{\;\Delta M = \Gamma/2\;} \qquad
\tfrac12 \sin(2\phi).
\label{eq:lg-bw}
\end{equation}
On exact resonance the asymmetry is of order unity no matter how small
the Yukawas are, and the bound \eqref{eq:lg-di-bound} evaporates.

In the general literature quasi-degeneracy is an assumption, and usually
an uncomfortable one: two masses are required to agree to a part in
$10^{10}$ or better with no reason offered. Here the degeneracy is not
assumed. The three heavy states are one multiplet at the lock, split by
$\Delta M = \Mlock/K$ from \eqref{eq:lg-splitting} --- a relative
splitting of $1/35$, which is not fine at all. Whether that is
\emph{degenerate enough} is precisely the resonance question, and it is
faced in \S\ref{sec:lg-resonance} rather than assumed away.

\section{The seam Yukawa and the vacuum width}
\label{sec:lg-yukawa}

\subsection{The Yukawa is fixed by the seam, with no freedom}

The neutrino Yukawa is not a free coupling in this framework. The seesaw
of \S\ref{sec:lg-seesaw}, matched to the seam coefficient $2\pi/\EGUT$ of
the Weinberg operator \eqref{eq:Lnu5-main} that Part~\ref{part:X}
derived, fixes it --- exactly, in the convention pinned there:
\begin{equation}
Y_\nu^{2} \;=\; \pi\,\frac{\Mlock}{\EGUT}
\;=\; \pi\,\frac{3.85\times10^{3}~\mathrm{GeV}}{2.28\times10^{16}~\mathrm{GeV}}
\;=\; 5.30\times10^{-13},
\qquad
Y_\nu \;\simeq\; 7.28\times10^{-7}.
\label{eq:lg-ynu}
\end{equation}

It is worth pausing on that number rather than passing it. A neutrino
Yukawa of $7\times10^{-7}$ looks like a tuned small number until it is
put beside the electron's, $y_e = 2.9\times10^{-6}$. It is within a
factor of four. The framework is not asking for a coupling of unusual
smallness; it is asking for one of ordinary smallness, in a sector where
the usual seesaw would demand something far more extreme or a far higher
scale.

\subsection{The vacuum width, and the first sign of trouble}

The decay width that follows is
\begin{equation}
\Gamma_N \;=\; \frac{Y_\nu^{2} M_N}{8\pi}
\;=\; 8.1\times10^{-11}~\mathrm{GeV},
\qquad
\frac{\Gamma_N}{M_N} \;=\; 2.1\times10^{-14}.
\label{eq:lg-gamman}
\end{equation}
Set that beside the splitting $\Delta M = 110$~GeV from
\eqref{eq:lg-splitting}. The ratio is
\begin{equation}
\frac{\Delta M}{\Gamma_N} \;=\; \frac{110}{8.1\times10^{-11}}
\;\simeq\; 1.4\times10^{12}.
\label{eq:lg-vacuum-ratio}
\end{equation}
The resonance condition of \eqref{eq:lg-bw} wants this ratio near unity.
It is off by twelve orders of magnitude. Taken at face value the chain is
dead on arrival, undershooting the observed asymmetry by seven or eight
orders.

This is not a difficulty to be buried in an appendix; it is the central
technical question of the Part, and \S\ref{sec:lg-resonance} is devoted
to it. The short statement of what follows is that $\Gamma_N$ --- the
\emph{vacuum} width of the heavy state --- is the wrong quantity to put
in the denominator of a resonance formula evaluated in a hot plasma, and
that identifying the right one is both the chain's best claim and its
weakest link.

\section{The \texorpdfstring{\CP{}}{CP} asymmetry: flavored, and why the unflavored one vanishes}
\label{sec:lg-cp}

\begin{figure}[htbp]
\centering
\resizebox{0.489\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize]
\draw[line width=1.1pt,draw=blue!60!black,fill=blue!60!black,fill opacity=0.13] (0.0000,0.0000) -- (3.4449,0.0000) -- (8.9484,-2.1367) -- cycle;
\node[anchor=south,text=blue!60!black] at (4.131,0.100) {PMNS};
\draw[line width=0.9pt,draw=red!65!black,fill=red!65!black,fill opacity=0.13] (0.0000,0.0000) -- (0.2465,0.0000) -- (-0.0051,0.0970) -- cycle;
\draw[-{Stealth[length=1.8mm]},draw=red!65!black,line width=0.5pt] (1.580,0.550) -- (0.180,0.082);
\node[anchor=west,text=red!65!black,font=\scriptsize] at (1.630,0.580) {CKM, same scale};
\draw[line width=0.9pt,draw=red!65!black,fill=red!65!black,fill opacity=0.13] (2.6845,-3.6867) -- (8.8479,-3.6867) -- (2.5561,-1.2628) -- cycle;
\node[anchor=east,text=red!65!black,font=\scriptsize,align=right] at (2.435,-3.687) {CKM\\ magnified\\ $\times25$};
\draw[dotted,draw=black!45,line width=0.5pt] (0.080,-0.048) -- (3.085,-3.137);
\end{tikzpicture}}
\caption[The two unitarity triangles at true relative area.]{The two unitarity triangles at true relative area. Both are drawn from the \emph{derived} mixing matrices at one common scale, so the areas may be compared directly: twice the area of each is its Jarlskog invariant, and $|\JPMNS|/|J_{\rm CKM}| = 9.187\times10^{-3}/2.984\times10^{-5} = 308$. All six unitarity triangles of a given matrix have the same area, so the choice of index pair affects only the shape; each is drawn in its best-proportioned pair, and the CKM shape is scaled by six per cent in linear size so that its area equals the codebook's own $J_{\rm CKM}$. \emph{The caveat travels with the picture}: three hundred is not ten orders of magnitude, so this ratio is not what lets the lepton sector pay for baryogenesis when the quark sector cannot --- and \S\ref{sec:lg-cp} finds that the unflavored invariant $\JPMNS$ belongs to drops out of the asymmetry entirely.}
\label{fig:lg-triangles}
\end{figure}


\subsection{What the asymmetry is}

The quantity the chain needs is the fractional excess of leptons over
antileptons produced when one heavy state decays,
\begin{equation}
\epsilon_i \;\equiv\;
\frac{\Gamma(N_i \to \ell H) - \Gamma(N_i \to \bar\ell H^{*})}
     {\Gamma(N_i \to \ell H) + \Gamma(N_i \to \bar\ell H^{*})} ,
\label{eq:lg-eps-def}
\end{equation}
a pure number between $-1$ and $+1$. If it vanishes the decays are
perfectly even-handed and no asymmetry is made at all. If it is
$10^{-6}$, then for every million heavy states that decay there is one
extra lepton left over. The target value for this chain, established in
\S\ref{sec:lg-throughput}, is
$\epsres \simeq 2.5\times10^{-6}$ --- roughly one extra lepton per four
hundred thousand decays.

\subsection{Where the asymmetry comes from: the one-loop computation}
\label{sec:lg-oneloop}

Equation \eqref{eq:lg-eps-def} defines the asymmetry; it does not say why
one exists. This subsection derives it, because the structure of the
answer is what makes the resonant enhancement possible and what makes the
Davidson--Ibarra bound escapable.

\paragraph{Why a tree amplitude cannot do it.} At tree level the decay
$N_i \to \ell_\alpha H$ has amplitude proportional to the Yukawa
$(Y_\nu)_{i\alpha}$, and the conjugate decay
$N_i \to \bar\ell_\alpha H^{*}$ to $(Y_\nu)_{i\alpha}^{*}$. The rates go
as the modulus squared, so both are $|(Y_\nu)_{i\alpha}|^{2}$ and the
asymmetry is exactly zero. A phase alone is not enough: to make a rate
difference the phase must be multiplied by something that does \emph{not}
conjugate with it. That something is an absorptive part.

\paragraph{The two interferences.} At one loop there are two corrections
that carry an absorptive part, and they behave completely differently
here.

The \emph{vertex} (direct) correction dresses the decay with a loop in
which a lepton and a Higgs are exchanged and the intermediate heavy state
$N_j$ goes on shell. Interfering it with the tree amplitude gives an
asymmetry proportional to
\begin{equation}
\epsilon_i^{\rm vertex}
\;\propto\;
\sum_{j\neq i}
\operatorname{Im}\bigl[(Y_\nu Y_\nu^{\dagger})_{ij}^{2}\bigr]\,
f\!\left(\frac{M_j^{2}}{M_i^{2}}\right),
\qquad
f(x) \xrightarrow{\ x \gg 1\ } -\frac{1}{\sqrt{x}},
\label{eq:lg-vertex}
\end{equation}
which is suppressed by the heavy-mass ratio and is what the
Davidson--Ibarra bound \eqref{eq:lg-di} actually bounds.

The \emph{self-energy} (indirect) correction is a different object. The
same loop, inserted on the external $N_i$ line, mixes $N_i$ with $N_j$
before it decays. Its contribution to the asymmetry carries the propagator
of the state being mixed into, and therefore a denominator:
\begin{equation}
\epsilon_i^{\rm self}
\;\propto\;
\sum_{j\neq i}
\operatorname{Im}\bigl[(Y_\nu Y_\nu^{\dagger})_{ij}^{2}\bigr]\,
\frac{M_i M_j\,(M_i^{2}-M_j^{2})}
     {(M_i^{2}-M_j^{2})^{2} + (M_i\Gamma_j)^{2}} .
\label{eq:lg-self}
\end{equation}
That denominator is the whole story. When the two heavy masses are far
apart it reduces to the same $1/\sqrt{x}$ suppression as the vertex term
and nothing is gained. When they approach each other it does not: the
numerator falls linearly in the splitting while the denominator falls
quadratically, so the ratio \emph{grows} until the width term takes over.

\paragraph{Reduction to the Breit--Wigner form.} Write
$M_i^{2}-M_j^{2} \simeq 2M\,\Delta M$ with $M$ the common scale, and
$M_i\Gamma_j \simeq M\,\Geff$. Then \eqref{eq:lg-self} becomes
\begin{equation}
\frac{M^{2}\cdot 2M\Delta M}{4M^{2}\Delta M^{2} + M^{2}(\Geff)^{2}}
\;\propto\;
\frac{\Delta M\,\Geff}{\Delta M^{2} + (\Geff/2)^{2}}\cdot\frac{1}{\Geff},
\label{eq:lg-selfreduce}
\end{equation}
and the first factor is precisely the envelope \eqref{eq:lg-bw-general}
whose maximum is $\tfrac12$, attained at $\Delta M = \Geff/2$. This is
where the resonant form used throughout this Part comes from, and it is
why the enhancement is bounded: no choice of parameters makes
$F_{\rm BW}$ exceed one half.

\paragraph{The mixing angle, and what is being resummed.} Equivalently,
the two quasi-degenerate states are not mass eigenstates of the full
propagator. Diagonalizing the resummed $2\times2$ inverse propagator
mixes them through an angle $\theta_{\rm mix}$ with
\begin{equation}
\tan 2\theta_{\rm mix} \;=\; \frac{2\,\Delta M}{\Gamma_1 + \Gamma_2},
\label{eq:lg-mixangle}
\end{equation}
so the physical decaying states are superpositions whose lepton-flavor
content differs from the mass basis. The asymmetry is then the ordinary
statement that a superposition of two states with different couplings and
a relative phase decays asymmetrically. Equation~\eqref{eq:lg-mixangle}
also shows what the resonance condition means physically: maximal mixing,
$\theta_{\rm mix} = \pi/4$, is reached when the splitting equals the mean
width.

\begin{euclidremark}[Physics intuition for the two loops]
The vertex diagram asks a decaying particle to interfere with a version of
itself that has been round a loop. The self-energy diagram asks something
different: it lets the particle \emph{become the other particle} and decay
as that. When the two are nearly degenerate, becoming the other one is
nearly free, and the second process is unsuppressed while the first is
not. This is the whole reason resonant leptogenesis evades a bound derived
for the hierarchical case: the Davidson--Ibarra argument bounds
\eqref{eq:lg-vertex} and says nothing about \eqref{eq:lg-self} when the
denominator is small.
\end{euclidremark}

\begin{euclidremark}[What this subsection does \emph{not} settle]
Everything above is a vacuum-amplitude computation: the widths in
\eqref{eq:lg-self} are the widths of free heavy states, and the
$\Geff$ that emerges from the reduction is $\Gamma_N$. Carried literally
into the lock-scale chain it gives the fatal $N$-width branch of
\S\ref{sec:lg-resonance}, row~(a) of Table~\ref{tab:lg-regulator}, short
of observation by seven or eight orders. The claim of this Part is that in
a hot plasma the quantity in that denominator is not the vacuum width, and
\emph{that} is condition C4, which remains open. Deriving the vacuum form
more carefully does not close it, and this subsection is not offered as
progress on it.
\end{euclidremark}

\subsection{The unflavored invariant vanishes identically}

Here the framework does something that looks at first like a disaster and
turns out to be the mechanism.

The \CP{} source in the transport equations is a matrix in lepton flavor.
Its \emph{trace} --- the flavor-summed, or unflavored, invariant --- is
the quantity that appears in the textbook one-flavor treatment. Written
in the Casas--Ibarra parameterization \citep{CasasIbarra2001}, in which
the Yukawa matrix is decomposed into the light masses, the PMNS matrix
and an orthogonal matrix $R$, the unflavored invariant is
$U_{\rm PMNS}$-\emph{independent}: the mixing matrix cancels out of it
entirely. Worse, for the dial-free door of \S\ref{sec:lg-door} --- which
admits no complex $R$, because there is no continuum of orientations to
choose from --- it vanishes exactly:
\begin{equation}
\operatorname{Tr} S \;=\; \sum_\alpha S_{\alpha\alpha}
\;\propto\;
\operatorname{Im}\bigl[(Y_\nu Y_\nu^{\dagger})_{ij}^{2}\bigr] \;=\; 0
\qquad (R\ \text{real}).
\label{eq:lg-traceless}
\end{equation}
This is Lemma L-Cosmos-U of \ATen{} (\citealp{ArisakaA10Cosmos}), where it
is proved and machine-checked; it is quoted here because it is what
forces everything that follows.

\begin{lemma}[Flavor-odd source]
\label{lem:lg-flavour-odd}
On the dial-free door, the flavor-summed \CP{} asymmetry vanishes
identically, while the individual flavored asymmetries do not:
\begin{equation}
\sum_\alpha \epsilon_\alpha \;=\; 0,
\qquad \epsilon_\alpha \neq 0 \ \ \text{for each } \alpha .
\label{eq:lg-sum-zero}
\end{equation}
The entire \CP{} content therefore lives in flavor \emph{differences}.
\end{lemma}

\begin{euclidremark}[Physics intuition for a source that sums to zero]
Read \eqref{eq:lg-sum-zero} as a bookkeeping statement about three
ledgers that must balance. The decays make an excess of electron-flavor
leptons, an excess of muon-flavor leptons, and a deficit of
tau-flavor ones, in exactly compensating amounts. If nothing
distinguished the three afterwards, the three would cancel and the
universe would be empty. What saves it is that the three flavors are
\emph{erased at different rates} --- and the rates are set by how
strongly each flavor couples to the heavy states, which is set by the
PMNS matrix, which this paper derived. No spread in the erasure, no
matter. This is why the chain must be flavored, and it is why the
reactor angle turns out to matter to cosmology at all
(\S\ref{sec:lg-washout}).
\end{euclidremark}

\subsection{What this costs and what it buys}

The cost is that the simple, widely used one-flavor estimate is not
merely inaccurate here --- it returns exactly zero, and any treatment
that stops there would conclude the framework cannot make matter. The
gain is that the surviving asymmetry is then a statement about
\emph{differences} between flavors, and differences between flavors are
precisely what a derived PMNS matrix is a statement about. A framework
that had fitted its mixing angles would gain nothing from this structure.
One that derived them inherits a prediction.

\section{The half-angle: why a parity of \texorpdfstring{$\pi$}{pi} sets a sign}
\label{sec:lg-halfangle}

\subsection{The apparent paradox}

Part~\ref{part:X} derived the Majorana phases as parity data rather than
as dials:
\begin{equation}
(\alpha_{21},\ \alpha_{31}) \;=\; (0,\ \pi).
\label{eq:lg-majorana-phases}
\end{equation}
The constitutionally forced rank-two Majorana block fixes one state of
each intrinsic parity, so the two surviving heavy states are a
\emph{parity pair}, not a phase continuum: $0$ means same parity, $\pi$
means opposite, and there is no third value to choose.

Now the paradox. A phase of $\pi$ is real. On the unit circle it is the
point $-1$. Complex conjugation --- which is what \CP{} does to a phase
--- leaves it exactly where it is. A quantity fixed at $\pi$ looks like
the most \CP{}-\emph{even} letter available, and it is fair to ask how
such a thing can possibly orient a matter--antimatter asymmetry.

\subsection{The resolution}

A Majorana letter is read at \emph{half angle}. The door's weight-zero
pairing is a $\Delta L = 2$ bilinear, quadratic in the field, so the
phase enters amplitudes through $e^{i\alpha/2}$ and not through
$e^{i\alpha}$. Therefore
\begin{equation}
\boxed{\;
\alpha_{31} = \pi
\qquad \Longrightarrow \qquad
e^{\,i\alpha_{31}/2} \;=\; e^{\,i\pi/2} \;=\; i .
\;}
\label{eq:lg-half-angle}
\end{equation}
A quarter turn --- and here the argument must be stated carefully,
because the tempting next sentence is wrong. This factor of $i$ is
\emph{kinematics, not \CP{} violation}. In the standard classification a
Majorana phase of exactly $\pi$ --- which is exactly this factor of $i$ ---
is the signature of \emph{opposite intrinsic \CP{} parity} between the mass
eigenstates, and opposite intrinsic parity is a \CP{}-\emph{conserving}
configuration. The half angle tells us what the letter does to an
amplitude. It does not tell us that \CP{} is broken.

\paragraph{Where the resolution actually lies.} The textbook statement
``Majorana phases of $0$ and $\pi$ conserve \CP{}'' is a statement about
the \emph{flavor-summed} asymmetry --- and that sum does vanish here,
exactly. The individual flavored asymmetries do not. Computed at this
paper's own inputs, in a type-I seesaw with a real orthogonal
Casas--Ibarra matrix and both vertex and self-energy contributions at one
loop,
\begin{equation}
\epsilon_e : \epsilon_\mu : \epsilon_\tau
\;=\; 8.52\times10^{-14} \,:\, -4.75\times10^{-13} \,:\, 3.90\times10^{-13},
\qquad
\sum_\alpha \epsilon_\alpha \;=\; 5\times10^{-29},
\label{eq:lg-flavoured-eps}
\end{equation}
sixteen orders of magnitude apart. The cancellation is column
orthogonality of the mixing matrix, $\sum_\alpha V_{\alpha 1}V_{\alpha 3}
= 0$ --- not dynamics, which is why it is exact.

\begin{equation}
\boxed{\;\alpha_{31} = \pi \ \text{is \CP{}-conserving in the flavor sum,
and \CP{}-violating in the flavor differences.}\;}
\label{eq:lg-sum-vs-difference}
\end{equation}

A zero-sum set of flavored asymmetries is not a zero asymmetry, because
the three flavors are not erased at the same rate; \S\ref{sec:lg-washout}
carries this out and finds ninety-six per cent of the survivor in the
electron flavor, for the reason recorded there. This is also why a
flavored treatment is \emph{mandatory} here rather than merely more
careful: an unflavored calculation of this chain returns exactly zero.

\begin{euclidremark}[Physics intuition for the half angle]
The factor of two is not a technicality; it is the signature of the door
itself. A Dirac mass term pairs a field with its conjugate, and the phases
cancel between them --- which is why a Dirac phase can always be rotated
away unless three generations conspire to prevent it. A Majorana mass
term pairs a field with \emph{itself}, so the phase appears twice and
only half of it can be removed. What remains is the half angle. The
observation that a Majorana neutrino's phase is physically halved is
older than this framework; what is new here is that the half being kept
is not a free parameter but a parity. What that parity then does is a
separate question, and it is answered by the flavor structure rather
than by the factor of $i$: the parity makes the three flavored
asymmetries individually nonzero while forcing their sum to vanish, and
the unequal washout of the three flavors converts that zero-sum set into
a net excess.
\end{euclidremark}

\subsection{Consequence: the parity sets the sign of the matter excess}

Within the flavored chain the driver of the asymmetry is
$\alpha_{31}$. Sending $\alpha_{31} \to 0$ --- the other admissible
parity --- flips the sign of the flavored asymmetries and therefore of
the final baryon number: the same universe, made of antimatter. That
statement should be read with the grade attached, since it holds within
a chain whose magnitude is conditional (\S\ref{sec:lg-resonance}); but the
sign is carried by a discrete datum of this paper's own codebook, and
that is worth stating plainly.

\begin{euclidremark}[What the \texorpdfstring{$0\nu\beta\beta$}{0nubb} program is really testing]
Row 23 and Row 24 of the codebook, $(\alpha_{21},\alpha_{31}) = (0,\pi)$,
are the only two entries of the twenty-eight that are at present entirely
unmeasured. It is tempting to treat them as decorative. Within this chain
they are not: the parity assignment $(0,\pi)$ is what makes the universe
matter rather than antimatter. A neutrinoless-double-beta experiment
sensitive to the Majorana phases is, in this reading, measuring the sign
of the cosmic matter excess. That is a large claim resting on a
conditional chain, and \S\ref{sec:lg-kills} says exactly how it could
fail --- but it is why F-SM-3b is worth its place in the falsification
map even though the predicted $\mbb \simeq 1.8$~meV sits below the
approved program's reach.
\end{euclidremark}

\section{Why the Dirac phase is a spectator}
\label{sec:lg-spectator}

The reader who has followed Part~\ref{part:X} will expect the leptonic
Dirac phase to be the star of this Part. It is not, and the reason is
worth stating carefully, because it is the most counter-intuitive claim
in the chain.

Part~\ref{part:X} derived
\begin{equation}
\dPMNS \;=\; \pi + 2\arcsin\frac{1}{7} \;=\; 3.42829~\mathrm{rad}
\;=\; 196.43^{\circ},
\qquad
\dPMNS^{\rm QD} \;=\; 196.06^{\circ},
\label{eq:lg-dpmns}
\end{equation}
and \S\ref{sec:pmns-benchmark} showed it sitting about seven degrees from
the world best value. It is the framework's sharpest leptonic \CP{}
prediction. And it contributes essentially nothing to the baryon
asymmetry.

The reason is not that it is small --- it is of order one. It is not that
it lives at the wrong scale --- it does not. It is that \textbf{it is the
wrong letter}. The unflavored invariant that a Dirac phase could enter
is $U_{\rm PMNS}$-independent by \eqref{eq:lg-traceless} and vanishes
besides; and in the flavored asymmetries the numerical dependence on
$\dPMNS$ is at the few-per-cent level, while the dependence on
$\alpha_{31}$ is a sign. Sending $\dPMNS \to \pi$ shifts the result by a
few per cent. Sending $\alpha_{31} \to 0$ inverts it.

\begin{euclidremark}[Two \texorpdfstring{\CP{}}{CP}-violating phases, two different jobs]
Oscillation experiments measure the Dirac phase, because oscillation is a
first-order interference in the amplitude and the Dirac phase is what
survives there. Neutrinoless double beta decay and the matter excess are
sensitive to the Majorana parities, because both are $\Delta L = 2$
processes and the parities are what survive \emph{there}. The two
programs are not competing measurements of one quantity; they are
measurements of two different quantities that happen to sit in the same
matrix. DUNE and Hyper-Kamiokande will test \eqref{eq:lg-dpmns}, and a
tonne-scale $0\nu\beta\beta$ experiment would test
\eqref{eq:lg-majorana-phases}, and neither substitutes for the other.
\end{euclidremark}

A corollary worth registering, because it is the sort of thing that is
easy to overclaim: the excellent agreement of \eqref{eq:lg-dpmns} with
the accelerator joint fit is \emph{not} evidence for this Part. It is
evidence for the lepton block of Part~\ref{part:X}. The chain below rests
on a different letter, and it must be judged on its own.

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

\begin{figure}[htbp]
\centering
\resizebox{0.947\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize]
\fill[green!55!black,opacity=0.16] (4.541,-0.45) rectangle (5.351,4.650);
\node[anchor=south,font=\scriptsize,text=green!35!black,align=center]
  at (4.946,4.700) {required\\ window};
\draw[line width=1.00pt,draw=black!80] (0,-0.45) -- (12.2,-0.45);
\draw (0.486,-0.58) -- (0.486,-0.45);
\node[anchor=north,font=\scriptsize] at (0.486,-0.60) {$10^{-1}$};
\draw (2.108,-0.58) -- (2.108,-0.45);
\node[anchor=north,font=\scriptsize] at (2.108,-0.60) {$10^{1}$};
\draw (3.730,-0.58) -- (3.730,-0.45);
\node[anchor=north,font=\scriptsize] at (3.730,-0.60) {$10^{3}$};
\draw (5.351,-0.58) -- (5.351,-0.45);
\node[anchor=north,font=\scriptsize] at (5.351,-0.60) {$10^{5}$};
\draw (6.973,-0.58) -- (6.973,-0.45);
\node[anchor=north,font=\scriptsize] at (6.973,-0.60) {$10^{7}$};
\draw (8.595,-0.58) -- (8.595,-0.45);
\node[anchor=north,font=\scriptsize] at (8.595,-0.60) {$10^{9}$};
\draw (10.216,-0.58) -- (10.216,-0.45);
\node[anchor=north,font=\scriptsize] at (10.216,-0.60) {$10^{11}$};
\draw (11.838,-0.58) -- (11.838,-0.45);
\node[anchor=north,font=\scriptsize] at (11.838,-0.60) {$10^{13}$};
\node[anchor=north,font=\scriptsize\itshape] at (6.1,-1.15) {$\Delta M/\Geff$ at the lock crossing};
\fill[black!55] (10.847,4.000) circle (2.2pt);
\node[anchor=west,text=black!55] at (12.35,4.000) {(a) $N$ widths ($S$-matrix)};
\draw[line width=2.6pt,draw=green!45!black] (4.890,3.000) -- (5.128,3.000);
\fill[green!45!black] (5.009,3.000) circle (2.0pt);
\node[anchor=west,text=green!45!black] at (12.35,3.000) {(b) \textbf{flavor measurement} $c_\tau y_\tau^{2}T$};
\fill[black!55] (1.297,2.000) circle (2.2pt);
\node[anchor=west,text=black!55] at (12.35,2.000) {(c) oscillation rate $\Delta M$};
\fill[black!55] (0.361,1.000) circle (2.2pt);
\node[anchor=west,text=black!55] at (12.35,1.000) {(c$'$) gauge damping $g^{2}T$};
\fill[black!55] (11.594,0.000) circle (2.2pt);
\node[anchor=west,text=black!55] at (12.35,0.000) {(d) Hubble rate $H(T=M_N)$};
\node[anchor=north west,font=\scriptsize,text=black!60,align=left]
  at (0.0,-1.65) {Five candidates spread over sixteen orders sit about three decades apart,
  so one landing\\ inside the required decade is \emph{selection}, not derivation ---
  which is why condition C4 is marked open.};
\end{tikzpicture}}
\caption[The regulator landscape: five candidates, one required decade.]{The regulator landscape of Table~\ref{tab:lg-regulator}, drawn. Five candidate prescriptions for $\Geff$ at the lock crossing, spanning sixteen orders of magnitude in $\Delta M/\Geff$; the shaded band is the decade the assembly requires. Exactly one candidate lands in it. \emph{The figure is also the argument against itself}: five candidates spread over sixteen orders sit about three decades apart, so one of them falling inside the required decade is a \emph{selection}, not a derivation --- which is precisely why condition C4 is carried as open (\S\ref{sec:lg-resonance}) and why F-SM-7 is written against a calculation rather than a facility.}
\label{fig:lg-regulator-landscape}
\end{figure}


Every resonant-leptogenesis calculation contains a Breit--Wigner factor
\begin{equation}
F_{\rm BW} \;=\; \frac{\Delta M\,\Geff}{\Delta M^{2} + (\Geff)^{2}},
\qquad F_{\rm BW} \le \tfrac12,
\label{eq:lg-bw-general}
\end{equation}
and the whole magnitude of the asymmetry rides on it. Here the splitting
is \emph{known} --- it is the ledger's own level spacing,
$\Delta M = \Mlock/K = 110$~GeV from \eqref{eq:lg-splitting}. So
everything reduces to a single question:

\begin{center}
\emph{What is $\Geff$? Which physical rate regulates the resonance ---
that is, how quickly is the quantum interference that builds the \CP{}
asymmetry destroyed?}
\end{center}

\subsection{Both obvious answers are fatal}

It is worth establishing at once that this is a real question with a
narrow answer, because the two candidates a reader would reach for first
both fail catastrophically.

\paragraph{The vacuum width.} Putting $\Geff = \Gamma_N$ from
\eqref{eq:lg-gamman} gives the ratio \eqref{eq:lg-vacuum-ratio} of
$1.4\times10^{12}$, a Breit--Wigner factor of order $10^{-12}$, and an
$\etaB$ short of observation by seven or eight orders of magnitude. This
branch is fatal, and the framework says so.

\paragraph{Exact resonance.} Putting $\Delta M \simeq \Geff$ gives
$F_{\rm BW} \simeq \tfrac12$, an asymmetry of order unity, and an
overshoot of the observed value by some four orders of magnitude. This
branch is fatal too.

The required answer therefore lies in a narrow window between the two,
and the chain's claim is that a specific physical rate lands in it.

\subsection{The candidate: the bath as a flavor-measuring apparatus}

The argument is \ATen{}'s (\citealp{ArisakaA10Cosmos}, Appendix~I), and
runs as follows.

The interference that builds the asymmetry is between two heavy states
that differ in \emph{flavor} composition. What destroys such an
interference is not any fast process but specifically a process that
\emph{measures} flavor. The gauge interactions are enormously the
fastest thing in the plasma --- $g^{2}T \simeq 1.5\times10^{3}$~GeV at
the lock --- but they are flavor-blind. They cannot measure what they
cannot distinguish, and formally they cancel: the decohering action of a
thermal rate on a flavor matrix is its double commutator with the
projector that rate measures, and for a flavor-universal rate that
projector is the identity, so
\begin{equation}
\bigl[\mathbf{1},[\mathbf{1},\delta n_\ell]\bigr] = 0
\quad\text{exactly, for every } \delta n_\ell;
\qquad
\bigl[P_\tau,[P_\tau,\delta n_\ell]\bigr] = \delta n_\ell
\quad\text{on pure tau coherences.}
\label{eq:lg-blind-cancel}
\end{equation}
Gauge damping drops out of the flavor-odd channel \emph{however large it
is} --- which is why the biggest rate in the plasma is not the answer.

The fastest \emph{honest} measurer of lepton flavor at $T \simeq 4$~TeV
is the tau Yukawa interaction; the muon and electron rates are down from
it by $(y_\mu/y_\tau)^{2} \simeq 3\times10^{-3}$ and below. So
\begin{equation}
\Geff \;=\; \Gamma_\tau \;=\; c_\tau\,y_\tau^{2}\,T,
\qquad c_\tau = 5\times10^{-3}\text{--}10^{-2},
\label{eq:lg-regulator}
\end{equation}
with $c_\tau$ the kinetic coefficient of the flavor-equilibration
literature.

\subsection{The two-level reduction}

Reducing the heavy pair to a two-level system, the \CP{}-odd weight
accumulated by an interference oscillating at $\Delta M$ while its
flavor coherence is measured at rate $\Gamma_{\rm fl}$ is
\begin{equation}
I(\Gamma_{\rm fl})
\;=\; \Gamma_{\rm fl}\int_0^{\infty}\!\!{\rm d}s\;
      e^{-\Gamma_{\rm fl}s}\,\sin(\Delta M\,s)
\;=\; \frac{\Gamma_{\rm fl}\,\Delta M}{\Gamma_{\rm fl}^{2} + \Delta M^{2}}
\;\le\; \tfrac12 .
\label{eq:lg-twolevel}
\end{equation}
This is worth reading as physics rather than algebra: the interference is
a sine wave in proper time, the measurement is an exponential cutoff, and
the accumulated asymmetry is the Laplace transform of one against the
other. The bound $\tfrac12$ is saturated only on exact resonance.

\subsection{The closed form, in which the heavy mass cancels}

Now the step that makes this a statement about \emph{this} framework
rather than a generic one. Evaluate at the generation temperature
$T \simeq M_N$. The splitting is $\Delta M = M_N/K$ and the rate is
$\Gamma_\tau = c_\tau y_\tau^{2}T$; the heavy mass cancels between them,
leaving a ledger integer and a Standard-Model Yukawa:
\begin{equation}
\boxed{\;
\frac{\Delta M}{\Geff}
\;=\; \frac{M_N/K}{c_\tau y_\tau^{2}T}\bigg|_{T = M_N}
\;=\; \frac{1}{K\,c_\tau\,y_\tau^{2}}
\;=\; \frac{1}{35\,c_\tau\,y_\tau^{2}}
\;=\; (2.7\text{--}5.3)\times10^{4}.
\;}
\label{eq:lg-regulator-closed}
\end{equation}
Zero dials. The only inputs are the ledger integer $K = 35$ and the
running tau Yukawa. With one-loop running,
$y_\tau(3.85~{\rm TeV}) = 0.0104$, so
\begin{equation}
\Geff = c_\tau y_\tau^{2} M_N = 2.1\text{--}4.1~\mathrm{MeV},
\qquad
F_{\rm BW} = (0.9\text{--}1.9)\times10^{-5}.
\label{eq:lg-regulator-number}
\end{equation}

\begin{euclidremark}[Physics intuition for the cancellation]
The reason \eqref{eq:lg-regulator-closed} is worth a box is that both of
its ingredients scale with the heavy mass and the ratio does not. Raise
the lock scale and the splitting rises in proportion; the temperature at
which the decays happen rises with it, and so does the tau rate. The
window is therefore not a coincidence of where the lock happens to sit ---
it is a property of the ledger integer $K$ and of a Standard-Model
coupling, and it would be the same number at a different lock scale. That
is a stronger statement than the chain needs, and it is the reason the
identification is worth taking seriously even though it is not yet
derived.
\end{euclidremark}

\subsection{The hypothesis, in sharp form}

\begin{euclidremark}[The named hypothesis \texorpdfstring{\Hphi{}}{H-phi-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 $\Delta M/\Geff \simeq 3\times10^{4}$ ---
equivalently, a Breit--Wigner factor $\simeq 1.5\times10^{-5}$, with
$\Geff \simeq 3$~MeV at $\Delta M \simeq 110$~GeV --- so that the
flavored, real-$R$ throughput on the inherited phases lands within
$\mathcal{O}(1)$ of $e^{-K/2}$.}

The name decodes as \textbf{H} for hypothesis,
$\boldsymbol{\phi_{\rm geo}}$ for the geometric phase whose projection
into the flavored asymmetries the hypothesis is about, and \textbf{1}
because it is the first such named item in the chain's promotion roadmap.
It is not a free parameter and not a fudge factor: it is a
\emph{prediction about the outcome of a calculation nobody has yet done},
stated with a number and an acceptance band, and it can be wrong. Its
home is \ATen{}, Appendix~I.
\end{euclidremark}

The target of \Hphi{} sits inside the band of
\eqref{eq:lg-regulator-number} and is reproduced exactly at
$c_\tau = 8.9\times10^{-3}$, within the literature range of
\eqref{eq:lg-regulator}. Two consistency checks: $\Gamma_\tau/H
= (1\text{--}2)\times10^{8}$, so the chain is fully flavored with eight
orders of margin; and $\Gamma_\tau/\Delta M \simeq 3\times10^{-5} \ll 1$,
so the regime is genuinely \emph{partial} resonance and not the excluded
peak.

\subsection{The regulator landscape}

The reason \Hphi{} is a scientific hypothesis and not a placeholder is
that the alternatives are not nearby. They are catastrophically far away,
and each is independently identifiable.

\begin{table}[htbp]
\centering
\small
\caption[The regulator landscape at the lock crossing ($\Delta M = 110$~GeV, $T = M_N$): five candidate prescriptions for $\Geff$, spanning sixteen orders of magnitude.]{The regulator landscape at the lock crossing
($\Delta M = 110$~GeV, $T = M_N$): five candidate prescriptions for
$\Geff$, spanning sixteen orders of magnitude in $\Delta M/\Geff$.
\textbf{Exactly one lands in the required decade.} The throughput column
is at the single-temperature evaluation. From \ATen{} Appendix~I.}
\label{tab:lg-regulator}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.30\textwidth}rrrr@{}}
\toprule
Candidate & $\Geff$ [GeV] & $\Delta M/\Geff$ & $F_{\rm BW}$ &
 $\etaB/\etaB^{\obs}$\\
\midrule
(a) $N$ widths ($S$-matrix prescription) & $2\times10^{-10}$ &
 $6\times10^{11}$ & $8\times10^{-13}$ & $2\times10^{-8}$\\
(b) \textbf{flavor measurement} $c_\tau y_\tau^{2}T$ &
 $(2\text{--}4)\times10^{-3}$ & $(2.7\text{--}5.3)\times10^{4}$ &
 $(0.9\text{--}1.9)\times10^{-5}$ & $\mathbf{0.28\text{--}0.55}$\\
(c) oscillation rate $\Delta M$ itself & $1.1\times10^{2}$ & $1$ &
 $0.40$ & $1.4\times10^{4}$\\
(c$'$) gauge damping $g^{2}T$ & $1.5\times10^{3}$ & $0.07$ & $0.14$ &
 $1.7\times10^{4}$\\
(d) Hubble rate $H(T = M_N)$ & $2.1\times10^{-11}$ & $5\times10^{12}$ &
 $1\times10^{-13}$ & $3\times10^{-9}$\\
\bottomrule
\end{tabular}
\end{table}

\subsection{The six conditions, and which one is the crux}

The candidate identification decomposes into six conditions. Three are
established, one is plausible, one is computed with its closure open, and
one is the crux. They are stated here in full because a reader cannot
weigh the chain without them.

\begin{table}[htbp]
\centering
\small
\caption[The six conditions C1--C6 underlying the regulator identification.]{The six conditions C1--C6 underlying the regulator
identification. \textbf{C4 is the crux}: its outstanding content is a
full Kadanoff--Baym resummation, and it is the single most valuable
calculation anyone could do to this chain, in either direction. Home:
\ATen{} Appendix~I; carried in the roadmap of \PaperPFive{}.}
\label{tab:lg-c1c6}
\begin{tabular}{@{}ll>{\raggedright\arraybackslash}p{0.52\textwidth}@{}}
\toprule
 & Condition & Status\\
\midrule
C1 & flavor-odd source & $\operatorname{Tr}S = 0$ exactly for real $R$,
 Eq.~\eqref{eq:lg-traceless}. \emph{Established} (algebraic;
 machine-checked)\\
C2 & flavor-blind cancellation & gauge damping enters through the
 identity projector and drops from the flavor-odd channel,
 Eq.~\eqref{eq:lg-blind-cancel}. \emph{Established} within the
 density-matrix framework; its persistence under full resummation folds
 into C4\\
C3 & two-level reduction & the Lorentzian envelope and the
 $|\mathcal{R}| \le \tfrac12$ bound survive for every regulator value,
 Eq.~\eqref{eq:lg-twolevel}. \emph{Established} at reduction level, given
 C4\\
C4 & \textbf{regulator identification (the crux)} & in the regime
 $\Gamma_\tau/\Gamma_N \sim 10^{7}$ with the flavor-summed source zero,
 the interference time of the flavor-odd source is $1/\Gamma_\tau$, not
 $1/\Gamma_N$: the vacuum $N$-width regulator is superseded in-medium.
 \textbf{Open at derivation level} --- structurally supported, but the
 full Kadanoff--Baym computation has not been carried out, and the
 standard fully-flavored Boltzmann limit would return the fatal
 $N$-width class instead\\
C5 & the coefficient & $c_\tau \in [5\times10^{-3},10^{-2}]$ at
 $T \simeq M_N$; the kinetic-estimate, regime-boundary and equilibration
 classes of the flavor-equilibration literature agree within a factor
 two. \emph{Plausible}; the precise value open\\
C6 & window throughput & the temperature-integrated window factor is part
 of the honest chain, and the priced $\mathcal{O}(1)$ prefactors must
 supply a remaining one-sided factor \emph{from below}. \emph{Computed}
 at Boltzmann level (\S\ref{sec:lg-boltzmann}); closure open\\
\bottomrule
\end{tabular}
\end{table}

\begin{euclidremark}[What is being claimed, and what is not]
It would be easy to read \eqref{eq:lg-regulator-closed} as a derivation.
It is not one. What is derived is that \emph{if} the flavor-measuring
rate is the regulator, then the ledger integer and the tau Yukawa fix the
Breit--Wigner factor with no freedom, and the number that comes out is
the one the chain needs. What is not derived is the antecedent. The
honest grade for the magnitude of the asymmetry is therefore
derived-conditional, and it stays that way for the rest of this Part.
After all, a framework's value lies partly in knowing which of its
statements are load-bearing: here it is one, it has a name, and it has a
calculation attached that would settle it either way.
\end{euclidremark}

\section{The washout, flavor by flavor: the electron door}
\label{sec:lg-washout}

\subsection{Why anything survives at all}

By Lemma~\ref{lem:lg-flavour-odd} the flavor-summed asymmetry is exactly
zero. Whatever survives to become matter survives because the three
flavors are washed out at \emph{different} rates. This section computes
those rates, and the answer turns on a number this paper derived for an
entirely different reason.

The strength with which flavor $\alpha$ is erased is set by how strongly
the heavy states couple to it, summarized by the flavor-projected
effective mass:
\begin{equation}
\widetilde m_\alpha \;=\; \sum_i m_i\,\bigl|U_{\alpha i}\bigr|^{2},
\qquad
K_\alpha \;=\; \frac{\widetilde m_\alpha}{m_\star^{\rm wash}},
\qquad
m_\star^{\rm wash} = 1.07\times10^{-3}~\mathrm{eV}.
\label{eq:lg-Kalpha}
\end{equation}
Read this physically: $|U_{\alpha i}|^{2}$ is the probability that mass
state $i$ ``is'' flavor $\alpha$, so $\widetilde m_\alpha$ is the share
of the total seesaw coupling that flavor $\alpha$ owns, and $K_\alpha$
compares that flavor's erasure rate with the expansion rate.

\subsection{The equilibrium mass, and why it is not mysterious}

The scale in the denominator of \eqref{eq:lg-Kalpha} is fixed by
comparing the decay rate with the Hubble rate at the moment the decays
matter:
\begin{equation}
\Kwash \;\equiv\; \frac{\Gamma_N}{H}\bigg|_{T = M_N}
\;=\; \frac{\widetilde m}{m_\star^{\rm wash}},
\qquad
m_\star^{\rm wash}
\;=\; \frac{16\pi^{5/2}\sqrt{\gstar}}{3\sqrt{5}}\,
      \frac{\bar v^{2}}{M_{\rm Pl}}
\;=\; 1.07\times10^{-3}~\mathrm{eV}.
\label{eq:lg-kwash}
\end{equation}
The second equality is worth reading twice, because it is what makes
$m_\star^{\rm wash}$ meaningful rather than mysterious:
\emph{$m_\star^{\rm wash}$ is the value of the effective neutrino mass at
which the decay rate exactly equals the Hubble rate.} Above it the decays
are fast compared with the expansion and the plasma sits near
equilibrium, erasing what it makes; below it the decays are slow and the
asymmetry survives.

With the light spectrum of Part~\ref{part:X},
$\widetilde m \sim m_3 = 50.5$~meV, so
\begin{equation}
\Kwash \;=\; \frac{50.5}{1.07} \;\simeq\; 47,
\qquad
\kappa \;\simeq\; \frac{1}{\Kwash} \;=\; 2.1\times10^{-2},
\label{eq:lg-kappa}
\end{equation}
and the chain adopts $\kappa = 1.0\times10^{-2}$ within the uncomputed
order-unity factor. $\Kwash \simeq 47 \gg 1$ is the \emph{strong washout}
regime: about ninety-eight per cent of the asymmetry that was ever made
was subsequently erased.

\subsection{A bookkeeping convention that must be declared}

Two different effective masses appear in this chain and they are not the
same number. The heavy-sector resonance width is evaluated at the
seam-matched one-generation Yukawa, a property of the seam-localized
texture, whose effective mass is the seam scale itself,
$\widetilde m \simeq 8.4$~meV. The washout strength is evaluated at the
light-neutrino mass the thermal bath actually sees,
$\widetilde m \sim m_3 \simeq 50.5$~meV. The factor of $\simeq 13$ this
induces in $Y_\nu^{2}$ is declared here, is accounted for, and is priced
within the order-of-magnitude register of \ATen{}'s leptogenesis
appendix. At the seam convention the numbers would instead read
$\Kwash \simeq 7.9$ and $\kappa \simeq 0.13$. This paper's coefficient
$\pi/\EGUT$ is the upstream input of both.

\subsection{The flavored washouts, and the electron door}

Computing \eqref{eq:lg-Kalpha} on the derived masses and the derived
mixing matrix of Part~\ref{part:X}:

\begin{table}[htbp]
\centering
\small
\caption[The flavored washouts, computed from the derived masses and the derived PMNS matrix, against the values \ATen{} Appendix~I obtains independently on the.]{The flavored washouts, computed from the derived masses and the
derived PMNS matrix, against the values \ATen{} Appendix~I obtains
independently on the same ledger. The last column is the efficiency each
flavor's asymmetry enjoys: the electron flavor survives an order of
magnitude better than the other two.}
\label{tab:lg-Kalpha}
\begin{tabular}{@{}lrrrlr@{}}
\toprule
Flavor & $\widetilde m_\alpha$ [meV] & $K_\alpha$ (here) &
 $K_\alpha$ (\ATen{}) & regime & $\kappa(K_\alpha)$\\
\midrule
$e$    & $3.7$  & $3.5$  & $3.4$  & mild washout   & $0.137$\\
$\mu$  & $31.3$ & $29.3$ & $28.0$ & strong washout & $0.0084$\\
$\tau$ & $24.1$ & $22.5$ & $22.4$ & strong washout & $0.0116$\\
\midrule
\multicolumn{2}{@{}l}{sum $= \Summnu = 59.2$~meV} & $55.3$ & & &\\
\bottomrule
\end{tabular}
\end{table}

The electron flavor is washed out an order of magnitude more weakly than
the other two. The reason is arithmetic and it is this paper's:
\begin{equation}
\widetilde m_e \;=\; \sum_i m_i |U_{ei}|^{2},
\qquad
|U_{e3}|^{2} \;=\; \sin^{2}\theta_{13} \;=\; \frac{N}{4K} \;=\; \frac{3}{140}
\;=\; 0.0214 .
\label{eq:lg-Ue3}
\end{equation}
The heaviest light state $m_3 = 50.5$~meV dominates
$\widetilde m_\mu$ and $\widetilde m_\tau$ because $|U_{\mu 3}|^{2}$ and
$|U_{\tau3}|^{2}$ are of order one half. It barely touches
$\widetilde m_e$, because $|U_{e3}|^{2}$ is small --- and it is small
because the reactor angle is $N/4K$.

\begin{euclidremark}[The electron door]
The flavor-summed asymmetry is exactly zero. What survives is the
difference between how fast the flavors are erased, and that difference
exists because $\sin^{2}\theta_{13} = 3/140$ decouples the electron
flavor from the heavy third mass state. The matter in the universe leaks
out through the electron flavor --- the numerical integration of
\S\ref{sec:lg-boltzmann} finds \emph{ninety-six per cent} of the final
asymmetry in it --- at a rate set by the same ledger integers that fix
the reactor angle.

Stated at its strongest, and then immediately fenced: in this framework
\emph{the reactor angle being small is why there is something rather than
nothing.} The fence is that the statement holds within a chain whose
magnitude is conditional on \Hphi{}. But the structure --- flavor-summed
zero, survival by unequal erasure, unequal erasure by a derived mixing
angle --- does not depend on that hypothesis at all.
\end{euclidremark}

\section{The Boltzmann system, integrated}
\label{sec:lg-boltzmann}

\begin{figure}[htbp]
\centering
\resizebox{0.773\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize]
\draw[line width=0.90pt,draw=black!80] (0,-0.1) -- (12.7,-0.1);
\draw[line width=0.90pt,draw=black!80] (0,-0.1) -- (0,9.6);
\node[anchor=north,font=\scriptsize\itshape] at (6.3,-0.85) {$z=M_N/T$ \quad (time runs to the right)};
\node[rotate=90,anchor=south,font=\scriptsize\itshape] at (-1.05,4.7) {abundance $N_X=n_X/n_\gamma$};
\draw (0.697,-0.1) -- (0.697,-0.22);
\node[anchor=north,font=\scriptsize] at (0.697,-0.24) {$0.2$};
\draw (4.598,-0.1) -- (4.598,-0.22);
\node[anchor=north,font=\scriptsize] at (4.598,-0.24) {$1$};
\draw (8.499,-0.1) -- (8.499,-0.22);
\node[anchor=north,font=\scriptsize] at (8.499,-0.24) {$5$};
\draw (12.400,-0.1) -- (12.400,-0.22);
\node[anchor=north,font=\scriptsize] at (12.400,-0.24) {$25$};
\draw (0,0.000) -- (-0.12,0.000);
\node[anchor=east,font=\scriptsize] at (-0.14,0.000) {$10^{-11}$};
\draw (0,1.621) -- (-0.12,1.621);
\node[anchor=east,font=\scriptsize] at (-0.14,1.621) {$10^{-9}$};
\draw (0,3.241) -- (-0.12,3.241);
\node[anchor=east,font=\scriptsize] at (-0.14,3.241) {$10^{-7}$};
\draw (0,4.862) -- (-0.12,4.862);
\node[anchor=east,font=\scriptsize] at (-0.14,4.862) {$10^{-5}$};
\draw (0,6.483) -- (-0.12,6.483);
\node[anchor=east,font=\scriptsize] at (-0.14,6.483) {$10^{-3}$};
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\node[anchor=west,font=\scriptsize,text=black!85] at (5.6,7.992) {$N_N$};
\node[anchor=west,font=\scriptsize,text=black!50] at (2.3,8.984) {$N_N^{\rm eq}$};
\node[anchor=west,font=\scriptsize,text=blue!60!black] at (9.2,2.097) {$|N_{\Delta e}|$};
\node[anchor=west,font=\scriptsize,text=red!65!black] at (9.2,1.105) {$|N_{\Delta\tau}|$};
\node[anchor=west,font=\scriptsize,text=orange!75!black] at (9.2,0.530) {$|N_{\Delta\mu}|$};
\draw[dotted,draw=black!45] (6.278,-0.1) -- (6.278,9.2);
\draw[dotted,draw=black!45] (7.958,-0.1) -- (7.958,9.2);
\node[anchor=south,font=\scriptsize,text=black!55,align=center] at (7.120,9.2) {the window};
\end{tikzpicture}}
\caption[The Boltzmann system integrated, with an independent check.]{The Boltzmann system integrated. Heavy abundance $N_N$ against its equilibrium value $N_N^{\rm eq}$ (the two are indistinguishable on this scale --- that \emph{is} the strong-washout regime), and the three flavored asymmetries, from \eqref{eq:lg-boltz-N}--\eqref{eq:lg-boltz-D} with the paper's own parameters at $c_\tau=7.5\times10^{-3}$. Each asymmetry is built up around $z\simeq1$, peaks near $z\simeq2$, and is then cut down by inverse decays until they switch off; the electron flavor survives an order of magnitude better than the other two because its washout $K_e=3.5$ is mild, which is the electron door of \S\ref{sec:lg-washout}. \emph{Independent check:} integrating this system afresh, with no further freedom, gives $\etaB=(0.23$--$0.45)\times10^{-10}$ across $c_\tau\in[5,10]\times10^{-3}$, against Level~B's $0.86^{+0.52}_{-0.32}\times10^{-10}$ --- the two agree once the declared pattern-acceptance and washout-kernel factors of \S\ref{sec:lg-errorbar} are applied, and not before.}
\label{fig:lg-boltzmann}
\end{figure}


The estimates so far are single-temperature snapshots. The asymmetry is
made over a range of temperatures, against a washout that is itself
changing, with a resonance factor that changes too --- and the honest
question is what the full time integration costs. This section states the
system, its validation, and its three results. The integration is
\ATen{}'s, Appendix~I; it is reported here because a reader cannot
otherwise tell how much of the chain is arithmetic and how much is
dynamics.

\subsection{The system}

Two coupled equations, in the variable $z = M_N/T$ --- time runs with
$z$, the early universe at small $z$ and the late at large --- and in
photon-normalized abundances $N_X = n_X/n_\gamma$:
\begin{align}
\frac{{\rm d}N_N}{{\rm d}z}
&= -\,D(z)\,\bigl(N_N - N_N^{\rm eq}\bigr),
\label{eq:lg-boltz-N}\\[2pt]
\frac{{\rm d}N_{\Delta\alpha}}{{\rm d}z}
&= \underbrace{\epsilon_\alpha(z)\,D(z)\,
   \bigl(N_N - N_N^{\rm eq}\bigr)}_{\text{source}}
 \;-\;
   \underbrace{\tfrac14\,K_\alpha\,z^{3}K_1(z)\,N_{\Delta\alpha}}_{\text{washout}} .
\label{eq:lg-boltz-D}
\end{align}

Term by term. Equation~\eqref{eq:lg-boltz-N} says the heavy abundance
relaxes toward its equilibrium value $N_N^{\rm eq}(z) =
\tfrac34\cdot\tfrac12 z^{2}K_2(z)$ at a rate $D(z) = K_{\rm tot}\,
zK_1(z)/K_2(z)$; the ratio of Bessel functions is time dilation, tending
to $z/2$ at small $z$ because a relativistic heavy state decays slowly in
the bath frame. In \eqref{eq:lg-boltz-D} the source is proportional to
the \emph{departure} from equilibrium --- Sakharov's third condition
appearing explicitly as a factor --- and the washout term erases the
asymmetry already present at a rate proportional to that flavor's
$K_\alpha$ from Table~\ref{tab:lg-Kalpha}.

The asymmetry per decay is itself temperature-dependent, because the
regulator of \S\ref{sec:lg-resonance} is thermal:
\begin{equation}
\epsilon_\alpha(z) \;=\; \bar\epsilon_\alpha\cdot 2R(z),
\qquad
R(z) \;=\; \frac{2\,\Delta M\,\Geff(z)}{4\Delta M^{2} + \Geff(z)^{2}},
\qquad
\Geff(z) \;=\; c_\tau y_\tau^{2}\,\frac{M_N}{z},
\label{eq:lg-eps-z}
\end{equation}
with $\bar\epsilon_\alpha = (+3.4,\,+3.1,\,-6.6)\times10^{-3}$ the flavored
pattern computed on the ledger inputs (peak-normalized). The sum
$\sum_\alpha\bar\epsilon_\alpha$ vanishes \emph{exactly}, by
Lemma~\ref{lem:lg-flavour-odd}; a reader who adds the three printed
values will get $-1\times10^{-4}$ rather than zero, which is the rounding
to two significant figures and not a failure of the lemma. The zero is
what makes the asymmetry a matter of flavor \emph{differences}, so it is
worth not leaving the reader to wonder. Since $\Geff \propto T$, the resonance
factor $R(z)$ \emph{falls} as the universe cools --- the interference is
measured more slowly while the splitting stays put --- and that decline
is the physical origin of the window suppression the integration
measures.

For the reader who wants the efficiency in closed form before the
numerics, the standard strong-washout fit is
\begin{equation}
\kappa(K) \;\simeq\; \frac{2}{K\,z_B(K)}
\Bigl(1 - e^{-K z_B(K)/2}\Bigr),
\qquad
z_B(K) \;\simeq\; 2 + 4K^{0.13}e^{-2.5/K},
\label{eq:lg-bdp}
\end{equation}
\citep{BuchmullerDiBariPlumacher2005}, whose physical content is one
sentence: the asymmetry that survives is the asymmetry made in the
\emph{last} moment before the inverse decays shut off, at $z = z_B$, and
by then only a fraction $\sim 1/(Kz_B)$ of the heavy states remain to make
it. Everything earlier is erased; $\kappa$ is the fraction made late.

\subsection{Where the two right-hand sides come from}
\label{sec:lg-collision}

The system \eqref{eq:lg-boltz-N}--\eqref{eq:lg-boltz-D} is usually quoted.
It is short enough to derive, and the derivation explains two features
that otherwise look arbitrary: the Bessel functions, and the factor of a
quarter on the washout.

\paragraph{The collision integral.} For a decay $N \to \ell H$ the
Boltzmann equation for the heavy abundance is, in comoving form,
\begin{equation}
\frac{{\rm d}n_N}{{\rm d}t} + 3Hn_N
\;=\; -\,\bigl\langle \Gamma_N \bigr\rangle
      \bigl(n_N - n_N^{\rm eq}\bigr),
\label{eq:lg-collision}
\end{equation}
where the two terms on the right are the decay and its inverse. The
inverse decay carries $n_N^{\rm eq}$ rather than $n_N$ because the
incoming leptons and Higgses are themselves in equilibrium: detailed
balance fixes the rate at which they reconstitute a heavy state, and that
rate depends on the bath, not on how many heavy states happen to be
present. This is why the right-hand side is proportional to the
\emph{departure} from equilibrium --- Sakharov's third condition is not
imposed here, it is a factor in the equation.

\paragraph{Where the Bessel functions come from.} The bracket in
\eqref{eq:lg-collision} is a thermal average of the decay rate over the
heavy states' momentum distribution, and a moving particle decays slowly:
\begin{equation}
\bigl\langle \Gamma_N \bigr\rangle
\;=\; \Gamma_N \Bigl\langle \frac{M_N}{E} \Bigr\rangle
\;=\; \Gamma_N \frac{K_1(z)}{K_2(z)},
\qquad z \equiv \frac{M_N}{T},
\label{eq:lg-timedilation}
\end{equation}
because for a Maxwell--Boltzmann distribution the average of $M/E$ is
exactly the ratio of the two modified Bessel functions. At small $z$ ---
that is, at high temperature --- $K_1/K_2 \to z/2$, which is the
statement that a highly relativistic heavy state is time-dilated into near
immortality. At large $z$ the ratio tends to $1$ and the decay proceeds at
its rest-frame rate. Changing the independent variable from $t$ to $z$ and
normalizing to the photon density turns \eqref{eq:lg-collision} into
\eqref{eq:lg-boltz-N} with
$D(z) = K_{\rm tot}\,zK_1(z)/K_2(z)$.

The equilibrium abundance is the same integral without the rate:
$N_N^{\rm eq}(z) = \tfrac38 z^{2}K_2(z)$, which is the Maxwell--Boltzmann
number density of a species of mass $M_N$ at temperature $M_N/z$, divided
by the photon density.

\paragraph{The washout term, and its factor of a quarter.} The asymmetry
equation has a source and a sink. The source is the same decay term
weighted by the asymmetry per decay, which is the first term of
\eqref{eq:lg-boltz-D}. The sink is inverse decay acting on the asymmetry
already present: a lepton excess is depleted by $\ell H \to N$ at a rate
set by the same Yukawa.

The washout rate is \emph{not} equal to the source rate, and the ratio is
where the quarter comes from. The inverse-decay rate involves the
equilibrium density of the incoming leptons rather than of the heavy
state, which supplies one factor of $\tfrac12$ after the spin and
particle--antiparticle averaging that the decay term does not carry; and
the asymmetry is depleted only through the difference between $\ell$ and
$\bar\ell$ inverse decays, which supplies another. Collecting the thermal
average for the inverse process --- which weights $K_1$ rather than
$K_1/K_2$, because the incoming particles are massless and their
distribution supplies the remaining $z^{2}$ --- gives the washout term
\begin{equation}
\Gamma_\alpha^{\rm wash}(z) \;=\; \tfrac14\,K_\alpha\,z^{3}K_1(z),
\label{eq:lg-washterm}
\end{equation}
which is the second term of \eqref{eq:lg-boltz-D}. Its $z^{3}K_1(z)$ rises
from zero, peaks near $z \simeq 2$, and then falls exponentially: inverse
decays switch off once the bath is too cold to make a heavy state at all,
and that switch-off is what allows any asymmetry to survive.

\begin{euclidremark}[Physics intuition for the shape of the answer]
Two competing exponentials decide everything. The source needs the heavy
states to be out of equilibrium, which happens only once the bath has
cooled below their mass --- so it turns on late. The washout needs the
bath to be hot enough to run the decay backwards --- so it turns off late
too, but not quite as late. The asymmetry that survives is made in the
narrow window between the two, at $z \simeq 2$--$4$, which is exactly
where the numerical integration of \S\ref{sec:lg-boltzmann} finds it. And
because the resonance factor $R(z)$ is \emph{falling} through that window
--- $\Geff \propto T$ while $\Delta M$ does not --- the asymmetry is made
just after the best moment for making it. That mismatch, and not any
uncertainty in the inputs, is the physical origin of the factor
$E = 7.1$ in \eqref{eq:lg-E}.
\end{euclidremark}

\paragraph{Scope, declared.} The system \eqref{eq:lg-boltz-N}--%
\eqref{eq:lg-boltz-D} carries decays and inverse decays only. Lepton-number
-violating scatterings, spectator processes, thermal-mass corrections and
the relativistic edge are deliberately omitted and separately priced in
\S\ref{sec:lg-errorbar}. Every one of them points the same way.

\subsection{Three results}

\paragraph{First: the sign is an output, not an input.} The flavored
pattern fed in sums to zero. The integration nonetheless returns a net
positive asymmetry, because the tau-dominated negative contributions sit
in strong washout and are erased faster than the electron-dominated
positive one. Ninety-six per cent of the surviving asymmetry is in the
electron flavor. The universe is made of matter rather than antimatter
in this chain because of the ordering of three erasure rates, and that
ordering traces to \eqref{eq:lg-Ue3}.

\paragraph{Second: the window, measured.} At central parameters the
integration gives
\begin{equation}
\etaB^{\rm Boltzmann} \;=\; 1.0\times10^{-10},
\qquad
\frac{\etaB^{\rm Boltzmann}}{\etaB^{\obs}} \;=\; 0.16 .
\label{eq:lg-boltz-central}
\end{equation}
The deficit against observation --- the quantity condition C6 named but
could not compute --- is now a measured number with an error bar rather
than a bracket spanning a decade:
\begin{equation}
\boxed{\;
E \;\equiv\; \frac{\etaB^{\obs}}{\etaB^{\rm B}} \;=\; 7.1,
\qquad 68\%\ \text{interval } [4.4,\ 11.4],
\;}
\label{eq:lg-E}
\end{equation}
where $\etaB^{\rm B} = 0.86^{+0.52}_{-0.32}\times10^{-10}$ is the Level-B
value of \eqref{eq:lg-levelB} --- the integration above, scanned over the
four remaining freedoms. The distinction is easy to lose and worth
keeping: at \emph{strictly} central parameters the deficit is
$6.12/1.0 = 6.1$ and carries no band, whereas $E$ carries one because the
scan does. Every digit of $E$ is Level B's: $6.12/0.86 = 7.1$, and the
interval is $6.12/1.38 = 4.4$ and $6.12/0.54 = 11.4$.

\paragraph{Third: the deficit is one-sided.} The asymmetry is generated
at $z \simeq 1.5$--$4$, which is off the best part of the resonance
because $\Geff \propto 1/z$. Every effect omitted from the scope
declaration above moves production \emph{earlier}, where $R(z)$ is
larger. The deficit therefore points up, not in an unknown direction ---
which is the difference between a discrepancy and a missing term.

\begin{euclidremark}[Why a factor of seven is not a small matter, and not a fatal one]
It would be dishonest to present \eqref{eq:lg-E} as agreement. The chain
as integrated makes about a seventh of the observed asymmetry. It would
be equally dishonest to present it as refutation: the integration omits
four standard transport effects by construction, each of which is known
to increase the answer, and \S\ref{sec:lg-errorbar} prices them. The
useful statement is the one in the middle --- the chain lands within an
order of magnitude of the observed baryon asymmetry from a scale fixed by
an anomaly count, with no fitted parameter, and the residual factor is
now measured rather than bracketed.
\end{euclidremark}

\section{Sphalerons and the transfer to baryon number}
\label{sec:lg-sphaleron}

Everything so far has made a lepton asymmetry. The universe is short of
\emph{baryons}, and something must convert.

The converter is the electroweak sphaleron
\citep{KuzminRubakovShaposhnikov1985}: a non-perturbative Standard-Model
process, unsuppressed at temperatures above the electroweak scale, which
violates $B + L$ while conserving $B - L$. It cannot change $B - L$, so
it cannot create the asymmetry; but it is free to \emph{redistribute}
whatever $B - L$ exists between the baryon and lepton sectors, and it
does so until the chemical potentials balance.

Solving the equilibrium conditions --- sphaleron equilibrium, Yukawa
equilibrium, hypercharge neutrality --- for the Standard Model with three
generations and one Higgs doublet gives \citep{HarveyTurner1990}
\begin{equation}
B \;=\; \frac{28}{79}\,(B-L),
\qquad
\csph \;=\; \frac{28}{79} \;=\; 0.35443 .
\label{eq:lg-2879}
\end{equation}

\paragraph{Grade.} This factor is \emph{textbook, not derived here}. The
$28/79$ is a Standard-Model equilibrium computation that this framework
inherits unchanged, and it is flagged as inherited rather than presented
as a result. Its accuracy is not in question; its provenance is simply
not this paper's.

There is an interesting tension worth naming. The same framework that
forbids $\Delta B = 1$ and $\Delta B = 2$ outright --- and thereby makes
$\tau_p = \infty$ a theorem --- relies on $\Delta B \neq 0$ sphaleron
processes to make baryons at all. The resolution is the selection rule of
\S\ref{subsec:fsm4}: sphalerons change baryon number by
$N_{\rm gen} = 3$ units at a time, which is admissible, while proton
decay would require $\Delta B = 1$, which is not. The framework is
therefore not merely consistent on this point; it is consistent for a
reason that is itself a prediction.

\section{The six factors, and the assembly}
\label{sec:lg-assembly}

\begin{figure}[htbp]
\centering
\resizebox{0.663\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize,
  bx/.style={draw=black!65,rounded corners=1.6pt,inner sep=3.2pt,align=center,fill=black!3},
  drv/.style={bx,draw=blue!55!black,fill=blue!5},
  opn/.style={bx,draw=red!60!black,fill=red!6,align=center},
  ar/.style={-{Stealth[length=2.0mm]},draw=black!60,line width=0.7pt}]
\node[drv] (K)  at (0,0)      {$K=LJ=35$\\ \scriptsize anomaly audit};
\node[drv] (M)  at (3.1,1.15) {$\Mlock=\tfrac12\EPl e^{-K}$\\ \scriptsize $3.85$~TeV};
\node[drv] (dm) at (3.1,0)    {$\Delta M=\Mlock/K$\\ \scriptsize $110$~GeV};
\node[drv] (Y)  at (3.1,-1.15){$Y_\nu^{2}=\pi\Mlock/\EGUT$\\ \scriptsize $7.28\times10^{-7}$};
\node[opn] (C4) at (6.5,0.60) {$\epsres$ (resonance)\\ \scriptsize \textbf{C4: open}};
\node[bx]  (kap)at (6.5,-1.15){$\kappa$ (washout)\\ \scriptsize $K_\alpha=(3.5,29.3,22.5)$};
\node[drv] (thr)at (3.1,-2.85){$\epsres\kappa\simeq e^{-K/2}$\\ \scriptsize \textbf{postulated}};
\node[bx]  (six)at (6.5,-2.85){$\times\;\tfrac{s}{n_\gamma}\,\csph\,\tfrac{1.25}{\gstar}$\\ \scriptsize textbook};
\node[drv] (eta)at (9.6,-2.85){$\etaB=7.34\times10^{-10}$\\ \scriptsize $+20\%$ on observation};
\foreach \a/\b in {K/M,K/dm,K/Y,M/C4,dm/C4,Y/C4,Y/kap}
  \draw[ar] (\a) -- (\b);
\draw[ar] (C4) -- (7.9,0.60) -- (7.9,-3.55) -- (3.1,-3.55) -- (thr);
\draw[ar] (kap) -- (6.5,-1.9) -- (2.2,-1.9) -- (2.2,-2.85) -- (thr);
\draw[ar] (thr) -- (six);
\draw[ar] (six) -- (eta);
\end{tikzpicture}}
\caption[The chain from $K=35$ to $\etaB$, with the open link marked.]{The chain end to end. Blue: quantities that descend from the ledger integer $K=35$ with no freedom left. Red: the single open link. Every quantity downstream of C4 inherits its conditionality, which is why the Part reports its result at three levels (Fig.~\ref{fig:lg-three-levels}) rather than as a number --- and why the falsification channel F-SM-7 is written so that one finite-temperature calculation settles the whole diagram in either direction.}
\label{fig:lg-chain}
\end{figure}


\subsection{The bookkeeping variable}

Number densities dilute with expansion, so the useful variable is a
ratio. The entropy density of a relativistic bath is
\begin{equation}
s \;=\; \frac{2\pi^{2}}{45}\,g_{*s}(T)\,T^{3},
\label{eq:lg-entropy}
\end{equation}
with $g_{*s}$ counting relativistic degrees of freedom weighted for
entropy (bosons $1$, fermions $7/8$). Its virtue is a conservation law:
for adiabatic expansion the total comoving entropy $S = sa^{3}$ is
constant, which absorbs every reheating event automatically --- when a
species annihilates, $g_{*s}$ falls and $T$ rises to compensate. So define
the \emph{yield} $Y_X \equiv n_X/s$, a pure number that does not change
with expansion.

\subsection{The six factors, one at a time}

\paragraph{1. How many heavy states there were: $Y_N^{\rm eq}$.} At
$T \gg M_N$ the heavy states are relativistic and in equilibrium, so they
are one more species in the bath. With
$n_N^{\rm eq} = (3\zeta(3)/4\pi^{2})g_N T^{3}$ and $g_N = 2$ for a
Majorana fermion, dividing by \eqref{eq:lg-entropy}:
\begin{equation}
Y_N^{\rm eq}
\;=\; \frac{3\zeta(3)}{4\pi^{2}}\,\frac{45}{2\pi^{2}}\,\frac{g_N}{g_{*s}}
\;=\; \frac{0.42}{\gstar}
\qquad \text{per species.}
\label{eq:lg-YNeq}
\end{equation}
There are three quasi-degenerate heavy states, so the total is
$3\times 0.42/\gstar = 1.25/\gstar$. This factor of $1.25$ is carried
explicitly rather than absorbed into an order-unity allowance, because it
is a definite number and it points up.

\paragraph{2. How biased each decay was: $\epsres$.} The resonant \CP{}
asymmetry per decay, Eq.~\eqref{eq:lg-eps-def}, at
$\epsres \simeq 2.5\times10^{-6}$. \emph{Derived-conditional} on \Hphi{}.

\paragraph{3. How much survived: $\kappa$.} The washout efficiency,
$\kappa \simeq 1.0\times10^{-2}$ from \eqref{eq:lg-kappa}.
\emph{Derived-conditional.}

\paragraph{4. How much became baryons: $\csph = 28/79$.} From
\eqref{eq:lg-2879}. \emph{Inherited.}

\paragraph{5. How much else was in the bath: $\gstar$.} Above the
electroweak scale the whole Standard Model is relativistic. Counting:
eight gluons $\times$ two helicities $= 16$; the four electroweak bosons
before symmetry breaking, $4\times2 = 8$; the complex Higgs doublet, $4$
--- bosons total $28$. Then six quark flavors $\times$ three colors
$\times$ two spins $\times$ two (particle/antiparticle) $= 72$; three
charged leptons $\times2\times2 = 12$; three neutrinos $\times1\times2
= 6$ --- fermions total $90$. Hence
\begin{equation}
\gstar \;=\; 28 + \tfrac78\times 90 \;=\; 28 + 78.75 \;=\; 106.75 .
\label{eq:lg-gstar}
\end{equation}
It enters twice: once in the entropy denominator, and once through
$H \propto \sqrt{\gstar}\,T^{2}/M_{\rm Pl}$ inside
$m_\star^{\rm wash}$.

\paragraph{6. From entropy to photons: $s/n_\gamma$.} Astronomers measure
baryons per photon, not per unit entropy. The conversion is the ratio of
the two reference densities today:
\begin{equation}
\frac{s}{n_\gamma}\bigg|_{T_0}
\;=\; \frac{(2\pi^{2}/45)\,g_{*s}(T_0)}{2\zeta(3)/\pi^{2}}
\;=\; \frac{0.43865\times 3.909}{0.24359}
\;=\; 7.04,
\qquad \etaB \;=\; 7.04\,Y_B .
\label{eq:lg-704}
\end{equation}
Here $g_{*s}(T_0) = 2 + \tfrac78\times6\times\tfrac{4}{11} = 3.909$: two
photon polarizations plus six neutrino states, colder than the photons by
$(4/11)^{1/3}$ because electron--positron annihilation heated the photons
after the neutrinos had decoupled.

\subsection{The assembly}

Each decay left a fractional excess $\epsres$, a fraction $\kappa$ of the
total survived washout, and the heavy abundance was $1.25/\gstar$:
\begin{equation}
Y_{B-L} \;\simeq\; \epsres\,\kappa \sum_i Y_{N_i}^{\rm eq}
\;\simeq\; \frac{1.25\,\epsres\,\kappa}{\gstar}.
\label{eq:lg-YL}
\end{equation}
Sphalerons move $28/79$ of it into baryons and
\eqref{eq:lg-704} converts the yield to the observed ratio:
\begin{equation}
\boxed{\;
\etaB \;\simeq\; 7.04\;\csph\;\frac{1.25\,\epsres\,\kappa}{\gstar}
\;=\; 7.04\left(\frac{28}{79}\right)\frac{1.25\,e^{-K/2}}{\gstar}.
\;}
\label{eq:lg-etaB-pred}
\end{equation}

\begin{table}[htbp]
\centering
\small
\caption[The chain arithmetic, factor by factor, with the running product.]{The chain arithmetic, factor by factor, with the running
product. Read down the last column to watch $10^{-8}$ become
$7\times10^{-10}$. Every factor is either a number this paper derives or a piece of Standard-Model arithmetic; none is fitted, and the running product makes the size of each individual step visible.}
\label{tab:lg-arithmetic}
\begin{tabular}{@{}llrl@{}}
\toprule
Step & Factor & Value & Running product\\
\midrule
1 & throughput $\epsres\,\kappa = e^{-K/2}$ & $2.511\times10^{-8}$ &
 $2.511\times10^{-8}$\\
2 & $\times$ triplet yield, $\sum_i Y_{N_i}^{\rm eq}\cdot\gstar$ &
 $1.249$ & $3.137\times10^{-8}$\\
3 & $\div$ degrees of freedom, $\gstar$ & $106.75$ &
 $2.939\times10^{-10}$\\
4 & $\times$ sphaleron transfer, $28/79$ & $0.35443$ &
 $1.042\times10^{-10}$\\
5 & $\times$ entropy-to-photon, $s/n_\gamma$ & $7.04$ &
 $\mathbf{7.334\times10^{-10}}$\\
\midrule
\multicolumn{3}{@{}l}{observed \citep{Planck2018}} &
 $(6.12\pm0.04)\times10^{-10}$\\
\multicolumn{3}{@{}l}{central deviation} & $+20\%$\\
\multicolumn{3}{@{}l}{declared theory band} & $\times 3^{\pm1}$\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{How this number is to be quoted.} At headline precision the
prediction of this chain is
\begin{equation}
\boxed{\;
\etaB \;\approx\; 7\times10^{-10},
\qquad \text{derived band } (2.4\text{--}22)\times10^{-10}.
\;}
\label{eq:lg-etaB-headline}
\end{equation}
The exact product $7.334\times10^{-10}$ of
Table~\ref{tab:lg-arithmetic} belongs to the worked arithmetic and should
not be carried into summary statements: three significant figures on a
quantity with a factor-three band would misrepresent what is known. The
band is derived in \S\ref{sec:lg-errorbar} and is not reverse-engineered
to cover the datum --- as \eqref{eq:lg-E} makes plain, the chain's own
integration sits below observation, not on it.

\section{The throughput identity, and the exponent that is postulated}
\label{sec:lg-throughput}

Two of the six factors are not simply inherited: the \CP{} asymmetry and
the washout efficiency. Their product is the chain's central claim:
\begin{equation}
\boxed{\;
\epsres\,\kappa
\;\simeq\; (2.5\times10^{-6})(1.0\times10^{-2})
\;\simeq\; 2.51\times10^{-8}
\;=\; e^{-K/2} \;=\; e^{-17.5}.
\;}
\label{eq:lg-throughput}
\end{equation}
An explicit seesaw-plus-washout chain, built from a Yukawa fixed by the
seam and a washout fixed by the light spectrum, \emph{lands on} the
ledger's own throat-depth exponential rather than asserting it.

That is a striking statement and it must be fenced immediately, in three
ways.

\begin{enumerate}[leftmargin=2.1em]
\item \textbf{The agreement uses the order-unity freedom in $\kappa$.}
At the unadjusted $\kappa = 2.1\times10^{-2}$ of \eqref{eq:lg-kappa} the
left-hand side would read $5.0\times10^{-8}$, a factor of two high.
\item \textbf{The exponent $K/2$ is a postulate of the compression, not a
derivation.} It is geometrically suggestive --- $e^{-K/2}$ is
$\sqrt{e^{-K}}$, the square root of the lock's own suppression --- and it
is not derived. Deriving it is the second clause of the chain's open
conjecture.
\item \textbf{$\epsres$ is derived-conditional on \Hphi{}}, which
\S\ref{sec:lg-resonance} states in sharp form and marks open at
derivation level.
\end{enumerate}

\begin{euclidremark}[What would make this identity mean something]
An identity between an assembled chain and a ledger exponential is
either a coincidence or a theorem, and at present the paper cannot tell
which. Two calculations would settle it: a finite-temperature transport
computation discharging \Hphi{}, and a derivation of the exponent $K/2$
from the descent rather than from reconstruction. Until both are done the
correct reading of \eqref{eq:lg-throughput} is that a chain with no
fitted parameters lands within an order-unity factor of a number the
geometry already contained --- which is interesting, and is not proof.
\end{euclidremark}

\section{The error bar, derived}
\label{sec:lg-errorbar}

\begin{figure}[htbp]
\centering
\resizebox{0.950\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize]
\fill[red!60!black,opacity=0.13] (7.496,-0.5) rectangle (7.525,3.05);
\draw[line width=1.00pt,draw=red!60!black] (7.510,-0.5) -- (7.510,3.05);
\node[anchor=south,font=\scriptsize,text=red!55!black,align=center] at (7.510,3.07)
  {observed\\ $6.12\times10^{-10}$};
\draw[line width=1.00pt,draw=black!80] (0,-0.5) -- (11.2,-0.5);
\draw (0.000,-0.5) -- (0.000,-0.63);
\node[anchor=north,font=\scriptsize] at (0.000,-0.65) {$0.2$};
\draw (2.012,-0.5) -- (2.012,-0.63);
\node[anchor=north,font=\scriptsize] at (2.012,-0.65) {$0.5$};
\draw (3.533,-0.5) -- (3.533,-0.63);
\node[anchor=north,font=\scriptsize] at (3.533,-0.65) {$1$};
\draw (5.055,-0.5) -- (5.055,-0.63);
\node[anchor=north,font=\scriptsize] at (5.055,-0.65) {$2$};
\draw (7.066,-0.5) -- (7.066,-0.63);
\node[anchor=north,font=\scriptsize] at (7.066,-0.65) {$5$};
\draw (8.588,-0.5) -- (8.588,-0.63);
\node[anchor=north,font=\scriptsize] at (8.588,-0.65) {$10$};
\draw (10.110,-0.5) -- (10.110,-0.63);
\node[anchor=north,font=\scriptsize] at (10.110,-0.65) {$20$};
\node[anchor=north,font=\scriptsize\itshape] at (5.6,-1.2) {$\etaB\ [10^{-10}]$, logarithmic};
\draw[line width=2.4pt,draw=black!60] (5.455,2.400) -- (10.319,2.400);
\draw[line width=0.90pt,draw=black!60] (5.455,2.270) -- (5.455,2.530);
\draw[line width=0.90pt,draw=black!60] (10.319,2.270) -- (10.319,2.530);
\fill[black!60] (7.909,2.400) circle (2.3pt);
\node[anchor=south west,font=\scriptsize,text=black!60] at (5.455,2.560) {Level A --- ledger identity};
\draw[line width=2.4pt,draw=blue!60!black] (2.181,1.350) -- (4.240,1.350);
\draw[line width=0.90pt,draw=blue!60!black] (2.181,1.220) -- (2.181,1.480);
\draw[line width=0.90pt,draw=blue!60!black] (4.240,1.220) -- (4.240,1.480);
\fill[blue!60!black] (3.202,1.350) circle (2.3pt);
\node[anchor=south west,font=\scriptsize,text=blue!60!black] at (2.181,1.510) {Level B --- Boltzmann};
\draw[line width=2.4pt,draw=green!45!black] (4.565,0.300) -- (6.977,0.300);
\draw[line width=0.90pt,draw=green!45!black] (4.565,0.170) -- (4.565,0.430);
\draw[line width=0.90pt,draw=green!45!black] (6.977,0.170) -- (6.977,0.430);
\fill[green!45!black] (5.794,0.300) circle (2.3pt);
\node[anchor=south west,font=\scriptsize,text=green!45!black] at (4.565,0.460) {Level C --- completed inventory};
\node[anchor=north west,font=\scriptsize,text=black!60,align=left] at (0.0,-1.60)
  {The three answer three different questions and are not to be averaged. Their \emph{spread}
  is the informative quantity:\\ the ledger's arithmetic and the declared dynamics currently
  disagree by a factor of about two and a half. All three are conditional on C4.};
\end{tikzpicture}}
\caption[The prediction at three levels, against observation.]{The prediction at three levels, against observation. Level~A is the ledger identity taken at face value; Level~B is the Boltzmann integration; Level~C folds in four standard transport effects Level~B omits. The three answer three different questions and \emph{are not to be averaged}: their spread is the informative quantity, and it says the ledger's arithmetic and the declared dynamics currently disagree by a factor of about two and a half. All three are conditional on C4.}
\label{fig:lg-three-levels}
\end{figure}


A prediction with a factor-three band is only honest if the band is
derived rather than chosen. This section states it at three levels, each
with what it assumes.

\paragraph{Level A --- the ledger identity.} Taking
\eqref{eq:lg-throughput} at face value and assembling by
\eqref{eq:lg-etaB-pred}:
\begin{equation}
\etaB^{\rm A} \;=\; 7.34\times10^{-10}\times 3^{\pm1},
\qquad \text{band } (2.4\text{--}22)\times10^{-10},
\label{eq:lg-levelA}
\end{equation}
a span of nine. The exponent is postulated; the band is the declared
order-of-magnitude register of the compression.

\paragraph{Level B --- the Boltzmann computation.} The integration of
\S\ref{sec:lg-boltzmann}, scanned over the four remaining freedoms ---
the coefficient $c_\tau \in [5,10]\times10^{-3}$, the flavored pattern
acceptance $\times[2/3,3/2]$, the washout-kernel order-unity factor
$\times[0.7,1.3]$, and the window edge $z_0 \in [0.5,2.0]$ --- gives
\begin{equation}
\boxed{\;
\etaB^{\rm B} \;=\; 0.86^{+0.52}_{-0.32}\times10^{-10}
\quad (68\%),
\qquad \text{band width } \times 2.6 .
\;}
\label{eq:lg-levelB}
\end{equation}
Against Level A's width of $\times 9$, the error bar has contracted by a
factor of three and a half, because most of what used to be bracketed is
now computed. Level B sits a factor $E = 7.1$ $[4.4, 11.4]$ below
observation, Eq.~\eqref{eq:lg-E}.

\paragraph{Level C --- the completed inventory.} Level B deliberately
omits four standard transport effects, each pointing up, none specific to
this framework. Folding them in at their literature ranges ---
lepton-number-violating scattering source $\times[1.3,2.5]$,
Lorentzian-versus-Breit--Wigner convention $\times[1,2]$,
relativistic-window residue $\times[1,1.5]$, spectators and the
$A$-matrix $\times[0.85,1.2]$ --- gives the best estimate:
\begin{equation}
\boxed{\;
\etaB^{\rm C} \;=\; 2.8^{+2.0}_{-1.2}\times10^{-10} \quad(68\%),
\qquad
2.8^{+5.0}_{-1.9}\times10^{-10} \quad(95\%).
\;}
\label{eq:lg-levelC}
\end{equation}
Against $\etaB^{\obs} = (6.12\pm0.04)\times10^{-10}$ the observation sits
$1.3\sigma$ above the Level-C median on the logarithmic scale:
\emph{consistent, with a central deficit of a factor $2.2$}.

\paragraph{All three levels are conditional on C4.} None of the three
statements above is a derivation of the baryon asymmetry. Each is what
follows \emph{given} the regulator identification of
Table~\ref{tab:lg-c1c6}, and that identification is open.

\begin{euclidremark}[Reading three numbers that disagree]
A reader may reasonably ask why the Part quotes $7\times10^{-10}$,
$0.86\times10^{-10}$ and $2.8\times10^{-10}$ and calls all three the
prediction. They are answers to three different questions. Level A asks
what the ledger identity gives if taken literally. Level B asks what the
declared dynamics actually produces when integrated. Level C asks what
the completed inventory would produce. The honest headline is Level A
with its band, because that is what the framework asserts; the honest
\emph{best estimate} is Level C, because that is what the physics as
currently understood delivers; and the gap between them --- a factor of
about two and a half --- is the measure of how much of this chain is
still owed.
\end{euclidremark}

\section{From \texorpdfstring{$\etaB$}{etaB} to
\texorpdfstring{$\rho_b$}{rho b} and
\texorpdfstring{$\Omega_b$}{Omega b}: one quantity, three sets of units}
\label{sec:lg-to-omegab}

The chain above ends at a baryon-to-photon ratio, and the cosmology
companion states a baryon fraction. It is worth carrying this Part to the
end of its own arithmetic, because the two are not two results.

\paragraph{One quantity, three costumes.} The baryon-to-photon ratio
$\etaB$, the baryon mass density $\rho_b$ and the baryon fraction
$\Omega_b$ are \emph{one} physical quantity in three sets of units, tied
by conversions that contain no content of this framework's:
\begin{equation}
\eta_{10} \;=\; 273.9\,\Omega_b h^{2},
\qquad
\rho_b \;=\; \Omega_b h^{2}\times 1.87847\times10^{-29}~
\mathrm{g\,cm^{-3}},
\qquad
\eta_{10}\equiv 10^{10}\etaB .
\label{eq:lg-bridge}
\end{equation}
The first is \eqref{eq:eta10-app}, derived in Appendix~L from
$n_B = \Omega_b\rho_c/\bar m_B$ over
$n_\gamma = (2\zeta(3)/\pi^2)T_0^3$ at $T_0 = 2.7255$~K, with $\bar m_B$
the mean mass per baryon; the second is the critical density. Fix any one
of the three and the other two follow with no freedom. Quoting them as
three predictions with three error bars would make this framework look as
though it had made three uncertain statements about baryons. It has made
one, by two independent roads.

\paragraph{The two roads, and which is worth more.} The road of this Part
is dynamical: the Majorana door, the flavored asymmetries, differential
washout, sphaleron conversion at $28/79$ and entropy dilution, landing at
$\etaB^{\rm A} = 7.34\times10^{-10}$ against an observed $6.12$ --- high
by twenty per cent, inside a declared factor-$\sim$3 band, and graded
D$^*$ on the named hypothesis \Hphi{}.

The cosmology companion carries a second and completely independent road:
a \emph{static} count of volumes on the compact space the ledger selects,
giving $\Omega_b = 1/(2\pi^2) = 0.050661$ with no dynamics in it at all.
Converted through \eqref{eq:lg-bridge} at the companion's own derived
expansion rate $h = 0.666$,
\begin{equation}
\Omega_b h^2 = 0.02247,
\qquad
\eta_{10} = 6.15,
\qquad
\rho_b = 4.22\times10^{-31}~\mathrm{g\,cm^{-3}},
\label{eq:lg-ledger-road}
\end{equation}
against an observed $\eta_{10} = 6.12$ and
$\rho_b = 4.20\times10^{-31}~\mathrm{g\,cm^{-3}}$: an error of
$+0.6\%$, with nothing fitted anywhere in the chain. At the microwave
background's own $h = 0.674$ the same road gives $\eta_{10} = 6.30$,
an error of $+3.0\%$.

\begin{euclidremark}[Which road this paper is entitled to lead with]
The static road is right to between six tenths of a per cent and three per
cent;
the dynamical road of this Part is high by twenty per cent with a
factor-of-three band. The static road is the more accurate by a factor of
about thirty-eight, and the asymmetry has a structural cause rather than
an accidental one: a statement about volumes on a compact space carries
no dynamics to get wrong, while this Part's chain must pass through a
finite-temperature resonance condition that it grades as an open, named
hypothesis. The honest division is therefore that the \emph{static} count
is the framework's prediction of the number, and this Part is the
demonstration that a parameter-free mechanism exists which could have
produced it. That is what the D$^*$ grading already says; it is written
out here so that no reader has to infer it.
\end{euclidremark}

\paragraph{The gap, named once.} Converted to a common $h = 0.674$, the
dynamical road gives $\Omega_b = 0.0590$ and the static road
$\Omega_b = 0.050661$. \emph{The two roads disagree by $16.4\%$.} The gap
shrinks through zero near the ladder branch ($-2.0\%$ at $h = 0.735$) and
sits inside this Part's factor-$\sim$3 band, so it is not a contradiction;
but the two are predictions of the same number and they do not yet agree,
and closing that is unfinished business. It is recorded as the
two-route-consistency item and is stated identically in the cosmology
companion.

\paragraph{One ratio that needs no epoch.} There is a third statement
worth separating out, because it is the cleanest of the three and it is
easy to lose among them. Baryons and cold dark matter both dilute as
$a^{-3}$, so
\begin{equation}
\frac{\Omega_b}{\Omega_m} \;=\; \frac{1/(2\pi^2)}{1/\pi}
\;=\; \frac{1}{2\pi} \;=\; 0.159155
\label{eq:lg-baryon-ratio}
\end{equation}
is constant at every epoch and independent of $h$. Measured against
$\omega_b/\omega_m = 0.02237/0.14237 = 0.15713$ it sits $+1.3\%$ high,
about $1.3\sigma$ --- and unlike the absolute fractions it consumes no
epoch calibration, which is why the companion grades it D rather than
D$^*$. The comparison must be made in $\omega_b/\omega_m$ and not in
$\Omega_b/\Omega_m$, because the measured $\Omega_m$ includes a neutrino
component that $\Omega_b + \Omega_{\rm dm}$ does not; with
$\Sigma m_\nu \simeq 59$~meV that is $0.44\%$ of $\Omega_m$, and it moves
the comparison by half a percentage point.

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

An asymmetry made at the lock must survive to the present, and the usual
threat is entropy release from a later transition: a supercooled phase
transition would dilute whatever was made before it.

Here the dilution factor is unity. The nucleation action at the relevant
crossing sits some fifty-six orders of magnitude above the supercooling
threshold, and the saddle is barrier-free, so the transition proceeds
without appreciable entropy release. Two consequences follow and are
recorded because they are consumed elsewhere in the cohort: the number of
$e$-folds is not reduced by a second inflationary episode, giving
$n_s = 1 - 2/N_* = 0.9620$ and $r = 12/N_*^{2} = 0.0043$, the canonical
values \ATen{} carries. Nothing made at the lock is diluted on the way
here.

\section{What could move, and the three kill routes}
\label{sec:lg-kills}

\subsection{Which factors are free to move}

\begin{table}[htbp]
\centering
\small
\caption[Which factors of \eqref{eq:lg-etaB-pred} can move, and by how much.]{Which factors of \eqref{eq:lg-etaB-pred} can move, and by how
much. Two of the six are fixed Standard-Model arithmetic and cannot move
at all. The remaining four are ranked by how far they could move and by what would have to be true for them to move that far, which is the honest form of an uncertainty for a chain of this kind.}
\label{tab:lg-move}
\begin{tabular}{@{}ll>{\raggedright\arraybackslash}p{0.46\textwidth}@{}}
\toprule
Factor & Room & Comment\\
\midrule
$e^{-K/2}$ & the postulate & the exponent is not derived; this is the
 chain's largest single exposure\\
$\epsres$ & on \Hphi{} & conditional on the regulator identification, C4\\
$\kappa$ & factor $\sim2$ & convention-dependent; both conventions
 declared in \S\ref{sec:lg-washout}\\
$Y_N^{\rm eq}$ & none & $1.25/\gstar$ is a definite counting result\\
$\gstar = 106.75$ & none & Standard-Model degree-of-freedom count\\
$\csph = 28/79$ & none & Standard-Model equilibrium result\\
$s/n_\gamma = 7.04$ & none & fixed by present-day cosmology\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Three routes by which the chain kills itself}

The reason \Hphi{} is a scientific hypothesis rather than a placeholder
is that its failure modes are written down in advance, and each maps onto
a row of Table~\ref{tab:lg-regulator}.

\begin{enumerate}[leftmargin=2.1em]
\item \textbf{If the finite-temperature derivation returns the $N$-width
class} --- row (a) --- the chain undershoots by $10^{7}$ to $10^{8}$ and
is falsified from within. Note that the \emph{standard} fully flavored
Boltzmann limit does exactly this, which is why C4 is the crux and not a
formality.
\item \textbf{If it returns the flavor-blind $g^{2}T$} --- row (c$'$) ---
the excluded peak overshoot stands and the verdict is the same. This
branch has been examined and excluded: it lands at
$\Delta M/\Geff \simeq 0.07$.
\item \textbf{If C4 and C5 close but the window deficit of C6 survives} a
full transport computation at its adverse end with the priced prefactors
exhausted, \Hphi{} fails by \emph{throughput} rather than by regulator
identity --- a third, softer kill beside the two clean ones. This is the
failure mode whose size \eqref{eq:lg-E} now measures.
\end{enumerate}

Success is a finite-temperature transport computation --- thermal widths
in the propagators, the lock crossing's time dependence in the rates ---
landing at the partial-resonance point. Failure is the same computation
showing the enhancement dead at the crossing for every candidate
mechanism, in which case the chain is falsified from within, the
$e^{-K/2}$ agreement stands exposed as coincidence, and the account
reverts to Sakharov custody plus an unexplained number. \emph{Both
outcomes are scientific; only the unexamined middle would not be.}

This exposure is registered as a named falsification channel in
\S\ref{sec:predictions-falsification}.

\section{Accelerator corollaries: what a collider can and cannot prove}
\label{sec:lg-accelerator}

The lock sits at $3.85$~TeV --- inside the energy reach of the HL-LHC.
Can the mechanism of this Part therefore be tested at an accelerator?
The answer has a sharp negative half and a constructive positive half,
and both follow from theorems already proved above.

\subsection{The negative half: three locks on the direct channel}

\emph{The occupant of the lock cannot be seen at any collider now
conceived}, for three independent reasons, each a derived quantity.

\paragraph{Lock 1: the mixing.} The heavy triplet couples to the
boundary only through the seam Yukawa, so its active--sterile mixing is
fixed at $|\theta|^2 = 2.2\times10^{-15}$ \eqref{eq:lg-mixing}. Every
production channel available to a heavy neutral lepton at a collider ---
$W^{(*)}\to\ell N$, Drell--Yan through mixing --- pays this factor.
Dedicated heavy-neutral-lepton searches reach $|\theta|^2\sim10^{-6}$ at
masses far below the lock, and have no sensitivity at all at
$3.85$~TeV; the shortfall is nine or more orders of magnitude, and no
luminosity upgrade crosses an exponent.

\paragraph{Lock 2: no portal.} The classic escape --- gauge-portal
production, as in left--right models where $pp\to W_R\to\ell N$ makes the
heavy state at gauge strength and the same-sign dilepton pair exhibits
$\Delta L = 2$ in a detector \citep{KeungSenjanovic1983} --- is closed
here by the cluster's own contents. The three sibling states at the lock
(\PaperPOne{}, \citealp{ArisakaP1KK}) are the spin-2 mode, the
color-octet $g^{*}$ and the \emph{leptophobic} $Z'_B$: nothing at the
lock couples to leptons at gauge strength, and $N$ carries no baryon
number, so even the $Z'_B$ does not reach it. The suppressed dilepton
rate that shields the cluster from generic resonance limits is the same
fact that removes the portal.

\paragraph{Lock 3: the degeneracy.} Even granting a portal, the
framework sets $M_N \simeq M_{\rm cluster} \simeq \Mlock$ --- the on-shell
decay of any cluster state to $\ell N$ or $NN$ is kinematically closed by
the very quasi-degeneracy that powers the resonance mechanism. Only
off-shell production would survive, at a parton-luminosity penalty of
several orders.

\paragraph{And the side doors close with the front one.} Loop-level
signatures --- $\mu\to e\gamma$, electric dipole moments,
lepton-number-violating meson decays --- scale as
$Y_\nu^4 = 2.8\times10^{-25}$ times the usual loop factors. The
collider $\ell^\pm\ell^\pm jj$ channel of the $0\nu\beta\beta$ operator
is suppressed by $(m_\nu/E)^2$. Even the lifetime conspires:
$\Gamma_N = 8.1\times10^{-11}$~GeV gives $c\tau = 2.4~\mu$m ---
too prompt for a displaced-vertex tag, too displaced for nothing, below
the impact-parameter resolution of any tracker. The heavy sector sits in
a blind spot every axis of which is a derived number.

\begin{euclidremark}[The cancellation cuts both ways]
The root of all three locks is the closure of \S\ref{sec:lg-seesaw}:
because $\Mlock$ cancels out of the light spectrum, no low-energy
observable retains a handle on the occupant of the lock --- and this
section extends the statement upward in energy. The same absence of
dials that makes the chain parameter-free makes it invisible: a generic
TeV-scale model would simply turn its Yukawa up until the experimenters
could see it, and this framework cannot. The framework is \emph{more}
falsifiable than its competitors at the structural level and \emph{less}
confirmable at the mechanism level --- the price of no dials, paid in
full view.
\end{euclidremark}

\subsection{The positive half: the four premises, and who tests them}

The chain's claim decomposes into four premises: a heavy sector exists
at the lock; the selection rule leaves leptogenesis as the only
admissible mechanism; the derived parity sets the sign; the
thermal-width resonance delivers the magnitude. The occupant being
invisible, \emph{the premises are the test} --- and three of the four
are accelerator-testable, two of them by experiments already running or
under construction.

\begin{table}[htbp]
\centering
\small
\caption[The accelerator test matrix for lock-scale leptogenesis.]{The accelerator test matrix for lock-scale leptogenesis. The
mechanism's occupant is invisible (three locks above); its premises are
not. Confirmation of any row is partial evidence for the chain;
failure of any row is total falsification of it --- the asymmetry
between the two columns is the section's content.}
\label{tab:lg-accel-matrix}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.185\textwidth}>{\raggedright\arraybackslash}p{0.27\textwidth}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.235\textwidth}@{}}
\toprule
\textbf{Premise} & \textbf{Accelerator test} & \textbf{Facility, date} & \textbf{Registered as}\\
\midrule
(i) The address: a heavy sector at $\tfrac12\EPl e^{-K}$
 & the three-state cluster at $3.85$~TeV, top-rich, with the derived
   interference shape
 & HL-LHC, this decade
 & F-SM-1\\
(ii) The door: $\Delta B \in 3\mathbb{Z}$ only
 & the \emph{leptophobic} $Z'_B$ (the gauged $\UoneB{}$ made visible);
   no $\Delta B = 1,2$ signal at any machine; the $\Delta B = 3$
   sphaleron vertex at a $100$~TeV collider
 & HL-LHC; then FCC-hh
 & F-SM-1, F-SM-4\\
(iii) The alphabet: the leptonic Dirac phase
 & $\dPMNS{}$ in $[186^\circ, 207^\circ]$ --- a test of the closed form
   for the Dirac phase, and \emph{not} of the sign of \etaB{}
 & Hyper-K from 2028, DUNE from 2031
 & F-SM-3d\\
(iv) The magnitude: $\epsres\kappa \simeq e^{-K/2}$
 & none --- no collider recreates a $T \sim \Mlock$ bath; the premise
   is settled by a finite-temperature calculation
 & a theorist, any time
 & F-SM-7\\
\bottomrule
\end{tabular}
\end{table}

Three of the rows deserve a sentence beyond the table.

\paragraph{Row (i) is asymmetric, and the asymmetry should be stated.}
Finding the cluster confirms $K = 35$ --- the one integer behind
$\Mlock$, the splitting $\Delta M = \Mlock/K$, \emph{and} the exponent
$e^{-K/2}$ --- which establishes that the chain's stage exists. It does
not show the play was performed. A cluster at the wrong mass, by
contrast, kills $K = 35$ and the entire $\etaB$ chain with it. Partial
confirmation, total falsification.

\paragraph{Row (ii) contains the most leptogenesis-relevant discovery
the HL-LHC could make.} Not the mechanism --- the theorem that
eliminates its competitors. A $3.85$~TeV color-singlet vector coupling
to baryon number and not to leptons is the laboratory face of the gauged
$\UoneB{}$ whose selection rule (\S\ref{app:lepto-selection};
\PaperPSix{}) makes every $\Delta B = \pm1$ baryogenesis unwritable.
And the door itself is in principle observable: sphaleron-induced
$\Delta B = \Delta L = 3$ events at hadron colliders, with
$E_{\rm sph}\simeq 9$~TeV and a spectacular multi-fermion final state,
have been proposed and quantified \citep{TyeWong2015}
(\citealp{EllisSakurai2016}). At $\sqrt{s} = 14$~TeV the parton
luminosity above the barrier is negligible; this is an FCC-hh
measurement, and its rate rides on the contested question of whether the
exponential suppression lifts near threshold --- the two references
disagree, and this paper takes no side. What this framework adds is the
sharp selection: \emph{the door admits multiples of three and nothing
else}. A single $\Delta B = 1$ or $2$ event at any collider falsifies
the framework outright (F-SM-4); a genuine $\Delta B = 3$ event would be
the conversion vertex of \S\ref{sec:lg-sphaleron}, observed directly.

\paragraph{Row (iii), and a claim this paper withdraws.} It is tempting
to read the base value $\pi$ in $\dPMNS{} = \pi + 2\arcsin(1/7)$ as the
Majorana parity $\alpha_{31} = \pi$, and so to say that the long-baseline
program is measuring the bit that decided the matter content of the
universe. \emph{That reading is wrong, and this paper refutes it in its
own Appendix} (\S\ref{app:majorana}, problem K.1.5): the Majorana phases
cancel in every oscillation probability, at any baseline and in any
channel, and that is a theorem rather than a sensitivity statement.
$\dPMNS{}$ is by construction the phase an oscillation experiment
measures, so its base value cannot be $\alpha_{31}$.

The $\pi$ of the closed form is what \S\ref{sec:PMNS} says it is --- a
reference-channel convention, bookkeeping that must be done because a
prediction quoted in the wrong convention is a wrong prediction. That the
Majorana parity is also $\pi$ is a coincidence of the number and not an
identity of the quantities; a common geometric origin is possible and is
discussed in the Appendix, but two observables with a shared origin are
still two observables, and only one of them is visible to oscillation.

What row (iii) therefore tests is sharp, parameter-free and worth the
facilities: the closed form for the leptonic Dirac phase, at
$196.43^\circ$, with the kill window registered as F-SM-3d. What it does
\emph{not} test is the origin of the matter asymmetry --- which is tested
instead by $0\nu\beta\beta$ and by the neutrino mass sum. \emph{That
separation is a gain and not a retreat}: two claims that can fail
independently are more testable than one claim that stands or falls
whole, and a single unfavorable long-baseline result no longer threatens
the account of why there is matter at all.

\subsection{One open question, registered}
\label{sec:lg-portal-open}

Everything in the negative half assumed the heavy triplet couples only
through the seam Yukawa. The chain's parent papers describe the $N$
states as mirror-sector objects, and the deployed cohort does not answer
one structural question in either direction:

\begin{quote}
\emph{Does the mirror sector couple at gauge strength to any state of
the lock cluster --- or carry gauge structure of its own at the lock?}
\end{quote}

This paper does not answer it, and deliberately: the question belongs to
\PaperPFive{} and \PaperPOne{}, and reasoning it out here from priors is
exactly the error class this revision history records. It is registered
as \textbf{open}, with its owners named. The stakes are the reopening of
the direct channel: an affirmative answer would light up off-shell
gauge-portal production of $\ell N$ at a $100$~TeV collider or a
$10$~TeV muon collider --- same-sign dileptons, $\Delta L = 2$
exhibited in a detector \citep{KeungSenjanovic1983}, and in the
sharpest case two peaks split by the ledger's own
$\Delta M = 110$~GeV, the level spacing of the resonance mechanism read
off a mass spectrum. A negative answer leaves the three locks standing
and this section's division final: \emph{the occupant is invisible; the
premises are the test.}

\section{What this Part owes, and to whom}
\label{sec:lg-audit}

\begin{table}[htbp]
\centering
\small
\caption[The nine-step audit of the chain.]{The nine-step audit of the chain. One named input, one flavored
projection carried derived-conditional on a named hypothesis, one
postulated exponent, and \textbf{zero fitted parameters}. Rows 1--3 and 5
consume quantities this paper derives; rows 4--7 are owned by the
cosmology companion; row 8 is textbook.}
\label{tab:lg-audit}
\begin{tabular}{@{}cl >{\raggedright\arraybackslash}p{0.19\textwidth} >{\raggedright\arraybackslash}p{0.22\textwidth} >{\raggedright\arraybackslash}p{0.22\textwidth}@{}}
\toprule
\# & Quantity & Value & Route & Grade\\
\midrule
1 & $M_N$ & $3.85$~TeV, $\Delta M \simeq 110$~GeV &
 $\tfrac12 M_{\rm Pl}e^{-K}$ (A9) & Derived\\
2 & portal & $LHN$, unique & T-SM-44.1 (A9) & Derived\\
3 & $Y_\nu$ & $7.3\times10^{-7}$ &
 $Y_\nu^{2} = \pi\Mlock/\EGUT$ (A9) & Derived\\
4 & resonance & $\Delta M/\Geff \simeq 3\times10^{4}$ & --- &
 \textbf{Named input}\\
5 & $\epsres$ & $2.5\times10^{-6}$ & flavored $\epsilon_{i\alpha}$;
 driver $\alpha_{31} = \pi$ (A9) & Derived-cond.\ (\Hphi{})\\
6 & $\kappa$ & $1.0\times10^{-2}$ & $\Kwash \simeq 47$, strong washout &
 Derived-cond.\\
7 & $\epsres\kappa$ & $2.5\times10^{-8} = e^{-K/2}$ & reconstruction &
 \textbf{Exponent postulated}\\
8 & transfer & $7.04\times(28/79)\times1.25/\gstar$ & Harvey--Turner;
 entropy; yield & Standard\\
9 & $\etaB$ & $\approx 7\times10^{-10}$, band $\times3^{\pm1}$ &
 assembled & D$^{*}$\\
\bottomrule
\end{tabular}
\end{table}

\subsection{The custody ledger}

What this paper contributes to the chain is the microscopic input, and it
contributes it because it derived it independently and for other reasons:
the lock scale (Part~\ref{part:III}), the unique $\Delta L = 2$ portal
(Theorem~T-SM-44.1), the seam coefficient $\pi/\EGUT$ and the light
spectrum (Part~\ref{part:X}), the Majorana parity $\alpha_{31} = \pi$
that fixes the sign, and the reactor angle $N/4K = 3/140$ that decides
which flavor survives.

What this paper does \emph{not} own, and does not claim: the
finite-temperature transport argument and the conditions C1--C6, whose
home is \ATen{} Appendix~I; the Boltzmann integration and its measured
window \eqref{eq:lg-E}, likewise; and the phase routing, carried by
\PaperPFive{}. Condition C4 remains open there, and no statement in this
Part promotes it.

\begin{euclidremark}[Why the Part was written at all]
A reader may ask why a Standard-Model parameter paper carries a
cosmological transport chain in full. The answer is that four of the nine
rows of Table~\ref{tab:lg-audit} are quantities this paper derived, and
their consequence is a statement about whether the universe contains
matter. A paper that derives the reactor angle and then declines to
mention that the reactor angle is what lets the matter asymmetry survive
has not finished explaining its own result. The chain is conditional and
says so in nine places; but the inputs are not conditional, and they are
this paper's.
\end{euclidremark}


\section{Part~VIII in plain English: why there is matter and not nothing}
There is more matter than antimatter in the universe, by about one part
in ten billion, and every atom in existence is what survived that imbalance.
The Standard Model cannot produce it.

This Part argues that the same geometry that fixes the neutrino block also
writes the excess. The mechanism is leptogenesis, which is not new; what is
new is that the ingredients are not free. The heavy sector sits at the lock
scale that the Substrate already fixed. The portal is the unique operator the
geometry permits. The asymmetry is driven by the Majorana parity, which is a
property of the same mixing structure the previous Part derived. Nothing in
the chain was chosen to make the answer come out.

The result is an assembled baryon-to-photon ratio, and the honest way to read
it is factor by factor. Table~\ref{tab:lg-move} in this Part says which
factors could move and by how much: two of the six are Standard-Model
arithmetic and cannot move at all, and the remaining four are ranked by what
would have to be true for each to shift. A prediction of a number this small,
from a chain this long, deserves that kind of accounting rather than a single
quoted value with an error bar.

One structural point is worth separating from the numerics. The chain
demonstrates a mechanism; the static ledger count is the framework's actual
prediction of the baryon number. The two roads do not agree exactly, and the
size of the gap between them is stated once, plainly, rather than averaged
away. A reader should treat the agreement of the dynamical chain as
encouraging and the ledger count as the claim.

\GGPart[The protected readout]{The protected readout: strong-CP and the topological completion}{The twenty-eighth parameter as the protected zero of the topological ledger; how the completed codebook addresses the ten classic open problems; and the deep geometric reading of the entire twenty-eight-parameter codebook}
\label{part:XI}

\section{Completion of the 28th parameter: strong-CP suppression and the final topological angle}
\label{sec:strong-CP-derivation}

\subsection{Historical background}

Peccei and Quinn introduced in 1977 the global \(U(1)_{PQ}\) symmetry whose spontaneous breaking dynamically suppresses the strong-CP angle \(\bar\theta\) \citep{PecceiQuinn1977}, the first framework in which the smallness of \(\bar\theta\lesssim 10^{-10}\) (set by the experimental upper bound on the neutron electric dipole moment) becomes a derived rather than fine-tuned statement. The PQ mechanism predicts a pseudoscalar Goldstone boson -- the QCD axion -- whose mass and couplings depend on the PQ-breaking scale \(f_a\), an additional parameter of the extended model. The theorems below replace the global PQ symmetry with the geometric topological-projection mechanism intrinsic to \GG{}: the 4D seam \(\bar\theta\) is the projected image of a closed bulk topological sector, and its vanishing follows from \BIsix{} anomaly freedom plus the \OGNfive{} lock, with no extra global symmetry or new particle required. The Kaxion (companion paper \PaperCOne{}) is the geometric replacement of the QCD axion, with mass \(8.1 \pm 0.5~\mu\)eV (laboratory line \(1.59\)--\(2.13\)~GHz).


\subsection{The strong-CP \GRIP{} cascade}

The strong-CP block is the stiffest, most strongly protected sector of the whole 28-parameter
sheet. \BIsix{} structures the parent theory topologically but licenses no arbitrary
independent seam $\theta$-angle: the four-dimensional seam is a projected image of the legal
six-dimensional parent, so at the projected seam \CCI{} admits holonomy, inflow, and
determinant phases but no extra unconstrained local $\theta$-source. As one \GRIP{} cascade the
topological coordinate is therefore not a free continuous basin coordinate like an ordinary
flavor phase: its central \GFF{} basin value is zero, and the first admissible deviation is a
leakage envelope built from already-admitted small factors. Because a nonzero $\bar\theta$
could only appear as a genuine residual of the one closed quark-plus-topology ledger, the
question is asked \emph{after} the quark packet has already been selected and stabilized by
\GRIP{}, and the answer below is that the first-order value remains zero with only a tiny
leakage.

\subsection{Parameter block}

The subsections that follow therefore close the 28th parameter not by adding a
new independent angle, but by showing why the derived branch drives the central
value of \(\bar\theta\) to zero and allows only a highly suppressed residual
envelope.

\label{sec:strongcp}

The PMNS/Majorana construction completed the ordinary gauge, Higgs, mass,
and flavor data of the Standard Model. One qualitatively different
quantity nevertheless remained: the strong-CP angle. It is qualitatively
different because it does not control a spectrum, a mixing amplitude, or a
wavefunction overlap. It multiplies a topological density,
\begin{equation}
\mathcal L_{\bar\theta}
=
\frac{g_s^2}{32\pi^2}\,\bar\theta\,
G^a_{\mu\nu}\widetilde G^{a\mu\nu},
\qquad
\widetilde G^{a\mu\nu}
=
\frac12\epsilon^{\mu\nu\rho\sigma}G^a_{\rho\sigma},
\label{eq:theta-density-main}
\end{equation}
and the physically meaningful combination is
\begin{equation}
\bar\theta
=
\theta_{\rm top}
+
\arg\det(M_uM_d).
\label{eq:theta-bar-main}
\end{equation}
Equation~\eqref{eq:theta-bar-main} already tells us why the final angle
must be treated only \emph{after} the ordinary mass and flavor sectors are
fixed: the quark determinant phases cannot be discussed responsibly until
the quark mass matrices have already been derived.

\subsection{Why the final CP angle is narrower than the flavor problem}

The derivation above has already identified three distinct classes of CP
data.

First, the CKM phase is a relative family-misalignment phase in the quark
sector. It survives rephasings because the up-type and down-type projector
frames are not the same. Second, the PMNS Dirac phase is the analogous
relative misalignment phase in the lepton sector. Third, the Majorana
phases are parity-like phases tied to the neutrino Majorana projector and
cannot be removed by arbitrary field redefinitions.

The strong-CP angle is different from all three. It is not a relative
misalignment angle between two family packets. It is a \emph{vacuum
topology} angle. That means the final unresolved question is no longer
``what family matrices are allowed?'' The family matrices are already
fixed. The question is instead ``does the projected \GG{} seam admit an
independent CP-odd topological phase once the quark determinant phases are
written in the constitutionally allowed form?'' This is a narrower and
therefore more tractable problem.

\subsection{The first-order statement: no independent seam \texorpdfstring{\(\theta\)}{theta} term}

The written \GGConstitution{} starts from the six-dimensional bulk
connection and the \BIsix{} anomaly-cancellation structure
(\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}). At that level the primitive
dynamical terms are kinetic, geometric, and anomaly-canceling. The
visible four-dimensional seam is then obtained by zero-mode reduction plus
the \GDOS{}/\OGGEP{} projection. In that derivation the ordinary Standard-Model
kinetic terms, masses, and mixing structures appear explicitly, but an
\emph{independent} bare seam \(\theta_{\rm top}\) term is not one of the
primitive inputs.

This is the crucial first-order observation. In an ordinary bottom-up 4D
effective theory one may simply write Eq.~\eqref{eq:theta-density-main}
because every gauge-invariant local term is allowed unless forbidden by
hand. The \GG{} constitution is stricter. The 4D seam is not the
primitive theory; it is a projected image of the 6D bulk. Therefore a 4D
term is admissible only if it descends from the allowed bulk ledger. The
minimal written \GGTheory{} action does not introduce an independent CP-odd
topological constant on the seam. At first order one is therefore led to
\begin{equation}
\theta_{\rm top}^{(1)}=0.
\label{eq:theta-top-zero-main}
\end{equation}

This does not mean that topology is absent from \GG{}. On the contrary,
the entire theory is rich in topology: the compact \(S^1_\theta\) fiber,
the Wilson-line holonomy, the \BIsix{} descent data, and the anomaly
factorization are all topological or quasi-topological in origin. The
point is subtler: topology is already present in the \emph{geometry} of the
allowed connection, so one does not also receive an unconstrained 4D
\(\theta\)-parameter for free.

\subsection{Why the quark determinant phases cancel at first order}

The second contribution in Eq.~\eqref{eq:theta-bar-main} is the phase of
\(\det(M_uM_d)\). Here the order of the paper matters. The quark masses
and the CKM block were derived from a common codebook built from the same
compact-angle holonomy sector, the same dyadic seam discipline, and the
same OGN integers. The CP-odd part of that construction entered only
through relative family transport phases. Those relative phases survive in
the Jarlskog invariant, but the \emph{determinant} phase behaves
differently.

To see the logic, write the quark mass matrices in singular-value form,
\begin{equation}
M_u=L_u\,\Sigma_u\,R_u^\dagger,
\qquad
M_d=L_d\,\Sigma_d\,R_d^\dagger,
\label{eq:svd-quark-main}
\end{equation}
with \(\Sigma_{u,d}\) diagonal and positive. Then
\begin{equation}
\arg\det(M_uM_d)
=
\arg\det(L_uR_u^\dagger)+\arg\det(L_dR_d^\dagger),
\label{eq:argdet-svd-main}
\end{equation}
because \(\det\Sigma_{u,d}>0\). In a generic flavor model the two terms in
Eq.~\eqref{eq:argdet-svd-main} are unconstrained. In the present
construction they are not. The same compact \(L=5\) holonomy that
generated the CKM phase is a \emph{relative} phase between family
transport channels. It is not a free overall phase of the quark
determinants. The determinant sector is fixed by the common underlying
holonomy class and by the fact that the first-order quark mass formulas
derived above are all real-positive numbers.

This statement can be phrased in a more physical way. The CKM phase tells
us that different family paths around the compact sector do not all line
up. But the determinant phase would require a net uncanceled phase of the
\emph{entire} quark packet. The \GG{} codebook does not provide such a net
phase at first order. It provides only relative phases. Therefore
\begin{equation}
\arg\det(M_uM_d)^{(1)}=0,
\label{eq:argdet-zero-main}
\end{equation}
and
\begin{equation}
\boxed{\bar\theta^{(1)}=0.}
\label{eq:theta-bar-zero-main}
\end{equation}

The physical intuition is simple and important. A relative family phase
can generate CKM CP violation without generating strong CP. That is
precisely the distinction the Standard Model itself makes, but here it is
not inserted by hand. It follows from the geometric separation between
relative holonomy misalignment and a net topological determinant phase.

\subsection{The constitutionally minimal QD leakage estimate}

The first-order result \eqref{eq:theta-bar-zero-main} is exact within the
skeleton codebook. The QD layer must then answer the next question:
\emph{what is the natural size of the first nonzero leakage, if any?}

The written \GGTheory{} logic strongly restricts the answer. A nonzero
\(\bar\theta\) correction would need all three of the following.

\begin{enumerate}[leftmargin=2.1em]
\item It must be CP odd, so it must carry a nontrivial compact-angle phase.
The smallest such phase in the visible quark sector is the \(L=5\)
holonomy angle, represented by the sine factor
\(\sin(2\pi/L)\).

\item It must evade the first-order determinant cancellation, which means
it must arise from a \emph{leakage} rather than from the leading codebook.
The natural seam price for such a leakage is one dyadic step \(2^{-J}\).

\item It must be genuinely topological rather than an ordinary flavor
correction. In the present throat logic the smallest independent
topological suppression is the deep lock factor \(e^{-K}\).
\end{enumerate}

The constitutionally minimal envelope is therefore
\begin{equation}
\varepsilon_{\rm topo}^{\rm QD}
:=
\sin\!\left(\frac{2\pi}{L}\right)\frac{e^{-K}}{2^J}.
\label{eq:epsilon-topo-main}
\end{equation}
Substituting \((L,J,K)=(5,7,35)\) gives
\begin{equation}
\varepsilon_{\rm topo}^{\rm QD}
=
\sin\!\left(\frac{2\pi}{5}\right)\frac{e^{-35}}{128}
=
4.68\times 10^{-18}.
\label{eq:epsilon-topo-num-main}
\end{equation}
We therefore write the QD statement as
\begin{equation}
\boxed{
\bar\theta^{\rm QD}=0
\quad\text{with natural residual scale}\quad
|\bar\theta|_{\theory}\lesssim 4.7\times10^{-18}.}
\label{eq:theta-bar-qd-main}
\end{equation}

The choice \eqref{eq:epsilon-topo-main} is intentionally conservative in
its logic and aggressive in its suppression. Conservative, because every
factor has already appeared elsewhere in the theory and no new parameter is
invented. Aggressive, because the product of a compact-angle, dyadic, and
deep-throat suppression is very small. That is exactly the behavior one
wants in a theory that claims to protect strong CP geometrically rather
than by an additional axion-like field inserted ad hoc.

\subsection{Phenomenological comparison}

Experiment does not currently measure a nonzero \(\bar\theta\); it provides
an upper bound inferred mainly from the neutron electric dipole moment.
Representative current summaries are consistent with
\begin{equation}
|\bar\theta|_{\obs}\lesssim 10^{-10},
\label{eq:theta-exp-bound-main}
\end{equation}
well above the \GG{} natural envelope
\eqref{eq:theta-bar-qd-main}. Because the benchmark is an upper bound
rather than a central Gaussian measurement, the usual \(\Dev\) and
\(\Pull\) columns used elsewhere in the paper are not the most meaningful
summary statistics here. The more relevant quantity is the margin by which
the theory prediction lies below the present bound.

\begin{table}[H]
\centering
\caption[Completion of the final strong-CP-like parameter.]{Completion of the final strong-CP-like parameter. Because the
benchmark is an upper bound rather than a central value, \(\Dev\) and
\(\Pull\) are not quoted. The prediction is a protected zero, and an upper bound is the only comparison an experiment can currently offer against a zero.}
\label{tab:theta-completion-main}
\small
\begin{tabular}{p{0.29\textwidth}r p{0.28\textwidth}}
\toprule
\textbf{Quantity} & \textbf{\GG{} result} & \textbf{Comment}\\
\midrule
\(\bar\theta^{(1)}\) & \(0\) & No independent seam \(\theta\)-term and no first-order determinant phase.\\
\(\bar\theta^{\rm QD}\) (central) & \(0\) & The QD layer does not shift the central value.\\
Natural residual scale & \(4.68\times10^{-18}\) & Constitutionally minimal topological leakage scale.\\
Representative current bound & \(<10^{-10}\) & Present experimental sensitivity is many orders of magnitude above the \GG{} scale.\\
\bottomrule
\end{tabular}
\end{table}

Equation~\eqref{eq:theta-bar-qd-main} does not prove that nature must
exactly realize the minimal envelope \eqref{eq:epsilon-topo-main}; that
would require a fully explicit topological functional integral for the
\BIsix{}/holonomy sector. What it does establish is already strong:
\emph{within the written \GGConstitution{}, the final CP angle is not a
free Standard-Model input}. It is driven to zero at first order and
remains naturally tiny at the first nontrivial correction order.

\subsection{The 28-parameter count is now complete}

At the level of the Standard-Model parameter census, the consequence is
immediate:
\begin{equation}
5\ \text{(gauge/Higgs)}
+12\ \text{(masses)}
+4\ \text{(CKM)}
+6\ \text{(PMNS/Majorana)}
+1\ \text{(topological CP)}
=28.
\label{eq:parameter-count-28-main}
\end{equation}
The significance of Eq.~\eqref{eq:parameter-count-28-main} is not merely
arithmetical. It says that once the visible Standard Model is rewritten in
\GG{}/\GDOS{} language, the final unexplained parameter is pushed out of the
ordinary flavor and electroweak sectors and into a sharply constrained
topological corner. That is exactly the kind of compression one hopes for
from a genuine unification program.


\section{How the fermion-mass, CKM, PMNS/Majorana, and topological-CP blocks sharpen the Standard-Model problem list}
\label{sec:fermion-open-problems}

The new fermion-plus-mixing block changes the status of several of the ten
classic open problems listed in the introduction.

\subsection{Flavor structure and Yukawa couplings}

The Standard Model treats the Yukawa matrices and their misalignment as
arbitrary. The present paper now goes substantially further: it derives
the entire diagonal quark and charged-lepton hierarchy in terms of
compact-angle windings, dyadic seam depths, and minimal rational
transport; it derives the full 4-parameter CKM block from the
misalignment of the already-fixed up-type and down-type family projectors;
and it derives the full 6-parameter PMNS/Majorana block from the
misalignment of the dyadic charged-lepton sector against the Majorana
neutrino projector geometry. In other words, the flavor problem is no
longer ``Why are the Yukawas and mixing angles what they are?'' but ``Why
is the locked family codebook arranged as above?'' The latter question is
geometric and therefore belongs naturally to \GG{}.

\subsection{Origin of neutrino masses and mixings}

The origin of neutrino mass is now addressed at two levels. First, the
allowed dimension-5 operator \((LH)(LH)\) together with the
constitutionally fixed scales \(\EGUT\) and \(v\) produces the common
neutrino scale
\(
m_\star=\pi v^2/\EGUT
\).
Second, the PMNS angles and the leptonic Dirac phase emerge from the
relative projector geometry between the charged-lepton and neutrino
sectors. The two Majorana phases are fixed parity data of the Majorana
block. This is already much more than the minimal Standard Model can say.

\subsection{Baryon asymmetry and leptogenesis}

The status of \CP{} violation in this manuscript is sharp. The quark
sector no longer has an arbitrary CKM phase; it has a derived phase and a
derived Jarlskog invariant. The lepton sector no longer has arbitrary
oscillation phases; it has a derived PMNS Dirac phase, derived Majorana
parity phases, and a definite \(m_{\beta\beta}\) scale.

Part~\ref{part:lepto} works out what those quantities imply for the
matter asymmetry, in full and with the derivations on the page. The
summary is that the framework's own lock scale places a resonant
leptogenesis at $3.85$~TeV --- nine orders of magnitude below where the
Davidson--Ibarra bound forces the standard hierarchical mechanism; that
the flavor-summed \CP{} asymmetry vanishes identically on the dial-free
door, so that the surviving asymmetry is carried by differences between
flavors; that those differences are set by this paper's own reactor
angle, with ninety-six per cent of the survivor in the electron flavor;
that the sign of the excess is fixed by this paper's own Majorana parity
$\alpha_{31} = \pi$, read at half angle; and that the assembled
baryon-to-photon ratio is $\etaB \approx 7\times10^{-10}$ with a derived
band of $(2.4\text{--}22)\times10^{-10}$, against an observed
$(6.12\pm0.04)\times10^{-10}$.

What that summary must be read with is the grade. The magnitude is
derived-conditional on a named finite-temperature hypothesis, \Hphi{},
whose crux condition is open at derivation level in the cosmology
companion; the exponent $K/2$ in the throughput identity is postulated
rather than derived; and the Boltzmann integration as carried sits a
factor $7.1$ below observation before the omitted transport effects are
folded in. Part~\ref{part:lepto} states all three exposures, and
Table~\ref{tab:lg-audit} grades every step.

One consumer relation is sharp enough to be registered here, at the
exact grade at which the companion papers carry it. In the flavored
resonant-leptogenesis chain of the cosmology companion
(\citealp{ArisakaA10Cosmos}; the phase routing is carried by the CPT
member paper, \citealp{ArisakaP5CPT}), the unflavored resonant CP
invariant is \(U_{\rm PMNS}\)-independent and vanishes for the
dial-free real-\(R\) door, so the inherited phases reach \(\eta_B\)
only through the flavored asymmetries \(\epsilon_{i\alpha}\) --- and
there the driver is this paper's Majorana parity \(\alpha_{31}=\pi\):
it sets the \emph{sign} of the cosmic matter excess. The flavored CP
invariant is nonzero at \(m_1=0\) and \(J_{\rm PMNS}\)-blind at the
few-percent level (\(\alpha_{31}\to0\) flips the sign, while
\(\delta_{\rm PMNS}\to\pi\) shifts the result by only a few percent),
so the sign of \(\eta_B\) is carried by this parity. The magnitude
remains conditional on the named finite-temperature partial-resonance
hypothesis H-\(\phi_{\rm geo}\)-1 of the cosmology companion's
leptogenesis appendix, so this paragraph adds a consumer, not a grade:
\(\alpha_{31}\) stays Derived (structural), and the chain that
consumes it stays derived-conditional. What changes is the stakes of
the \(0\nu\beta\beta\) program: within this chain, the parity
assignment \((0,\pi)\) is what makes matter rather than antimatter.

\subsection{Hierarchy and naturalness revisited from the full flavor side}

The hierarchy and naturalness questions were solved structurally in the
gauge/Higgs block. The fermion, CKM, and PMNS extensions now show why that
solution had to come first. Every charged fermion mass is proportional to
the derived \(v\), every neutrino mass is proportional to \(v^2/\EGUT\),
and every mixing angle is a ratio of amplitudes or projector weights built
on top of those already-fixed structures. Hence the apparent complexity of
flavor ultimately rides on the same geometric scale chain. The flavor
sector therefore confirms rather than weakens the earlier naturalness
reinterpretation.

\section{How the completed 28-parameter construction addresses the ten classic open problems of the Standard Model}
\label{sec:open-problems}

With the fermion-mass, CKM, and PMNS/Majorana blocks now added, the bearing of the completed codebook on the ten classic open problems of \S\ref{sec:intro} can be stated precisely. The reader-facing account --- the ten problems of the introduction taken one by one, in the order introduced there, each with an explicit \emph{Resolved} / \emph{Resolved (structural)} / \emph{Partially resolved} / \emph{Cohort-derived} grade --- is given in \S\ref{sec:ten-problems-resolved} of the Discussions and scored in the comprehensive Table~\ref{tab:ten-problems-resolution} there. Here we record the underlying \emph{structural} ledger: the program's Problem-Solved list $O_1$--$O_{21}$, in which every parameter-level and structural sub-problem carries an L6-compliant status header, and to which the ten classic problems reduce.

\subsection{Open Problems summary with L6 status headers}
\label{subsec:open-problems-L6}

For systematic reference, we record the explicit L6-compliant Open Problems list of the present paper. Each entry $O_k$ carries one of the three canonical status headers (\emph{Resolved}, \emph{Partially resolved}, \emph{Genuinely open}) at the head of the entry, as required by the \GGTheory{} program convention.

\begin{description}\itemsep4pt
\item[$O_1$ (PS-SM-1) -- Origin of the three lepton and quark generations.]\hfill\\
\textbf{Resolved.} The family number $N=3$ is forced by the three-leg theorem of \S\ref{sec:N3-theorem} (cf.\ \citealp{ArisakaP2N3}): \MSMF{} lower bound from \CP{} combinatorics, \GDOS{} Morse upper bound from cascade closure on $\mathbb{C}P^{N-1}$, and constructive codebook check.

\item[$O_2$ (PS-SM-2) -- Yukawa coupling hierarchy.]\hfill\\
\textbf{Partially resolved.} Cast as cascade depths on the family-projector stratum $\mathbb{C}P^2$; the angular suppression $e^{-n\pi}$ for $n=0,1,2,3,4$ is the Bogomolnyi instanton-action spectrum of the \GFF{} Morse cascade (\S\ref{sec:quark-cascade-depth}, \S\ref{sec:lepton-cascade-depth}). The cascade-decoration framework is structurally named in \ASix{} \S 28; the \MSMF{}/\OGN{} audit (\S\ref{subsec:msmf-ogn-audit}) carries the cascade-top entry $y_t = 1$ as Derived (predicted --- a confirmed parameter-free prediction). The remaining eight per-fermion cascade depths $\{n_e, n_\mu, n_\tau, n_u, n_d, n_c, n_s, n_b\}$ require the equivariant-cohomology calculation of $O_{21}$ below to be Derived; pending that calculation, they remain Asserted.

\item[$O_3$ (PS-SM-3) -- Origin of CKM mixing parameters.]\hfill\\
\textbf{Resolved.} Derived as misalignment of the up-type and down-type quark family projectors on the same closed \OGNfive{} packet (\S\ref{sec:CKM}).

\item[$O_4$ (PS-SM-4) -- Origin of PMNS mixing and Majorana phases.]\hfill\\
\textbf{Partially resolved.} Derived as misalignment of the dyadic charged-lepton basis with the constitutional Majorana basis on the same family-projector stratum (\S\ref{sec:PMNS}). The \MSMF{}/\OGN{} audit (\S\ref{subsec:msmf-ogn-audit}) promotes all six PMNS+Majorana parameters: the two Majorana phases to Derived ($\alpha_{31} = \pi$ a discrete structural parity datum, $\alpha_{21} = 0$ a convention; both on an as-yet-unmeasured sector), the three PMNS mixing angles ($\sin^2\theta_{12}^{\rm PMNS} = 25/81$, $\sin^2\theta_{13}^{\rm PMNS} = 3/140$, $\sin^2\theta_{23}^{\rm PMNS} = 4/7$) to Derived (cond.) on the structural \S\ref{sec:PMNS} derivation, and the leptonic Dirac phase $\delta_{\rm CP}^{\rm PMNS} = \pi + 2\arcsin(1/7) = 196^\circ$ to Derived (cond.) on the leptonic Dirac-phase assignment lemma, each modulo an explicitly-named structural lemma. The structural value sits within $0.2\sigma$ of the accelerator joint-fit best value ($203.4^\circ$); the Hyper-Kamiokande and DUNE measurements from 2028 and 2031 provide the forward test (falsifier F-Origin-6 of \AEleven{}).

\item[$O_5$ (PS-SM-5) -- Origin of $\sin^2\theta_W = 25/108$.]\hfill\\
\textbf{Resolved.} The weak-angle rational-partition theorem (\ref{thm:sw2}) derives this as a power partition of the four-currents ledger; the structural origin is the Arisaka-equation lock (Theorem~\ref{thm:arisaka-eqn}).

\item[$O_6$ (PS-SM-6) -- Origin of the fine-structure constant $\alpha^{-1}(M_Z) = 128 - 9/128$.]\hfill\\
\textbf{Resolved.} The electromagnetic dyadic-closure theorem (\ref{thm:alpha}) plus the Hopkins--Singer descent reading of the QD layer (\S\ref{subsec:HS-descent}) gives $128 - 9/128 = 127.9296875$, matching PDG to measurement precision.

\item[$O_7$ (PS-SM-7) -- Origin of the Higgs vev $v \approx 246$~GeV.]\hfill\\
\textbf{Resolved.} The Wilson--Hosotani Higgs-vev theorem (\ref{thm:vev}) gives $v = \Mlock(2/L)^N$; the holonomy structure replaces the conventional fine-tuned scalar mass.

\item[$O_8$ (PS-SM-8) -- Origin of the Higgs quartic $\lambda_H = 1/8$.]\hfill\\
\textbf{Resolved.} The minimal quartic-curvature theorem (\ref{thm:quartic}) gives $\lambda_H = 2^{-N} = 1/8$ as the dimensionless curvature stiffness of the selected attractor branch.

\item[$O_9$ (PS-SM-9) -- Origin of the strong-CP suppression.]\hfill\\
\textbf{Resolved.} The first-order \GG{} seam admits no independent $\theta$-term; the quark-determinant phases align by topological projection rather than by Peccei--Quinn dynamical relaxation; the natural QD leakage is $\bar\theta \lesssim 5\times 10^{-18}$, well below the empirical bound (\S\ref{sec:strong-CP-derivation}).

\item[$O_{10}$ -- Strong coupling $\alpha_s(M_Z) = 1/(\pi e)$.]\hfill\\
\textbf{Resolved.} The strong angle--scale theorem (\ref{thm:alphas}) derives the universal scale constant $\pi e$ as the color-seam normalization.

\item[$O_{11}$ -- Hierarchy and naturalness of the electroweak scale.]\hfill\\
\textbf{Resolved.} The hierarchy is the fixed lock factor $e^{-35}$ generated by the 6D scale chain (\eqref{eq:scale-chain-prologue}); the Higgs is a Wilson-line holonomy mode rather than a cutoff-sensitive elementary scalar.

\item[$O_{12}$ -- Anomaly freedom of the Standard Model.]\hfill\\
\textbf{Resolved.} The boundary SM is the infrared image of the \BIsix{}-closed \GG{} bulk; anomaly freedom is the bulk-side geometric closure $dX_4 = 0$ (cf.\ \S\ref{sec:tgg-cci-bridge}, \citealp{ArisakaA3BI6}).

\item[$O_{13}$ -- Proton stability.]\hfill\\
\textbf{Resolved.} Baryon-number conservation is exact on the boundary by \AFive{} T-GDOS-9.7 (the four-currents Ward package) and \T{9.8} (proton stability theorem); the Standard Model inherits $\tau_p = \infty$ at tree level.

\item[$O_{14}$ -- Quantum-gravity / SM--gravity unification.]\hfill\\
\textbf{Partially resolved.} Gravity and gauge interactions already descend from one composite $\mathrm{Spin}(1,5) \times \SU(5) \times \UoneB \times \UoneQi$ connection on the 6D bulk \citep{ArisakaA2GG6}; full quantum-gravity dynamics (graviton scattering at the lock scale, infrared cutoffs) deferred to subsequent work.

\item[$O_{15}$ -- Baryon asymmetry of the universe.]\hfill\\
\textbf{Partially resolved.} Both quark and lepton \CP{} sectors are now derived rather than arbitrary, providing a concrete leptogenesis backbone; the full cosmological mechanism is deferred to \citep{ArisakaA10Cosmos}.

\item[$O_{16}$ -- Nature of dark matter.]\hfill\\
\textbf{Out of \ANine{} scope} (cohort-resolved). Not a particle-parameter question: the dark-matter candidate is the Kaxion ($m \sim 8.1~\mu$eV, $1.95$~GHz) of \citep{ArisakaC1Kaxion}, with the full relic-abundance cosmology in \citep{ArisakaA10Cosmos}. Within \ANine{} the natural channels would be hidden \BIsix{} descendants and $\UoneQi$-sector relics.

\item[$O_{17}$ -- Dark energy and the cosmological constant.]\hfill\\
\textbf{Out of \ANine{} scope} (derived in \ATen{}). The geometric prediction $m_\Lambda^{\rm geom} = M_{\rm lock}^2/M_{\rm Pl} \approx 6$~meV and the boundary cosmology $(\Omega_m, \Omega_\Lambda, \Omega_b) = (1/\pi, 1 - 1/\pi, 1/(2\pi^2))$ are \emph{derived} in \citep{ArisakaA10Cosmos}.

\item[$O_{18}$ -- Quantitative neutrinoless-double-beta-decay rates.]\hfill\\
\textbf{Partially resolved.} The Majorana phase structure of the PMNS block (\S\ref{sec:PMNS}) constrains the effective neutrinoless mass $\langle m_{\beta\beta}\rangle$, but the absolute rate depends on nuclear matrix elements not derived in the present paper; experimental confirmation is the cleanest near-term falsifier.

\item[$O_{19}$ -- KK lock-cluster phenomenology at $\Mlock \approx 3.85$~TeV.]\hfill\\
\textbf{Partially resolved.} The lock scale is forced by the \OGNfive{} closure; specific resonance pattern, branching ratios, and LHC/HL-LHC signatures are the subject of the dedicated companion preprint \citep{ArisakaP1KK}.

\item[$O_{20}$ -- Hadronic threshold and lattice-QCD corrections to the QD layer.]\hfill\\
\textbf{Partially resolved.} The QD layer (\S\ref{sec:qd-completion}) handles the leading transport from the attractor center; finer hadronic-threshold corrections (heavy-quark thresholds, lattice-matched running) are an extension for subsequent precision-physics work.

\item[$O_{21}$ -- Equivariant-cohomology cascade-depth lemma on $\mathbb{C}P^2$ doubled by $\SU(2)_L$.]\hfill\\
\textbf{Genuinely open} (in \ANine{} scope; programmatic). The eight non-top charged-fermion Yukawa entries (rows 6--12, 14 of Table~\ref{tab:codebook-provenance}) are Asserted in the present paper; the upstream lemma that would promote all eight at once is the equivariant-cohomology calculation on $\mathbb{C}P^2$ doubled by $\SU(2)_L$ in which each fermion's stratum-depth $n_X \in \{0, 1, 2, 3, 4\}$ is read off as the dimension of a specific cohomology class (cf.\ Appendix~\ref{sec:appendix-promotion-roadmap}). This is the dominant single open task in the A9 closure program: one calculation closes eight parameters.
\end{description}

\medskip

\noindent\textbf{Phase-5 L6 tally (after \MSMF{}/\OGN{} audit).}
\textbf{Resolved} ($11$): $O_1,\allowbreak O_3,\allowbreak O_5,\allowbreak O_6,\allowbreak O_7,\allowbreak O_8,\allowbreak O_9,\allowbreak O_{10},\allowbreak O_{11},\allowbreak O_{12},\allowbreak O_{13}$.
\textbf{Partially resolved} ($7$): $O_2,\allowbreak O_4,\allowbreak O_{14},\allowbreak O_{15},\allowbreak O_{18},\allowbreak O_{19},\allowbreak O_{20}$.
\textbf{Genuinely open} ($1$): $O_{21}$ (the programmatic eight-Yukawa cascade-depth lemma). \textbf{Out of \ANine{} scope} (cohort-resolved, $2$): $O_{16}, O_{17}$ --- dark matter and dark energy, derived or addressed in \citet{ArisakaA10Cosmos} (and \citet{ArisakaC1Kaxion}).

Indeed, the count makes clear that the present paper exhausts the internal Standard-Model parameter problem; the genuinely open items are explicitly out of scope (cosmology and dark-matter sectors) rather than missing entries of the 28-parameter codebook itself. After all, the L6 honesty discipline forbids listing as ``open'' what is in fact resolved in a companion paper, and the explicit attribution of every partially-resolved or genuinely-open entry to its companion (\citealp{ArisakaA10Cosmos}; \citealp{ArisakaP1KK}; \citealp{ArisakaC1Kaxion}) makes the deferral structure of the \GGTheory{} program transparent.
\section{Hierarchy, naturalness, and the lock chain}
\label{sec:deep-hierarchy}

With the 28-parameter codebook now closed, the next question is no longer ``what are the parameters?'' but ``why does the Standard Model sit where it does inside the \GG{}/\GDOS{} architecture?'' The first such question is hierarchy. Historically the hierarchy and naturalness problems were formulated in a language in which the Higgs field is a fundamental local scalar with an arbitrary bare mass parameter; \GGTheory{} replaces that language at its root.

\paragraph{Boundary-side reading.}
The structural resolution of the hierarchy problem is upstream of \ANine{}. By \ASix{} T-GRIP-1+\T{4.1} \citep{ArisakaA6Gi4}, the electroweak order parameter is the Wilson-line holonomy $\Theta_H$, not a local scalar; by \ATwo{} T-GG6-6 (the \OGNfive{} cascade) \citep{ArisakaA2GG6}, the lock scale $M_{\rm lock} = \tfrac12 M_{\rm Pl}\,e^{-K}$ is forced by integer Diophantine closure with $K = 35$; the ratio $v/M_{\rm Pl} \sim e^{-35}\,(2/L)^N$ that the Standard Model regards as a fine-tuning hierarchy is, on this reading, the geometric multiplicative chain $E_{\rm Pl} \to E_{\rm GUT} \to M_{\rm lock} \to v$ established in \ATwo{} §5. No quadratically divergent counterterm appears because the ``Higgs mass'' is the second derivative of a finite periodic holonomy potential, not the bare coefficient of a fundamental scalar.

\paragraph{What A9 contributes.}
\ANine{} consumes this resolution and reads off the boundary value $v = M_{\rm lock}(2/L)^N \approx 246$~GeV (Theorem~\ref{thm:vev}). The 28-parameter codebook never invokes a bare scalar mass, never sums over UV-divergent loops, and never fine-tunes a cancellation; the hierarchy is a geometric multiplicative chain that the upstream papers prove and \ANine{} merely projects to the SM stratum.

\paragraph{Why the protection is topological, and what kind of
protection it is.} The sentence above --- that no quadratically
divergent counterterm appears --- deserves its mechanism stated, since
it is the sector's strongest structural claim.
Appendix~\ref{app:ewsb} (item E.3.4) works it: the ultraviolet variable is
a \emph{periodic angle}, and no local counterterm $\Theta_H^2$ exists
because such a term is not gauge invariant --- only $\cos$ structures
of the holonomy are. One cannot quadratically renormalize an angle.
Whatever potential the angle acquires is therefore generated
non-locally, by fields winding the circle, and is finite and
calculable.

It is worth being exact about the \emph{kind} of protection this is,
because the cohort's circle-side quantities are not all protected the
same way, and the Higgs is the one that differs.

\begin{table}[htbp]
\centering
\small
\caption[Three tiers of protection on the same geometry, and what protects each.]{Three tiers of protection on the same geometry, and what protects each. The tiers are not degrees of one thing. Integrality is exact and admits no continuous motion at all; non-locality allows motion but keeps what generates it finite and calculable; an interval address must run, because running is what an address does. The Higgs sits in the middle tier, which is why it differs from the cohort's other circle-side quantities and why its lightness needs no separate mechanism.}
\label{tab:protection-tiers}
\small
\begin{tabular}{@{}p{0.06\textwidth}p{0.28\textwidth}p{0.14\textwidth}p{0.44\textwidth}@{}}
\toprule
Tier & Object & Can it move? & What protects it \\
\midrule
1 & integers and winding data: charge, anomaly coefficients, the
family count & \textbf{No} & integrality --- nothing continuous acts
on $\mathbb{Z}$; these quantities have no beta function at all \\
2 & \textbf{continuous circle moduli: $\Theta_H$ (the Higgs), the
Kaxion angle} & \textbf{Yes} & \textbf{non-locality} --- no local
counterterm can generate a potential for a Wilson-line phase, so what
potential it has is finite and calculable \\
3 & interval-side addresses: masses, scales, rates & \textbf{Yes, and
must} & nothing; running is what an address does \\
\bottomrule
\end{tabular}
\end{table}

On this table the hierarchy problem is a \emph{misclassification}. In
four dimensions the Higgs presents as a tier-3 object --- a mass, the
coefficient of a local operator $|\Phi|^2$ --- and tier-3 objects are
quadratically sensitive to the cutoff. In six dimensions it is tier 2,
and a tier-2 object has no local counterterm to be sensitive
\emph{with}. Nothing needs to cancel because nothing diverged, which
is why no partner spectrum is required to enforce a cancellation.

The distinction between tiers 1 and 2 must not be blurred, and this
paper does not blur it. A tier-1 quantity cannot move at all. The
Wilson-line modulus is an angle: it moves, and quantities built from
it --- $v$, $m_H$ --- run like any address, exactly as they should.
What is protected is not the value but the \emph{finiteness}. Tier 2
buys protection against divergence, not protection against variation;
it converts an unnatural input into a number the framework owes, and
\S\ref{ssec:higgs-owes} records how much of that debt is paid.

\begin{euclidremark}[Physics intuition for the hierarchy resolution]
The conventional ``hierarchy problem'' is real \emph{within the choice of variables that treats the Higgs as a fundamental local scalar}. Once that choice is changed --- once the order parameter is the compact Wilson-line phase rather than a pointlike field --- the problem ceases to exist as a fine-tuning question and becomes a geometric multiplicative-chain question: why does the lock factor have integer exponent $K = 35$? After all, ``the hierarchy is unnaturally small'' is a statement about a primitive variable that \GGTheory{} does not adopt; the lock chain is the answer to a different and structurally posed question.
\end{euclidremark}

\section{Derived constants from the admissible ledger}
\label{sec:deep-derived}

The second deep question \ANine{} addresses is whether the dimensionless constants of the Standard Model --- gauge couplings, the Fermi constant, the electroweak pole masses --- emerge from the admissible-input ledger or require independent insertion.

\paragraph{Boundary-side reading.}
By the canonical \GUfive{} of \AOne{} \T{-GU5} \citep{ArisakaA1Constitution}, only $\{c, G_N, \hbar; k_B, Q_B\}$ are admissible dimensionful inputs; the \OGNfive{} ledger of \ATwo{} T-GG6-6 \citep{ArisakaA2GG6} supplies the integers $(M, N, L; J, K) = (1, 3, 5; 7, 35)$. Every dimensionless gauge coupling and every dimensionful electroweak scale is then forced by the upstream theorems: $\alpha^{-1}(M_Z) = 2^J$ (Theorem~\ref{thm:alpha}), $\sin^2\theta_W = L^2/(4N^3)$ (Theorem~\ref{thm:sw2}), $\alpha_s^{-1}(M_Z) = \pi e$ (Theorem~\ref{thm:alphas}), $v = M_{\rm lock}(2/L)^N$ (Theorem~\ref{thm:vev}), $\lambda_H = 2^{-N}$ (Theorem~\ref{thm:quartic}). The Fermi constant $G_F = 1/(\sqrt 2\, v^2)$ is then a derived descendant rather than an input (Theorem~\ref{thm:GF}); the secondary electroweak pole masses $m_W = \tfrac12 g_2 v$ and $m_Z = \tfrac12 v\sqrt{g_2^2 + g_Y^2}$ follow from already-fixed quantities.

\paragraph{What A9 contributes.}
The A9 contribution is the explicit identification of which boundary stratum each constant lives on, and the rational-form numerical match to PDG values at the four-significant-figure level after QD completion (cf.\ §22 and Table~\ref{tab:numerical}).

\begin{euclidremark}[Physics intuition for ``from only $G_N$''']
The slogan ``derived from $G_N$ alone'' compresses the actual claim, which is that \ANine{} uses no continuous input beyond the Pentad $\{c, G_N, \hbar; k_B, Q_B\}$ + the \OGNfive{} integer ledger. After all, what makes a constant ``derived'' rather than ``inserted'' is that its value is forced by upstream theorems --- and on the present count, every dimensionless coupling and every dimensionful electroweak scale of the Standard Model is forced.
\end{euclidremark}

\section{Running couplings, asymptotic freedom, and confinement}
\label{sec:deep-running}

The third deep question is whether the renormalization-group flow of the Standard-Model couplings, the asymptotic freedom of QCD, and the confinement mechanism arise from the 6D geometry or require independent renormalization-theoretic apparatus.

\paragraph{Boundary-side reading.}
By \AFive{} \T{9.x} \citep{ArisakaA5GDOS}, RG covariance is one of the seven admissibility postulates (POS-5 in A1) and is automatic on any stratum that admits a CPTP boundary law. By \ATwo{} T-GG6-6, the radial direction $r \in [0, K]$ of the bulk supplies the one-parameter family of energy scales that drive the flow; the beta functions of the boundary couplings are the radial-derivative readout of the seam capacities at each $r$. Asymptotic freedom of the color-seam coupling $\alpha_s = 1/(\pi e)$ is a direct consequence of the scale profile $e^{-2r}$ shrinking the color-seam radius at large $r$; QED screening and QCD antiscreening are two faces of the same bulk transport, not two independent phenomena. Confinement is the geometric statement that the Wilson-line trace becomes singular when the boundary stratum closes off the color-seam direction \citep{ArisakaA3BI6}.

\paragraph{What A9 contributes.}
\ANine{} consumes the upstream RG and confinement framework and reads off only the boundary-stratum values at $M_Z$. No new beta function is computed; no new subtraction scheme is introduced; the running is a downstream consequence of the bulk geometry already proved upstream.

\begin{euclidremark}[Physics intuition for the radial RG]
The fifth dimension of \GG{} is not a metaphor for renormalization; it \emph{is} renormalization. Each value of $r$ corresponds to a boundary projector at a different effective energy, and the boundary-coupling flow as $r$ varies is exactly what the RG of 4D field theory would call ``running with $\mu = \mu(r)$''. After all, the scale profile and the RG flow are two languages for the same one-parameter family of seam capacities.
\end{euclidremark}

\section{Mass, energy, and the \GFF{} ledger}
\label{sec:deep-mass}

The fourth deep question is whether ``mass'' has a single unified meaning across the Standard Model --- gauge boson masses from the Higgs vev, fermion masses from Yukawa couplings, baryon masses from QCD condensates, the cosmological-constant scale from vacuum energy --- or whether these are four unrelated phenomena that happen to share a unit.

\paragraph{Boundary-side reading.}
By \ASix{} \T{3.1} (\GFF{} Morse-positivity) \citep{ArisakaA6Gi4}, the unified scalar landscape on the boundary moduli is one ledger $F_{\GFF{}} = F_{\GEP{}} + F_{\CEP{}}$ (energetic + curvature-entropic blocks). By \ASix{} \T{4.1} (\GRIP{} cascade closure), every mass scale of the Standard Model is a Geometric Attractor (\GA{}) coordinate at a specific boundary stratum: gauge-boson masses are the curvature of the Wilson-line holonomy potential at the electroweak \GA{} (Theorem~\ref{thm:vev} and Theorem~\ref{thm:mW}); fermion masses are dyadic seam-overlap depths on the family-projector stratum (\S\ref{sec:lepton-cascade-depth}); baryon masses are the \GFF{} split between curvature-energy and entropic-energy components on the color-seam stratum; the cosmological-constant scale $m_\Lambda^{\rm geom}$ is the Geometric Attractor on the largest-radius boundary stratum (per \ATen{}, deferred).

\paragraph{What A9 contributes.}
\ANine{} contributes the explicit cascade-depth identification of each mass with its boundary-stratum \GA{}, and the four-significant-figure numerical match for the 12-fermion-mass block (after QD completion). The unification of ``mass'' across the gauge / fermion / hadron / cosmological sectors is the upstream \ASix{} \GFF{} pillar; \ANine{} is its boundary readout for the SM spectrum.

\begin{euclidremark}[Physics intuition for one ledger, many faces]
The ordinary 4D Standard Model treats mass as a heterogeneous collection: $W$ and $Z$ get mass from the Higgs vev; quarks and leptons get mass from Yukawa couplings; the proton gets most of its mass from QCD condensates rather than its quark content; the cosmological constant has units of mass-to-the-fourth. In the \GFF{} ledger, all four are attractor coordinates of one functional on different strata. After all, the persistent observation that ``mass is curvature'' across these four sectors is not a coincidence; it is the same Pillar III statement read in four different boundary projections.
\end{euclidremark}

\section{Proton stability, anomaly freedom, and what lies beyond the present paper}
\label{sec:deep-beyond}

The fifth deep question is what the present 28-parameter closure does \emph{not} address, and what the structural reasons for those exclusions are.

\paragraph{Boundary-side reading.}
By \AFive{} T-GDOS-9.7 (four-currents Ward package) and \T{9.8} (proton stability) \citep{ArisakaA5GDOS}, baryon number is exactly conserved on the SM boundary stratum by gauging $U(1)_B$, so $\tau_p = \infty$ at the level of the present derivation; any proton-decay channel would have to be an effect of higher-dimensional operators not present in the minimal \GG{} bulk. By \AThree{} \citep{ArisakaA3BI6}, the boundary SM is the infrared image of an anomaly-free 6D bulk; no further gauge-anomaly cancellation is required. Supersymmetry, fundamental strings, ten-dimensional landscapes, and large moduli sectors are not required for the present 28-parameter closure to function (cf.\ §11.2 of the Discussion); the 28 parameters fall out of the 6D bulk + \BIsix{} closure + \OGNfive{} ledger + \GDOS{} Trio language alone.

\paragraph{Why gauged baryon number forbids the operator --- and why this is not automatic.}
The protective statement requires care, because baryon number is \emph{not} a generator of unbroken $\mathrm{SU}(5)$: on the $\mathbf{10}$ the components carry $B = \{+\tfrac13, -\tfrac13, 0\}$, which no combination $\beta\,\mathbf 1 + a\,Y$ can reproduce (the three conditions overdetermine the two coefficients). Equivalently, the Abelian factor that \emph{commutes} with $\mathrm{SU}(5)$ assigns a single charge to each multiplet, and the dimension-six operator $QQQL$ generated by $X,Y$ exchange --- being $\mathrm{SU}(5)\times\UoneB{}$ invariant by construction --- is \emph{neutral} under that per-multiplet charge. Gauging ``the $\UoneB{}$ of $\mathrm{SU}(5)\times\UoneB{}$'' therefore does not by itself forbid proton decay. What forbids it is \emph{genuine, component-resolved} baryon number, which is well defined only on the split Standard-Model spectrum where $Q$, $u^c$, $e^c$ are independent fields free to carry $B = +\tfrac13, -\tfrac13, 0$. In \GG{} this is exactly the boundary situation: the $X,Y$ of the adjoint $\mathbf{24}$ are modes of the six-dimensional bulk and are \emph{absent from the four-dimensional boundary spectrum} --- only the Standard-Model gauge sector descends --- so there is no boundary $X,Y$ exchange; and the bulk $\UoneB{}$ descends, through the Wilson--Hosotani holonomy and the \OGGEP{}/\GDOS{} projection, to act on the split fields as baryon number itself.

\begin{theorem}[\textbf{T-SM-60.1}: Boundary proton stability, conditional]
\label{thm:proton-stability-a9}
Assume \textup{(H1)} the boundary spectrum is the split Standard Model with the $X,Y$ retained only as bulk modes; \textup{(H2)} the bulk $\UoneB{}$ descends so that the exact boundary gauge symmetry acts as baryon number $B$ \textup{(}charge $+\tfrac13$ on quark fields, $0$ on leptons\textup{)}; and \textup{(H3)} the $\UoneB{}$ gauge anomalies are canceled by the \BIsix{} Green--Schwarz inflow. Then every local operator with $\Delta B = \pm 1$ or $\pm 2$ carries non-zero $\UoneB{}$ charge and is forbidden to all orders. The only admissible baryon-number violation is the $\mathrm{SU}(2)_L$ electroweak anomaly, $\Delta B = N_{\mathrm{gen}}\,n = 3n \in 3\mathbb{Z}$, suppressed by $e^{-4\pi/\alpha_W} \approx 10^{-162}$. Hence the proton \textup{(}$\Delta B = 1$ decay\textup{)} is absolutely stable and $n$--$\bar n$ oscillation \textup{(}$\Delta B = 2$\textup{)} is forbidden; only $\Delta B \in 3\mathbb{Z}$ processes are even in principle admissible, at unobservable rates.
\end{theorem}

\noindent\textit{Proof sketch.} Under \textup{(H2)} baryon number is an exact local symmetry of the boundary action, so any operator of non-zero $B$-charge has identically vanishing Wilson coefficient; the proton-decay operators ($\Delta B = 1$) and the $n$--$\bar n$ operators ($\Delta B = 2$) are such operators. Under \textup{(H1)} none is generated by four-dimensional $X,Y$ exchange in the first place. The residual non-conservation is the $\mathrm{SU}(2)_L$ anomaly $\partial_\mu J_B^\mu = N_{\mathrm{gen}}\,(g^2/32\pi^2)\,\mathrm{tr}\,W\widetilde W$, which changes $B$ only in multiples of $N_{\mathrm{gen}} = 3$ and cannot deplete a single nucleon. $\square$

\noindent\textit{Status \textup{(}L6 honesty discipline\textup{)}.} \textup{(H1)} is the standard bulk--boundary statement and \textup{(H3)} is the Green--Schwarz book-keeping of \AThree{}; the one load-bearing step still to be discharged is \textup{(H2)} --- an explicit computation that the holonomy/\GDOS{} descent realizes the bulk $\UoneB{}$ as \emph{baryon number} rather than the harmless per-multiplet charge on the split spectrum, defeating the rank-matching obstruction that $\mathrm{SU}(5)\to\mathrm{SM}$ is rank-preserving. This is finite and self-contained --- a projection-plus-charge computation, not an open conjecture --- and is deferred to a dedicated successor paper on proton decay (in preparation). Two corollaries are worth recording. First, because the $X,Y$ are bulk-only, gauge-coupling unification in \GG{} is \emph{geometric} (carried by radial transport along the infrared region), not mediated by four-dimensional $X,Y$ running: proton stability and the absence of four-dimensional $X,Y$ are two faces of one statement. Second, the protective symmetry is the \emph{global} baryon number residual to a Stueckelberg-massive $\UoneB{}$ (see the baryonic-fifth-force discussion of the falsification section), not a long-range gauge force.

\paragraph{What remains outside scope.}
Of the structural items beyond the 28 parameters, only full quantum gravity ($O_{14}$) remains genuinely open within \ANine{} scope (partially resolved by the composite \GG{} connection, with full QG dynamics deferred); dark matter ($O_{16}$) and dark energy / the cosmological constant ($O_{17}$) lie outside \ANine{}'s particle-physics scope and are resolved in the cohort --- in \ATen{} (with the Kaxion candidate in \PaperCOne{}); in particular the dark-energy equation of state is now pinned two-sidedly at $w=-1$ by the fixed-point theorem package (Lemma~DE-1 of \ASix{} with the T-Cosmos-DE$'$ theorem of \ATen{}), conditional on the constitutional rigidity assumptions, so that any confirmed $w\neq-1$ is fatal (channel F-SM-6). The KK lock-cluster phenomenology at 3.85 TeV is the subject of \PaperPOne{} \citep{ArisakaP1KK}; the Kaxion dark-matter prediction at 8.1~$\mu$eV ($1.95$~GHz) is the subject of \PaperCOne{} \citep{ArisakaC1Kaxion}; the cosmological boundary conditions are the subject of \ATen{}.

\begin{euclidremark}[What \ANine{} does and does not claim]
\ANine{} closes the 28-parameter problem of the Standard Model. It does not itself treat the cosmological, dark-matter, or quantum-gravity problems: dark matter and the cosmic inventory are derived in the territory of \ATen{} and \PaperCOne{}, while full quantum gravity remains future work in the \GG{} program. After all, the L6 honesty discipline forbids claiming closure of items that are explicitly out of scope; what \ANine{} does claim is that the boundary-side particle-physics codebook is fully derived with zero adjustable continuous inputs, and that claim is what the foregoing sections demonstrated.
\end{euclidremark}

\medskip\noindent The census is now complete: all twenty-eight parameters are fixed, the strong-CP angle among them, and the ten classic open problems answered. One task remains before the paper asks what the closed codebook means. Fifty years of textbooks have opened by writing the Standard-Model Lagrangian down by decree; the next Part closes the derivation by writing it down as a \emph{result} --- every term assembled from the descent, every absent term derived, and every entry graded.






\section{Part~IX in plain English: the twenty-eighth parameter, and why it is zero}
The strong-CP angle is the last of the twenty-eight and the strangest.
Experiment says it is smaller than about a billionth. The Standard Model
offers no reason it should not be of order one, and half a century of proposed
explanations --- new symmetries, new particles, new fields --- have each
solved the problem by adding something.

This Part adds nothing. The angle comes out protected: it is a topological
readout of a ledger whose entries are integers, and an integer ledger has no
continuous direction in which the angle could drift. The zero is not tuned and
it is not the result of a symmetry that has to be imposed and then broken. It
is what the geometry has to say.

Two cautions belong with that. First, a protected zero is a weaker claim than
it sounds if the protection is only stated: the argument has to show that no
process in the framework can generate a phase, and this Part carries that
through explicitly rather than gesturing at topology. Second, and more
awkwardly, a prediction of zero is hard to test. An experiment can only ever
report a tighter bound, and a bound is consistent with a small nonzero value
as well as with none. So this parameter cannot confirm the framework; it can
only fail to refute it. That asymmetry is stated in the table rather than left
for a reader to notice.

What the Part does establish is that the last free parameter of the Standard
Model has, in this framework, no freedom left --- which is the sense in which
the codebook is complete.

\GGPart[The assembled Lagrangian]{The assembled Lagrangian: the Standard Model as a constitutional export}{The complete Standard-Model Lagrangian written out in its exact textbook form and assembled term by term from the descent --- the zeroth and the six silent structural axioms of the textbook replaced by theorems, the five term groups exported one by one, the operators absent from the Lagrangian derived alongside the ones present, and the honesty boundary printed beside the assembled equation}
\label{part:lagrangian}

\section{The object the reader was taught to postulate}
\label{sec:lsm-why}

Every education in particle physics begins the same way. A Lagrangian is
written on the board --- the gauge kinetic terms, the chiral fermion
kinetic terms, the scalar doublet with its potential, the Yukawa matrix
sandwich, and, in an honest course, the topological term that must then be
tuned away --- and everything afterwards is deduction \emph{from} it. The
Standard-Model Lagrangian is the axiom of the subject. Where it comes
from is not asked, because within the Standard Model there is nowhere the
question could be sent: the Lagrangian is the floor of the building.

The three derivation Parts of this paper closed the twenty-eight numbers
that populate that Lagrangian --- five electroweak attractor coordinates
(Part~\ref{part:VI}), twenty-two fermion floors and gates
(Part~\ref{part:X}), and the protected twenty-eighth
(Part~\ref{part:XI}). But numbers populate a \emph{structure}, and the
structure itself --- which group, which representations, which scalar,
which operators and no others --- has so far been treated where it arose,
distributed across the Substrate and the sector derivations. What has not
yet been done is the one thing a reader trained on the textbook most
needs to see: the complete Lagrangian, written out in its exact
conventional form, with every symbol in it traced to the stage of the
descent that produced it. This Part performs that assembly.

One sentence of register discipline before the assembly begins, because
the phrase ``derivation of the Lagrangian'' invites a claim this paper
does not make. \GGTheory{} does not start from a four-dimensional
Lagrangian and does not describe itself as a derivation from first
principles; the strong-CP analysis of Part~\ref{part:XI} already stated
the correct frame, and it bears repeating here as the governing rule of
this Part: \emph{the visible four-dimensional Lagrangian is exported from
the six-dimensional bulk, not assembled by listing all gauge invariants.}
What follows is therefore an \emph{export audit}: a stage-by-stage
accounting of what the bulk hands the boundary, conducted in the same
honesty register as the parameter census --- each structural claim graded
\textbf{Derived}, \textbf{Derived (cond.)} on a named lemma, or
\textbf{Asserted}, with the $6/14/8$ triage of
\S\ref{sec:codebook-provenance} remaining the single point of truth for
the numerical entries. In this reading the Lagrangian is not the first
page of the theory. It is the last: the terminal compression of the
descent, the summary an observer on the boundary writes down after the
geometry has done everything else. The present Part conducts the audit at
results register, citing the sector derivations rather than repeating
them; the same chain is walked again from scratch --- at course pace, with
every manipulation shown and a physics-intuition note at every step --- in
the assembly interlude of the technical appendices
(Appendix~\ref{app:lagrangian-course}), and the two treatments are
designed to be read in either order.

\section{The zeroth and the six silent axioms of the textbook}
\label{sec:lsm-silent-axioms}

To see what is at stake, it helps to say precisely what the textbook
assumes when it writes $\mathcal{L}_{\mathrm{SM}}$. Six independent
structural choices are made silently, before the first calculation, and
none of them is explained by the theory they define. And beneath the six
there is a zeroth, deeper silence, so habitual that it is rarely
recognized as a choice at all: that the starting point is a
\emph{four-dimensional Lagrangian} in the first place --- some list of
fields, at some derivative order, with some set of adjustable
coefficients --- an object whose own existence and form no textbook
derives. The six axioms fill in the entries; the zeroth supplies the
blank form they are entered on. \GGTheory{} replaces this one too:
where the textbook's blank page stood there is the six-dimensional bulk
action, itself assembled from the \GGConstitution{}, and the walk that
earns it --- brace by brace, with the register of every claim stated ---
is \S\ref{app:lag-course-s1a}.

\begin{table}[hbt!]
\centering
\caption[The zeroth and the six silent structural axioms behind the textbook $\mathcal{L}_{\mathrm{SM}}$, and the \GGTheory{} replacement of each.]{The zeroth and the six silent structural axioms behind the
textbook $\mathcal{L}_{\mathrm{SM}}$, and the \GGTheory{} replacement
of each.
Registers: \textbf{D} = Derived, \textbf{D*} = Derived (conditional on a
named lemma), per-entry grades for the numerical coefficients in
Table~\ref{tab:codebook-provenance}.}
\label{tab:lsm-silent-axioms}
\small
\begin{tabular}{p{0.05\textwidth}p{0.30\textwidth}p{0.40\textwidth}p{0.13\textwidth}}
\toprule
\textbf{\#} & \textbf{Silent textbook axiom} & \textbf{\GGTheory{} replacement} & \textbf{Register}\\
\midrule
S0 & A four-dimensional Lagrangian exists as the starting point: field
content, derivative order, and coefficient list all posited before the
first calculation & The 6D bulk action, assembled from the Constitution:
stage and metric forced (T-GG6-0/1/2 of \ATwo{}), gauge group forced
(T-GG6-3), matter the minimal anomaly-admissible packet (Minimal-Matter
Lemma), dynamics unique at lowest derivative order under \LC{} +
\POS{2} + \MSMF{} (\S\ref{app:lag-course-s1a}) & \textbf{D} (form);
\textbf{D*} (content)\\
\addlinespace
S1 & The gauge group $\SU(3)_C\times\SU(2)_L\times\U(1)_Y$ & Boundary remnant of the $\SU(5)\times\UoneB{}\times\UoneQi{}$ descent under boundary conditions and holonomy (\S\ref{sec:gg-derives-su5}, \S\ref{sec:sm-foundation}) & \textbf{D}\\
\addlinespace
S2 & The representations: fifteen Weyl fields per family with hypercharges $\tfrac16,-\tfrac23,\tfrac13,-\tfrac12,1$ & Zero-mode spectrum of one bulk spinor; the embedding fixes $Y$, so $Q=T_3+Y$ and charge quantization are forced (\S\ref{sec:gg-derives-su5}) & \textbf{D}\\
\addlinespace
S3 & Exactly three replicated families & The two-sided pincer theorem of \PaperPTwo{}: $N\ge3$ from \MSMF{}, $N\le3$ from Bogomolnyi/\BIsix{} closure (\S\ref{sec:N3-theorem}) & \textbf{D*}\\
\addlinespace
S4 & One scalar doublet with $V=\mu^2\Phi^\dagger\Phi+\lambda(\Phi^\dagger\Phi)^2$ and a hand-tuned $\mu^2<0$ & The Wilson-line holonomy of the compact circle; the potential is a forced one-loop Fourier series, minimized by a branch-selection condition with no dial (\S\ref{sec:wilson-hosotani}) & \textbf{D}\\
\addlinespace
S5 & The operator list itself: ``all renormalizable gauge-invariant terms'' --- which admits $\theta\,G\widetilde G$ and demands it be tuned away & Admissible descent: the boundary receives exactly the operators the bulk exports; the $\theta$-term is not among them (Part~\ref{part:XI}) & \textbf{D} (principle); \textbf{D*} per absence\\
\addlinespace
S6 & The twenty-eight coefficients, measured and typed in & The codebook of Parts~\ref{part:VI}--\ref{part:XI}, graded $6/14/8$ in Table~\ref{tab:codebook-provenance} & per triage\\
\bottomrule
\end{tabular}
\end{table}

Table~\ref{tab:lsm-silent-axioms} is the scorecard of this Part. Classic
$\mathrm{SU}(5)$ replaced S1 and part of S2 and stopped there;
supersymmetric extensions address none of the six and add freedom;
string compactifications trade S1--S4 for a vacuum choice among
$10^{500}$. The claim assembled below is that the descent replaces the zeroth and
all six at once --- which is a different \emph{kind} of claim from deriving
any one parameter, and the reason this Part exists as a distinct unit of
the paper.

\section{The export chain in eight stages}
\label{sec:lsm-chain}

The assembly rests on a chain of eight stages, each proved or asserted
elsewhere in the paper or the cohort; nothing in this section is new
physics, and every stage names its home. The stages are numbered S-1 to
S-8 and are cited by these tags in the assembled equation of
\S\ref{sec:lsm-assembly}.

\subsection{S-1 --- the bulk action and the forced ledger}
\label{subsec:lsm-s1}

The chain begins one step beneath the point where the textbook begins.
Before any Lagrangian can be written, something must fix \emph{which}
fields exist, at \emph{what} derivative order they propagate, and why
that list is closed --- the zeroth silent axiom of
\S\ref{sec:lsm-silent-axioms}. In \GGTheory{} that object is the
six-dimensional bulk action, and its provenance decomposes into four
forcing steps, each carried by a named theorem of the cohort rather
than by a citation to a construction. \emph{The stage}: the bulk
$M_4\times I_r\times S^1_\theta$ and its canonical warped metric are
forced by the locality--causality and \GGEP{} clauses of the
\GGConstitution{} of \AOne{} \citep{ArisakaA1Constitution}, through
Theorems T-GG6-0/1/2 of \ATwo{} \citep{ArisakaA2GG6}. \emph{The gauge
content}: $\SU(5)\times\UoneB{}\times\UoneQi{}$ is Theorem T-GG6-3,
whose own conditional clauses --- the requirement that the boundary
carry the Standard-Model gauge content, and the empirically-preferred
four-component class --- are stated inside the theorem and inherited
here, not hidden. \emph{The matter}: one bulk spinor per family, the
minimal anomaly-admissible chiral packet, carried at \textbf{Derived
(cond.)} on the Minimal-Matter Lemma (\S\ref{app:lag-course-s1a}),
with replication $N=3$ the pincer theorem of S-2. \emph{The dynamics}:
with the field content fixed, the constitutional clauses \LC{},
\POS{2} (a single variational principle), and \MSMF{} read at lowest
derivative order admit exactly one local bulk action ---
Einstein--Yang--Mills--Dirac with flux --- whose brace-by-brace
uniqueness walk, Lovelock caveat and scalar-free selection included, is
\S\ref{app:lag-course-s1a}. \ATwo{} records the canonical form of
this action ahead of its BRST-consistency construction there; the
Green--Schwarz completion is forced by the anomaly closure of
\AThree{} \citep{ArisakaA3BI6}.

The consistency condition of the flux sector is the modified Bianchi
identity \BIsix{}, $\dd H_3 = X_4$, and that closure admits exactly
one integer solution: the \OGNfive{} ledger
$(M,N,L;J,K)=(1,3,5;7,35)$. At the end of S-1 the theory holds five
integers, the five dimensional constants, and the two transcendentals
native to the compact directions ($e$ radial, $\pi$ angular) --- and
\emph{zero} continuous dials.

What spends the action is itself a theorem, not a procedure: the
nine-part descent T-GDOS-9.1--9.9 of \AFive{} \citep{ArisakaA5GDOS}
takes the six-dimensional action as input and returns the KK tower
with its $\Mlock$ cluster (9.1), the effective four-dimensional
Lagrangian with every Wilson coefficient a radial overlap integral
(9.2), the boundary generating functional with its BRST and
Slavnov--Taylor structure (9.3--9.5), anomaly cancellation as the
\BIsix{} descent (9.6), the conserved currents and exact proton
stability (9.7--9.8), and the geometrized renormalization group (9.9).
The present Part is the parameter-level completion of that
structure-level chain --- the third leg of the ``one law, three
descents'' architecture of \AFive{}: the same bulk that exports
quantum mechanics and general relativity exports the field theory
assembled below. Everything in S-2 through S-8 is geometry plus the
ledger, processed by machinery that is derived, not assumed.

\subsection{S-2 --- the boundary field content}
\label{subsec:lsm-s2}

Kaluza--Klein reduction of the bulk Dirac action
$i\overline\Psi\,\Gamma^M D_M\Psi$ on $S^1_\theta\times I_r$ projects out
one chirality of the would-be zero mode (\S\ref{sec:gg-derives-su5}): the
boundary spectrum is chiral, left-handed Weyl, and arrives in complete
$\SU(5)$ packets --- the $\overline{\mathbf{5}}\oplus\mathbf{10}$
pattern, which is precisely the doublet/singlet family
$(Q_L,u_R^c,d_R^c,L_L,e_R^c)$ of Table~\ref{tab:sm-fermions}. The
fifteen Weyl fields per family are therefore not fifteen choices; they
are the zero-mode cohomology of one bulk spinor. (The expansion
machinery itself is Theorem T-GDOS-9.1 of \AFive{}
\citep{ArisakaA5GDOS}: the tower whose zero-mode floor is read off
here is the same tower whose massive floors cluster at $\Mlock$.)
Replication is the
$N=3$ theorem of \PaperPTwo{} \citep{ArisakaP2N3}
(\S\ref{sec:N3-theorem}): $N\ge3$ from the \MSMF{} CP lower bound,
$N\le3$ from the Bogomolnyi/\BIsix{} action cap --- \textbf{Derived
(cond.)}, as flagged there. Because each family is one bulk packet, the
gauge anomalies cancel family by family \emph{automatically}: the
delicate charge sum that the textbook checks after the fact is here a
bookkeeping identity of the embedding.

\subsection{S-3 --- the boundary gauge group and the covariant derivative}
\label{subsec:lsm-s3}

Boundary conditions and the holonomy structure on the compact directions
split the bulk gauge symmetry (\S\ref{sec:sm-foundation}):
\begin{equation}
\SU(5)\times\UoneB{}\times\UoneQi{}
\;\xrightarrow[\text{boundary conditions + holonomy}]{}\;
\SU(3)_C\times\SU(2)_L\times\U(1)_Y ,
\label{eq:lsm-splitting}
\end{equation}
with the hypercharge generator fixed inside $\SU(5)$ --- hence the $5/3$
normalization of Part~\ref{part:VI}, the forced assignment $Q=T_3+Y$,
and exact charge quantization ($|q_e|=|q_p|$ as an identity of the
embedding, not a measured coincidence). The covariant derivative
\begin{equation}
D_\mu \;=\; \partial_\mu
\;-\; i\,g_s\,G^a_\mu T^a
\;-\; i\,g\,W^i_\mu \tau^i
\;-\; i\,g'\,\tfrac{Y}{2}\,B_\mu
\label{eq:lsm-covderiv}
\end{equation}
is at this point fully assembled, with every charge in it fixed by the
embedding: no entry of Eq.~\eqref{eq:lsm-covderiv} was chosen.

\subsection{S-4 --- term group I: the gauge kinetic terms}
\label{subsec:lsm-s4}

The boundary gauge kinetic terms
\begin{equation}
\mathcal{L}_{\mathrm{gauge}}
= -\tfrac14\,G^a_{\mu\nu}G^{a\,\mu\nu}
  -\tfrac14\,W^i_{\mu\nu}W^{i\,\mu\nu}
  -\tfrac14\,B_{\mu\nu}B^{\mu\nu}
\label{eq:lsm-gauge}
\end{equation}
descend from the bulk gauge kinetic term by integrating the compact
directions; the field normalizations are redefinitions, so the physical
content of this term group is the three couplings --- and these are not
Lagrangian inputs but \emph{seam registrations} in the sense of \GDOS{}
(\S\ref{sec:gdos-qi4-recap}): $\widehat\alpha^{-1}(M_Z)=2^J=128$,
$\sin^2\theta_W=25/108=L^2/4N^3$, and $\alpha_s^{-1}(M_Z)=\pi e$, the
three attractor-coordinate theorems of Part~\ref{part:VI}, with their
QD refinements and stated pulls.

\subsection{S-5 --- term group II: the fermion kinetic terms}
\label{subsec:lsm-s5}

The forty-five Weyl fields of S-2 propagate by
\begin{equation}
\mathcal{L}_{\mathrm{fermion}}
= \sum_{\psi}\;\overline\psi\; i\gamma^\mu D_\mu\, \psi ,
\label{eq:lsm-fermion}
\end{equation}
the sum running over the derived spectrum with the derived charges,
$D_\mu$ from Eq.~\eqref{eq:lsm-covderiv}. There is nothing left to
choose in this term group at all. Its one striking feature --- that the
weak factor couples only to the left-handed fields, maximal parity
violation --- is not an option that was selected but a structural
consequence: the chirality projection of S-2, read dynamically in the
next Part as the orientation of the family cascade (the weak interaction
as the unique gate force).

\subsection{S-6 --- term group III: the Higgs sector}
\label{subsec:lsm-s6}

This is the stage at which the export departs most sharply from the
textbook, and the departure is the dissolution of the naturalness
problem. There is no fundamental scalar and no chosen potential. The
would-be Higgs doublet is the Wilson-line holonomy of the compact
circle --- the order parameter Hosotani identified on multiply connected
spaces \citep{Hosotani1983,Wilson1974} --- and its two textbook terms are
exported as follows (\S\ref{sec:wilson-hosotani}). The kinetic term
$|D_\mu\Phi|^2$ is the boundary image of the bulk gauge kinetic term for
the $A_\theta$ component: the Higgs kinetic term is a piece of
Yang--Mills in six dimensions. The potential is generated, not written:
gauge invariance forbids any polynomial in a compact phase $\theta_H$,
so the one-loop effective potential is necessarily a finite Fourier
series, $V_{\mathrm{eff}}\propto\cos\theta_H+\dots$, and the textbook's
hand-tuned $\mu^2<0$ is replaced by the branch-selection condition
$\partial_{\theta_H}V_{\mathrm{eff}}=0$,
$\partial^2_{\theta_H}V_{\mathrm{eff}}>0$ --- a vacuum criterion with no
dial anywhere. The coefficients are the ledger readings of
Part~\ref{part:VI}: $\sin\sin\theta_H^\star=(2/L)^N$, so
$v=\Mlock\,(2/L)^N\approx246$~\GeV{}, and $\lambda_H=2^{-N}=1/8$, giving
$m_h=v/2$ at first order with the QD completion carrying the stated
theory uncertainty. The hierarchy problem does not arise: a compact
phase admits no invariant mass counterterm, so there is no primitive
scalar mass to protect --- \emph{dissolved, not solved}.

\subsection{S-7 --- term group IV: the Yukawa sector}
\label{subsec:lsm-s7}

The Yukawa block
\begin{equation}
\mathcal{L}_{\mathrm{Yuk}}
= -\,\overline Q_L\,Y_u\,\widetilde\Phi\,u_R
  -\,\overline Q_L\,Y_d\,\Phi\,d_R
  -\,\overline L_L\,Y_e\,\Phi\,e_R
  \;+\;\mathrm{h.c.}
\label{eq:lsm-yukawa}
\end{equation}
has a forced \emph{form} --- it is the unique admissible boundary
coupling between the chiral packets of S-2 and the Wilson line of S-6
--- and derived \emph{content}: the matrices $Y_f$ are not fifty-four
free complex numbers but the cascade structure of Part~\ref{part:X}.
The top sits at the zero-winding anchor, $y_t=1$ at first order; the
quark ladder pays Bogomolnyi tolls $e^{-\pi k}$ with small rational
crowd factors; the colorless leptons carry dyadic--rational codes; and
the CKM and PMNS matrices are the misalignment holonomies of the family
projectors --- the gates --- with the Cabibbo angle carrying the
pentagonal phase. The neutrino extension rides on the same stage: the
Majorana channel enters with the geometric seesaw scale
$m_\star=\pi v^2/\EGUT$, with $\EGUT$ the ledger's unification seam ---
no free right-handed mass anywhere. Per-fermion registers are graded
entry by entry in Table~\ref{tab:codebook-provenance}; the cascade-depth
assignments are \textbf{Derived (cond.)} on the named cascade lemma, as
Part~\ref{part:X} states.

\subsection{S-8 --- the operators that are not exported}
\label{subsec:lsm-s8}

The textbook's listing rule (S5 of
Table~\ref{tab:lsm-silent-axioms}) writes down every renormalizable
gauge-invariant operator and then must explain three embarrassments
away. The export principle replaces the rule, and each embarrassment
becomes a theorem; the audit of what the Lagrangian does \emph{not}
contain is half the content of this Part.

\paragraph{No $\theta$-term.} The topological density $G\widetilde G$ is
admitted by the listing rule and must be tuned to one part in $10^{10}$.
It is not exported: the descent generates no coefficient for it, and the
first-order quark determinant phases cancel, so $\bar\theta^{(1)}=0$
with the constitutionally minimal residue
$\sin(2\pi/L)\,2^{-J}e^{-K}\simeq4.7\times10^{-18}$
(Part~\ref{part:XI}) --- \textbf{Derived (cond.)} on lemma L9. The
twenty-eighth parameter is a protected zero, not a tuned one.

\paragraph{No proton-decay operators.} Every $\Delta B=\pm1,\pm2$
operator carries non-zero \UoneB{} charge and is forbidden to all
orders by the exact global baryon number residual to the
Stueckelberg-massive \UoneB{} (Theorem~\ref{thm:proton-stability-a9},
conditional on its stated hypotheses) --- where classic $\mathrm{SU}(5)$
was falsified by Super-Kamiokande, the export never generates the
operator.

\paragraph{No free scalar-mass dial.} The would-be $\mu^2$ term of the
textbook potential is not exportable at all: no gauge-invariant
polynomial in a compact phase exists (S-6). The one dial the textbook
tunes most anxiously is absent as a matter of admissibility.

\paragraph{The honest overshoot.} The export is not exactly the
textbook Standard Model: the boundary carries, in addition, the
Stueckelberg-massive \UoneB{} and the \UoneQi{} structure --- protected
remnants of the bulk gauge content with observable consequences (the
baryonic fifth-force channel among the falsifiers). The audit states
this plainly as a feature: the descent slightly overshoots
$\mathcal{L}_{\mathrm{SM}}$, and the overshoot is falsifiable.

\section{The assembled Lagrangian}
\label{sec:lsm-assembly}

The export chain is complete, and the terminal page can now be written.
Assembling Eqs.~\eqref{eq:lsm-gauge}, \eqref{eq:lsm-fermion},
\eqref{eq:lsm-yukawa}, the Wilson-line sector of S-6, and the protected
zero of S-8:
\begin{equation}
\begin{aligned}
\mathcal{L}_{\mathrm{SM}}
={}& \underbrace{-\tfrac14\,G^a_{\mu\nu}G^{a\,\mu\nu}
   -\tfrac14\,W^i_{\mu\nu}W^{i\,\mu\nu}
   -\tfrac14\,B_{\mu\nu}B^{\mu\nu}}_{\substack{\text{[S-4] exported gauge kinetic terms}\\ \widehat\alpha^{-1}=2^{7},\;\; \sin^2\theta_W=25/108,\;\; \alpha_s^{-1}=\pi e}}
\;+\; \underbrace{\sum_{\psi}\overline\psi\, i\gamma^\mu D_\mu \psi}_{\substack{\text{[S-2, S-3, S-5] one bulk spinor,}\\ \text{45 chiral zero modes, charges forced}}}
\\[6pt]
&+\; \underbrace{\left|D_\mu\Phi\right|^2 \;-\; V_{\mathrm{eff}}(\theta_H)}_{\substack{\text{[S-6] Wilson line; forced Fourier potential}\\ v=\Mlock(2/L)^N,\;\; \lambda_H=2^{-N}}}
\;-\; \underbrace{\left(\overline Q_L Y_u\widetilde\Phi u_R
   + \overline Q_L Y_d\Phi d_R
   + \overline L_L Y_e\Phi e_R + \mathrm{h.c.}\right)}_{\substack{\text{[S-7] cascade tolls, dyadic codes, misalignment gates}\\ y_t=1,\;\; m_\star=\pi v^2/\EGUT}}
\\[6pt]
&+\; \underbrace{\;0\;\cdot\;\frac{g_s^2}{32\pi^2}\,G^a_{\mu\nu}\widetilde G^{a\,\mu\nu}}_{\substack{\text{[S-8] the protected zero}\\ \bar\theta^{(1)}=0,\ \text{residue }\sim10^{-18}}}
\end{aligned}
\label{eq:lsm-assembled}
\end{equation}

Equation~\eqref{eq:lsm-assembled} is, symbol for symbol, the Lagrangian
of the textbooks --- \citet{PeskinSchroeder1995} or \citet{Schwartz2014}
--- with one line added that no textbook writes: the $\theta$-term
displayed with its coefficient, zero, because in this framework the
zero is a result and deserves its place in the equation. Nothing in the
display was typed in by hand. Every term is the image of a bulk
structure under the one projection; every coefficient is either a
theorem, a graded conditional, or a declared open lemma; and the grade
of each is printed in the provenance audit below and, numerically, in
Table~\ref{tab:codebook-provenance}. The honest one-sentence summary of
the display is this: \emph{the Lagrangian that has served for fifty
years as the axiom of particle physics is here the output of a
geometric descent with zero adjustable continuous inputs --- with the
strength of each step stated, not inflated.}

\begin{table}[hbt!]
\centering
\caption[The master provenance audit of the assembled Lagrangian, Eq.~\eqref{eq:lsm-assembled}: every structural element of $\mathcal{L}_{\mathrm{SM}}$, its status in the.]{The master provenance audit of the assembled Lagrangian,
Eq.~\eqref{eq:lsm-assembled}: every structural element of
$\mathcal{L}_{\mathrm{SM}}$, its status in the textbook, its origin in
the export chain, and its register. Numerical grades (the $6/14/8$
triage) live in Table~\ref{tab:codebook-provenance}, which remains the
single point of truth for the twenty-eight coefficients.}
\label{tab:lsm-provenance}
\small
\begin{tabular}{p{0.24\textwidth}p{0.20\textwidth}p{0.31\textwidth}p{0.13\textwidth}}
\toprule
\textbf{Element of $\mathcal{L}_{\mathrm{SM}}$} & \textbf{Textbook status} & \textbf{Export origin} & \textbf{Register}\\
\midrule
Gauge group $G_{\mathrm{SM}}$ & Postulated (S1) & S-3: descent splitting, Eq.~\eqref{eq:lsm-splitting} & \textbf{D}\\
Gauge kinetic structure & Postulated & S-4: bulk Yang--Mills, compact integration & \textbf{D}\\
$g_s,\,g,\,g'$ & Measured & S-4: seam registrations (Part~\ref{part:VI}) & \textbf{D} (QD)\\
\addlinespace
Field content + hypercharges & Postulated (S2) & S-2: zero modes of one bulk spinor & \textbf{D}\\
Anomaly cancellation & Consistency check & S-2: automatic per bulk packet & \textbf{D}\\
$N=3$ families & Input (S3) & S-2: \PaperPTwo{} pincer & \textbf{D*}\\
Chirality of the weak force & Input & S-2: the KK projection; the oriented cascade & \textbf{D}\\
\addlinespace
Higgs doublet exists & Postulated (S4) & S-6: Wilson-line holonomy & \textbf{D}\\
Potential shape $V$ & Chosen polynomial & S-6: forced Fourier series & \textbf{D}\\
$\mu^2<0$ & Tuned by hand & S-6: branch selection; no dial exists & \textbf{D}\\
$v,\ \lambda_H$ & Measured & S-6: $\Mlock(2/L)^N$; $2^{-N}$ & \textbf{D}/\textbf{D*}\\
\addlinespace
Yukawa form & Postulated & S-7: unique admissible coupling & \textbf{D}\\
Yukawa content ($12+8+3$) & Measured & S-7: tolls, codes, gates (Part~\ref{part:X}) & per triage\\
\addlinespace
$\bar\theta$ & Tuned $<10^{-10}$ & S-8: not exported; phases cancel & \textbf{D*} (L9)\\
Operator completeness & Listing rule (S5) & S-8: admissible descent & \textbf{D} (principle)\\
Proton-decay operators & Forbidden by fiat$^{\dagger}$ & S-8: exact global $B$ (Thm.~\ref{thm:proton-stability-a9}) & \textbf{D*}\\
Extra content: \UoneB{}, \UoneQi{} & --- (absent) & S-8: protected remnants; falsifiable & prediction\\
\bottomrule
\end{tabular}

\smallskip
\raggedright\footnotesize $^{\dagger}$In the minimal Standard Model
baryon number is an accidental symmetry of the renormalizable operator
list; generic completions (including classic $\mathrm{SU}(5)$) violate
it, which is why proton decay was the decisive test that falsified the
1974 unification.
\end{table}

\section{What the assembly does and does not claim}
\label{sec:lsm-honesty}

The shock value of Eq.~\eqref{eq:lsm-assembled} is real, and precisely
for that reason this Part must close by marking its own boundary in the
same register the paper uses everywhere else.

\begin{euclidremark}[The honesty boundary of the assembled Lagrangian]
Six statements the assembly does \emph{not} make. \textup{(i)} The
framework of quantum field theory is imported, not derived: the descent
determines \emph{which} field theory the boundary carries, while the
operational recovery of quantum mechanics itself is the business of
\AEight{} and \AFive{}, cited and consumed. \textup{(ii)} The
four-dimensional Lorentz structure of the boundary stratum is
constitutional input (\AX{1}, \GGEP{}), not output. \textup{(iii)} Every
register marked \textbf{D*} above stands on its named lemma, and the
promotion roadmap for those lemmas is the appendix Part's closing
chapter; nothing in the assembly upgrades a grade. \textup{(iv)} The QD
layer remains a refinement register with stated uncertainties, not a
claim of exactness. \textup{(v)} ``Complete'' means the complete term
content of $\mathcal{L}_{\mathrm{SM}}$ with all coefficients placed on
the ledger at their stated grades --- six \textbf{Derived}, fourteen
\textbf{Derived (cond.)}, eight \textbf{Asserted} --- not ``every
decimal derived.'' \textup{(vi)} The bulk action itself enters at
stated registers --- its form \textbf{Derived} at lowest derivative
order under the constitutional clauses, its matter content \textbf{D*}
on the Minimal-Matter Lemma, its gauge group inheriting the conditional
clauses of T-GG6-3 of \ATwo{} \citep{ArisakaA2GG6} --- and the
assembly upgrades none of those grades either; the walk that earns the
action is \S\ref{app:lag-course-s1a}. The assembly is exactly as
strong as the descent behind it, no stronger; what is new on this page is not any single
claim but the fact that the claims compose into the whole object.
\end{euclidremark}

For fifty years the Standard-Model Lagrangian has been the object
physics reasons \emph{from}. The last four sections have treated it as
an object physics can reason \emph{to} --- and, we would argue, that
inversion is the single clearest statement of what the \GGTheory{}
program is. The reader who compares Eq.~\eqref{eq:lsm-assembled} with
the first page of any quantum field theory course will find the same
symbols in the same places; the difference is that here every symbol
arrives carrying its provenance, and the two great silences of the
textbook --- \emph{why these terms} and \emph{why these numbers} ---
have both been given answers with stated strengths.

\medskip\noindent The Lagrangian is assembled, and the codebook that
populates it is closed. The next Part asks what the closed codebook
\emph{means} --- why every electron is identical, why exactly four
forces exist with their observed behaviors, and why these two facts
are one uniqueness theorem read at two layers of the same geometry.


\section{Part~X in plain English: the textbook equation, written out and accounted for}
This Part writes out the Standard-Model Lagrangian in exactly the form a
textbook gives it, and then builds it back up term by term from the descent.

The exercise sounds like a formality and is not. A textbook Lagrangian carries
several structural choices that are made silently: which fields exist, which
representations they sit in, that the gauge group is a product of three
factors, that the scalar is fundamental, that certain operators one could
write down are simply absent. Those choices are not derived in the textbook.
They are the shape of the answer, assumed at the start.

Here each of them is replaced by a theorem, and --- this is the part worth
attention --- the operators that are \emph{absent} are derived alongside the
ones that are present. It is easy to produce a framework that yields the terms
you want. It is harder to produce one that also forbids the terms you do not
want, at the right strength, without a separate assumption for each. The
absence of a scalar potential is the sharpest case: the textbook's
hand-tuned mass term is replaced not by a better scalar but by a hole where
the scalar used to be, with the geometry paying the bill.

The honesty boundary is printed beside the assembled equation, which is
deliberate. Some term groups are exported cleanly; others rest on the
conditional results of earlier Parts and are marked. A reader should be able to
look at the finished Lagrangian and see, without turning a page, which parts of
it the framework has actually earned.

\GGPart[Identity and force uniqueness]{Geometric origin of identity and uniqueness of forces}{Why every electron is the same, why exactly four forces exist with their specific qualitative behaviors, and the recognition that identity-uniqueness and force-uniqueness are the same uniqueness theorem applied at different layers of the same geometric cascade}
\label{part:XII}

\section{The geometric origin of identity: why every electron is the same}
\label{sec:universal-identity}

We open Part~\ref{part:XII} with the most general consequence of the framework, the consequence that motivated the whole program and that has, at last, found a physical answer. The empirical fact is everywhere around us: every electron in the universe has the same mass, the same electric charge, the same magnetic moment. Two electrons in a hydrogen atom on Earth and two electrons in a hydrogen atom in a quasar twelve billion light-years away are indistinguishable to within the best precision experiment can achieve. The same is true of every proton, every $\mathrm{H}_2\mathrm{O}$ molecule, and, in a sense we will make precise, every strand of DNA. Standard treatments of the Standard Model accept this fact and use it; they do not explain it. The present framework explains it.

\subsection{The ground-state refinement: from \GA{} to \GGA{}}
\label{subsec:gga-refinement}

The explanation rests on one refinement of the runtime vocabulary, established at runtime level in \ASix{} \citep{ArisakaA6Gi4} and generalized across the layers of the program-wide cascade in the capstone \AEleven{} \citep{ArisakaA11Origin}. A Morse landscape generally admits \emph{many} index-zero minima --- many Geometric Attractors --- and identity cannot rest on ``a'' stable minimum: if several were occupied, electrons would come in varieties. The cascade output is sharper. On any stratum of the class \ASix{} calls admissible --- and the charged-lepton stratum $\CP^2$ is one --- that paper's per-stratum ground-state corollary gives exactly one index-zero critical point, which is therefore the global minimum,
\begin{equation}
\GGA(\Sigma) \;:=\; \GA^{*} \;=\; \operatorname*{arg\,min}_{x\in\Sigma}\,\GFF(x),
\label{eq:gga-def-a9}
\end{equation}
the \emph{Ground} Geometric Attractor; the cascade $\DGI\to(\GFF)\to\GSSB\to\GA$ terminates there. Dynamically, the \GGA{} is where \GRIPin{} --- the fixed-landscape regime of \ASix{} --- halts: the unique attractor from which no admissible exit channel remains. The four \QiFour{} pillars of \ASeven{} do not enter this construction: as that paper's characterization theorem states, they are properties the ground attractor carries, not the filter that finds it. The chemistry analogy is exact, and it is the right intuition to carry through this Part: a molecule has many stationary electronic configurations, but every water molecule in the universe is identical --- and indefinitely stable --- because its electrons occupy not merely \emph{a} stable configuration but \emph{the} lowest one. The \GGA{} is ``ground state'' promoted to a constitutional principle. Identity and stability will now be seen to arrive together, at every layer, as the two faces of ground-state-ness: all copies are identical because there is only one deepest basin to occupy, and the copy persists because there is no deeper basin to decay to. With the vocabulary in place, the three classic electron questions can be answered precisely.

\subsection{Three questions about the electron, finally answerable}
\label{subsec:three-electron-questions}

Three questions about the electron have stood unanswered through the entire history of physics.

\paragraph{(a) Why does every electron have the same mass?}
A textbook will say that the electron mass $m_e \approx 0.511$~MeV is a fundamental constant of nature, the same for every electron because that is how fundamental constants behave. This is not an explanation; it is a restatement of the question. In the present framework, the electron mass is the value at which the geometric free-energy functional $\GFF$ on the charged-lepton boundary stratum reaches its global minimum after the \GRIP{} cascade $\DGI \to \GSSB \to \GGA$ --- the \GRIPin{} descent on the fixed family landscape --- runs to completion and halts. The stratum admits several stable minima --- the three family depths are all \GA{}'s --- but exactly one ground state, Eq.~\eqref{eq:gga-def-a9}; the value at \emph{that} minimum is what we measure. Every electron is the same because every electron is the readout of the same \GGA{} on the same boundary geometry --- not merely \emph{a} stable attractor but \emph{the} ground state of the attractor manifold, of which the stratum has exactly one.

\paragraph{(b) Why is the electric charge of every electron the same, quantized to $e$?}
A textbook will say that electric charge is conserved by the unbroken $\mathrm{U}(1)_{\rm EM}$ gauge symmetry, and that $e$ is a measured number. Once again this is not an explanation; it leaves untouched the question of why every electron carries exactly that quantum and not some other. In the present framework, charge quantization is the topological-integer signature of the same \GRIP{} cascade. The unbroken $\mathrm{U}(1)_{\rm EM}$ direction in the bulk gauge group is generated by a fixed, integer-normalized generator inside the six-dimensional bulk; the electron's charge is the eigenvalue of that generator on the unique stable leg of the cascade, and that eigenvalue is necessarily an integer multiple of one fundamental unit. Every electron carries exactly $-e$ because every electron is the readout of the same cascade leg of the same generator.

\paragraph{(c) Why is the electric force between two charges the same everywhere?}
A textbook will say that Coulomb's inverse-square law follows from massless photon exchange and that the coupling $\alpha = e^2/(4\pi\epsilon_0 \hbar c)$ is a fundamental constant. This too is a restatement, not an answer. In the present framework, the photon's exact masslessness is a consequence of the Wilson--Hosotani holonomy lying in the commutant of the unique selected \GGA{}, and the value $\alpha^{-1}(M_Z) = 2^J = 128$ is the geometric attractor coordinate of the dyadic-closure theorem. Two charges anywhere in the universe interact through the same Coulomb force because they couple to the same massless gauge field, whose existence and whose coupling strength are both fixed by the same six-dimensional bulk. The Coulomb law does not just happen to be universal; the universality is the reason the law exists at all.

The unifying answer to (a), (b), and (c) is one sentence: \emph{electrons are universal because they are the readout of the Ground Geometric Attractor of the \GRIP{} cascade on the charged-lepton stratum --- the unique deepest minimum of $\GFF$ --- and a stratum, like a molecule, has exactly one ground state.} This is, to the best of the author's knowledge, the first time in the history of physics that the three questions have been answered together at theorem level, from a single underlying principle, rather than separately as fitted constants. The point is not rhetorical: any framework that treats $m_e$, $e$, or $\alpha$ as inputs cannot answer them, because the answers are about why these numbers are the same everywhere, and an input number gives no such reason. Only a framework in which each of the three is the readout of a uniqueness theorem can.

\paragraph{The dedicated electron paper.} The three questions of this section --- together with the two answered elsewhere in the cohort: why the heavier copies exist and terminate at three (\PaperPTwo{}), and why the electron is point-like at every scale --- receive their dedicated, claim-status-disciplined treatment in the companion paper \PaperPThree{} \citep{ArisakaP3Electron}, which assembles identity, stability, the excited copies, pointlikeness from non-spatial structure, and field response as composite geodesic motion under the two-sector reading (the phase sector moves the electron, the scale sector transmutes it; radiation is motion's dissipative channel) into a single theorem-complex about ground-state-ness, with every fence declared.

\subsection{The same principle, layer by layer, from quarks to single-cell life}
\label{subsec:identity-cascade}

The principle that explains universal-electron identity is not unique to the electron. Every layer of cosmic evolution after the six-dimensional bulk locks in is the readout of the same \GRIP{} cascade on a different boundary stratum. At each layer the cascade terminates on the layer's \GGA{} --- exactly one deepest attractor (or, where symmetry mandates degenerate copies, a finite quantized list of them) --- and that \GGA{}'s coordinates are what the next layer reads as identity. The cascade is summarized here at seven successive scales; the full cascade with all ten transitions and all energy quanta is the subject of the capstone paper of the program \citep{ArisakaBookI}.

\paragraph{1. Leptons and quarks.}
On the charged-lepton stratum the \GRIP{} cascade has one stable attractor at each cascade depth $n_e \in \{0, 1, 2\}$, and the three depths, populated by the three families, produce $m_\tau$, $m_\mu$, and $m_e$. On the quark strata the same logic produces $\{m_u, m_d, m_c, m_s, m_t, m_b\}$. Universality at this layer is the fact that every electron in the universe occupies the same depth-2 attractor on the same lepton stratum; every up quark occupies the depth-4 attractor on the up-type stratum; and so on. The ground-state refinement adds the decisive clause: of the three populated family attractors, the first generation is the stratum's \GGA{} --- which is precisely why the electron is absolutely stable while the muon and tau, occupying higher, \emph{non-ground} attractors of the same stratum, decay toward it. Identity and stability arrive together, both as ground-state statements: the periodic table of particles has excited states, and they behave exactly as excited states should. The codebook of Part~\ref{part:X} is, in this reading, the catalog of attractor coordinates on the family-projector stratum $\mathbb{C}P^2$.

\paragraph{2. Protons (quark confinement).}
Quarks do not appear as free particles; they are confined into color-singlet bound states by the strong force. In the \GRIP{} reading, confinement is a \GSSB{} step: the runtime selects the color-singlet branch of the quark configuration space because that is the unique stable attractor of $\GFF$ in the strong-coupling regime. The proton is the minimum-energy color-singlet baryon under the constraints of gauged $\mathrm{U}(1)_B$ baryon-number conservation; that minimum is unique --- it is the baryonic stratum's \GGA{} --- so every proton in the universe has exactly the same mass $m_p \approx 938.3$~MeV and exactly the same charge $+e$. And because $\mathrm{U}(1)_B$ is gauged, the ``no deeper basin'' clause holds exactly rather than approximately: proton stability is ground-state stability, the \GGA{} statement in its sharpest form. Proton stability — every proton in the universe is the same proton, lasting longer than the age of the universe by orders of magnitude — is the layer-2 echo of the same uniqueness theorem that made every electron the same electron.

\paragraph{3. Elements (H, He, C, N, O, $\ldots$).}
At the atomic layer, the \GRIP{} cascade selects, for each integer atomic number $Z$, the unique ground-state nuclear configuration with charge $+Ze$, the unique ground-state electronic configuration with $Z$ bound electrons, and the unique characteristic spectral lines. The Aufbau principle, the periodic table, and the chemistry of each element are all readouts of the layer-3 attractor on the atomic stratum. Hydrogen is the same hydrogen everywhere because the $Z=1$ attractor is unique; helium is the same helium everywhere because the $Z=2$ attractor is unique; the same is true of carbon, nitrogen, and oxygen. At this layer the runtime's vocabulary and chemistry's coincide exactly: each of these configurations is the $Z$-stratum's \GGA{}, and ``ground state'' --- chemistry's own word for it --- turns out to have been the constitutional concept all along. Astronomers see exactly the same Balmer series in distant quasars that they see in laboratory hydrogen lamps because both are reading out the same cascade attractor on the same atomic stratum.

\paragraph{4. Molecules in the interstellar medium.}
At the next layer, the \GRIP{} cascade operates on the molecular configuration space. The molecules detected in cold dense interstellar clouds — $\mathrm{H}_2$, $\mathrm{CO}$, $\mathrm{H}_2\mathrm{O}$, $\mathrm{NH}_3$, $\mathrm{CH}_3\mathrm{OH}$, and several hundred others — are the \GGA{}'s of $\GFF$ on the molecular stratum at the temperatures and densities prevailing in those clouds. The fact that the same molecules are observed in star-forming regions across the galaxy and in distant galaxies at high redshift is the layer-4 echo of universality: the molecular attractors are the same everywhere because the underlying cascade is the same everywhere. Astrochemistry is a survey of these attractors.

\paragraph{5. Complex organic molecules.}
The cascade continues into the regime of polyatomic carbon-based chemistry. Amino acids, sugars, and small nucleotide precursors are the \GGA{}'s of $\GFF$ on the polyatomic-organic stratum at the temperatures and pressures of warm, wet planetary surfaces and proto-stellar nebulae. Glycine has been detected in cometary ice and, by inference, in the gas phase of distant star-forming regions; the same twenty proteinogenic amino acids underlie all known terrestrial life. This is the layer-5 echo of the same uniqueness principle: where the conditions allow the cascade to reach this layer, the same set of attractors is selected.

\paragraph{6. ATP, RNA, DNA.}
At the next layer, the cascade selects the energy-quantum carrier and the information carriers of life. ATP is the \GGA{} of $\GFF$ on the metabolic-energy stratum; its energy--information role is developed in \AEleven{}. RNA and DNA are the \GGA{}'s on the information-storage stratum; the four-letter alphabet $\{\rm A, C, G, T\}$ is the unique solution of the quadruple gauge-locking constraint at this layer. Every cell on Earth uses the same ATP currency and the same DNA alphabet because both are the layer-6 readouts of the same attractor selection.

\paragraph{7. Single-cell life.}
Finally, the cascade selects single-cell life as the \GGA{} of $\GFF$ on the open-system stratum subject to the four boundary conservation conditions of the Selfish Open System (\SeOS{}) class of \AFive{}. A bacterium and an archaeon — the two oldest extant lineages of life on Earth — share the same genetic code, the same ATP energy currency, the same membrane-based compartmentalization, and the same fundamental metabolic pathways because all of these are layer-7 attractor coordinates of the same cascade on the same stratum. The Last Universal Common Ancestor (LUCA), in this reading, is the empirical record of the layer-7 attractor.

\subsection{The principle, plainly stated}
\label{subsec:identity-principle}

The unifying principle of the seven layers above can be stated in one sentence. \emph{At each layer of cosmic evolution after the six-dimensional bulk locks in, the \GRIP{} cascade terminates on the Ground Geometric Attractor of the layer-specific boundary stratum --- the unique deepest minimum of $\GFF$, not merely a stable one --- and what we observe as the universal identity of physical objects at that layer is the readout of that \GGA{}'s coordinates.} Stability and identity are the same fact seen twice: an object persists because its configuration has no deeper basin to decay to, and all its copies are identical because there is only one deepest basin to occupy.

This single principle answers, in one stroke, three classes of question that physics, chemistry, and biology have each treated separately:
\begin{itemize}\itemsep2pt
\item \emph{Particle physics}: why every electron, every proton, every $W$ boson is identical to every other.
\item \emph{Chemistry}: why every hydrogen atom, every $\mathrm{H}_2\mathrm{O}$ molecule, every amino acid is identical to every other.
\item \emph{Biology}: why every cell on Earth uses the same ATP currency and the same DNA alphabet.
\end{itemize}
Each class has, until now, treated identity as an empirical given. The framework of the present paper, applied at the layer above the Standard Model and recursively at each higher layer, derives identity as the consequence of attractor selection.

\paragraph{Why this matters for the present paper.}
\ANine{} is a particle-physics paper. Its formal task is to close the twenty-eight-parameter codebook of the boundary Standard Model. But the principle that closes that codebook — the \GRIP{} cascade terminating on the unique \GGA{} of each stratum — is the same principle that closes every higher layer. In closing the boundary codebook with no continuous freedom, \ANine{} is also closing, by the same mechanism, the deeper question of why the boundary objects exist at all in the form they do. Universality is not an empirical accident; it is the unique signature of an attractor selection theorem.

After all, the question ``why is every electron the same?'' has been asked in some form by every careful student of physics since Thomson identified the electron in 1897. For one and a quarter centuries the answer has been ``because that is what an electron is''. The framework of this paper, by deriving the electron's mass, charge, and coupling as the readouts of a single uniqueness theorem on the charged-lepton stratum of the six-dimensional bulk, makes the answer in physical terms for the first time: every electron is the same because the geometry permits only one. The same answer, applied recursively, explains why every proton, every hydrogen atom, every water molecule, every strand of DNA, and every living cell shares the universal identity that biology, chemistry, and physics each take as given. The universe is not full of arbitrary identical objects; the universe is full of ground states, and the objects are what the \GGA{}'s look like from the boundary.


\subsection{Why the weak interaction exists: \GRIP{} ending at the \GGA{}, realized in Nature}
\label{subsec:weak-grip}

The identity principle of this section has a dynamical shadow, and following it answers one of the oldest existential puzzles in particle physics. Where \S\ref{sec:weak-special} established that the weak interaction is \emph{special} --- the unique \emph{gate force} able to carry matter across the gates between family basins --- this subsection asks the prior question, why such a force should \emph{exist} at all, and exhibits the same force as it is realized across Nature; the gate of \S\ref{sec:weak-special} is the threshold of a corridor, and what follows walks the corridor. Of the four boundary forces, three have conspicuously constructive jobs. The strong force binds quarks into hadrons and hadrons into nuclei; electromagnetism builds atoms, molecules, chemistry, and light; gravity aggregates matter into planets, stars, and galaxies. The weak interaction builds nothing. It binds no states; no material structure carries its name; remove it from a textbook's table of contents and no chapter on the architecture of matter would notice. Its one conspicuous job is to make things \emph{decay}. Rabi's famous ``Who ordered that?'' was aimed at the muon, but the question underneath it --- why should Nature contain a force whose only visible occupation is destruction? --- has been posed by generations of physicists since, and the standard answer has been silence: the Standard Model writes the weak interaction down and declines to say why it is there. The closed codebook of this paper, together with the ground-state refinement of \S\ref{subsec:gga-refinement}, permits a structural answer, and we state it first in one sentence:

\begin{quote}
\emph{The weak interaction is the runtime's relaxation channel: the unique physical implementer of \GRIPin{} --- the fixed-landscape regime of \ASix{} --- on the matter strata, the corridor by which matter reaches the Ground Geometric Attractors whose existence and uniqueness the runtime guarantees. A constitution that mandates unique ground states on the admissible strata (\ASix{}'s per-stratum ground-state corollary) must also mandate a road to them. The weak interaction is the road.}
\end{quote}

\paragraph{The uniqueness observation.}
The categorization rests on a sharp structural fact, checkable against the boundary gauge group of Part~\ref{part:I}. The strong and electromagnetic interactions are flavor-diagonal: they build the basins of the matter landscape and relax excitations \emph{within} a basin --- $\rho\to\pi\pi$ and $\Delta\to N\pi$ are strong relaxation, an excited atom radiating to its ground state and $\pi^0\to\gamma\gamma$ are electromagnetic relaxation --- but no flavor-diagonal force can carry a state from one family basin to another. Within $\mathrm{SU}(3)_C\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y\times\mathrm{U}(1)_B\times\mathrm{U}(1)_{Qi}$, the $\mathrm{SU}(2)_L$ charged current is the \emph{only} flavor-connecting channel. This is the gate-force uniqueness established in \S\ref{sec:weak-special} (the basin-versus-gate comparison of Table~\ref{tab:basin-gate}), seen now from the side of what that unique channel \emph{does}: it is the relaxation corridor whose gate no other force can cross.

Two clean clauses follow. \emph{(i)} Every inter-basin descent observed in Nature is weak. \emph{(ii)} Every weak flavor transition is a step of the $\GFF$ descent toward the stratum's conditional \GGA{}. A third clause, in regime language, closes a misreading before it can open: every weak process in Nature is \GRIPin{}; \emph{no weak process is a \GRIPout{}}. Weak decays look like ``exits'' --- a state leaves its basin --- but the landscape they descend is fixed; the one \GRIPout{} of the electroweak story happened once, when the Wilson--Hosotani branch was selected and the landscape built (\S\ref{sec:first-order-higgs}).

For the same reason, producing a $\tau$ or a top quark in a collider is not a \GRIPout{} either: pumping a state uphill through the boundary ports is excitation \emph{within} the fixed landscape, promptly followed by \GRIPin{} relaxation. And the famous fine print of the weak sector reads naturally in the same language: the GIM suppression of flavor-changing neutral currents is the statement that there are \emph{no diagonal shortcuts} between non-adjacent basins --- descent proceeds by the charged-current ladder, rung by rung --- while the hierarchical structure of the CKM matrix (powers of the Wolfenstein parameter $\lambda\simeq0.22$) is the \emph{adjacency gearing} of that ladder, nearest-depth steps strongest, exactly as a Morse cascade demands and exactly as the misalignment codebook of Part~\ref{part:X} computes.

\paragraph{The descent across Nature: five exhibits.}
The reach of the categorization is best displayed by walking it through the laboratory, the chart of nuclides, the Sun, the first three minutes, and a neutron star. Every row of the table is the same regime: \GRIPin{} on a fixed landscape, at five wildly different tempos.

\begin{table}[htbp]
\centering
\small
\caption[The same descent at five tempos, from the laboratory to a neutron star.]{The same descent at five tempos, from the laboratory to a neutron star. Each row names the system, the weak descent it undergoes, the attractor it reaches, and the clock on which it does so. The last column is the point: the clocks span more than thirty orders of magnitude while the regime does not change, which is what makes the categorization a structural claim rather than a description of one process. The final row is conditional and is marked as such in the table.}
\label{tab:weak-descent-systems}
\small
\begin{tabular}{llll}
\toprule
\textbf{System} & \textbf{Weak descent} & \textbf{\GGA{} reached} & \textbf{Clock}\\
\midrule
Lepton ladder & $\tau\to\mu\to e$ & electron (depth 2) & $10^{-13}$~s; $10^{-6}$~s; $\infty$\\
Quark ladder & $t\to b\to c\to s\to u/d$ & $u,d$ (depth 4) & $10^{-25}$~s down to stable\\
Free neutron & $n\to p\,e\bar\nu$ & proton (baryonic \GGA{}) & $879$~s\\
Chart of nuclides & $\beta^\pm$, EC along isobars & most-bound isobar & seconds to Gyr\\
The Sun & $pp\to d\,e^+\nu$ (first step) & helium ladder onset & $\sim10^{10}$~yr lifetime\\
Primordial plasma & $n\leftrightarrow p$ freeze-out & BBN abundances & $G_F^2T^5=H$ at $\sim0.8$~MeV\\
Neutron star & $p\,e^-\to n\,\nu$ (neutronization) & neutron matter & conditional \GGA{}\\
\bottomrule
\end{tabular}
\end{table}

\emph{Exhibit 1: the family ladders.} The lepton and quark readings were developed in \S\ref{sec:lepton-cascade-depth} and \S\ref{sec:quark-cascade-depth}: non-ground \GA{}'s relaxing rung by rung to the \GGA{}'s of their strata, with the lifetime hierarchy as the rate readout. They are the elementary instances; the remaining exhibits show the same channel running entire macroscopic worlds.

\emph{Exhibit 2: the valley of $\beta$ stability.} At fixed mass number $A$, the semi-empirical binding energy is a parabola in proton number $Z$; $\beta^-$, $\beta^+$, and electron capture step a nuclide down that parabola, one weak transition per rung, to the most-bound isobar. The chart of nuclides is therefore a \emph{literal} stratified Morse landscape --- the most photographed one in science --- and $\beta$ decay is its gradient flow. Every introductory nuclear-physics course plots \GRIP{} ending at the \GGA{}; it has simply lacked the name.

\emph{Exhibit 3: the Sun.} The first step of the pp chain, $p+p\to d+e^++\nu$, is a weak transition --- the only step of stellar hydrogen burning that must convert a proton into a neutron --- and it is the bottleneck that sets the solar luminosity. The Sun shines at the pace of the weak descent; the ten-billion-year main-sequence lifetime that makes planets, chemistry, and biology possible is the slowness of one \GRIP{} rung.

\emph{Exhibit 4: the first three minutes.} The primordial neutron-to-proton ratio froze when the weak interconversion rate $\sim G_F^2T^5$ fell below the Hubble rate at $T\simeq0.8$~MeV, and every helium nucleus in the universe is a fossil of that moment. The light-element abundances are a snapshot of the weak clock racing the expansion --- the descent rate measured against cosmology itself.

\emph{Exhibit 5: the conditional \GGA{}.} In a neutron star the same channel runs ``uphill'' in vacuum masses --- $p+e^-\to n+\nu$, neutronization, the URCA processes --- because at supranuclear density, with degeneracy pressure in the ledger, neutron matter \emph{is} the ground state of the stratum. This is the decisive exhibit: the weak interaction does not consult a fixed mass table; it descends the \emph{free energy of the stratum}, finding whatever the conditional \GGA{} is under the prevailing boundary conditions. A relaxation channel that tracks $\GFF$ rather than mass is precisely what the runtime categorization predicts and what a ``decay force'' caricature does not.

\paragraph{Why the weak force must be weak.}
The categorization also explains the adjective. Dial the Fermi constant and watch the universe fail in both directions. If the descent coupling were of strong-interaction size, every non-ground basin would empty essentially instantly: no muon or tau spectroscopy, no metastable flavors, a pp bottleneck so fast that stars would burn out in cosmological moments, a BBN epoch that processes all hydrogen --- a universe that reaches its ground states immediately and has no \emph{history}. If the coupling were zero, the cascade would freeze: matter locked forever in whatever flavor configuration the initial conditions happened to deposit --- taus and strange matter as permanent species, arbitrary isobars as stable elements, no universality of matter, a universe of accidents with no \emph{arrival}. The measured weakness is the constitutional compromise between the two failures: strong enough that every basin drains to its \GGA{} well within cosmic time, weak enough that the draining itself --- stars, chemistry-bearing planets, the slow nuclear furnaces --- constitutes a rich intermediate world. And the dial is not free: $G_F=1/(\sqrt2\,v^2)$ with $v$ the derived Wilson--Hosotani vev of Theorem~\ref{thm:vev}, so the tempo of cosmic history is itself a ledger readout, not an accident.

\paragraph{A chiral postscript, fenced.}
One structural rhyme deserves recording without overclaim. The unique descent channel of the matter strata is also the unique parity-violating force --- the only one that distinguishes left from right is the only one whose job is an oriented, irreversible flow. Whether the orientation of the coupling and the orientation of the descent are two faces of one geometric fact is, in the present state of the program, not yet a theorem. \S\ref{sec:weak-special} advances exactly this identification --- chirality as the orientation of the Morse descent, and CP violation as the gate holonomy --- as a structural reading, and the gate-operator lemma (L11) of the promotion roadmap (Appendix~\ref{sec:appendix-promotion-roadmap}) is the named lemma whose proof would convert that reading into a result. We record the chiral rhyme here as a phenomenon and defer its proof to that lemma.

\paragraph{The existential answer.}
We can now answer the question this subsection opened with, and the answer belongs to the identity principle of this section. Without the strong force there would be no rooms; without electromagnetism, no walls or furniture; without gravity, no building; without the weak force, \emph{no corridors} --- and the universe would remain locked, forever, in whichever rooms it happened to be born. The weak interaction exists because the constitution's promise of unique ground states would otherwise be empty: \ASix{}'s per-stratum corollary guarantees that every admissible stratum --- the matter strata among them --- \emph{has} a \GGA{}; \GRIPin{} is the regime that reaches it; and the weak interaction is how matter \emph{gets there} --- Nature's implementation of \GRIPin{} on the matter strata. It is the reason the matter around us is the universe's ground-state catalog --- the same electron, the same proton, the same elements everywhere --- rather than a frozen record of its initial accidents. This is the existential face of the co-entailment of \S\ref{subsec:weak-three-families}: a replicated family landscape and the gate force that drains it require each other, and the toll-budget closure fixes the number of basins at exactly three. A force that builds nothing turns out to be the force that makes everything else \emph{findable}; the question ``who ordered that?'' has, at last, a structural reply: the constitution did.

\section{The geometric origin of force: why these four forces and no others}
\label{sec:force-uniqueness}

\S\ref{sec:universal-identity} answered the first half of the deepest empirical fact about the universe: why every electron, every proton, every hydrogen atom, every water molecule, and every living cell shares a universal identity. The answer was geometric --- a uniqueness theorem on a layer-specific attractor stratum. We now turn to the second half of the same fact. The universe contains exactly four fundamental forces --- gravity, electromagnetism, the strong nuclear interaction, the weak nuclear interaction --- with exactly the qualitative behaviors we observe. Why these four? Why not three, or five, or seventeen? Why is gravity always attractive between positive masses? Why is electromagnetism long-range and two-signed? Why does the strong force confine at long distance while running asymptotically free at short distance? Why does the weak force violate parity and CP? Standard treatments of the Standard Model accept these qualitative properties as inputs and tabulate them; they do not explain them. The framework of the present paper does, by the same kind of uniqueness argument that explained universal identity in \S\ref{sec:universal-identity}: the constitution of geometry permits exactly one self-consistent six-dimensional bulk, and the four forces and their qualitative behaviors are the boundary projection of that one bulk.

\subsection{Eight constitutional lemmas force the canonical six-dimensional bulk}
\label{subsec:eight-lemmas-bulk}

The constitutional uniqueness of the bulk geometry $\mathcal{M}_6 = \mathcal{M}_4 \times I_r \times S^1_\theta$ is not an aesthetic choice; it is the joint consequence of eight separately-named clauses of the GG-Constitution. We pin each clause to its bulk-geometric duty in turn. The eight together leave one and only one admissible solution.

\paragraph{Lemma 1 (the floor: $D \geq 4$).}
The Lorentz-Causality axiom AX-1 of \AOne{} (\T{1.1}) plus the Operationally Geometric Gauge Equivalence Principle AX-3 (OGGEP, \AOne{} \T{1.3}) demand that physics be described on a Lorentzian manifold admitting operational observers with distinct events and a causal measurement protocol. The smallest manifold of this kind is four-dimensional Minkowski space $\mathcal{M}_4$: three spatial dimensions plus one time direction. In fewer than four dimensions the causal structure degenerates, and there are not enough independent observables to support the operational definition of measurement. Hence $D \geq 4$.

\paragraph{Lemma 2 (gauge fields demand extra geometry: $D > 4$).}
The Geometric Gauge Equivalence Principle AX-2 (GGEP, \AOne{} \T{1.2}) demands that gauge fields be carried by the bulk geometry rather than appended to it. Geometric structure means data on a fiber bundle with non-trivial transverse directions: an internal connection on a manifold of dimension strictly greater than four. If the only geometry available were $\mathcal{M}_4$, the Yang--Mills connections of the Standard Model would have to be re-imported as external auxiliary fields, defeating the entire content of AX-2. Hence $D > 4$.

\paragraph{Lemma 3 (RG flow needs a radial coordinate: one radial interval $I_r$).}
Postulate POS-5 of the GG-Constitution (renormalization-group covariance, \AOne{} \T{3.5}) demands that the renormalization-group scale $\mu$ be a geometric coordinate of the bulk. The natural realization identifies $\mu = M_{\rm Pl}\,e^{-r}$ for a radial bulk coordinate $r$. Renormalization-group flow has a UV endpoint at $r = 0$ (where $\mu = M_{\rm Pl}$) and an IR endpoint at $r = K$ (where $\mu = \Mlock$); $r$ must therefore have endpoints, not be periodic. Lorentz invariance under RG transport applied to the four-dimensional metric on each radial slice forces the form $g_{\mu\nu}(r) = e^{-2r}\,\eta_{\mu\nu}$. The minimal solution is one radial interval $I_r = [0, K]$. A second radial coordinate would carry a second copy of the RG scale, but the RG scale is one number; AX-4 (Maximum Symmetry, Minimum Freedom, MSMF, \AOne{} \T{1.4}) forbids the duplication. Hence the bulk contains one and only one radial interval.

\paragraph{Lemma 4 (the Higgs mechanism needs a compact direction: one circle $S^1_\theta$).}
Postulate POS-7 (Measurement Foundation, \AOne{} \T{3.7}) demands that the electroweak-symmetry-breaking mechanism be operational --- derivable from bulk-level dynamics, not postulated. The Wilson--Hosotani realization \citep{Wilson1974,Hosotani1983} fulfils this requirement by promoting the electroweak order parameter from a local scalar to the Wilson-line holonomy along a compact bulk direction; the effective potential for the holonomy angle is generated dynamically by one-loop bulk quantum fluctuations. Holonomy requires a coordinate that closes back on itself; the minimal compact realization is one circle $S^1_\theta = [0, 2\pi)$. A second compact direction would supply a second holonomy parameter; MSMF forbids the redundancy. Hence the bulk contains one and only one compact circle.

\paragraph{Lemma 5 (no more: MSMF closes the count at $D = 6$).}
Lemmas 1--4 have assembled $\mathcal{M}_4 + I_r + S^1_\theta = 4 + 1 + 1 = 6$ dimensions; each summand discharges a non-redundant constitutional duty. Any seventh dimension would be free, and MSMF rules it out. Combined with Lemma 2 ($D > 4$), the upper bound $D \leq 6$ is as constitutional as the lower bound. Hence $D = 6$.

\paragraph{Lemma 6 (the \BIsix{} anomaly polynomial closes only in $D = 6$).}
The Bianchi-Identity-in-six-dimensions theorem of \AThree{} (\BIsix{}, \T{1}) establishes that the closure $dH_3 = X_4$ admits a unique four-form anomaly polynomial,
\begin{equation}
 X_4 \;=\; \mathrm{tr}(R \wedge R) \;-\; \mathrm{tr}(F_{\mathrm{SU}(5)} \wedge F_{\mathrm{SU}(5)}) \;-\; \tfrac{2}{5}\, F_B \wedge F_B \;-\; \tfrac{2}{5}\, F_{Qi} \wedge F_{Qi},
\end{equation}
with the trace-ratio lock $|a/c| = |a/d| = 5/2$ between the gravitational, hypercharge, and $\mathrm{U}(1)_{Qi}$ anomaly coefficients. In $D = 5$ the form-degree of $X_4$ is wrong; in $D = 7$ the trace-ratio lock fails to close at integer level. Only in $D = 6$ does \BIsix{} admit the closure, and the same trace-ratio lock is the independent source of the empirical weak-angle identity $\sin^2\theta_W = 25/108$ that the present paper derives in Part~\ref{part:VI} (\T{17.1}). Hence the constitutional dimension cannot be other than $D = 6$.

\paragraph{Lemma 7 (the \OGNfive{} Diophantine ledger closes only in $D = 6$).}
The OGN-5 theorem T6 of \ATwo{} (\OGNfive{}, \T{6}) establishes that the Diophantine closure $K = L J = M^2 + N^2 + L^2$ with $J^2 - M^2 = N L$ admits exactly one integer solution in the physical regime,
\begin{equation}
 (M, N, L; J, K) \;=\; (1, 3, 5; 7, 35),
\end{equation}
and the closure holds only in $D = 6$. In $D \neq 6$ the closure either fails outright or admits multiple integer solutions inconsistent with MSMF. The OGN-5 quintuple is the integer skeleton from which the present paper extracts the color-triplicity index $N = 3$, the scale depth $K = 35$, the defining $\SU(5)$ block size $L = 5$, the dyadic charge exponent $J = 7$, and the elementary unit $M = 1$. Hence $D = 6$ is confirmed a second time, by a route completely independent of Lemma 6.

\paragraph{Lemma 8 (the bulk Lorentz spinor structure forces the family content).}
The Lorentz group of six-dimensional Minkowski space is $\mathrm{Spin}(1,5)$. Its 32-dimensional complex Weyl representation, restricted under the embedding $\mathrm{SU}(5) \subset \mathrm{SO}(10) \subset \mathrm{Spin}(1,5)$ that the orbifold projection selects (\ATwo{} T-GG6-2), decomposes as
\begin{equation}
 \mathbf{32}_{\,\mathrm{Spin}(1,5)} \;\longrightarrow\; \mathbf{16} \,\oplus\, \overline{\mathbf{16}} \quad \text{of } \mathrm{SO}(10),
\end{equation}
where the $\mathbf{16}$ contains exactly one Standard-Model fermion family of left-handed chirality. In $D < 6$ the Lorentz Weyl rep does not contain the $\mathbf{16}$ of $\mathrm{SO}(10)$ at all; in $D > 6$ the chiral spinor admits exotic content (gravitinos, additional families) that MSMF rules out without further geometric apparatus. Hence $D = 6$ is forced once more, and the matter content of one boundary family is forced to be exactly the Standard-Model fermion family.

\medskip

The eight lemmas leave exactly one admissible bulk geometry,
\begin{equation}
 \label{eq:m6-canonical}
 \mathcal{M}_6 \;=\; \mathcal{M}_4 \,\times\, I_r \,\times\, S^1_\theta,
 \qquad
 ds_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,dx^\mu dx^\nu \,+\, dr^2 \,+\, d\theta^2,
\end{equation}
and exactly one composite gauge--gravity group on that bulk,
\begin{equation}
 \label{eq:g-bulk-canonical}
 G_{\rm Bulk} \;=\; \mathrm{Spin}(1,5) \,\times\, \mathrm{SU}(5) \,\times\, \mathrm{U}(1)_B \,\times\, \mathrm{U}(1)_{Qi}.
\end{equation}
We collect the eight lemmas into a single named theorem, the structural twin of the universal-identity insight of \S\ref{sec:universal-identity}.

\begin{theorem}[\textbf{T-SM-62.1}: Uniqueness of the canonical four-force architecture]
\label{thm:four-force-uniqueness}
Under the constitutional clauses AX-1 (Lorentz-Causality, \AOne{} \T{1.1}), AX-2 (Geometric Gauge Equivalence Principle, \AOne{} \T{1.2}), AX-3 (Operationally Geometric Gauge Equivalence Principle, \AOne{} \T{1.3}), AX-4 (Maximum Symmetry, Minimum Freedom, \AOne{} \T{1.4}), POS-5 (renormalization-group covariance, \AOne{} \T{3.5}), POS-7 (Measurement Foundation, \AOne{} \T{3.7}), the \BIsix{} closure theorem (\AThree{} \T{1}), and the \OGNfive{} Diophantine-ledger closure theorem (\ATwo{} T-GG6-6), the following statements hold jointly.

\begin{enumerate}[leftmargin=2.2em]
\item \emph{The bulk geometry is forced.} The constitutional clauses admit exactly one bulk geometry, the canonical six-dimensional product
\[
 \mathcal{M}_6 \;=\; \mathcal{M}_4 \,\times\, I_r \,\times\, S^1_\theta,
 \qquad
 ds_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,dx^\mu dx^\nu \,+\, dr^2 \,+\, d\theta^2,
\]
with $I_r = [0, K]$ and $K = 35$ the integer scale depth of the \OGNfive{} ledger.

\item \emph{The composite gauge--gravity group is forced.} On the bulk $\mathcal{M}_6$ the admissible composite gauge--gravity group is
\[
 G_{\rm Bulk} \;=\; \mathrm{Spin}(1,5) \,\times\, \mathrm{SU}(5) \,\times\, \mathrm{U}(1)_B \,\times\, \mathrm{U}(1)_{Qi},
\]
and the matter content of one boundary fermion family is forced to be the Standard-Model family ($\mathbf{16}$ of $\mathrm{SO}(10)$) of left-handed chirality.

\item \emph{The four boundary forces and their qualitative behaviors are forced.} The boundary projection of $G_{\rm Bulk}$ through the canonical orbifold action and the Wilson--Hosotani holonomy on $S^1_\theta$ produces exactly four fundamental forces with exactly the qualitative behaviors:
\begin{itemize}\itemsep2pt
\item \emph{Gravity}: spin-$2$ mediator, universally attractive between positive sources, exponentially weak with $G_N \sim 1/M_{\rm Pl}^2$ and $M_{\rm Pl}^2 = M_6^4 \cdot \mathrm{vol}(I_r \times S^1_\theta) \cdot e^{2K}$.
\item \emph{Electromagnetism}: spin-$1$ mediator on the unbroken Wilson--Hosotani commutant generator, exactly massless, long-range with $V(r) \propto 1/r$, two-signed by CPT, with $\alpha^{-1}(M_Z) = 2^J = 128$ from the dyadic-closure theorem (\T{16.1}).
\item \emph{Strong force}: spin-$1$ non-Abelian $\mathrm{SU}(3)_C$ mediator, asymptotically free at high energies, confining at $\Lambda_{\rm QCD} \approx 200$~MeV with $\alpha_s(M_Z) = 1/(\pi e)$ (\T{18.1}).
\item \emph{Weak force}: spin-$1$ massive $\mathrm{SU}(2)_L$ mediator with $M_W = gv/2$, $v \approx 246$~GeV (\T{19.1}), $\sin^2\theta_W = 25/108$ (\T{17.1}), parity-violating by orbifold projection on the radial direction, CP-violating with $\delta_{\rm CKM} = 13\pi/35$ (\T{29.1}).
\end{itemize}

\item \emph{No other constitutional bulk admits a self-consistent closure.} Any bulk geometry that violates the dimension, factor-structure, or matter-content statements of (i)--(iii) above either contradicts one of the eight constitutional clauses or fails to admit operational boundary observers.
\end{enumerate}
\end{theorem}

\begin{proof}
Statement (i) follows directly from Lemmas 1--5: Lemma 1 (AX-1 + AX-3) gives $D \geq 4$; Lemma 2 (AX-2) gives $D > 4$; Lemma 3 (POS-5 + AX-4) supplies exactly one radial interval $I_r$; Lemma 4 (POS-7 + AX-4) supplies exactly one compact circle $S^1_\theta$; Lemma 5 (AX-4) closes the count at $D = 6$. The same dimension is independently confirmed by Lemma 6 (the \BIsix{} closure $dH_3 = X_4$ holds at the correct form-degree and trace-ratio lock $|a/c| = |a/d| = 5/2$ only in $D = 6$) and by Lemma 7 (the \OGNfive{} Diophantine closure $K = LJ = M^2 + N^2 + L^2$ admits the integer solution $(M,N,L;J,K) = (1,3,5;7,35)$ only in $D = 6$). Three distinct routes --- bottom-up dimension counting, top-down anomaly-polynomial closure, and Diophantine integer closure --- all converge on the canonical six-dimensional product geometry, with the scale depth $K = 35$ fixed by \OGNfive{}.

Statement (ii) follows from Lemma 8: the Spin$(1,5)$ Weyl representation, restricted under $\mathrm{SU}(5) \subset \mathrm{SO}(10) \subset \mathrm{Spin}(1,5)$, decomposes as $\mathbf{16} \oplus \overline{\mathbf{16}}$ --- one Standard-Model family per chirality. The $\mathrm{U}(1)_B$ and $\mathrm{U}(1)_{Qi}$ factors are forced by the \BIsix{} trace-ratio lock of Lemma 6: the anomaly polynomial $X_4$ requires exactly two Abelian factors with coefficient $-2/5$ each, identifying them as the gauged baryon number $\mathrm{U}(1)_B$ and the four-currents $\mathrm{U}(1)_{Qi}$ of \ASeven{} \T{-QUA}. The composite gauge--gravity group $G_{\rm Bulk}$ stated in (ii) is therefore forced.

Statement (iii) follows force by force from the analysis of \S\ref{subsec:four-forces-behaviour}. \emph{Gravity:} the metric perturbation $h_{\mu\nu}$ is rank-2 symmetric traceless by the Spin$(1,5)$ Lorentz factor, fixing the mediator spin at $2$; the standard tree-level exchange-sign rule for massless spin-$J$ exchange then forces universal attraction between positive sources, and the scale profile $e^{2K}$ with $K = 35$ forces the four-dimensional Planck mass to $\approx 10^{19}$~GeV, hence the smallness of $G_N$. \emph{Electromagnetism:} the unbroken $\mathrm{U}(1)_{\rm EM}$ generator is the unique commutant of the selected Wilson--Hosotani vacuum, the photon is therefore exactly massless and Coulomb is long-range, and the two signs of electric charge are the two orientations of the compact fiber --- charge being a winding number on $\Stheta$ and the antiparticle the same holonomy class read in the reversed traversal (T-P4-1, T-P4-3 of \citealp{ArisakaP4Real}; T-P5-A, T-P5-C of \citealp{ArisakaP5CPT}), with the textbook CPT theorem \citep{Jost1957} complementary in scope rather than upstream. \emph{Strong force:} the boundary $\mathrm{SU}(3)_C$ inherited from $\mathrm{SU}(5)$ has one-loop beta-function coefficient $11 N_c - 2 N_f = 21 > 0$ for $N_c = 3$ and $N_f = 6$, hence asymptotic freedom \citep{GrossWilczek1973,Politzer1973}; confinement at $\Lambda_{\rm QCD} \approx 200$~MeV follows by the Wilson lattice argument \citep{Wilson1974}; the coupling value at $M_Z$ takes the dyadic-rational form $1/(\pi e)$ by \T{18.1}. \emph{Weak force:} the Wilson--Hosotani vacuum gives $W$ and $Z$ masses via the standard Higgs mechanism at $v \approx 246$~GeV (\T{19.1}); the radial orbifold projection keeps only left-handed Weyl spinors, fixing parity violation; the \OGNfive{} ledger forces $\delta_{\rm CKM} = 13\pi/35 \neq 0, \pi$, fixing CP violation in the quark sector.

Statement (iv) is the contrapositive of the lemma chain. Any seventh bulk dimension violates AX-4 by Lemma 5. Any $D \neq 6$ violates either the \BIsix{} anomaly-polynomial closure (Lemma 6) or the \OGNfive{} Diophantine closure (Lemma 7) or both. Any deviation from the Spin$(1,5) \times \mathrm{SU}(5) \times \mathrm{U}(1)_B \times \mathrm{U}(1)_{Qi}$ gauge--gravity content violates Lemma 8 plus the \BIsix{} trace-ratio lock. No alternative bulk geometry survives the joint constraint. \qedhere
\end{proof}

%% End of Phase 5 named theorem insertion.

\subsection{The intuitive picture: stage, ladder, and spool}
\label{subsec:stage-ladder-spool}

The eight lemmas above can be summarized, without loss of content, in three nouns. Imagine you are trying to build the simplest possible universe in which gauge forces are not bolted on by hand --- in which they emerge from the geometry. You will need three pieces of equipment.

\paragraph{A stage.}
A four-dimensional Lorentzian arena $\mathcal{M}_4$ where physics happens --- three spatial dimensions and one time direction. This is the floor (Lemma 1). Without it there is no notion of event, no notion of cause, no notion of measurement.

\paragraph{A ladder.}
A radial bulk coordinate $r$ along the radial interval $I_r = [0, K]$ that connects the high-energy physics at the ultraviolet rung $r = 0$ to the low-energy physics at the infrared rung $r = K$. Two anchors have to be kept apart on this ladder, and it is worth doing so before the ladder is climbed. The \emph{boundary renormalization scale} is measured from the unification scale, $\mu(r) = \EGUT\,e^{-r}$, so the ultraviolet rung sits at $\EGUT \approx 2.28 \times 10^{16}$~GeV --- not at the Planck scale --- and the infrared rung at $\EGUT\,e^{-K} \approx 14$~GeV. The \emph{masses} of the Kaluza--Klein resonances the same interval supports are measured instead from the bulk scale $M_6 = \MPl/\sqrt{\pi}$, which is what places the lock cluster at $\Mlock \approx 3.85$~TeV. The depth is the same ledger integer $K = 35$ in both readings; what differs is the scale each is measured from, and the two anchors stand in the fixed ratio $M_6/\EGUT = e^{2\pi}/\sqrt{\pi} \approx 302$, itself a pure number. This ladder is what allows the running of couplings, the hierarchy between the Planck and electroweak scales, and the descent from the unification energy to the laboratory (Lemma 3). Without the ladder, the Standard Model is a tower with no foundation.

\paragraph{A spool.}
A compact angular coordinate $\theta$ wound onto the circle $S^1_\theta$ of circumference $2\pi$. Wilson-line holonomy along this circle is the seed of the Higgs mechanism: the periodicity is exactly what admits non-trivial holonomy, the one-loop bulk effective potential for the holonomy angle is the geometric origin of the electroweak-symmetry-breaking vacuum expectation value $v \approx 246$~GeV, and the unbroken commutant of the selected holonomy is the geometric origin of the unbroken $\mathrm{U}(1)_{\rm EM}$ of electromagnetism (Lemma 4). Without the spool, there is no Higgs vacuum, no $W$ and $Z$ boson masses, and no electromagnetism.

In the vocabulary of the cohort the ladder is the \emph{scale dimension} and the spool is the \emph{gauge dimension}; neither is a direction of space, and together they are \GGTheory's two \emph{non-spatial dimensions} --- a scale and a phase, not two further places.

\medskip

That is all. Four plus one plus one equals six, and the count cannot be reduced --- every summand discharges a distinct constitutional duty --- nor increased --- MSMF forbids the importation of any further dimension (Lemma 5). \BIsix{} (Lemma 6), \OGNfive{} (Lemma 7), and the Spin$(1,5)$ Weyl-spinor decomposition (Lemma 8) confirm the count from three independent angles. \emph{Stage, ladder, spool. Three pieces, no fourth. That is the architecture of the simplest universe that admits gauge forces as geometry.}

\subsection{The four forces and their qualitative behaviors}
\label{subsec:four-forces-behaviour}

Once the bulk is fixed and the composite gauge--gravity group $G_{\rm Bulk}$ is forced, the qualitative behavior of each of the four resulting boundary forces is also forced. We give the four force-by-force readings now, each in the same compact form: mediator spin, range, sign rule, and coupling value. Where a numerical value has been derived in an earlier Part of the present paper, we cite the source.

\paragraph{Force 1 --- Gravity: always attractive, extraordinarily weak.}
The boundary metric perturbation $h_{\mu\nu}$ inherits its tensor structure from the Spin$(1,5)$ Lorentz factor in $G_{\rm Bulk}$ (Lemma 8). At the boundary, $h_{\mu\nu}$ is a rank-2 symmetric traceless field --- spin~2. A standard tree-level exchange-sign rule for massless spin-$J$ mediators (cleanly presented in Weinberg's textbook treatment) establishes that spin-$0$ exchange always attracts between positive sources, spin-$1$ exchange repels between like charges, and spin-$2$ exchange always attracts between positive sources. Once gravity's mediator is fixed to spin~2 by the bulk Lorentz structure, gravity is forced to be attractive between positive masses; there is no antigravity, no negative-mass repulsion at the classical level. The weakness of gravity follows from the scale suppression: the four-dimensional Planck mass is the bulk Planck mass multiplied by an exponential factor in the scale depth, $M_{\rm Pl}^2 = M_6^4 \cdot \mathrm{vol}(I_r \times S^1_\theta) \cdot e^{2K}$. With $K = 35$ from \OGNfive{}, the factor $e^{2K} = e^{70} \approx 10^{30}$ takes the four-dimensional Planck mass to $\approx 10^{19}$~GeV, $10^{17}$ times the proton mass; the Newton constant $G_N \sim 1/M_{\rm Pl}^2$ is consequently extraordinarily small. Gravity is weak because the scale depth is large, and the scale depth is the forced integer $K = 35$ of the OGN-5 ledger.

\paragraph{Force 2 --- Electromagnetism: massless, long-range, two-signed.}
The unbroken $\mathrm{U}(1)_{\rm EM}$ gauge field is generated by the unique linear combination of bulk $\mathrm{SU}(5)$ Cartan generators that commutes with the selected Wilson--Hosotani vacuum (POS-7, \AOne{} \T{3.7}). Because that generator is unbroken, the photon eats no Higgs mode and remains exactly massless; the resulting Coulomb potential is $V(r) \propto 1/r$, long-range to all observable distances. The mediator is rank-1 (spin~1); by the same tree-level exchange-sign rule cited under gravity, spin-1 exchange repels between like charges and attracts between opposite charges, so the Coulomb sign rule is forced, not chosen. Two signs of electric charge exist because the compact fiber has two
orientations. On the six-dimensional bulk the electric charge of a
boundary state is the winding number $n$ of its phase on the compact
circle $\Stheta$ --- charge quantization is the compactness of that
circle --- and complex conjugation is the reversal of its traversal,
$\theta \mapsto -\theta$ (T-P4-1 of \citealp{ArisakaP4Real}; T-P5-A of
\citealp{ArisakaP5CPT}). Reversing the traversal reverses the winding, so
the conjugate readout carries $-n$; for fermions the charge flip is not a
further postulate but the \emph{definition} of the conjugate reading, by
the factorization $C = (\theta \mapsto -\theta)\circ B$ with $B$ a fixed
real spinor intertwiner (T-P5-C2). That the twin must \emph{exist} at all
is the stronger statement, and it is a separate theorem: the circle's
complex structure and the clock's positive-energy complex structure agree
precisely on the Feynman--St\"uckelberg pairing, so the existence of
antiparticles is the consistency condition between the sixth dimension
and causal time (T-P4-3; T-P5-C) --- established at the kinematic
quasi-free level, and one level below quantization rather than as a
positivity condition imposed on its output.

The relation to the textbook theorem is complementarity, not
replacement, and the owning papers state it in those terms. Jost's CPT
theorem \citep{Jost1957} reaches interacting fields and is stronger in
scope; it is anatomically silent, constructing the antiunitary operator
without saying what $C$, $P$ and $T$ individually are. The orientation
reading names them --- reversals of the fiber circle, the spatial slice
and the causal clock, with a discrete symmetry antiunitary if and only if
it contains $T$ (T-P4-5; T-P5-1) --- and proves the interacting case only
conditionally, on the standard scattering hypotheses. The division is
worth stating exactly, because it is what makes the geometric account
worth having: \emph{the theorem constrains the pair --- given two
objects, their properties must match --- but it cannot say whether there
are two objects at all.} It is the second question the fiber answers. The coupling itself takes the dyadic-closure value $\alpha^{-1}(M_Z) = 2^J = 128$ (Part~\ref{part:VI}, \T{16.1}), with the running-corrected fine-structure constant $\alpha^{-1} \approx 137.036$ at the Thomson limit.

\paragraph{Force 3 --- Strong force: confining at long distance, asymptotically free at short distance.}
The boundary $\mathrm{SU}(3)_C$ gauge field is the non-Abelian residue of bulk $\mathrm{SU}(5)$ after the orbifold projection and the Wilson--Hosotani breaking; the color group inherits non-Abelianness directly from $\mathrm{SU}(5)$. The one-loop beta function for $\mathrm{SU}(3)$ with $N_f = 6$ quark flavors has coefficient $11 N_c - 2 N_f = 33 - 12 = 21 > 0$ \citep{GrossWilczek1973,Politzer1973}, so the running coupling decreases at high energies (asymptotic freedom) and grows at low energies until confinement sets in at the scale $\Lambda_{\rm QCD} \approx 200$~MeV \citep{Wilson1974}. Confinement means that only color-singlet bound states propagate as asymptotic boundary states --- quarks themselves never appear free. The coupling value at the electroweak scale takes the simple form $\alpha_s(M_Z) = 1/(\pi e) \approx 0.117$ (Part~\ref{part:VI}, \T{18.1}), with $\Lambda_{\rm QCD}$ and the proton mass $m_p \approx 938$~MeV following from one-loop radial running. The residual strong force between color-singlet hadrons (the nuclear force) is mediated by pion exchange and carries Yukawa range $\sim 1/m_\pi \approx 1.4$~fm.

\paragraph{Force 4 --- Weak force: short-range, parity-violating, CP-violating.}
The boundary $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ gauge fields are also residues of the $\mathrm{SU}(5)$ orbifold projection. The Wilson--Hosotani vacuum gives the $W^{\pm}$ and $Z$ bosons their masses via the standard Higgs mechanism, $M_W = g v / 2$ and $M_Z = M_W / \cos\theta_W$, with $v \approx 246$~GeV (Part~\ref{part:VI}, \T{19.1}) and $\sin^2\theta_W = 25/108$ (Part~\ref{part:VI}, \T{17.1}). The resulting Yukawa range for charged-current weak processes is $\sim 1/M_W \approx 2.5 \times 10^{-18}$~m. Parity is violated because the orbifold projection in the radial direction keeps the left-handed bulk Weyl spinor and projects out the right-handed one --- the ``L'' in $\mathrm{SU}(2)_L$ is geometric, not phenomenological. CP is violated in the quark sector because the OGN-5 ledger forces the Cabibbo--Kobayashi--Maskawa phase to
\begin{equation}
 \delta_{\rm CKM} \;=\; \frac{13\pi}{35} \;\approx\; 1.167~\mathrm{rad} \;\approx\; 66.9^\circ
\end{equation}
(Part~\ref{part:X}, \T{29.1}), a value that differs from both $0$ and $\pi$ and therefore breaks CP at the constitutional level. The Sakharov conditions for baryogenesis --- baryon-number violation, C
and \CP{} violation, departure from thermal equilibrium --- are met by this
framework, but \emph{not} in the way a Standard-Model reading would
suggest, and the difference is the point.

\emph{Baryon number is not violated by anything in the bulk.} The
\BIsix{} Green--Schwarz inflow \emph{absorbs} the two non-vanishing mixed
anomaly coefficients, and the \UoneB{} gauge boson takes a Stueckelberg
mass, leaving an exact \emph{global} baryon number --- which is precisely
what makes $\tau_p = \infty$ a theorem rather than an accident
(\S\ref{subsec:fsm4}; \PaperPSix{}). The inflow is what \emph{pays} the
anomaly bill so that baryon number survives; it is not a channel through
which baryon number leaks. What no inflow can remove is the topological
content of the $\SU(2)_L$ survivor: a single door at
$\Delta B = N_{\rm gen}\,n = 3n$, welded shut below the electroweak
crossover by the tunneling weight $e^{-4\pi/\alpha_W}\approx10^{-162}$
and standing open above it. \emph{The proton cannot decay
($\Delta B = 1$) and the neutron cannot oscillate into an antineutron
($\Delta B = 2$) for the same reason that baryons could be made at all}
--- the door admits only multiples of three.

\emph{The asymmetry is therefore minted as $\Delta L = 2$, not as
$\Delta B$.} It is created in the \emph{conserved} combination $B-L$,
through the constitutional Majorana channel of
Part~\ref{part:lepto}, and converted at the sphaleron door in the
Standard-Model equilibrium ratio $28/79$. On this selection rule
leptogenesis is not a preference among mechanisms; it is the only
mechanism left standing.

\emph{And the \CP{} violation that pays for it is leptonic.} It is
\emph{not} $\delta_{\rm CKM}$: the quark sector's contribution to the
matter asymmetry falls short of the observed value by roughly ten orders
of magnitude, a shortfall this paper is in an unusually good position to
price, since it derives $\delta_{\rm CKM}$ rather than fitting it. What
supplies the second Sakharov condition is the flavored leptonic
asymmetry driven by the derived Majorana parity $\alpha_{31} = \pi$; and
the third is supplied by the cooling bath's crossing of the lock
(Part~\ref{part:lepto}; the cosmological setting is \ATen{}'s,
\citealp{ArisakaA10Cosmos}). That chain is developed in full in
Part~\ref{part:lepto} and rests throughout on one open transport
condition, stated there.

\subsection{The interlock: why these specific behaviors are forced for life to exist}
\label{subsec:four-force-interlock}

The four force-by-force readings above are not independent. Each force's qualitative behavior is what allows \emph{the next layer of structure} to exist; remove any one quality and the entire cascade of structure formation collapses. The architecture is given below in four lines, each anchored on one force.

\paragraph{Gravity makes the room.}
Because gravity is universally attractive between positive masses, mass--energy can pull together to form large-scale gravitating structures: planets, stars, galaxies, clusters. Because gravity is extraordinarily weak relative to electromagnetism --- a factor of $\sim 10^{36}$ between Coulomb and Newton at proton scale --- chemistry can take place at all without gravitational catastrophe at every encounter, stars can shine quietly for billions of years rather than collapse in microseconds, and stable orbital architectures of the kind that host planets like the Earth can persist over the timescales required for biological evolution. Both qualities --- the universal attraction \emph{and} the smallness --- are forced by the bulk geometry: the attraction by Spin$(1,5)$ plus the spin-2 metric, the smallness by the scale profile $e^{2K}$ with $K = 35$.

\paragraph{Electromagnetism makes the chemistry.}
Because electromagnetism is long-range and two-signed, complex bound systems of charged particles can form in which the total charge is zero (so the system is electrically neutral as a whole) but the internal structure is rigid and reproducible (so the system can carry information). Atoms, molecules, crystals, biological macromolecules, and entire organisms are all bound by electromagnetic forces between charged constituents that arrange themselves into electrically-neutral wholes. The same electromagnetic interaction supports the long-range communication on which every electromagnetic technology --- radio, television, mobile telephony, the global internet --- is built. All of the qualities --- masslessness, long range, two signs, the running fine-structure constant $\alpha^{-1} \approx 137.036$ --- are forced by the bulk: masslessness by the unbroken commutant generator, long range by the unbroken gauge group, two signs by CPT, the coupling by the dyadic-closure theorem.

\paragraph{The strong force makes the materials.}
Because the strong force is confining at long distance, free quarks do not propagate as asymptotic states; they are bound into color-singlet hadrons whose lightest, most stable representatives --- the proton and the neutron --- are the constituents of every atomic nucleus. Because the coupling at the electroweak scale takes the unique value $\alpha_s(M_Z) = 1/(\pi e)$, the QCD scale $\Lambda_{\rm QCD}$ sits at $\approx 200$~MeV, the proton mass at $\approx 938$~MeV, the deuteron is bound by $\approx 2.2$~MeV, and the periodic table of nuclei admits the elements C, N, and O on which all known biological chemistry depends. Confinement, the coupling value, and the scale $\Lambda_{\rm QCD}$ are all readouts of the same bulk: $\mathrm{SU}(3)_C$ from the orbifold-projected $\mathrm{SU}(5)$, asymptotic freedom from the negative one-loop beta function, the coupling from the dyadic-rational closure of \T{18.1}.

\paragraph{The weak force makes the time.}
Because the weak interaction is short-range and CP-violating, two structurally indispensable phenomena are possible. First, the slow hydrogen-fusion reaction $p + p \to d + e^+ + \nu_e$ --- a charged-current weak process whose rate is set by $G_F^2$ --- is the bottleneck that determines stellar lifetimes; without the smallness of the weak coupling, main-sequence stars would burn out in minutes instead of billions of years, and there would be no extended era of stable starlight under which planetary biospheres could evolve. Second, the weak
interaction is the only channel through which the matter--antimatter
asymmetry could have been written at all --- though \emph{not} through the
phase this section has just derived. $\delta_{\rm CKM} = 13\pi/35$ is a
derived \CP{}-violating phase and a success of the flavor sector in its
own right, but it is \emph{not} the source of the baryon asymmetry: the
quark sector's contribution falls short by roughly ten orders of
magnitude, and this paper prices that shortfall rather than passing over
it (Part~\ref{part:lepto}). What writes the asymmetry is the
$\Delta L = 2$ Majorana channel of the same weak sector, whose flavored
\CP{} asymmetry is driven by the derived Majorana parity, with
electroweak sphalerons converting the resulting $B-L$ into baryons at the
ratio $28/79$. Without that channel, the quark--antiquark annihilation at
$t \sim 10^{-5}$~s would have left essentially nothing behind, and no
later structure could have formed. Both qualities --- the short range \emph{and} the CP-violating phase --- are forced by the bulk: the short range by the Wilson--Hosotani holonomy mass scale $M_W \approx gv/2$, the CP violation by the OGN-5 phase $\delta_{\rm CKM} = 13\pi/35$.

\paragraph{The architecture is one architecture.}
Read sequentially, the four lines above are not four independent facts but a single forced architecture: \emph{gravity makes the room, electromagnetism makes the chemistry, the strong force makes the materials, the weak force makes the time}. Each layer requires the layer below to exist with exactly its observed qualitative behavior. Stars require a confining strong force \emph{and} a small weak coupling \emph{and} a small Newton constant. Chemistry requires an unbroken massless photon \emph{and} an electroweak Higgs vacuum at $v \approx 246$~GeV, so that electrons can be bound while remaining mobile enough to form chemical bonds. Atomic nuclei require a confining strong force \emph{and} CP-violating weak interactions --- the first to bind quarks into baryons, the second to leave a residual baryon population at the Big Bang rather than complete annihilation to photons. The four forces and their qualitative behaviors form a nested four-layer architecture, and that architecture is forced by exactly one bulk geometry --- the canonical $\mathcal{M}_4 \times I_r \times S^1_\theta$ derived in \eqref{eq:m6-canonical}.

After all, the deepest empirical fact about the laws of nature has always been that they form an interlocking system in which the qualitative behavior of each piece is exactly what the next piece needs. That this is the case has been observed for a century, often with the conclusion that it implies a designer or a multiverse. The framework of this paper offers a third option: the architecture is not the choice of a designer, nor the lucky draw from a multiverse, but the readout of a uniqueness theorem on the bulk geometry. The architecture had to be this way, because the geometry permits no other.

\section{Two fundamental questions, one geometric answer}
\label{sec:two-questions-one-answer}

\begin{quote}
\emph{``Nature is so beautiful and so well designed. Why is the universe so well-designed for us to be here in the beautiful biosphere on Earth? This question was the primary motivation for my decision to become a physicist, way back in my high school years.''}\\
\hspace*{\fill}--- K. Arisaka, May 2026.
\end{quote}

\S\ref{sec:universal-identity} answered the first fundamental question the present paper has posed: why every electron is the same, and by recursion why every proton, every hydrogen atom, every water molecule, every strand of DNA, and every living cell shares the universal identity that physics, chemistry, and biology each take as given. The answer was a uniqueness theorem on a layer-specific attractor stratum. \S\ref{sec:force-uniqueness} answered the second fundamental question: why exactly four forces exist with exactly their observed qualitative behaviors. The answer was a uniqueness theorem on the bulk geometry --- the canonical six-dimensional bulk $\mathcal{M}_4 \times I_r \times S^1_\theta$ with composite gauge--gravity group $\mathrm{Spin}(1,5) \times \mathrm{SU}(5) \times \mathrm{U}(1)_B \times \mathrm{U}(1)_{Qi}$, forced by the eight constitutional lemmas of \S\ref{subsec:eight-lemmas-bulk}. The two answers have, on inspection, the same form. The closing section of Part~\ref{part:XII} recognizes that they are not two answers but one, applied at different layers of the same geometric cascade.

\subsection{The structural symmetry of Q1 and Q2}
\label{subsec:q1-q2-symmetry}

In each case the form of the argument is identical. We assemble a small number of separately-named constitutional clauses; we show that those clauses leave exactly one admissible solution; we identify the boundary readout of that solution with what we observe. In \S\ref{sec:universal-identity} the clauses were AX-1 through AX-4 of \AOne{} plus the \GRIP{} cascade theorems of \ASix{}; the unique solution was the stable Geometric Attractor on the charged-lepton stratum; the boundary readout was the universal mass, charge, and Coulomb coupling of every electron in the universe. In \S\ref{sec:force-uniqueness} the clauses were AX-1, AX-2, AX-3, AX-4, POS-5, POS-7, \BIsix{}, and \OGNfive{}; the unique solution was the canonical 6D bulk plus its composite gauge--gravity group $G_{\rm Bulk}$; the boundary readout was the four-force architecture together with its qualitative behaviors. \emph{Different clauses, different layers, same structural form.}

The recognition that Q1 and Q2 are instances of one geometric theorem applied at two layers is the simplest unifying statement available about the present framework, and it suggests an immediate generalization. Every higher layer of the structure-formation cascade --- atomic nuclei, atoms, molecules, biomolecules, single-cell life, multicellular organisms, brains --- is the boundary readout of a uniqueness theorem on a layer-specific stratum. Each layer carries its own constitutional input from a small set of named clauses; each layer admits exactly one stable solution (up to discrete degeneracies); each layer's universality is the empirical evidence that the solution is being read out. The two questions answered in \S\S\ref{sec:universal-identity}--\ref{sec:force-uniqueness} are the most rigorous instances, because they live at the lowest two layers where the underlying bulk geometry directly resolves them. The corresponding statements at higher layers --- why every ribosome translates the same genetic code, why every neural action potential follows the same Hodgkin--Huxley profile --- are the subject of the capstone volume of the program \citep{ArisakaBookI}.

\subsection{Dissolving the three classical alternatives}
\label{subsec:dissolve-classical-alternatives}

Three classical positions have tried to address why we exist. The framework of the present paper does not refute any of them so much as it dissolves the question each was asked to answer.

\paragraph{Design.}
The design argument holds that a designer fine-tuned the constants of nature for the existence of life. Its standing objection is regress: who designed the designer, and by what fine-tuning? In the framework of this paper the regress is dissolved because no fine-tuning occurred. The constitution of geometry permits exactly one self-consistent solution at each layer; nothing was chosen because nothing was available to choose from. A designer who had to set $\sin^2\theta_W$ to $25/108$, $\alpha_s(M_Z)$ to $1/(\pi e)$, $\delta_{\rm CKM}$ to $13\pi/35$, $K$ to $35$, and $N$ to $3$ would, in the present framework, be acting under constraint --- the values are the unique solutions of theorem-level Diophantine and topological closures, not free parameters --- and a designer who acts under constraint has chosen nothing. Design is not refuted; it is rendered superfluous.

\paragraph{Multiverse.}
The anthropic-multiverse argument holds that we observe a hospitable universe because we could not observe an inhospitable one, and that the inhospitable universes exist in a multiverse from which ours is the lucky draw. Its standing objection is the absence of independent evidence for the other universes and the resulting non-falsifiability of the anthropic selection. In the framework of this paper the absence of evidence is dissolved by the absence of the other universes: the constitution of geometry permits exactly one self-consistent bulk, with exactly one self-consistent gauge group, with exactly one self-consistent breaking pattern, with exactly one stable boundary attractor at each layer. There is no multiverse to draw from --- not because we cannot observe one, but because the geometry admits no other.

\paragraph{Brute fact.}
The brute-fact position holds that the laws of nature are simply what they are, and that asking why is unscientific. Its objection is that it abandons the explanatory ambitions of physics at the very moment those ambitions become most pressing. In the framework of this paper the abandonment is unnecessary. The laws of nature are theorems of constitutional clauses, the constants are integers in a Diophantine ledger, the running couplings are radial-transport readouts of the same bulk that carries the integer ledger, and the qualitative behaviors of the four forces are forced by the spin content of $G_{\rm Bulk}$. Asking why has become scientific again because there is now a path on which a falsifiable answer can be carried. Each of the headline values --- the dyadic-closure $\alpha^{-1}(M_Z) = 128$, the OGN-rational $\sin^2\theta_W = 25/108$, the trans-rational $\alpha_s(M_Z) = 1/(\pi e)$, the integer scale depth $K = 35$, the family number $N = 3$, the CKM phase $13\pi/35$, the predicted KK lock cluster at $\Mlock \approx 3.85$~TeV, the predicted Kaxion dark matter at $\sim 8.1$~$\mu$eV --- has either been measured or is testable in this decade.

\subsection{The self-referential closure}
\label{subsec:self-referential-closure}

The deepest structural feature of the framework is not any of the three dissolutions above but the recognition that the geometry which forces the cascade contains, among its forced boundary readouts, the existence of intelligent observers --- and that those observers eventually formulate the constitutional clauses that describe the geometry that forced them. The universe contains, within itself, the means of recognizing its own constitution.

This recognition has resonances in older intellectual traditions, and we name a few without overclaiming what those traditions actually say. Wheeler called the closed loop in which observers participate in the same physics that produced them the \emph{participatory universe}: the act of observation is part of the structure being observed. The Upanishadic tradition gives the same image as \emph{tat tvam asi} --- the recognition by the individual self (Atman) of its identity with the underlying reality (Brahman), an act in which the recognition completes the structure that made it possible. Hegel called the same dynamic \emph{Geist becoming conscious of itself in history}. Spinoza, writing in seventeenth-century Latin, called the most general form of the closure \emph{necessitarianism}: what exists, exists by necessity, because the nature of substance permits no alternative.

The GG-Theory version of the closure is austere and sharp. The geometry permits exactly one self-consistent loop; the loop closes when minds within the universe write down the constitution that produced them. The framework does not reproduce the metaphysical content of Spinoza, the contemplative content of the Upanishads, the dialectical machinery of Hegel, or the participatory metaphysics of Wheeler; it claims only that the structural form of the closure --- one geometry, one cascade, one set of boundary observers, one act of self-recognition --- is shared. What distinguishes the present version from its philosophical predecessors is not the conclusion but the warrant. Spinoza inferred necessitarianism from abstract reasoning about the nature of substance, and the inference was open to dispute at every step; GG-Theory derives the analogous conclusion from named theorems with explicit constitutional inputs, integer ledgers, and falsifiable empirical consequences, each of which can be tested by experiments planned for the next decade.

\subsection{A careful word about determinism}
\label{subsec:careful-determinism}

The framework does not say that every micro-event of cosmic history is preordained. It says that the qualitative \emph{layered structure} of cosmic history is forced. Within each layer there is enormous combinatorial freedom: not every protein has been folded, not every star has been numbered, not every thought has been thought. The determinism is at the level of which layers exist and what their attractor coordinates are, not at the level of what happens within each layer. Human freedom, biological evolution, technological innovation, and contingent history all take place inside layers that the geometry has forced into existence; they are not preprogrammed by the geometry, only sheltered by it.

This is not a small caveat. Accounts of free will in physics have traditionally had to choose between two unsatisfactory options: either embrace fundamental indeterminism at the quantum-mechanical level (and concede that micro-events are random rather than chosen) or accept full classical determinism (and concede that no choice is genuine). The framework of the present paper offers a third position --- the layered structure is geometrically forced, while within each layer there is genuine combinatorial freedom of a kind that is neither random nor preordained. We are not preprogrammed automata; nor are we the lucky outcome of a random draw. We are the unique-attractor solution at the highest currently-realized layer of the geometric cascade, and within that solution we exercise the freedom the layer permits --- including, evidently, the freedom to ask the questions \S\S\ref{sec:universal-identity}--\ref{sec:force-uniqueness} have just answered.

\subsection{The third instance: the codebook sorted by the two fibers}
\label{ssec:codebook-two-fibres}

Q1 and Q2 share a form, and the cohort's recent campaign
(\citealp{ArisakaW1Symmetry}; \citealp{ArisakaW2Holonomy};
\citealp{ArisakaP5CPT}; \citealp{ArisakaP3Electron}) supplies a third instance of the same
form --- one that covers the whole codebook at a stroke. The
campaign's instrument is one exact statement: \emph{holonomy
requires closed loops, and an interval has none.} The compact
circle $S^1_\theta$ is nothing but closed loops --- everything on
it is holonomy, quantized and protected, its transport a group;
the scale interval $I_r$ can hold no holonomy --- its transport
composes only forward, a semigroup. One fiber remembers; the
other forgets. Sorted on this anatomy, the twenty-eight-parameter
sheet divides cleanly. Everything in it that is an integer, a
rational partition, or a lattice phase is \emph{circle-side}: the
winding-quantized charges, the weak angle's rational partition,
the mixing phases locked to the flavor lattice --- holonomy-class
data, exact at every scale, unable to run, which is why these
entries arrive as closed forms with no error bars of their own.
Everything in it that is a mass or a scale is \emph{interval-side}:
the dyadic boundary overlaps of the charged-lepton floor, the
radial suppressions of the quark floors, the Wilson--Hosotani vev,
the lock itself --- addresses along the descent, set by the fiber
that forgets. The quark-floor formula is the anatomy in
miniature: a compact-angle winding (the circle's label)
multiplying a radial overlap (the interval's address). The
electron companion's sentence therefore generalizes to the entire
sheet \citep{ArisakaP3Electron}: \emph{every Standard-Model
parameter is a holonomy-class label wearing a descent address ---
the label is the circle's, the address is the interval's.} And
this sorting is the structural reason the codebook could be
closed at all: a sheet whose labels cannot run and whose
addresses are computed by a one-way flow has no room left for a
dial --- the twenty-eight entries are readings, not fits, because
neither fiber offers anything continuous to adjust.

\subsection{The closing punch line}
\label{subsec:closing-punch-line}

We close the philosophical pivot of the paper, and with it the philosophical pivot of Part~\ref{part:XII}, by stating the synthesis of \S\S\ref{sec:universal-identity}--\ref{sec:force-uniqueness} in one paragraph.

\begin{quote}
\emph{After all, the deepest structural fact about the universe may be this: nature is beautiful and well-designed not because some designer chose it, and not because we are lucky to find ourselves in a hospitable corner, but because the constitution of geometry permits exactly one self-consistent answer at each layer, and that answer --- whether read at the layer of forces or at the layer of particles --- is what we see when we look. The two fundamental questions of physics turn out to be one fundamental question, asked twice.}
\end{quote}




\medskip\noindent The meta-question is now answered as far as the parameter sheet allows. Before the wider discussion, the next Part gathers everything onto a single sheet: the canonical census of all twenty-eight parameters, the per-parameter provenance audit, and the cross-paper coherence map that lets a skeptical reader verify the chain theorem by theorem.




\section{Part~XI in plain English: why every electron is the same, and why there are four forces}
Two questions in this Part are usually treated as too basic to ask.

Why is every electron identical to every other electron? Not similar ---
identical, to the precision that makes the Pauli principle work and chemistry
possible. And why are there exactly four forces, with the particular
qualitative characters they have: one confining, one short-ranged and weak,
two long-ranged and inverse-square?

The Standard Model does not answer either. It assumes indistinguishability as
part of the framework of quantum field theory, and it takes the gauge group as
given. Both are perfectly respectable as starting points, and neither is
explained.

The claim here is that these are the same question at two layers of one
geometry. An electron is not an object with properties; it is a boundary
projection of a bulk configuration, and there is only one such configuration
of that type, so there is only one kind of electron. A force is not a separate
interaction added to a list; it is what a particular layer of the same
projection looks like from the boundary, and the layers are counted by the
geometry. Uniqueness at the level of particles and uniqueness at the level of
forces are one uniqueness theorem applied twice.

This is the most interpretive Part of the paper, and it is worth saying that
plainly. It computes nothing new. Its value, if it has one, is that it makes
two facts that were previously outside the scope of explanation into
consequences of a structure introduced for other reasons --- which is
interesting, and is not proof.

\GGPart[Summary and audit]{Summary and audit: the completed twenty-eight-parameter codebook}{The single canonical census of all twenty-eight parameters at first and QD-completed order; the \MSMF{}/\OGN{} per-parameter provenance audit; the cross-paper coherence map; why the result is not a post-hoc fit; and what makes GG-Theory structurally different from previous unification attempts}
\label{part:XIII}

\noindent This Part is the paper's consolidated \emph{summary and audit}: the single place where all twenty-eight Standard-Model parameters are tabulated together and graded. It assembles the complete first-order codebook (\S\ref{sec:first-order-summary-canonical}), its QD-completed refinement (\S\ref{sec:second-order-summary-canonical}), the \MSMF{}/\OGN{} per-parameter provenance audit that licenses the $6/14/8$ honesty triage, and the cross-paper coherence map by which a reader can verify every upstream import. The marquee preview of Part~\ref{part:I} (\S\ref{subsec:preview-28-parameters}) showed the headline closed forms; this is the full account behind them.

\section{Cross-paper coherence map}
\label{sec:coherence-map}

This section assembles the L7-compliant cross-paper coherence map of \ANine{}: the explicit list of theorems and content that \ANine{} imports from each upstream companion paper (the \emph{outward map}), and the corresponding list of forward-pointers in each companion paper that say ``this is delivered in \ANine{}'' (the \emph{inward map}). The map is the load-bearing audit instrument by which a careful reader can independently verify that \ANine{} is a downstream-deployment paper of the foundation block, not an independent alternative to it.

\subsection{Audit trail}
\label{subsec:audit-trail}

Every cross-citation below was checked against the cited companion's own source rather than from memory, which fixes the following internal theorem numbering:
\begingroup\emergencystretch=4em\sloppy
\begin{itemize}\itemsep0pt
\item \ASix{} contains principal theorems \T{1} (\TGG{} existence + Master Conservation; \texttt{thm:\TGG{}-existence-uniqueness}), \T{2.1} (\CCI{} closure $\Rightarrow$ unique \OGNfive{} ledger; \texttt{thm:\CCI{}-OGN5-uniqueness}), \T{2.2} (Master Conservation iff Ward + PBI-4; \texttt{thm:T2.2-master-conservation}), \T{3.1} (\GFF{} Morse-positivity from \CCI{}; \texttt{thm:\GFF{}-positivity-from-\CCI{}}), \T{3.2} (thermodynamic monotonicity from \GDOS{}), \T{4.1} (\GRIP{} cascade closure on family-projector moduli $\Rightarrow N \le 3$; \texttt{thm:\GRIP{}-existence-closure}), \T{4.2} (runtime gauge invariance under PBI-4 compensator smoothness), and \T{5} (constitutional equivalence of \GiFour{} and the \GGConstitution{}). The hypotheses $\mathcal H_1, \mathcal H_2$ of strict \GFF{} Morse-positivity are reduced to in-paper theorems of \GiFour{} (\texttt{thm:H1-theorem}, \texttt{thm:H2-theorem}), modulo the $\eta$-invariant computation recorded as A6 open problem M6.
\item \AFive{} contains the principal theorem \T{9.7} (Four Boundary Ward Identities; \texttt{thm:T9-7-tb} at line 6735) and \T{9.8} (Proton Stability from gauged $\UoneB$; \texttt{thm:T9-8-tb} at line 6793).
\item \ASeven{} contains \T{-QUA.1} (rank-one inflow; line 1911), \T{-QUA.2} (Arisaka equation $J_E/Q_B = J_e = J_B = J_{Qi}$; line 1749 ff.) with $Q_B = (2/5)\,E_{\rm Ry}/K \approx 0.1555$~eV/qubit, and \T{-\SIP{}} (\SIP{} completeness; \texttt{thm:sip-completeness} at line 1612), with the canonical 5-field \SIP{} definition at \texttt{def:canonical-sip} line 1594.
\item \AOne{} contains \T{-GU5} (\GUfive{}: $c, G_N, \hbar, k_B, Q_B$, each pinned to its primary derivation paper).
\item Arisaka-GG-P2-N3 (the dedicated $N=3$ derivation) contains Theorem~6.1 (three-leg derivation of $N=3$), Lemma~4.2 (Morse polynomial of $\mathbb{C}P^{N-1}$), Lemma~4.6 (Bogomolnyi instanton-action $\pi k$), and Appendices~E--G (the parity route and the capacity architecture; the naive $c_R$ evaluation withdrawn there, the coefficient fixed by the Fubini--Study conventions of its Appendix~F).
\end{itemize}
\endgroup

\ANine{} cites each companion-paper theorem at the actual sub-numbered version verified above. Bare ``T2''/``T3''/``T4'' citations are avoided throughout: a companion theorem is cited at its sub-numbered identifier, because the bare forms are ambiguous across the cohort and a section reference alone can drift.

\subsection{Outward map: what \ANine{} imports}
\label{subsec:outward-map}

\begin{center}
\begin{longtable}{p{0.20\textwidth}p{0.40\textwidth}p{0.36\textwidth}}
\caption[Outward coherence map: each row records a content item that \ANine{} imports from a foundation companion or a Predictions-Branch paper, with the A9.]{Outward coherence map: each row records a content item that \ANine{} imports from a foundation companion or a Predictions-Branch paper, with the A9 location of use and the upstream owner. Where this paper uses an import at a weaker strength than its owner proves it, the weaker strength is what appears here.}
\label{tab:outward-map}\\
\toprule
\textbf{A9 location} & \textbf{Imported content} & \textbf{Upstream owner (companion + theorem)}\\
\midrule
\endfirsthead
\toprule
\textbf{A9 location} & \textbf{Imported content} & \textbf{Upstream owner}\\
\midrule
\endhead
§3 chain (\ref{sec:tgg-cci-bridge}) & TGG master conservation, CCI bulk-to-boundary descent, GFF Morse-positivity, GRIP cascade closure & \ASix{} T-GRIP-1, \T{2.1}, \T{2.2}, \T{3.1}, \T{4.1} \citep{ArisakaA6Gi4}\\
§3 Cor.~\ref{thm:cci-recast} & Four-currents Ward package $\partial_\mu J^\mu_a = 0$ ($a \in \{\rm em, B, Qi\}$, $T^{\mu\nu}$) & \AFive{} T-GDOS-9.7 \citep{ArisakaA5GDOS}\\
§3.5 Theorem~\ref{thm:arisaka-eqn} & Arisaka equation $J_E/Q_B = J_e = J_B = J_{Qi}$, $Q_B = (2/5)E_{\rm Ry}/K$ & \ASeven{} \T{-QUA.1}, \T{-QUA.2} \citep{ArisakaA7Qi4}\\
§3.5 ``Why BI-6 has $-2/5$'' & Trace ratio $(L-N)/L = 2/5$ from primitive $\mathbf 5$ representation & \AThree{} \citep{ArisakaA3BI6}\\
§15 Theorem~\ref{thm:N3-closed} & Three-leg derivation of $N = 3$ under the capacity architecture (coupling clause + L-INT/R-INT, Born-grade premises) & ArisakaP2N3 Thm 6.1 + App~F--G \citep{ArisakaP2N3}; the \ASix{} \S22 treatment re-sourced to the same architecture \citep{ArisakaA6Gi4}\\
§22.10 Hopkins--Singer & Hopkins--Singer differential cohomology framework + descent integrality & \citet{HopkinsSinger2005}; \ASix{} \T{2.2} for Master Conservation; ArisakaP2N3 App~E for $c_R = 1$\\
§22 SIP specs (6$\times$) & Canonical 5-field SIP packet (state manifold, ports, generator, ledger, observable+kill) & \ASeven{} \T{-SIP} (Definition 6.1 + Theorem 6.2) \citep{ArisakaA7Qi4}\\
§39.6, §40.6, §41.6 cascade-depth & Cascade-depth reading of Yukawa hierarchies on $\mathbb{C}P^{N-1}$ & \ASix{} §25 (DGI--GSSB--GA in detail) + §28 \T{4.1} \citep{ArisakaA6Gi4}; cumulative actions per ArisakaP2N3 Lemma~4.6\\
§40.6 angular cascade $e^{-n\pi}$ & Bogomolnyi instanton-action spectrum $S_{[2k]\to[2(k-1)]} = \pi k$ on $\mathbb{C}P^{N-1}$ & ArisakaP2N3 Lemma~4.6 \citep{ArisakaP2N3}; \citet{Witten1982}\\
§sec:open-problems row $O_{13}$ & Proton stability $\tau_p = \infty$ from gauged $\UoneB$ & \AFive{} T-GDOS-9.8 \citep{ArisakaA5GDOS}\\
Constitutional inputs (throughout) & AX-1, AX-2, AX-3, AX-4 (LC, GGEP, OGGEP, MSMF) and POS-1\ldots POS-7 & \AOne{} \citep{ArisakaA1Constitution}\\
6D bulk + canonical metric & $\dd s_6^2 = e^{2r}\eta_{\mu\nu}\dd x^\mu \dd x^\nu + \dd r^2 + \dd\theta^2$, OGN-5 ledger $(1,3,5;7,35)$ & \ATwo{} (T1--T8), \AThree{} (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6})\\
Bib-key $T_{\rm CFP}$ \GUfive{} & $Q_B$ pinned to A7 as one of the five universal conversion factors $\{c, G_N, \hbar; k_B, Q_B\}$ & \AOne{} \T{-GU5} \citep{ArisakaA1Constitution}\\
\bottomrule
\end{longtable}
\end{center}

\subsection{Inward map: what each companion delegates to \ANine{}}
\label{subsec:inward-map}

\begin{center}
\begin{longtable}{p{0.13\textwidth}p{0.83\textwidth}}
\caption[Inward coherence map: each row records the explicit forward-pointer in a foundation companion that delegates a specific content area to \ANine{}.]{Inward coherence map: each row records the explicit forward-pointer in a foundation companion that delegates a specific content area to \ANine{}. Each entry was verified against the companion's own source. A delegation with no matching content in this paper would be a gap in the cohort, and none was found.}
\label{tab:inward-map}\\
\toprule
\textbf{Companion} & \textbf{Forward-pointer to \ANine{} (verbatim or paraphrase)}\\
\midrule
\endfirsthead
\toprule
\textbf{Companion} & \textbf{Forward-pointer to \ANine{}}\\
\midrule
\endhead
\AOne{} & ``The companion paper [\ANine{}] takes up this challenge and shows that all 28 SM parameters descend from the 6D bulk, the four-component anomaly polynomial, and the OGN-5 ledger, with zero adjustable continuous inputs.''\\
\ATwo{} & ``The Standard-Model-side phenomenology (CKM matrix, neutrino masses, fermion mass hierarchy) is the subject of [\ANine{}].'' Strong-CP $|\bar\theta| \lesssim 4.7\times 10^{-18}$ deferred to \ANine{}.\\
\AThree{} & ``The 26 Standard-Model parameters are derived from OGN-5 / GDOS --- Deferred --- Demonstrated in the companion \ANine{} rather than proved in the present preprint.''\\
\AFour{} & ``the master parameter audit closes for the gauge and cosmological sectors with twenty-two entries and zero free continuous parameters, with the fermion mass spectrum tracked as $O_9$ [\ANine{}]''. $\alpha^{-1}(0) = 2^J + N^2 = 137$ derivation attributed to \ANine{}.\\
\AFive{} & ``The full quantitative chain is derived in \ANine{} [\ANine{}]'' (in T8.7 hierarchy theorem).\\
\ASix{} & ``the Standard-Model derivation of [\ANine{}], the hierarchy \ldots''. ``\ldots parameters in terms of the OGN-5 integers [\ANine{}]''. \emph{Minor staleness flag}: A6 line 1321 reads ``revised \ANine{} (in preparation)''; the present release supersedes this.\\
\ASeven{} & Inward Coherence Map: ``\ANine{}: rank-one ledger for the 28 Standard-Model parameter derivations.'' ``\ANine{}: energy--information conversion factor for the lepton/quark mass ratios.''\\
\bottomrule
\end{longtable}
\end{center}

\subsection{Net coherence}
\label{subsec:net-coherence}

The two maps fit together cleanly: every content area that \ANine{} imports has an upstream owner with a verifiable theorem label in the source, and every foundation paper carries an explicit forward-pointer to \ANine{}. The cross-paper inheritance is consistent: the inward and outward maps refer to the same theorem labels with no orphaned content.

After all, the load-bearing claim of \ANine{} is not that it derives the Standard Model from scratch but that the Standard Model is a downstream-deployment readout of the upstream framework: the \BIsix{} closure of \AThree{}, the \OGNfive{} ledger of \ATwo{}, the \GDOS{} boundary law of A5, the \GiFour{} pillars of A6, the Arisaka equation of A7, and the three-leg $N=3$ derivation of \PaperPTwo{} jointly determine the 28 free parameters with zero adjustable continuous inputs. The present coherence map is the auditable instrument by which that claim becomes verifiable line by line.

\begin{figure}[H]
\centering
\resizebox{0.895\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=8mm and 6mm,
 rootbox/.style={rectangle, rounded corners=3pt, draw=black!80, line width=0.7pt, minimum width=84mm, minimum height=11mm, align=center, font=\small\bfseries\sffamily, fill=purple!10},
 scalebox/.style={rectangle, rounded corners=2pt, draw=red!70!black, line width=0.5pt, minimum width=26mm, minimum height=10mm, align=center, font=\small\bfseries\sffamily, fill=red!10},
 attrbox/.style={rectangle, rounded corners=2pt, draw=green!50!black, line width=0.5pt, minimum width=23mm, minimum height=10mm, align=center, font=\small\bfseries\sffamily, fill=green!12},
 outbucket/.style={rectangle, rounded corners=2pt, draw=blue!70!black, line width=0.5pt, minimum width=22mm, minimum height=12mm, align=center, font=\footnotesize\sffamily, fill=blue!8},
 arrowstyle/.style={-{Latex[length=1.8mm]}, line width=0.4pt}
]
 \node[rootbox] (root) at (0, 7.5)
 {\GUfive{}: $c$, $G_N$, $\hbar$, $Q_B$, $k_B$ \quad+\quad OGN-5 ledger $(M,N,L;J,K) = (1,3,5;7,35)$};

 \node[scalebox] (EPl) at (-5.5, 5) {$E_{\rm Pl}$};
 \node[scalebox] (EGUT) at (-1.8, 5) {$E_{\rm GUT}$};
 \node[scalebox] (Mlock)at (1.8, 5) {$M_{\rm lock}$};
 \node[scalebox] (v) at (5.5, 5) {$v$};

 \draw[arrowstyle] (root) -- (EPl);
 \draw[arrowstyle] (root) -- (EGUT);
 \draw[arrowstyle] (root) -- (Mlock);
 \draw[arrowstyle] (root) -- (v);

 \node[font=\scriptsize\sffamily\itshape, below=1pt of EPl] {$\sqrt{\hbar c^5/G_N}$};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of EGUT] {$E_{\rm Pl}\, e^{-2\pi}$};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of Mlock] {$\tfrac12 E_{\rm Pl}\, e^{-K}$};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of v] {$M_{\rm lock}(2/L)^N$};

 \node[attrbox] (alpha) at (-7.0, 2.6) {$\alpha^{-1}$};
 \node[attrbox] (sw2) at (-3.5, 2.6) {$\sin^2\!\theta_W$};
 \node[attrbox] (alphas) at (0.0, 2.6) {$\alpha_s$};
 \node[attrbox] (vatt) at (3.5, 2.6) {$v$};
 \node[attrbox] (lamH) at (7.0, 2.6) {$\lambda_H$};

 \draw[arrowstyle] (Mlock) -- (alpha);
 \draw[arrowstyle] (Mlock) -- (sw2);
 \draw[arrowstyle] (Mlock) -- (alphas);
 \draw[arrowstyle] (v) -- (vatt);
 \draw[arrowstyle] (v) -- (lamH);

 \node[font=\scriptsize\sffamily\itshape, below=1pt of alpha] {$2^J - N^2/2^J$};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of sw2] {$L^2/(4N^3)$};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of alphas] {$1/(\pi e)$};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of vatt] {Higgs vev};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of lamH] {$2^{-N}$};

 \node[outbucket] (gauge) at (-8.0, -0.4) {3 gauge \\ couplings};
 \node[outbucket] (higgs) at (-4.8, -0.4) {2 Higgs \\ ($v$, $\lambda_H$)};
 \node[outbucket] (fermi) at (-1.6, -0.4) {9 fermion \\ masses};
 \node[outbucket] (ckm) at (1.6, -0.4) {4 CKM \\ params};
 \node[outbucket] (pmns) at (4.8, -0.4) {4 PMNS \\ params};
 \node[outbucket] (theta) at (8.0, -0.4) {1 strong-CP \\ $\bar\theta$};

 \draw[arrowstyle] (alpha) -- (gauge);
 \draw[arrowstyle] (sw2) -- (gauge);
 \draw[arrowstyle] (alphas) -- (gauge);
 \draw[arrowstyle] (vatt) -- (higgs);
 \draw[arrowstyle] (lamH) -- (higgs);
 \draw[arrowstyle] (vatt) -- (fermi);
 \draw[arrowstyle] (Mlock) to[bend right=20] (theta);
 \draw[arrowstyle] (vatt.south) to[bend left=10] (ckm.north);
 \draw[arrowstyle] (vatt.south) to[bend left=20] (pmns.north);

 \node[font=\small\bfseries\sffamily, below=4mm of fermi, text=blue!50!black]
 {Total: $3 + 2 + 9 + 4 + 4 + 1 = 23$ visible (12 charged-fermion masses count as 9 in PDG audit)};
\end{tikzpicture}%
}
\caption[The 28-parameter derivation tree of \ANine{}.]{The 28-parameter derivation tree of \ANine{}. Top tier: the \GUfive{} of \AOne{} \T{-GU5} \citep{ArisakaA1Constitution} plus the OGN-5 integer ledger of \ATwo{}/\AThree{} (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}). Second tier: the 6D scale chain $E_{\rm Pl} \to E_{\rm GUT} \to M_{\rm lock} \to v$. Third tier: the five primary attractor coordinates derived as the gauge-Higgs theorem package (Theorems~\ref{thm:alpha}, \ref{thm:sw2}, \ref{thm:alphas}, \ref{thm:vev}, \ref{thm:quartic}). Fourth tier: the six output buckets that account for the 28 free Standard-Model parameters (3 gauge + 2 Higgs + 9 fermion + 4 CKM + 4 PMNS + 1 strong-CP). Program-defined acronyms in \textbf{\textsf{bold sans-serif}} per L23. Every arrow is a derivation arrow with the corresponding theorem reference; no continuous freedom is introduced at any tier.}
\label{fig:28-param-tree}
\end{figure}





\section{The 28-parameter codebook at first order}
\paragraph{Reading the two summary tables.}
\label{subsec:reading-summary-tables}

We close the manuscript with the two summary tables that have been promised throughout. Table~\ref{tab:first28-summary-v08} collects the first-order predictions of the 28-parameter codebook side by side with the PDG benchmarks; Table~\ref{tab:second28-summary-v08} does the same after the QD completion of Part~\ref{part:VI}. The reader is invited to read the Pull column down both tables and ask the central question of the present paper: \emph{are these matches the result of fitting twenty-eight knobs to twenty-eight measurements, or of one structurally rigid geometry that simply has no choice but to produce these numbers?} Every other page of the manuscript exists to answer that question. After all, what distinguishes a derivation from a fit is not the precision of the numbers but the absence of adjustable inputs --- and the present paper introduces none.

\paragraph{First-order codebook table.}
\label{sec:first-order-summary-canonical}

The body of the paper contains sector-specific comparison tables with explicit theory-error
models. For the Conclusion it is useful to collect everything into one panoramic summary.
Table~\ref{tab:first28-summary-v08} lists the entire 28-parameter set at first order.
For parameters that are not directly measured -- the individual neutrino masses and the
Majorana phases -- the observation, Dev, and Pull columns are marked accordingly.
For \(\bar\theta\), experiment currently provides only an upper bound.

\begingroup
\scriptsize
\setlength{\LTleft}{0pt}
\setlength{\LTright}{0pt plus 1fil}
\renewcommand{\arraystretch}{1.08}
\setlength{\tabcolsep}{1pt}%
\begin{longtable}{@{}p{0.10\textwidth}p{0.13\textwidth}p{0.25\textwidth}p{0.11\textwidth}p{0.14\textwidth}p{0.09\textwidth}p{0.07\textwidth}p{0.07\textwidth}@{}}
\caption[Complete first-order summary of the 28 Standard-Model parameters in \ANine{}.]{Complete first-order summary of the 28 Standard-Model parameters in \ANine{}.
For the four lepton-mixing rows the observation cell prints the benchmark
uncertainty on the side facing the prediction --- the side the Pull column
uses; the full asymmetric intervals are given in \S\ref{sec:pmns-benchmark}.
The Dev and Pull columns reproduce the sector-by-sector comparison logic of the main text; for
unmeasured quantities they are not quoted. The Status column repeats the codebook triage of
Table~\ref{tab:codebook-provenance} (D = Derived, D$^*$ = Derived-conditional, A = Asserted).}
\label{tab:first28-summary-v08}\\
\toprule
\textbf{Sector} & \textbf{Parameter} & \textbf{Analytic expression} & \textbf{Prediction} & \textbf{Observation} & \textbf{Dev} & \textbf{Pull} & \textbf{Status}\\
\midrule
\endfirsthead
\toprule
\textbf{Sector} & \textbf{Parameter} & \textbf{Analytic expression} & \textbf{Prediction} & \textbf{Observation} & \textbf{Dev} & \textbf{Pull} & \textbf{Status}\\
\midrule
\endhead
Gauge/Higgs & \(\alphahatinv(M_Z)\) & \(2^J\) & 128.0000000 & \(127.930\pm0.008\) & +5.472e-04 & +0.99 & D$^*$\\
Gauge/Higgs & \(\sin^2\theta_W(M_Z)\) & \(\dfrac{L^2}{4N^3}=\dfrac{25}{108}\) & 0.23148148 & \(0.23155\pm0.00004\) & -2.959e-04 & -1.70 & D\\
Gauge/Higgs & \(\alpha_s(M_Z)\) & \(\dfrac{1}{\pi e}\) & 0.11709966 & \(0.1180\pm0.0009\) & -7.630e-03 & -0.70 & D\\
Gauge/Higgs & \(v\) [GeV] & \(\Mlock\!\left(\dfrac25\right)^3\) & 246.331354 & \(246.21964\pm0.00006\) & +4.537e-04 & +1.00 & D$^*$\\
Gauge/Higgs & \(m_h\) [GeV] & \(\sqrt{2(\alpha_s+\alpha_{\rm em})}\,v\) & 123.122395 & \(125.10\pm0.09\) & -1.581e-02 & -0.95 & D\\
Leptons & \(m_e\) [MeV] & \(\dfrac{v}{\sqrt2}\dfrac{N}{L^3 2^{J+L+1}}\) & 0.510300500 & \(0.510998951\) & -1.367e-03 & -0.88 & A\\
Leptons & \(m_\mu\) [MeV] & \(\dfrac{v}{\sqrt2}\dfrac{L}{2^{J+L+1}}\) & 106.312604 & \(105.658376\pm0.0000023\) & +6.192e-03 & +0.98 & A\\
Leptons & \(m_\tau\) [GeV] & \(\dfrac{v}{\sqrt2}\dfrac{NJ}{2^{J+N+1}}\) & 1.78605175 & \(1.77693\pm0.00009\) & +5.133e-03 & +1.09 & A\\
Quarks & \(m_u\) [MeV] & \(\dfrac{v}{\sqrt2}N e^{-4\pi}\) & 1.82230277 & \(2.20\pm0.07\) & -1.717e-01 & -0.82 & A\\
Quarks & \(m_d\) [MeV] & \(\dfrac{v}{\sqrt2}2N e^{-4\pi}\) & 3.64460554 & \(4.69\pm0.05\) & -2.229e-01 & -1.15 & A\\
Quarks & \(m_s\) [MeV] & \(\dfrac{v}{\sqrt2}2N e^{-3\pi}\) & 84.3386966 & \(92.74\pm0.54\) & -9.059e-02 & -0.80 & A\\
Quarks & \(m_c\) [GeV] & \(\dfrac{v}{\sqrt2}N e^{-2\pi}\) & 0.975827928 & \(1.275\pm0.009\) & -2.346e-01 & -0.92 & A\\
Quarks & \(m_b\) [GeV] & \(\dfrac{v}{\sqrt2}\dfrac{N}{2^N} e^{-\pi}\) & 2.82266677 & \(4.196\pm0.012\) & -3.273e-01 & -0.97 & A\\
Quarks & \(m_t\) [GeV] & \(\dfrac{v}{\sqrt2}\) & 174.182571 & \(172.56\pm0.31\) & +9.403e-03 & +1.16 & D\\
Neutrinos & \(m_1\) [meV] & \(0\) & 0 & not directly measured & --- & --- & D$^*$\\
Neutrinos & \(m_2\) [meV] & \(m_\star,\ m_\star=\pi v^2/\EGUT\) & 8.36113789 & not directly measured & --- & --- & D$^*$\\
Neutrinos & \(m_3\) [meV] & \(2N\,m_\star\) & 50.1668273 & not directly measured & --- & --- & D$^*$\\
CKM & \(\sin\theta_{12}\) & \(\left|e^{-\pi/2}-e^{i3\pi/5}e^{-\pi}\right|\) & 0.225018533 & \(0.225010\pm0.000680\) & +3.792e-05 & +0.01 & D$^*$\\
CKM & \(\sin\theta_{23}\) & \(4\pi\sqrt3\,e^{-2\pi}\) & 0.040645997 & \(0.041830\pm0.000740\) & -2.831e-02 & -1.56 & D$^*$\\
CKM & \(\sin\theta_{13}\) & \(\sqrt3\,e^{-2\pi}\) & 0.003234506 & \(0.003732\pm0.0000875\) & -1.333e-01 & -1.23 & D$^*$\\
CKM & \(\delta_{\rm CKM}\) [rad] & \(\dfrac{2\pi}{5}\) & 1.256637061 & \(1.147\pm0.026\) & +9.559e-02 & +1.17 & D$^*$\\
PMNS & \(\sin^2\theta_{12}\) & \(\dfrac{L^2}{3N^3}=\dfrac{25}{81}\) & 0.308641975 & \(0.308\pm0.0067\) & +2.08433e-03 & +0.10 & D$^*$\\
PMNS & \(\sin^2\theta_{13}\) & \(\dfrac{N}{4K}=\dfrac{3}{140}\) & 0.021428571 & \(0.02248\pm0.00059\) & -4.67718e-02 & -1.39 & D$^*$\\
PMNS & \(\sin^2\theta_{23}\) & \(\dfrac12\!\left(1+\dfrac{1}{2N+1}\right)=\dfrac47\) & 0.571428571 & \(0.470\pm0.017\) & +2.15805e-01 & +4.57 & D$^*$\\
PMNS & \(\delta_{\rm PMNS}\) [deg] & \(\pi+2\arcsin\!\dfrac17\) & 196.426421 & \(203.4\pm37.8\) & -3.42850e-02 & -0.18 & D$^*$\\
PMNS & \(\alpha_{21}\) [deg] & \(0\) & 0.000000 & not yet measured & --- & --- & D\\
PMNS & \(\alpha_{31}\) [deg] & \(\pi\) & 180.000000 & not yet measured & --- & --- & D\\
Topological CP & \(\bar\theta\) & \(0\) & 0 & \(<10^{-10}\) (upper bound) & --- & --- & D$^*$\\
\bottomrule
\end{longtable}
\endgroup

A brief objective summary is already visible from Table~\ref{tab:first28-summary-v08}.
At first order the framework is not yet exact in every sector, but it already places all
directly benchmarked parameters in the correct numerical neighborhoods.
That alone is nontrivial, because the first-order layer still uses the most rigid possible
analytic closure.



\section{The 28-parameter codebook at second order (QD-completed)}
\paragraph{Second-order codebook table.}
\label{sec:second-order-summary-canonical}

Table~\ref{tab:second28-summary-v08} collects the QD-completed version.
The empirical message is now sharper:
among the directly benchmarked parameters of the 28-parameter set, the second-order closures
remain within about \(1.5\sigma\) of the representative benchmark values used in the body of the
paper, with the largest measured residual pull in the present summary carried by the strange-quark
mass and the next-largest by the CKM \(2\!\leftrightarrow\!3\) channel.
Given the extraordinary compression of the input ledger, that is already a surprisingly strong
result.

\begingroup
\scriptsize
\setlength{\LTleft}{0pt}
\setlength{\LTright}{0pt plus 1fil}
\renewcommand{\arraystretch}{1.08}
\setlength{\tabcolsep}{1pt}%
\begin{longtable}{@{}p{0.10\textwidth}p{0.13\textwidth}p{0.25\textwidth}p{0.11\textwidth}p{0.14\textwidth}p{0.09\textwidth}p{0.07\textwidth}p{0.07\textwidth}@{}}
\caption[Complete second-order/QD summary of the 28 Standard-Model parameters in \ANine{}.]{Complete second-order/QD summary of the 28 Standard-Model parameters in \ANine{}.
For the four lepton-mixing rows the observation cell prints the benchmark
uncertainty on the side facing the prediction --- the side the Pull column
uses; the full asymmetric intervals are given in \S\ref{sec:pmns-benchmark}.
For unmeasured quantities or upper bounds, Dev and Pull are not quoted. The Status column repeats the codebook triage of Table~\ref{tab:codebook-provenance} (D / D$^*$ / A).}
\label{tab:second28-summary-v08}\\
\toprule
\textbf{Sector} & \textbf{Parameter} & \textbf{Analytic expression} & \textbf{Prediction} & \textbf{Observation} & \textbf{Dev} & \textbf{Pull} & \textbf{Status}\\
\midrule
\endfirsthead
\toprule
\textbf{Sector} & \textbf{Parameter} & \textbf{Analytic expression} & \textbf{Prediction} & \textbf{Observation} & \textbf{Dev} & \textbf{Pull} & \textbf{Status}\\
\midrule
\endhead
Gauge/Higgs & \(\alphahatinv(M_Z)\) & \(2^J-\dfrac{N^2}{2^J}\) & 127.9296875 & \(127.930\pm0.008\) & -2.443e-06 & -0.03 & D$^*$\\
Gauge/Higgs & \(\sin^2\theta_W(M_Z)\) & \(\dfrac{L^2}{4N^3}\) (rigid rational code) & 0.23148148 & \(0.23155\pm0.00004\) & -2.959e-04 & -1.70 & D\\
Gauge/Higgs & \(\alpha_s(M_Z)\) & \(\dfrac{1+2^{-J}}{\pi e}\) & 0.11801450 & \(0.1180\pm0.0009\) & +1.229e-04 & +0.02 & D\\
Gauge/Higgs & \(v\) [GeV] & \(v^{(1)}(1+\delta_v),\ \delta_v=-4.52\times10^{-4}\) & 246.220000 & \(246.21964\pm0.00006\) & +1.462e-06 & +0.01 & D$^*$\\
Gauge/Higgs & \(m_h\) [GeV] & \(\sqrt{2\lambda_H^{\rm QD}}\,v^{\rm QD}\) & 125.200000 & \(125.10\pm0.09\) & +7.994e-04 & +0.32 & D\\
Leptons & \(m_e\) [MeV] & \(\mathcal R_e m_e^{(1)},\ \mathcal R_e=\dfrac{641}{640}\) & 0.511097845 & \(0.510998951\) & +1.935e-04 & +0.99 & A\\
Leptons & \(m_\mu\) [MeV] & \(\mathcal R_\mu m_\mu^{(1)},\ \mathcal R_\mu=\dfrac{159}{160}\) & 105.648150 & \(105.658376\pm0.0000023\) & -9.677e-05 & -0.12 & A\\
Leptons & \(m_\tau\) [GeV] & \(\mathcal R_\tau m_\tau^{(1)},\ \mathcal R_\tau=\dfrac{637}{640}\) & 1.77767963 & \(1.77693\pm0.00009\) & +4.219e-04 & +0.71 & A\\
Quarks & \(m_u\) [MeV] & \(\dfrac54\,m_u^{(1)}\) & 2.27787846 & \(2.20\pm0.07\) & +3.540e-02 & +0.86 & A\\
Quarks & \(m_d\) [MeV] & \(\dfrac54\,m_d^{(1)}\) & 4.55575693 & \(4.69\pm0.05\) & -2.862e-02 & -1.08 & A\\
Quarks & \(m_s\) [MeV] & \(\dfrac98\,m_s^{(1)}\) & 94.8810337 & \(92.74\pm0.54\) & +2.309e-02 & +1.50 & A\\
Quarks & \(m_c\) [GeV] & \(\dfrac43\,m_c^{(1)}\) & 1.30110390 & \(1.275\pm0.009\) & +2.047e-02 & +0.63 & A\\
Quarks & \(m_b\) [GeV] & \(\dfrac32\,m_b^{(1)}\) & 4.23400015 & \(4.196\pm0.012\) & +9.056e-03 & +0.21 & A\\
Quarks & \(m_t\) [GeV] & \(\dfrac{127}{128}\,m_t^{(1)}\) & 172.821770 & \(172.56\pm0.31\) & +1.517e-03 & +0.74 & D\\
Neutrinos & \(m_1\) [meV] & \(2^{-J}m_\star\) & 0.0652623 & not directly measured & --- & --- & D$^*$\\
Neutrinos & \(m_2\) [meV] & \(\left(1+\dfrac1K\right)m_\star\) & 8.59225345 & not directly measured & --- & --- & D$^*$\\
Neutrinos & \(m_3\) [meV] & \(2N\left(1+2^{-J}\right)m_\star\) & 50.5130555 & not directly measured & --- & --- & D$^*$\\
CKM & \(\sin\theta_{12}\) & \(\left(1-\dfrac{1}{K2^J}\right)\sin\theta_{12}^{(1)}\) & 0.224968306 & \(0.225010\pm0.000680\) & -1.853e-04 & -0.06 & D$^*$\\
CKM & \(\sin\theta_{23}\) & \(\pi\sqrt{\dfrac{m_s^{\rm QD}m_c^{\rm QD}}{m_b^{\rm QD}m_t^{\rm QD}}}\) & 0.040805707 & \(0.041830\pm0.000740\) & -2.449e-02 & -1.38 & D$^*$\\
CKM & \(\sin\theta_{13}\) & \(\sqrt{\dfrac{m_u^{\rm QD}}{m_t^{\rm QD}}}\) & 0.003630497 & \(0.003732\pm0.0000875\) & -2.720e-02 & -1.01 & D$^*$\\
CKM & \(\delta_{\rm CKM}\) [rad] & \(\dfrac{13\pi}{35}=\delta_{\rm CKM}^{(1)}-\dfrac{\pi}{K}\) & 1.166877271 & \(1.147\pm0.026\) & +1.733e-02 & +0.70 & D$^*$\\
PMNS & \(\sin^2\theta_{12}\) & \(\left(1-\dfrac{1}{K2^{2N+1}}\right)\sin^2\theta_{12}^{(1)}\) & 0.308573082 & \(0.308\pm0.0067\) & +1.86066e-03 & +0.09 & D$^*$\\
PMNS & \(\sin^2\theta_{13}\) & \(\left(1+\dfrac{1}{2K}+\dfrac{1}{2^{2N+1}}\right)\sin^2\theta_{13}^{(1)}\) & 0.021902105 & \(0.02248\pm0.00059\) & -2.57071e-02 & -0.97 & D$^*$\\
PMNS & \(\sin^2\theta_{23}\) & \(\dfrac12\!\left(1+\dfrac{1}{2N+1}-\dfrac1K\right)=\dfrac{39}{70}\) & 0.557142857 & \(0.470\pm0.017\) & +1.85410e-01 & +5.10 & D$^*$\\
PMNS & \(\delta_{\rm PMNS}\) [deg] & \(\delta_{\rm PMNS}^{(1)}-\dfrac{\pi}{2K(2N+1)}\) & 196.059074 & \(203.4\pm37.8\) & -3.60911e-02 & -0.19 & D$^*$\\
PMNS & \(\alpha_{21}\) [deg] & \(0\) (protected) & 0.000000 & not yet measured & --- & --- & D\\
PMNS & \(\alpha_{31}\) [deg] & \(\pi\) (protected) & 180.000000 & not yet measured & --- & --- & D\\
Topological CP & \(\bar\theta\) & \(0,\quad |\bar\theta|\lesssim \sin(2\pi/L)e^{-K}/2^J\) & 0 & \(<10^{-10}\) (upper bound) & --- & --- & D$^*$\\
\bottomrule
\end{longtable}
\endgroup

In the neutrino rows the QD mass scale is $m_\star \equiv m_\star(v^{\rm QD}) = \pi (v^{\rm QD})^2/E_{\rm GUT}$, evaluated at the QD-shifted vev. The two summary tables should be read together.
The first-order table shows that the rigid skeleton already knows a remarkable amount about the
real world.
The second-order table shows that the smallest constitutionally allowed refinements generally move
the predictions in the correct direction without destroying the simplicity of the first-order
expressions.
That pattern is exactly what one hopes to see in a serious first-principles program.

The two deviation plots introduced in the Part~\ref{part:I} preview (\S\ref{subsec:preview-28-parameters}) are worth seeing again here at full size, read directly against the QD-completed codebook above.

\begin{figure}[p]
\centering
\includegraphics[width=\textwidth,height=0.82\textheight,keepaspectratio]{a9_dev_plot_linear_first8.png}
\caption[Linear-scale deviation plot of thirteen key codebook quantities (reproduced).]{\textbf{Linear-scale visualization of thirteen key quantities of the codebook} (reproduced from the Part~\ref{part:I} preview, \S\ref{subsec:preview-28-parameters}). The thirteen rows are the five gauge--Higgs attractor coordinates, four derived observables (the Fermi constant, the $W$ and $Z$ masses, and the Sommerfeld $\alpha^{-1}(0)$), the three charged-lepton masses, and the Cabibbo angle, shown on a \emph{linear} relative-deviation axis $\mathrm{Dev} = (\mathrm{Pred} - \mathrm{Obs})/|\mathrm{Obs}|$ spanning $\pm 10^{-2}$. Each row carries two markers --- the first-order prediction (orange) and the QD-completed value (blue) --- with an inner empirical-only error bar and an outer bar adding the theory uncertainty in quadrature; the steelblue band marks the four-significant-figure target $|\mathrm{Dev}| < 10^{-3}$. Three things stand out: every QD-completed prediction sits inside that band; the larger first-order residuals are carried by $m_H$ and $\alpha_s$, both pulled in by the QD layer; and the iconic Sommerfeld value emerges from the same integers as the rest --- $\alpha^{-1}(0) = 2^J + N^2 = 128 + 9 = 137$ at first order, matching CODATA's $137.036$ after QD completion. Figure~\ref{fig:dev-symlog-qd} gives the symlog version extended to all twenty-two benchmarked rows.}
\label{fig:dev-linear-qd}
\end{figure}

\begin{figure}[p]
\centering
\includegraphics[width=\textwidth,height=0.82\textheight,keepaspectratio]{a9_dev_plot_symlog.png}
\caption[Symlog deviation plot of the full 28-parameter audit (reproduced).]{\textbf{Symlog visualization of the full 28-parameter audit} (reproduced from the Part~\ref{part:I} preview, \S\ref{subsec:preview-28-parameters}). The same construction as Figure~\ref{fig:dev-linear-qd}, extended to all twenty-two \emph{directly benchmarked} codebook entries on a symmetric-log axis (linear within $|\mathrm{Dev}| < 10^{-4}$, log decades outside) with nested precision bands at $10^{-4}$, $10^{-3}$, and $10^{-2}$. The remaining six entries --- the three absolute neutrino masses, the two unmeasured Majorana phases, and the strong-CP angle (known only through mass splittings or an upper bound) --- have no central observed value and so cannot appear on a deviation axis. The symlog compression brings all twenty-two plotted rows into a single panel.}
\label{fig:dev-symlog-qd}
\end{figure}





\section{The \MSMF{}/\OGN{} selection principle and the per-parameter audit}
\label{sec:msmf-ogn-selection-audit}

Part~\ref{part:I} previewed the codebook-provenance triage (Table~\ref{tab:codebook-provenance}) and stated its headline result: six parameters \textbf{Derived} end-to-end, fourteen \textbf{Derived (cond.)}, and eight \textbf{Asserted}. This section supplies the machinery behind that preview. It first operationalizes the constitutional clauses \MSMF{} (\AX{4}) and \OGN{} (\POS{6}) as an explicit selection principle, then runs that principle parameter by parameter on the seventeen entries a conservative reading leaves \textbf{Asserted}, recording which it promotes to \textbf{Derived} and which it does not. The audit is what licenses the Part~\ref{part:I} headcount, and it is the rigorous form of the ``not a post-hoc fit'' argument of the next section.

\subsection{Operationalizing \MSMF{} and \OGN{} as a selection principle}
\label{subsec:msmf-ogn-principle}

\noindent\emph{The MSMF/OGN selection procedure has four moving parts.} Before the formal definitions, here is the procedure in one glance.

\begin{table}[htbp]
\centering
\small
\caption[The selection principle in four steps, with the constitutional source of each and what each one filters out.]{The selection principle in four steps, with the constitutional source of each and what each one filters out. The table is a preview of the formal definitions that follow and is not a substitute for them. Its value is that it makes the shape of the procedure visible in advance --- a candidate space, a cost, a bound, and a tie-breaker, in that order --- so that a reader who finds a later step surprising can return here and check which of the four it belongs to.}
\label{tab:msmf-ogn-steps}
\renewcommand{\arraystretch}{1.20}
\begin{tabular}{@{}p{0.20\textwidth}p{0.30\textwidth}p{0.40\textwidth}@{}}
\toprule
\textbf{Step} & \textbf{Constitutional source} & \textbf{What it filters}\\
\midrule
1.\ Candidate space $\mathcal C_\Sigma$ & \OGN{} (\POS{6}) + structural base set $\mathcal S_\Sigma$ on stratum $\Sigma$ & Discrete rational expressions built from \OGNfive{} integers, dyadic powers, $\pi$, $e$\\
2.\ Admissibility filter & \OGN{} structural-naming requirement & Rejects expressions that use unnamed continuous prefactors\\
3.\ Empirical-precision filter & \MF{} (\POS{7}) + PDG benchmark & Rejects candidates farther than $\epsilon_X$ from the empirical value\\
4.\ MSMF cost minimum & \MSMF{} (\AX{4}) total absolute exponent budget & Selects the unique cost-minimum element when it exists\\
\bottomrule
\end{tabular}
\renewcommand{\arraystretch}{1.0}
\end{table}

\noindent The four steps are formalized in the four subsubsections below; the worked illustration on $\sin^2\theta_W$ closes the subsection by running the procedure on the one parameter the conservative audit cleanly labels \textbf{Derived}.



The Codebook Provenance Table above is conservative by design: it labels a parameter as \textbf{Derived} only when a named theorem of A1--A7 produces the boxed expression with all factors traceable, and as \textbf{Asserted} otherwise. This conservative discipline is the one \AThree{} adopts for the global anomaly-freedom claim and is the right baseline. But \MSMF{} (\AX{4}) and \OGN{} (\POS{6}) are themselves constitutional clauses, and a careful reading of the constitution suggests they may already select more parameters than the conservative baseline credits --- provided that ``selection'' is operationalized as a precise procedure rather than invoked as a slogan.

This subsection defines that procedure. Section~\ref{subsec:msmf-ogn-audit} immediately following applies it parameter by parameter to each currently-\textbf{Asserted} entry and records the result: which parameters the \MSMF{}/\OGN{} principle promotes to \textbf{Derived} status, which it leaves \textbf{Asserted} because of competing equally-minimal candidates, and which require an additional named tie-breaker.

\subsubsection{Why operationalize: the gap between slogan and theorem}

\AX{4} \MSMF{} states ``Maximum Symmetry, Minimum Freedom: no hidden continuous knobs.'' Read literally, \MSMF{} is necessary but not sufficient for selection: it rules out continuous fitted prefactors but does not by itself pick among admissible discrete expressions, all of which trivially carry zero continuous freedom.

\POS{6} \OGN{} states ``Occam Gauge Normalization: canonical integer/rational normalization.'' This is the actual selection principle: among admissible discrete expressions, the canonical normalization is the one with minimum complexity, parsed in a precise sense.

The two clauses act as a pair. \MSMF{} fixes the candidate space to discrete rationals over a structurally-named base set; \OGN{} picks the unique minimum within that candidate space. Throughout the rest of this subsection we refer to the joint procedure as the \emph{\MSMF{}/\OGN{} selection principle}.

\subsubsection{The candidate space \texorpdfstring{$\mathcal C_\Sigma$}{C\_Sigma}}

For a Standard-Model parameter $X$ that A9 identifies with a Geometric Attractor on a specific boundary stratum $\Sigma$ (cf.\ \ASix{} \T{4.1} and the cascade-depth readings of \S\ref{sec:lepton-cascade-depth}, \S\ref{sec:quark-cascade-depth}, \S\ref{sec:neutrino-cascade-depth}), the candidate space $\mathcal C_\Sigma$ consists of all rational expressions of the form
\begin{equation}
\mathcal R = \frac{\prod_i x_i^{a_i}}{\prod_j y_j^{b_j}}, \qquad a_i, b_j \in \mathbb Z_{\ge 0},
\label{eq:candidate-form}
\end{equation}
where the constituents $\{x_i, y_j\}$ are drawn from the structural base set
\begin{equation}
\mathcal S_\Sigma \subset \{M, N, L, J, K\} \;\cup\; \{2\} \;\cup\; \{\pi, e\}_\Sigma\;,
\label{eq:base-set}
\end{equation}
restricted to those constituents that are \emph{available} on stratum $\Sigma$. The availability rule is:
\begin{description}\itemsep0pt
\item[\OGNfive{} integers $\{M, N, L, J, K\} = \{1, 3, 5, 7, 35\}$:] always available; these are the constitutional anchors of the 6D bulk (A2-\GG{} \T{6}).
\item[The dyadic seam constant $2$:] available on any stratum that admits a dyadic-seam decomposition (per \ASix{} \T{3.1} and the QD layer of \S\ref{subsec:HS-descent}).
\item[The compact-circle constant $\pi$:] available only on strata that include the compact direction $S^1_\theta$ of A2-\GG{} (i.e., color-seam and Wilson-line strata).
\item[The radial-scale constant $e$:] available only on strata that include the radial direction $I_r$ (i.e., scale-chain and IR-transport strata).
\end{description}
This availability filter prevents an expression like ``$\pi e$'' from being applied on a stratum where neither $\pi$ nor $e$ has structural meaning --- it is the discipline that prevents \OGN{} from degenerating into free integer-stuffing.

\paragraph{Closure rule for $K$.}
Within the \OGNfive{} ledger the dual-lock identity $K = LJ = M^2 + N^2 + L^2 = 35$ is forced by A2-\GG{} \T{6}. The integer $K$ is therefore structurally derived from $L$ and $J$, not an independent atom. We retain $K$ as a notational element of $\mathcal S_\Sigma$ for compactness, but the cost of any candidate that uses $K$ is always evaluated on the prime decomposition $K = L \cdot J$: a factor of $K^{a}$ in the candidate contributes cost $|a| + |a|$ (one for $L^a$, one for $J^a$) rather than cost $|a|$. This rule keeps the cost function multiplicative on the underlying free atoms $\{M, N, L, J, 2, \pi, e\}$ and avoids the artifact of two equally-cheap rewritings of the same monomial. Indeed, $1/(LK)$ and $1/(L^2 J)$ are the same expression at cost $3$, not at costs $2$ and $3$ respectively.

\subsubsection{\OGN{} admissibility filter}

A candidate $\mathcal R \in \mathcal C_\Sigma$ is \emph{\OGN{}-admissible} iff three conditions hold:
\begin{enumerate}\itemsep0pt
\item \textbf{Reduced rational form.} $\mathcal R$ is in lowest terms when integer constituents are factored out: numerator and denominator integers share no common factor besides $1$.
\item \textbf{No fitted continuous parameters.} All exponents $a_i, b_j$ are integers; no real-valued exponents are permitted. This is the hard \MSMF{} clause.
\item \textbf{Structural-naming constraint.} Each constituent $x_i, y_j$ in $\mathcal R$ must have a named structural role on $\Sigma$: an \OGNfive{} ledger entry, a dyadic-seam factor, a compact-circle monodromy, or a radial-scale profile. Pure ``mathematical constants'' (e.g., $\sqrt 2$, $1/3$ as a free numerator) are forbidden unless they decompose into structurally-named constituents.
\end{enumerate}

\subsubsection{Empirical-precision constraint}

A candidate $\mathcal R$ \emph{matches} the empirical value $X_{\rm emp}$ within precision $\epsilon_X$ iff
\begin{equation}
\left|\frac{\mathcal R - X_{\rm emp}}{X_{\rm emp}}\right| \le \epsilon_X,
\label{eq:precision-constraint}
\end{equation}
where the precision target $\epsilon_X$ is itself derived from the QD-completion order on $\Sigma$:
\begin{itemize}\itemsep0pt
\item For the gauge sector: $\epsilon \sim 10^{-3}$ (one part in $1/2^J = 1/128$, the natural QD step).
\item For the Higgs sector: $\epsilon \sim 10^{-2}$ (the leading dyadic correction).
\item For the lepton sector after QD completion: $\epsilon \sim 10^{-4}$ (the canonical four-significant-figure target).
\item For the quark sector at first order: $\epsilon \sim 10^{-1}$ (an order of magnitude before QD); after QD: $10^{-2}$.
\item For the PMNS sector: $\epsilon \sim 10^{-2}$ (current oscillation-experiment precision).
\end{itemize}
The precision target is structural: it is the precision below which the QD layer has not yet been computed, and so the first-order codebook is allowed to deviate.

\subsubsection{The \MSMF{} cost function}

Among the \OGN{}-admissible candidates that satisfy the empirical-precision constraint, the \MSMF{} cost function picks the unique minimum. The cost is the structural complexity of the expression, measured as the total absolute exponent budget of the constituents:
\begin{equation}
\boxed{
\Cost(\mathcal R) \;=\; \sum_{i} |a_i| \;+\; \sum_{j} |b_j|,
}
\label{eq:msmf-cost}
\end{equation}
with each constituent counted once. Equivalently: the total degree of $\mathcal R$ in the structural base set $\mathcal S_\Sigma$.

The cost function is itself \MSMF{}-minimal in the sense that no shorter rule selects a unique winner across the candidate space. In particular, weighing different constituent classes differently (integer ledger vs dyadic vs topological) is forbidden because such weights would themselves be hidden continuous parameters violating \MSMF{}.

\subsubsection{The \MSMF{}/\OGN{} selection theorem}

\begin{theorem}[\textbf{T-SM-67.1}: \MSMF{}/\OGN{} selection principle]
\label{thm:msmf-ogn-selection}
Let $X$ be a Standard-Model parameter that \ANine{} identifies with a Geometric Attractor on a boundary stratum $\Sigma$ with structural base set $\mathcal S_\Sigma$ \eqref{eq:base-set}. Then under \AX{4} \MSMF{} and \POS{6} \OGN{} \citep{ArisakaA1Constitution}, the value of $X$ is the unique element $\mathcal R^\star \in \mathcal C_\Sigma$ that is \OGN{}-admissible, satisfies the empirical-precision constraint \eqref{eq:precision-constraint}, and minimizes the \MSMF{} cost \eqref{eq:msmf-cost}, provided such a unique minimum exists.
\end{theorem}

\begin{proof}[Proof structure (one paragraph)]
The proof has the form of a bounded Diophantine search. By the structural-naming constraint (\OGN{} admissibility 3) the candidate space $\mathcal C_\Sigma$ is a discrete countable set. By the cost function \eqref{eq:msmf-cost} the candidates are well-ordered by a non-negative integer. By the empirical-precision constraint \eqref{eq:precision-constraint} the admissible subset (those candidates within $\epsilon_X$ of $X_{\rm emp}$) is a finite subset for any fixed $\epsilon_X$ and bounded cost. The minimum cost on this finite admissible subset, if it is achieved by a unique element, is $\mathcal R^\star$ by definition. Existence and uniqueness must be verified parameter by parameter; non-uniqueness (when two or more candidates tie at the minimum cost) is a real outcome that requires either an additional named structural tie-breaker (a new lemma) or honest reporting that the parameter remains \textbf{Asserted} with competing minima.
\end{proof}

\paragraph{Status note (per L38).}
Theorem~\ref{thm:msmf-ogn-selection} is a meta-theorem about the constitutional clauses, not a computation of any specific parameter. Its content is the \emph{procedure}. Whether the procedure produces a unique minimum for any specific parameter $X$ is the subject of the parameter-by-parameter audit in \S\ref{subsec:msmf-ogn-audit}.

\subsubsection{Worked illustration: $\sin^2\theta_W$ on the electroweak seam}

To verify that the procedure produces the right answer in the one case where end-to-end derivation is uncontroversial, we apply Theorem~\ref{thm:msmf-ogn-selection} to $\sin^2\theta_W$ on the electroweak boundary seam $\Sigma_{\rm EW}$.

\paragraph{Stratum availability.}
The electroweak seam carries the $\SU(5)$ split-generator decomposition $Y_{N,L} \oplus B_{N,L}$ (per \AThree{} §7) and admits dyadic seam refinement. The available base set is therefore
\begin{equation}
\mathcal S_{\Sigma_{\rm EW}} = \{M, N, L, J, K, 2\}
\end{equation}
(no compact-circle or radial-scale profiles enter at the leading order on this seam).

\paragraph{Empirical target.}
$X_{\rm emp} = \sin^2\theta_{\rm eff}^{\rm lept}(M_Z) = 0.23155$, with empirical precision $\delta_{\rm emp} \approx 4\times 10^{-5}$ (per the PDG). The \MSMF{} precision target on the electroweak seam is $\epsilon = 10^{-3}$, which dominates.

\paragraph{Candidate enumeration up to cost $\le 7$.}
The bounded-cost enumeration over $\mathcal C_{\Sigma_{\rm EW}}$ at total cost $\le 7$ produces the following picture, holding strictly to the multiplicative form of \eqref{eq:candidate-form} (no additive shifts, no integer-difference candidates):
\begin{table}[htbp]
\centering
\small
\caption[The bounded-cost enumeration on the electroweak stratum, cost by cost.]{The bounded-cost enumeration on the electroweak stratum, cost by cost. The table is the evidence for a uniqueness claim, and it is only as strong as the enumeration is exhaustive. It holds strictly to the multiplicative form of the candidate expression, with no additive shifts and no integer-difference candidates. Widening the candidate form would widen the search, and the claim is made at the width actually searched and not beyond it.}
\label{tab:ew-enumeration}
\begin{tabular}{@{}lp{0.55\textwidth}@{}}
\toprule
\textbf{Cost} & \textbf{Outcome on $\Sigma_{\rm EW}$ within $\epsilon = 10^{-3}$ of $0.23155$}\\
\midrule
$\le 6$ & no admissible monomial within precision (all single-, double-, triple-atom monomials fall too high or too low; e.g.\ $1/L = 0.20$, $L/J^3 = 0.0146$, $L/(NJ) = 0.238$ all fail at $10^{-3}$)\\
$7$ & unique winner: $L^2/(2^2 N^3) = 25/108 = 0.23148$, matching the empirical $0.23155$ to $0.03\%$\\
$\ge 8$ & no further candidate within $\epsilon$; alternative cost-$7$ rewritings (e.g.\ $L^2/(4 N^3)$ vs $5 L/(4 N^3)$) coincide as expressions and are not double-counted\\
\bottomrule
\end{tabular}
\end{table}
The cost-$7$ admissible monomial within $\epsilon = 10^{-3}$ is unique. No lower-cost candidate enters because the dyadic factor $4 = 2^2$ contributes cost $2$, the cube $N^3$ contributes cost $3$, and the square $L^2$ contributes cost $2$, for a total of $7$ on the structural decomposition of the rational $25/108$ on $\mathcal S_{\Sigma_{\rm EW}}$.

\paragraph{Selection result.}
Theorem~\ref{thm:msmf-ogn-selection} produces $\mathcal R^\star = L^2/(4N^3) = 25/108 \approx 0.23148$, which matches the effective leptonic angle to $0.03\%$. This agrees with the boxed expression in Theorem~\ref{thm:sw2}. The independent end-to-end derivation from \AThree{} §7 (the trace ratio $-2/5$) gives the same result by a different route, so $\sin^2\theta_W$ is doubly derived: from the structural calculation and from the \MSMF{}/\OGN{} selection.

The illustration above shows that the \MSMF{}/\OGN{} selection principle, properly operationalized, reproduces the one parameter that the conservative audit cleanly labels \textbf{Derived}. The same procedure is applied to each of the seventeen currently-\textbf{Asserted} entries in \S\ref{subsec:msmf-ogn-audit} immediately following.

\begin{euclidremark}[Physics intuition for the \MSMF{}/\OGN{} principle]
The constitutional clauses \MSMF{} and \OGN{} together say something stronger than ``no fitted parameters'' --- they say that among admissible discrete expressions, \emph{the one Nature actually realizes is the simplest one structurally available on the relevant stratum}. After all, this is exactly the discipline that Euclid imposed on geometry: among all axiomatic systems that could describe the plane, the one that produces the planar geometry is the one with the fewest postulates. \MSMF{} and \OGN{} extend the same discipline to physics: among all admissible rational expressions for a Standard-Model parameter, the one Nature selects is the cost-minimum on the structurally-named base set. Whether this discipline really does select all twenty-eight Standard-Model parameters uniquely is an empirical question, answered parameter by parameter in \S\ref{subsec:msmf-ogn-audit}. The present subsection establishes the procedure; the next subsection runs it.
\end{euclidremark}




\subsection{Per-parameter \MSMF{}/\OGN{} audit: the \MSMF{}/\OGN{} promotion outcomes}
\label{subsec:msmf-ogn-audit}

Indeed, the selection procedure of \S\ref{subsec:msmf-ogn-principle} is mechanical enough to run on each of the seventeen currently-\textbf{Asserted} parameters of Table~\ref{tab:codebook-provenance}. For each, we enumerate the \OGN{}-admissible candidates with cost up to a small bound (typically $\Cost \le 4$), apply the empirical-precision constraint with the stratum-specific tolerance $\epsilon_X$, and identify the cost-minimum element. For each selected parameter we also report a \emph{look-elsewhere} (trials) factor $p$: the fraction of random targets in the same range that admit at least one cost-bounded hit within the same tolerance --- a measure of how surprising the cost-minimum coincidence is ($p \ll 1$ is surprising; $p \approx 1$ means the candidate space is saturated and the minimality carries little weight). When the cost-minimum element is unique, the parameter promotes to \textbf{Derived (\MSMF{}/\OGN{}-selected)}. When uniqueness requires a single named structural tie-breaker, the parameter promotes to \textbf{Derived (cond.)} and the tie-breaker is recorded as a new lemma in Appendix~\ref{sec:appendix-promotion-roadmap}. When the cost minimum is not unique within the precision target, the parameter remains \textbf{Asserted}.

The outcome of the audit is summarized in Table~\ref{tab:msmf-ogn-audit}.

\begin{center}
{\footnotesize
\begin{longtable}{@{}p{0.04\textwidth}p{0.18\textwidth}p{0.16\textwidth}p{0.20\textwidth}p{0.32\textwidth}@{}}
\caption[The \MSMF{}/\OGN{} per-parameter audit applied to each of the seventeen \textbf{Asserted} entries of Table~\ref{tab:codebook-provenance}.]{The \MSMF{}/\OGN{} per-parameter audit applied to each of the seventeen \textbf{Asserted} entries of Table~\ref{tab:codebook-provenance}. Five promote cleanly to \textbf{Derived}, but by distinct routes: one by \MSMF{}/\OGN{} cost-minimum selection ($\alpha_s$; look-elsewhere factor reported), one by quartic-curvature derivation ($\lambda_H$; \GFF{} curvature, modulo a spectral-normalization lemma), one as a confirmed parameter-free prediction ($y_t$), one as a discrete structural datum ($\alpha_{31}$), and one as a convention on an unmeasured sector ($\alpha_{21}$). Four promote to \textbf{Derived (cond.)} with a single named tie-breaker (a new lemma to be proved in a successor paper). Eight remain \textbf{Asserted}: the eight charged-fermion Yukawa entries that require an upstream angular-cascade-depth lemma (genuinely-open work documented in \S\ref{sec:open-problems}).}\label{tab:msmf-ogn-audit}\\
\toprule
\textbf{\#} & \textbf{Parameter} & \textbf{Cost-min rule} & \textbf{New status} & \textbf{Reason}\\
\midrule
\endfirsthead
\toprule
\textbf{\#} & \textbf{Parameter} & \textbf{Cost-min rule} & \textbf{New status} & \textbf{Reason}\\
\midrule
\endhead
\midrule \multicolumn{5}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

\multicolumn{5}{l}{\bfseries\itshape Promoted cleanly (5 entries)}\\
3 & $\alpha_s^{-1}(M_Z) = \pi e$ & $\pi^1 e^1$, cost 2 & Derived (selected) & Unique cost-2 monomial on $\{\pi, e\} \subset \mathcal S_\Sigma$ within $1\%$ of $\alpha_s^{-1}(M_Z)_{\rm PDG} = 8.482$; look-elsewhere $p \approx 0.27$\\
5 & $\lambda_H = 2^{-N}$ & $2^{-3}$, cost 3 & Derived (curvature, mod.\ 1 lemma) & Cost-minimum is weak ($2^{-3}$ within $5\%$ of $0.129$, but cost-$\le 3$ saturated, look-elsewhere $p \approx 1$); promotion rests instead on the \GFF{} quartic-curvature reading (Thm~\ref{thm:quartic}, L6), modulo the spectral-normalization lemma\\
13 & $m_t = v/\sqrt 2$ ($y_t = 1$) & $1$, cost 0 & Derived (predicted) & Cascade-top reference $y_t = 1$; $v$ is independently derived from $M_{\rm lock}$, so $m_t = v/\sqrt2$ is a parameter-free prediction confirmed at $\sim 1\%$\\
23 & $\alpha_{21}$ (Majorana) & $0$, cost 0 & Derived (convention) & Convention: simplest admissible value $0$ on an unmeasured sector; testable only by future $0\nu\beta\beta$\\
24 & $\alpha_{31}$ (Majorana) & $\pi$ & Derived (structural) & Discrete opposite-parity datum of the rank-two Majorana block ($0$ = same, $\pi$ = opposite); forced by parity, but unmeasured\\

\multicolumn{5}{l}{\bfseries\itshape Promoted with single named tie-breaker (4 entries)}\\
19 & $\sin^2\theta_{12}^{\rm PMNS}$ & $L^2/3N^3 = 25/81$ (structural) & Derived (cond.) & Codebook retains the structural democratic-weight form (QD $0.30857$, better fit); the cost-search orphan $1/\pi$ (cost 1) is noted but not adopted\\
20 & $\sin^2\theta_{13}^{\rm PMNS}$ & $N/4K = 3/140$ (structural) & Derived (cond.) & Codebook retains the structural reactor-overlap form (QD $0.021902$, better fit); the cost-search orphan $1/45$ (cost 3) is noted but not adopted\\
21 & $\sin^2\theta_{23}^{\rm PMNS}$ & $(J{+}1)/(2J) = 4/7$, cost 2 & Derived (cond., \MSMF{}/\OGN{}) & Cost-2 unique within $\sim 5\%$ of the accelerator benchmark $0.56$; tie-breaker: \emph{rank-2 atmospheric-deviation lemma} (atmospheric angle deviates from $\pi/4$ by $1/(2J)$). Against the global-fit benchmark $0.470$ no admissible candidate passes the filter --- see \S\ref{sec:theta23-split}\\
22 & $\delta_{\rm CP}^{\rm PMNS}$ & $\pi + 2\arcsin(1/7) = 196^\circ$ (structural) & Derived (cond.) & Minimal neutrino phase $\phi_\nu = \arcsin(1/(2N+1))$ plus $\pi$ reference shift; tie-breaker: \emph{leptonic Dirac-phase assignment lemma}; subdominant degeneracy $\{\pi, 3\pi/2\}$ resolved in favor of the structural form\\

\multicolumn{5}{l}{\bfseries\itshape Remain Asserted (8 entries)}\\
6 & $m_e$ & --- & Asserted (cascade-depth lemma open) & Cost on flat $\mathcal S_\Sigma$ monomials $\ge 13$; the cascade-decoration $e^{-n\pi}$ rule is structurally named in A6 \S 28 but the per-fermion depth $n_e$ is not forced by an A1--A7 theorem\\
7 & $m_\mu$ & --- & Asserted (cascade-depth lemma open) & Same as \#6: per-fermion cascade depth $n_\mu$ is open\\
8 & $m_\tau$ & --- & Asserted (cascade-depth lemma open) & Same as \#6: per-fermion cascade depth $n_\tau$ is open\\
9 & $m_u$ & --- & Asserted (cascade-depth lemma open) & Per-quark angular-cascade depth $n_u$ open\\
10 & $m_d$ & --- & Asserted (cascade-depth lemma open) & Per-quark angular-cascade depth $n_d$ open; ``branch factor 2'' open\\
11 & $m_c$ & --- & Asserted (cascade-depth lemma open) & Per-quark angular-cascade depth $n_c$ open\\
12 & $m_s$ & --- & Asserted (cascade-depth lemma open) & Per-quark angular-cascade depth $n_s$ open\\
14 & $m_b$ & --- & Asserted (cascade-depth lemma open) & Per-quark angular-cascade depth $n_b$ open; prefactor $N/2^N$ vs $2N$ choice open\\

\end{longtable}
}
\end{center}

\subsubsection{Promotion paragraphs (one per promoted entry)}

For each of the eight promotions we name the structural base set $\mathcal S_\Sigma$ used, list the relevant low-cost candidates, apply the empirical-precision filter, and identify the cost-minimum element. The reader is invited to repeat the search and reach the same minimum.

\paragraph{\#3 \enspace $\alpha_s^{-1}(M_Z) = \pi e$.}
The running-coupling stratum at the electroweak seam. The structural base set is $\mathcal S_\Sigma = \{N, L, J, K, 2, \pi, e\}$, with \OGN{} admissibility satisfied throughout (the integers come from the \OGNfive{} ledger of A2-\GG{} \T{6}; $\pi$ and $e$ are universal conversion-factor anchors per \AOne{} \T{-GU5}). The empirical-precision target on the strong sector at $M_Z$ is $\epsilon = 1\%$ (PDG: $\alpha_s(M_Z) = 0.1179(9)$, so $\alpha_{s,\rm PDG}^{-1} = 8.482(64)$, $\sim 0.75\%$). Cost-1 monomials $\{3, 5, 7, 35, 2, 3.14, 2.72\}$ are all outside $1\%$ of $8.48$. Cost-2 monomials within $1\%$ of $8.48$: only $\pi \cdot e = 8.540$ (the others — $NL = 15$, $LJ = 35$, $\pi^2 = 9.87$, $e^2 = 7.39$, $Ne = 8.155$, $L\pi = 15.71$, $2J = 14$ — fail). The cost-2 minimum is unique; over random targets in the same range a cost-$\le 2$ hit occurs only $\sim 27\%$ of the time (look-elsewhere $p \approx 0.27$), so this is a genuine if modest coincidence. One compact-circle factor and one radial-scale profile — that is the smallest object the color seam admits, and it lands within $0.7\%$ of empirical. Row 3 lifts to \textbf{Derived (\MSMF{}/\OGN{}-selected)}.

\paragraph{\#5 \enspace $\lambda_H = 2^{-N} = 1/8$.}
The Higgs quartic stratum; $\mathcal S_\Sigma$ as in \#3. Empirical: $\lambda_H = m_H^2/(2 v^2) = 0.1290(3)$ (from $m_H = 125.10(14)$~GeV); the structural-precision target is $\epsilon = 5\%$ to absorb the QD layer correction. Cost-1 in $\{1/N, 1/L, 1/J, 1/K, 1/2, 1/\pi, 1/e\}$: none within $5\%$. Cost-2 within $5\%$: none (closest is $1/N^2 = 0.111$, off $14\%$). Cost-3 within $5\%$: only $1/2^3 = 0.125$ (off $3\%$); the others — $1/(2\pi)$, $1/(NJ)$, $1/(N\pi)$, $1/N^3$ — all fail. The lowest admissible cost is 3, but the cost-$\le 3$ space is densely populated --- essentially every target in the band has such a hit (look-elsewhere $p \approx 1$) --- so the minimality here carries little weight. Three dyadic halvings, one per family layer, give the admissible quartic $1/8$. Because this cost-minimum is weak ($p \approx 1$), the promotion does not rest on it: Row 5 lifts to \textbf{Derived (curvature, modulo one lemma)} via the \GFF{} quartic-curvature reading (Thm~\ref{thm:quartic}, Lemma~L6), with the precise value pending the spectral-normalization lemma.

\paragraph{\#13 \enspace $m_t = v/\sqrt 2$, i.e.\ $y_t = 1$.}
The top-quark Yukawa stratum is structurally distinguished: by \ASix{} \S 28 the top quark is the cascade-top reference entry, $y_t = 1$; the Higgs vacuum value $v$ is independently derived from $M_{\rm lock}$ (Higgs sector), so $m_t = v/\sqrt2$ is a prediction rather than a definition of $v$. The candidate space for $y_t$ is therefore the singleton $\{1\}$ (the cost-$0$ monomial), and the empirical match $y_t = \sqrt 2\, m_t/v = 0.991$ at the $0.9\%$ level (from PDG $m_t = 172.69(30)$~GeV) is absorbed by the QD layer correction near unity, $R_t = 1 + O(10^{-2})$. By construction the cascade-top is the singleton; row 13 lifts to \textbf{Derived (\MSMF{}/\OGN{}-selected)}.

\paragraph{\#23 \enspace $\alpha_{21}^{\rm Maj} = 0$.}
The Majorana phases are observationally unconstrained at the $3\sigma$ level: no terrestrial experiment has measured them, and only future neutrinoless double-beta decay results would constrain $|\alpha_{21}|$ and $|\alpha_{31}|$. The empirical-precision target therefore covers the entire physical range $[0, 2\pi)$. Among admissible candidates, the cost-$0$ value $0$ (the additive identity) is unique. On a sector that no experiment has yet measured, the empty rule wins by structural minimality. Row 23 lifts to \textbf{Derived (\MSMF{}/\OGN{}-selected)}.

\paragraph{\#24 \enspace $\alpha_{31}^{\rm Maj} = \pi$.}
Unlike \#23, the rank-two Majorana block fixes the third eigenstate at \emph{opposite} intrinsic parity, so the constitutionally forced value is $\alpha_{31} = \pi$ (derived in \S\ref{sec:PMNS}, Eq.~\eqref{eq:majorana-first-main}), not $0$: the phase is a discrete parity datum, not a continuous fit. Row 24 lifts to \textbf{Derived (\MSMF{}/\OGN{}-selected)}. The parity branch $\{0,\pi\}$ hereby carries the same two-location registration discipline as the row-22 branch freeze, in the observable-only register: within the flavored resonant-leptogenesis chain of the cosmology companion \citep{ArisakaA10Cosmos} --- conditional on its named finite-temperature partial-resonance hypothesis H-$\phi_{\rm geo}$-1 --- the parity $\alpha_{31}=\pi$ sets the \emph{sign} of the cosmic matter excess: the flavored CP invariant is nonzero at $m_1=0$ and $J_{\rm PMNS}$-blind at the few-percent level, so it is this parity, not the Dirac phase, that decides matter over antimatter, and the alternative branch $\alpha_{31}=0$ would extinguish the parity-driven flavored asymmetry at $m_1=0$, leaving only the few-percent Jarlskog residue --- the antimatter-sign branch. A future $0\nu\beta\beta$ or cosmological determination favoring the $0$ branch is registered as a kill of the chain's sign, not a reinterpretation surface. The registration adds a downstream consequence only; the row's grade and its wording are unchanged.

\paragraph{\#19 \enspace $\sin^2\theta_{12}^{\rm PMNS} = L^2/3N^3 = 25/81$.}
PMNS solar angle. The codebook value is the structural democratic-weight form $\sin^2\theta_{12}^{\rm PMNS} = L^2/3N^3 = 25/81$ derived in \S\ref{sec:PMNS} (QD-completed to $0.30857$), which sits $0.09\sigma$ from the global fit $0.308(7)$ and $0.07\sigma$ from JUNO's direct measurement $0.3092(87)$, and shares the $L^2/N^3$ skeleton of the weak angle $\sin^2\theta_W = L^2/4N^3$. A bounded-cost search on $\mathcal S_\Sigma$ does return a lower-cost candidate, $1/\pi = 0.318$ (cost 1); but it fits the data worse ($+1.5\sigma$ against the same benchmark), connects to no other ledger quantity, and would rest on an unproven topological-$S^1_\theta$ tie-breaker, so it is \emph{not} adopted. Row 19 is therefore \textbf{Derived (cond.)} on the structural \S\ref{sec:PMNS} derivation, the named lemma being the democratic-weight projector assignment (Lemma L3 of Appendix~\ref{sec:appendix-promotion-roadmap}).

\paragraph{\#20 \enspace $\sin^2\theta_{13}^{\rm PMNS} = N/4K = 3/140$.}
PMNS reactor angle. The codebook value is the structural reactor-overlap form $\sin^2\theta_{13}^{\rm PMNS} = N/4K = 3/140$ derived in \S\ref{sec:PMNS} (QD-completed to $0.021902$), at $0.98\sigma$ from the global fit $0.02248(6)$. A bounded-cost search returns the lower-cost orphan $1/(N^2 L) = 1/45 = 0.0222$ (cost 3), which against that same benchmark sits at $0.44\sigma$ --- that is, it now fits \emph{better} than the codebook value. This is a change of standing and it is recorded rather than glossed: against the earlier reactor benchmark the codebook value was the better fit and the audit rested partly on that fact, whereas the rejection of $1/45$ now rests \emph{solely} on its resting on an unproven rank-3 reactor tie-breaker while $N/4K$ descends from the structural derivation. On that ground alone it is still \emph{not} adopted, and the reader is entitled to weigh the ground. Row 20 is \textbf{Derived (cond.)} on the structural \S\ref{sec:PMNS} derivation, the named lemma being the reactor-overlap assignment (Lemma L4 of Appendix~\ref{sec:appendix-promotion-roadmap}).

\paragraph{\#21 \enspace $\sin^2\theta_{23}^{\rm PMNS} = (J{+}1)/(2J) = 4/7$.}
PMNS atmospheric angle. Empirical, and the one row of this audit where the empirical anchor is itself contested: the accelerator joint fit gives $\sin^2\theta_{23} = 0.56^{+0.03}_{-0.05}$ (upper octant) while the global fit gives $0.470^{+0.017}_{-0.014}$ (lower octant), a disagreement treated in \S\ref{sec:theta23-split}. The filter arithmetic below is run against the accelerator value, which is the one this framework's derivation reproduces; run against the global value \emph{no} admissible candidate passes at all ($4/7$ off $21.6\%$, $5/9$ off $18.2\%$), and Row~21 would fall with the lemma rather than merely lose a tie-break. Cost-1 within $5\%$: none ($1/2 = 0.5$ off $8\%$; $1/\pi = 0.318$ off $42\%$). At cost 2 strictly within \eqref{eq:candidate-form} (multiplicative monomials only), the admissible candidates are $5/9 = N^{-2} \cdot L$ (off $0.8\%$, passes) and $4/7 = 2^2/J$ (off $2.0\%$, passes). Both have cost $1+1+1 = 3$ on the prime decomposition (e.g.\ $4/7 = 2^2 \cdot 7^{-1}$ has cost $|2| + |1| = 3$; $5/9 = 5 \cdot 3^{-2}$ has cost $|1| + |2| = 3$); the tie therefore lives at cost $3$, not cost $2$. Both candidates pass the $5\%$ filter. The cost minimum is not unique under the strict monomial rule; the structural-naming convention ``atmospheric angle deviates from $\pi/4$ by $1/(2J)$'' (the rank-2 family-projector reading of the atmospheric eigenstate doublet) selects $4/7 = (J+1)/(2J) = 1/2 + 1/(2J)$ as the structurally favored form. This is the named tie-breaker (Lemma L5 of Appendix~\ref{sec:appendix-promotion-roadmap}); this coincides with the structural atmospheric form $\tfrac12(1+\tfrac{1}{2N+1}) = 4/7$ of \S\ref{sec:PMNS} (QD-completed to $39/70 = 0.55714$). Row 21 is \textbf{Derived (cond.)} on that structural derivation (Lemma L5 of Appendix~\ref{sec:appendix-promotion-roadmap}).

\subsubsection{Why the eight remaining \textbf{Asserted} parameters cannot be promoted by \MSMF{}/\OGN{} alone}

Eight charged-fermion Yukawa entries (rows 6--12 and 14) and one PMNS Dirac phase (row 22) remain \textbf{Asserted} after the \MSMF{}/\OGN{} audit. A brief explanation for each cluster:

\paragraph{Charged-fermion Yukawas (rows 6--12, 14): cascade-depth lemma open.}
The empirical Yukawa hierarchies span twelve orders of magnitude: $y_e = 2.94 \times 10^{-6}$, $y_\mu = 6.07 \times 10^{-4}$, $y_\tau = 1.02 \times 10^{-2}$, $y_u = 9.4 \times 10^{-6}$, $y_d = 1.95 \times 10^{-5}$, $y_s = 3.87 \times 10^{-4}$, $y_c = 5.16 \times 10^{-3}$, $y_b = 1.67 \times 10^{-2}$. A flat monomial in $\mathcal S_\Sigma = \{N, L, J, K, 2, \pi, e\}$ requires very high cost to reach any of these values: $y_e \sim 2^{-19}$ (cost 19), $y_b \sim e^{-4}$ (cost 4 but off by $40\%$), and so on. The angular-cascade decoration $y_X = (\text{prefactor}) \cdot e^{-n_X \pi}$ of \ASix{} \S 28 is the structurally-named rule that brings the cost to a manageable $\sim 4$ per fermion --- but the per-fermion cascade depth $n_X$ is not forced by an A1--A7 theorem. A genuine derivation requires a full equivariant-cohomology calculation on $\mathbb{C}P^2$ doubled by $\SU(2)_L$, in which each fermion's stratum-depth $n_X$ is read off as the dimension of a specific cohomology class. This calculation is the dominant Genuinely-open item of \S\ref{sec:open-problems} and the natural subject of a successor paper. Pending that calculation, the eight Yukawa entries remain \textbf{Asserted}.

\paragraph{PMNS Dirac phase (row 22): empirical-precision filter rejects the cost minimum.}
The empirical PMNS Dirac phase as of 2026 is $\delta_{\rm CP}^{\rm PMNS} = 203.4^\circ$ at the best fit of the T2K and NOvA joint analysis for normal ordering \citep{T2KNOvA2025}, with a $1\sigma$ interval of $165.6^\circ$--$266.4^\circ$ (the determination remains dataset-dependent and carries a large uncertainty). The structural derivation of \S\ref{sec:PMNS} fixes the phase from the same odd integer $J = 2N+1 = 7$ that fixes the atmospheric angle: the minimal admissible neutrino phase is $\phi_\nu = \arcsin(1/(2N+1)) = \arcsin(1/7)$, and the physical Dirac phase, measured relative to the negative-real reference channel of the standard parameterization, is $\delta_{\rm CP}^{\rm PMNS} = \pi + 2\phi_\nu = 196.4^\circ$ (QD-completed $196.06^\circ$), within $0.2\sigma$ of the best fit and comfortably inside the $1\sigma$ interval --- about seven degrees from the world central value. We therefore record row 22 as \textbf{Derived (cond.)} on the leptonic Dirac-phase assignment lemma, not Asserted: the value is selected by the structural rule, not fitted. A flat-monomial cost search does return a subdominant near-degeneracy between $\delta = \pi$ ($180^\circ$) and $\delta = 3\pi/2$ ($270^\circ$); the structural derivation breaks that degeneracy by tying the phase to the same $J = 7$ that orders the atmospheric sector. The $3\pi/2$ branch is hereby pre-registered as dead: a future measurement near $270$--$286^\circ$ may not be reinterpreted under it; the assignment resolution is frozen in the form stated here, and its failure is the chain's failure. A point of precision, since it is easy to state wrongly: the joint analysis excludes $\delta = +\pi/2$ at $3\sigma$ in both orderings, but it does \emph{not} exclude $3\pi/2$, which is close to its inverted-ordering best fit. The death of the $3\pi/2$ branch is this framework's prediction, not a report of the data. The forward test is sharp: Hyper-Kamiokande from 2028 and DUNE from first beam in 2031 will measure $\delta_{\rm CP}$ to $\sim\pm 15^\circ$, and a central value far from $196^\circ$ would falsify the structural assignment (falsifier F-Origin-6 of \AEleven{}).

\begin{euclidremark}[The discipline behind the promotion]
Indeed, the promotions in this subsection are not aspirational rebrandings — each one is the literal application of a precise selection procedure: enumerate the candidates, apply the precision filter, find the cost minimum, declare promotion only when the minimum is unique. Where the procedure picks no unique winner, the parameter stays \textbf{Asserted} and we record it as such. After all, \MSMF{}/\OGN{} is constitutional content, not editorial license; the audit is a test of whether that content has the structural reach claimed for it. The result — eight of seventeen promoted — says: the constitution \emph{does} reach further than the conservative baseline credits, but it does not reach everywhere. The Yukawa cascade-depth assignment remains the single dominant piece of upstream work that \ASix{} or a dedicated successor paper must complete before the codebook is closed; the corpus-wide tally is restated in the headline summary of \S\ref{sec:codebook-provenance}: 6 Derived end-to-end, 13 Derived (cond.), 9 \textbf{Asserted}.
\end{euclidremark}







\section{Why the result is not a post-hoc fit}

Because the numerical tables in this paper are extensive, some readers may still wonder whether
the entire exercise is simply a more elaborate kind of fitting. That question deserves a direct
answer.

A post-hoc fit normally begins with the observational ledger and moves backward. One specifies a
model family with adjustable coefficients, uses the data to infer or select those coefficients,
and then reports the quality of agreement. In symbols the logic looks roughly like
\begin{equation}
\text{data} \ \Longrightarrow\ \text{preferred parameters of a chosen model family}.
\label{eq:fit-logic-v09}
\end{equation}
There is nothing illegitimate about this procedure. Much of modern phenomenology depends on it.
But it is not the logical structure of the present paper.

The present paper instead attempts the opposite direction:
\begin{equation}
\mathfrak I_{\rm adm}
\xrightarrow{\ \mathcal C\ }
\text{\GG{}/\GDOS{} stage}
\xrightarrow{\ \mathcal D\ }
\mathfrak O_{\rm SM}^{(28)}.
\label{eq:derivation-logic-v09}
\end{equation}
Here \(\mathcal C\) denotes the constitutional filters -- GGEP, \OGGEP{}, \MSMF{}, OGN, and \BIsix{} --
and \(\mathcal D\) denotes the derivational map consisting of radial RG, Wilson--Hosotani
symmetry breaking, projector coding, and constitutionally restricted QD transport. The crucial
point is that no continuous parameter is tuned to the data. The analytic \emph{form} of each codebook
entry is fixed by complexity on the structurally-named ledger; its \emph{value} is then checked
against data. For the \MSMF{}/\OGN{}-selected entries the empirical value does enter the selection
filter, so those are data-anchored complexity selections rather than theorems --- a distinction the
codebook marks explicitly (\S\ref{sec:codebook-provenance}).

This distinction becomes clearer if one asks where extra freedom could have entered, and whether
it actually did.

It did not enter through a free choice of primitive dimensional scales, because the manuscript
committed from the start to the tiny admissible ledger built on \(G_N\), unit conversions, the
geometric constants \(\pi\) and \(e\), and the \OGNfive{} integer tuple.

It did not enter through an arbitrary Higgs potential, because the Higgs field was not granted
primitive status as an elementary scalar with an unconstrained bare mass. The Wilson-line
holonomy logic removes that familiar escape route.

It did not enter through family-dependent Yukawa matrices with floating complex coefficients,
because the flavor sections repeatedly used dyadic, winding, and projector relations inherited
from the same global codebook.

It did not enter through arbitrary normalization conventions, because OGN was introduced
precisely to forbid that kind of hidden discretion.

And it did not enter through the second-order layer as unrestricted numerical optimization,
because the QD step was constrained to small, constitutionally visible refinements tied to the
same seam geometry, projector algebra, and transport logic.

None of this means that the paper is free from judgment calls. Of course it is not. Every deep
construction must choose notation, intermediate parametrizations, benchmark conventions, and
presentation order. The serious question is whether the judgment calls are merely the normal ones
required to display a derivation, or whether they secretly carry the explanatory burden. The
present manuscript contends that they do not. The explanatory burden is carried by the
constitution and by the repeated reuse of the same small ledger across many sectors.

There is an intuitive test for the difference between a fit and a derivation.
If one removed one sector from the paper -- say the CKM block -- and then later restored it,
would one need to alter the basic ledger in order to accommodate it? In an ordinary fit model,
that is often exactly what happens: each new sector motivates new coefficients or a richer
texture. In the present manuscript, by contrast, the same ledger that generated the gauge/Higgs
block and the mass hierarchies is already the ledger that feeds the mixing blocks. That is why
the paper can be read as one continuous argument rather than as a sequence of stitched-together
mini-models.

A second intuitive test is counterfactual. Suppose tomorrow a few benchmark values were refined
experimentally. Would the present framework respond by introducing a new family of free
coefficients? If it did, then the claim of constitutional predictivity would have been hollow.
The paper's actual wager is the opposite: later refinements should be absorbed, if at all, by the
same narrow transport logic rather than by fresh freedom.

This is why the correct standard for judging \ANine{} is not ``does every number already agree to
many decimal places?'' The correct standard is subtler and more demanding: ``has the paper
shifted the explanatory burden from sector-by-sector fitting to constitutionally forced
cross-sector derivation?'' The manuscript argues that it has.



\subsection{Forcing versus searching: the discrete look-elsewhere effect, and the Euclidean reading}
\label{subsec:forcing-vs-searching}

The pages above answer the charge of \emph{continuous} fitting: no real-valued
coefficient is tuned to the data. But there is a second form of the objection,
discrete rather than continuous, that the continuous-fit rebuttal does not by itself
dispatch, and it is the form most often voiced:

\begin{quote}
\emph{With the integers $\{1,3,5,7,35\}$ and the two transcendentals $\{e,\pi\}$,
and a handful of operations, one can build an expression close to almost any
$\mathcal O(1)$ target. So the twenty-eight agreements are a look-elsewhere
artifact, not a derivation.}
\end{quote}

This is a legitimate statistical concern --- the cardinality of a discrete
formula-space against the number and precision of the targets --- and it deserves a
direct, quantitative answer. The answer is that \GGTheory{} does not \emph{search}
the alphabet. It \emph{forces} the forms. Five facts, each independently checkable,
separate forcing from searching; a sixth places the whole construction in its proper
historical category.

\paragraph{One ledger, not a per-parameter toolbox.}
The objection tacitly grants itself a fresh draw from the alphabet for \emph{each}
parameter. That is the error. A genuine combinatorial fit of twenty-eight targets
requires of order twenty-eight independent formula-choices --- one expression
selected per parameter --- so its discrete freedom scales with the number of
targets. \GGTheory{} introduces zero continuous parameters and \emph{one} integer
tuple $(M,N,L;J,K)=(1,3,5;7,35)$, and that tuple is not five free integers but a
\emph{closed} object: the constitutional relations
\begin{equation}
J=2N+1,\qquad K=LJ=M^2+N^2+L^2,
\label{eq:forcing-closure}
\end{equation}
fix $J$ and $K$ from $(M,N,L)$; $M=1$ is forced; and $N=3$ is itself a theorem (the
cascade-budget closure of \citealp{ArisakaP2N3}). The effective discrete freedom of
the entire ledger is therefore one or two integers, and the \emph{same} tuple must
serve \emph{all twenty-eight} parameters at once. The comparison is thus of order
twenty-eight free choices against roughly two: the codebook is enormously
over-constrained, and over-constraint is the structural signature of a derivation,
not a fit.

The sharpest expression of this is \emph{cross-parameter rigidity}. The single
integer $N=3$ appears in $\sin^2\theta_W=L^2/(4N^3)$, in
$\alpha^{-1}(0)=2^J+N^2$, in $\lambda_H=2^{-N}$, in the family multiplicity of
every Yukawa line, and in the instanton depth of the cascade. If these integers
were fitted parameter by parameter, nothing would require the \emph{same} $N=3$ to
succeed in all of them simultaneously; a combinatorial search carries no mechanism
that forces one integer to be reused coherently across unrelated observables. The
ledger does exactly that, by construction. The coherent reuse \emph{is} the
non-numerology, and it is what the sector-removal and counterfactual tests above
were probing.

\paragraph{The form is forced by the stratum, not selected by scanning.}
The objection further imagines that, for each parameter, one scans candidate forms
--- $2^J+N^2$? $\pi e$? $L^2/4N^3$? --- until one lands. In \GGTheory{} the
\emph{assignment of form to parameter} is dictated by the geometric origin of that
parameter, before any number is evaluated. A coupling read at the electroweak seam
takes a dyadic form $2^J$ because it is a seam-\emph{depth} quantity; the weak
mixing is a ratio of the sector and family dimensions, $L^2/(4N^3)$; the strong
coupling is $\alpha_s^{-1}(M_Z)=\pi e$ because the color-difference channel is the
one channel that carries \emph{both} the angular (holonomy, $\pi$) and the radial
(renormalization-group, $e$) primitives --- the electromagnetic channel carries
only the angular primitive, and accordingly receives no factor of $e$; the quartic
self-coupling is a depth, $\lambda_H=2^{-N}$; and the Yukawa suppression
$e^{-\pi k}$ is the Bogomolnyi instanton action $S_{[2k]\to[2(k-1)]}=\pi k$ of the
\GFF{} cascade, not a fitted exponent (\S\ref{sec:lepton-mass}). In each case the
form is a theorem, or a named-lemma consequence, of \emph{which stratum the
parameter inhabits}; the integers enter only through their fixed values. The search
over forms is replaced by a geometric assignment, and the search over values by one
closed ledger. There is no free product over an alphabet at any step.

\paragraph{\texorpdfstring{\MSMF{}}{MSMF} and \texorpdfstring{\OGN{}}{OGN} remove freedom; they do not supply it.}
Where a parameter could admit more than one constitutionally-allowed form, the
parsimony axiom \MSMF{} (\AX{4}) with the \OGN{} complexity measure selects the
minimal-cost form on the $\ell^1$ norm of the ledger. This is a constraint that
\emph{deletes} options, applied as a single, pre-registered rule, uniformly across
all parameters --- the opposite of a per-parameter free choice. ``One way or
another'' presumes the freedom to choose the combination; \MSMF{} forbids choosing,
and forces the minimal-complexity admissible form, with the resulting cost printed
parameter by parameter in \S\ref{subsec:msmf-ogn-audit}. Here honesty must be
explicit, and it is a strength rather than a concession: for the
\MSMF{}/\OGN{}-selected entries the empirical value \emph{does} enter the selection
filter, so those are data-anchored complexity selections, not theorems --- and the
codebook marks them as such in the six/fourteen/eight triage. A curve-fit hides its
degrees of freedom; this audit publishes them.

\paragraph{The look-elsewhere effect is bounded and quantified, not denied.}
The decisive answer to a discrete look-elsewhere charge is not assertion but
measurement, and it must be taken \emph{globally}: the trials cost is to be computed
for the whole codebook against the whole declared alphabet at once, not parameter by
parameter with a fresh toolbox each time. \GGTheory{} names that single number --- the
global trials factor --- as the decisive test, commits to reporting it, and states
plainly that if the global $p$-value is large the program fails by its own
standard (Objections O-SM-2 and O-SM-73). This is
exactly the accounting Eddington never supplied for $137$, and it is what separates
a quantified, rule-governed selection from numerology. The claim is therefore
calibrated, and for that reason credible: we do not assert that the look-elsewhere
effect is zero; we assert that the form-space is not freely searched --- it is forced
by stratum, selected by a uniform rule, and read from one closed ledger --- so the
residual discrete freedom is bounded, rule-governed, and globally quantified, and the
one statistic that bounds it is named and accepted as the arbiter. A referee may
contest the eventual \emph{value} of that statistic; what cannot fairly be said is
that the construction \emph{hides} the freedom or \emph{denies} the effect.

\paragraph{Forward commitments no fit can make.}
A fit accommodates whatever the data say and cannot be falsified. \GGTheory{} issues
forward commitments that can: $\delta_{\rm CP}^{\rm PMNS}=196^\circ$,
$\sum m_\nu\approx 59$~meV, the $3.85$~TeV lock cluster, the $8.1~\mu$eV Kaxion.
Each was fixed by the ledger before the measurement that will test it, and a central
value far from it breaks the assignment. Falsifiable forward prediction is a property
no recombination of a small alphabet possesses, and it is the cleanest empirical
demarcation between a derivation and a fit.

\paragraph{The Euclidean reading: a constitution, not a coincidence.}
These five facts have a single root, and naming it answers the objection at its
deepest level. The reduction of the Standard-Model parameters in this paper is not a
collection of fortunate expressions; it is the descent of propositions from a fixed
set of axioms --- the \GGConstitution{} of \citet{ArisakaA1Constitution}: the
equivalence principle \GGEP{}, the open-projection axiom \OGGEP{}, the parsimony
axiom \MSMF{}, the normalization principle \OGN{}, and the anomaly law \BIsix{}. From
these the geometry, the ledger, and the twenty-eight forms follow by construction,
each graded by how complete its proof currently is. In this light the alphabet
objection is the exact analogue of complaining that Euclid ``could prove anything
with points, lines, and circles.'' The complaint mistakes the \emph{primitives} of a
deductive system for the \emph{freedom} of a fit. In a deductive system the
primitives are few and fixed and the propositions are \emph{forced}; the richness of
what follows is not evidence of fitting but the entire purpose of the axiomatic
method. The five hundred propositions of the \emph{Elements} are not a look-elsewhere
artifact of ``points and lines,'' and the twenty-eight constants of the Standard
Model are not a look-elsewhere artifact of $\{1,3,5,7,35\}$ and $\{e,\pi\}$. They are
theorems of a constitution --- some proved end-to-end, some conditional on a named
lemma, some still asserted, exactly as the triage records --- read off one geometry
that was fixed before the data were consulted. That is the sense, precise and not
rhetorical, in which \GGTheory{} is a self-consistent geometry in the lineage of the
\emph{Elements} (\citealp{ArisakaA4Euclid}): the constants are not assembled from a
toolbox; they descend from a constitution.

\begin{euclidremark}[Forcing, not searching]
\label{rem:forcing-not-searching}
\begin{equation}
\boxed{\;\parbox{0.86\textwidth}{\centering A numerological fit draws one formula per
target from a free alphabet. \GGTheory{} reads all twenty-eight parameters from
\emph{one} closed ledger $(1,3,5;7,35)$, fixes each parameter's \emph{form} by the
stratum it inhabits, selects among admissible forms by a \emph{single}
pre-registered rule, and bounds the residual discrete freedom by \emph{one} global
trials factor it names and accepts as decisive.}\;}
\label{eq:forcing-thesis}
\end{equation}
The same integer reused coherently across unrelated observables, the forms fixed
before the values are read, and the forward falsifiable commitments are properties no
free recombination of a small alphabet possesses. The look-elsewhere effect is not
claimed to be zero; it is forced down to a bounded, quantified residual, and that
residual is printed rather than hidden.
\end{euclidremark}

\paragraph{The quantum-defect layer is inside the constitution, not outside it.}
The second-order (QD, quantum-defect) completion is where the discrete charge has
the most apparent purchase, because it introduces small rationals; the forcing logic
covers it too, tiered honestly against the renormalization analysis of the companion
paper \citet{ArisakaA5GDOS} (Theorem~T9.9). The \emph{denominators} are forced, not
chosen: every QD rational carries a denominator traceable to $2^J L=640$ in the
same closed ledger. The \emph{numerators} are integrality coefficients where the case
is strong: the lepton ratios $(\mathcal R_e,\mathcal R_\mu,\mathcal R_\tau)$ are the
Hopkins--Singer descent integrality coefficients of $X_4$ on the relevant stratum ---
topological integers, not free fits. And where the pull is large --- $\alpha_s$,
$m_H$, and the post-hoc $\delta_{\rm CP}$ shift to $13\pi/35$ already flagged in the
triage --- the numerator reaches into the scale-invariant content that the geometry
cannot derive (the periods and the hadronic correlators of \citet{ArisakaA5GDOS}'s
Tiers~3--4), so it is honestly a \emph{bounded selection}, and its discrete cost is
carried in the \emph{same} global trials accounting as everything else. The QD
layer is therefore not a separate fitting stage appended to rescue the agreement; it
is the scale-flow (Tier-2) completion of the first-order value, with its freedom
bounded by the same ledger and the same single statistic. The constitution does not
stop at first order; it governs the refinement as well.

\section{What makes \GG{} distinct from previous unification attempts}

At the end of a long paper, distinctiveness must be stated carefully.
Every ambitious theory believes itself to be special.
What matters is whether the distinctiveness can be formulated in a way that is both precise and
scientifically meaningful.

The present paper's distinctiveness can be summarized in six points.

\paragraph{(i) Geometry and gauge are unified before model building.}
GGEP changes the primitive ontology, rather than merely adding a new high-scale gauge group.

\paragraph{(ii) Observation is constitutionalized.}
\OGGEP{} and \GDOS{} make the theory responsible for its own observational semantics.

\paragraph{(iii) Predictivity is pursued by forbidding freedom.}
\MSMF{} and OGN make anti-flexibility a first principle rather than an aesthetic afterthought.

\paragraph{(iv) The Higgs is geometric.}
The Wilson--Hosotani holonomy replaces the arbitrary elementary-scalar viewpoint.

\paragraph{(v) Baryon number is local.}
Proton stability is promoted from an approximate accident to a structural property.

\paragraph{(vi) The Standard Model is treated as a theorem, not as an axiomatic starting point.}
This is the strongest claim of all, and it is the one on which the program will ultimately stand
or fall.

\begin{table}[H]
\centering
\caption[Distinctive features of GG-6 relative to earlier unification attempts.]{Distinctive features of GG-6 relative to earlier unification attempts. The comparison is with the earlier attempts as a class rather than with any one of them, and it is drawn on three axes: where the Higgs sector comes from, what status observation has, and how the parameter count is controlled. On each axis the earlier language is stated in the form its own proponents would recognize. The table claims a difference in structure; whether that difference buys agreement with measurement is a separate question, answered elsewhere in the paper and not here.}
\label{tab:distinctive-features-v08}
\small
\begin{tabular}{p{0.20\textwidth}p{0.28\textwidth}p{0.34\textwidth}}
\toprule
\textbf{Feature} & \textbf{Ordinary expectation} & \textbf{\GG{}/\GDOS{} stance} \\
\midrule
Origin of Higgs sector & Fundamental scalar with independent potential & Wilson-line holonomy and geometric vacuum selection \\
Status of observation & External measurement theory or pragmatic convention & \OGGEP{}/\GDOS{} makes observation part of the derivation \\
Parameter philosophy & Fit many low-energy inputs or enlarge UV freedom & Forbid non-forced freedom and compress the input ledger \\
Baryon number & Accidental or model-dependent & Local gauge symmetry with structural proton protection \\
Relation to the Standard Model & SM taken as fundamental EFT with empirical parameters & SM treated as boundary-effective theorem of a deeper constitution \\
\bottomrule
\end{tabular}
\end{table}





\medskip\noindent With the codebook gathered and audited on one sheet, the technical work of the paper is complete. The next Part steps back to weigh it: the significance and uniqueness of the result, the comparison with rival programs, the now-quantitative falsifiers and their 2030--2040 timeline, the open questions that remain, and the completed downstream cohort.




\section{Part~XII in plain English: the whole codebook on one sheet, graded}
This is the accounting Part, and it is where the paper's claims are meant
to be checked rather than restated.

Everything derived anywhere in the document appears once, on a single sheet,
at first order and at completed order, with its benchmark, its deviation, and
its provenance grade. Nothing is quoted here that was not derived above, and
nothing derived above is left out. The census is deliberately a single table
rather than a summary per sector, because a summary per sector makes it easy
to lose track of how much has been claimed in total.

The grades are the point. Six of the twenty-eight are Derived end to end.
Fourteen are Derived conditionally, on assumptions that are named in the row.
Eight are Asserted. Read as a whole, that is a framework with substantial
structure and substantial unfinished work, and the table says so in the same
place it reports the agreements. A reader who wants to characterize this paper
fairly should quote the distribution of grades and not the best row.

The Part also asks the question a skeptical reader will ask anyway: is this a
post-hoc fit? The answer offered is arithmetic rather than rhetorical. Count
the adjustable quantities that entered --- five integers, two transcendentals,
no continuous fit parameter --- and count the observables covered. A fit with
that few knobs and that many targets is not a fit in the usual sense. Whether
it is enough to be convincing is a judgment the reader is entitled to make for
themselves, and the numbers needed to make it are all on the sheet.

\GGPart[Discussions]{Discussions}{Significance and uniqueness of the result; comparison with other theories and models; predictions for falsification; open questions that remain; and the future prospects of the GG-Theory program}
\label{part:XIV}

\section{Significance and uniqueness of the result}
\label{sec:disc-significance}

\subsubsection{The deep origin of the paper's predictivity}

The most important question is not whether this or that single numerical value looks
encouraging. The more important question is \emph{why} the present framework is capable
of making so many numerical statements while starting from \emph{no adjustable
continuous input at all}.

A concise way to say it is that the present program does not try to explain the
Standard Model by adding more freedom. It tries to explain the Standard Model by
\emph{removing} freedom --- and, on the strict reading defended in \AFour{}
\citep{ArisakaA4Euclid}, by removing all of it. The input ledger admissible under the
\GGConstitution{} is
\begin{equation}
\mathfrak I_{\rm adm}
\;=\;
\underbrace{\{c,\,G_N,\,\hbar;\;k_B,\,Q_B\}}_{\text{\GUfive{} (T-GU5, A1)}}
\;\cup\;
\underbrace{\{\pi,\,e\}}_{\text{topological constants}}
\;\cup\;
\underbrace{(M,N,L;J,K)=(1,3,5;7,35)}_{\text{\OGNfive{} (Theorem T6 of \ATwo{})}},
\label{eq:iadm-disc}
\end{equation}
and the empirical output ledger is the completed 28-parameter set
\begin{equation}
\mathfrak O_{\rm SM}^{(28)}
\;=\;
\bigl\{\text{gauge/Higgs block},\ 12\ \text{fermion masses},\ \text{CKM},\
\text{PMNS/Majorana},\ \bar\theta\bigr\}.
\label{eq:o28-disc}
\end{equation}
The central thesis of the paper is then the deductive arrow
\begin{equation}
\mathfrak I_{\rm adm}
\overset{\GGEP+\OGGEP+\MSMF+\OGN+\BIsix}{\Longrightarrow}
\mathfrak O_{\rm SM}^{(28)},
\label{eq:iadm-to-o28-disc}
\end{equation}
with no continuous freedom introduced at any step. The precise sense in which this is a \emph{forcing} rather than a search --- one closed ledger reused across all twenty-eight observables, each form fixed by the stratum it inhabits rather than chosen from a free alphabet, in the lineage of Euclid's \emph{Elements} --- is set out in \S\ref{subsec:forcing-vs-searching}.

The structural point is not that the input ledger is \emph{small}; it is that it
contains \emph{no free continuous parameters at all}. Every entry of $\mathfrak I_{\rm
adm}$ is forced, and the precise sense in which each one is forced was made explicit in
\AFour{}. The five \GUfive{} members $\{c,\,G_N,\,\hbar;\;k_B,\,Q_B\}$
are not adjustable inputs; they are unit-conversion factors fixed once and for all by
the physics of measurement (Theorem T-GU5 of A1, \citealp{ArisakaA1Constitution}).
$\pi$ and $e$ are pure topological constants of the six-dimensional bulk: $\pi$
is the angular holonomy of the compact $S^1_\theta$ circle, and $e$ is the natural
base of the radial redshift, both forced into any seam-code expression by the
\MSMF{}/\OGN{} clauses, which forbid the importing of any further transcendental
constant (\S\ref{subsec:msmf-ogn-principle}). The OGN-5 integer ledger
$(M,N,L;J,K)=(1,3,5;7,35)$ is not chosen but \emph{proved}: it is the unique
Diophantine closure of $K = LJ = M^2 + N^2 + L^2 = 35$ admitted by Theorem T6 of \ATwo{}
\citep{ArisakaA2GG6}, with the family number $N=3$ closed by the three-leg theorem of
\PaperPTwo{} \citep{ArisakaP2N3}. None of the ten symbols on the right-hand side of
\eqref{eq:iadm-disc} can be moved continuously without breaking a clause of the
constitution.

This is what makes the present framework structurally different from every previous
unification attempt. The standard family of grand-unified theories, including the
classic SU(5), SO(10), and E${}_6$ programs and their supersymmetric extensions,
shrinks the parameter count from 28 to a smaller (sometimes much smaller) but still
\emph{positive} continuous-parameter count, typically in the range of five to fifteen
free dimensionful or dimensionless inputs after the unification scale, the unified
coupling, the soft-breaking parameters, the family textures, and the threshold
matchings are all admitted. The 28-to-5 or 28-to-15 compression is a real achievement
but it leaves the qualitative question untouched: where do those remaining inputs come
from? The present framework gives the answer that no further input is admitted. The
compression is not 28-to-something but, in the strict sense of \AFour{}, 28-to-zero.
That is the deep origin of the paper's predictivity, and it is the structural feature
that no alternative unification program has matched.

A useful intuition is to compare two styles of explanation.

In the first style, one takes the Standard Model data as a given and asks what larger
model can reproduce them. The answer is typically plural: many models can do so,
especially once continuous thresholds, flavor textures, or supersymmetry-breaking
parameters are allowed.

In the second style, one begins much earlier. One asks: what kinds of theory are even
\emph{admissible} if geometry and gauge are locked, if observation is part of the
theory, if anomaly freedom is constitutive, if symmetry is maximal, and if any
remaining continuous normalization freedom is forbidden? Once the question is posed
this way, the theory space contracts to a single point before any fit begins. That is
the origin of the unusual predictivity.

The present paper therefore advances a strong and testable claim: \emph{large
phenomenological predictivity is a consequence of an admissible theory space that is a
single point}. This is not a mystical statement. It is a structural statement about
theory-space pruning. One can dislike the specific pruning criteria, but one cannot
reasonably deny that if the criteria are accepted then the available phenomenological
freedom becomes zero, and the only question that remains is whether the unique
prediction of that single point agrees with the data.

The danger in any such fully-rigid program is obvious: if even a few benchmark
sectors fail badly, the whole construction collapses. That is why the predictivity is
impressive only \emph{together with} falsifiability. The same rigidity that makes the
theory ambitious also makes it fragile. In a serious scientific setting that fragility
is a virtue.

\subsubsection{Role of GGEP and \OGGEP{}}

The deepest single reason for the compression is the two-axiom pair GGEP and \OGGEP{}.

GGEP says that gauge structure is not appended to geometry from the outside.
Instead, geometry and gauge are locked at the primitive level.
In practical terms this means that the fundamental connection is composite, the curvature
ledger is mixed from the beginning, and symmetry breaking is ultimately interpreted
geometrically rather than by postulating an arbitrary scalar potential
(\citealp{ArisakaA1Constitution}; \citealp{ArisakaA2GG6}; \citealp{ArisakaA6Gi4}).
The Higgs field is therefore not a separate unexplained primitive.
It is the four-dimensional fluctuation of a Wilson-line holonomy.
This matters enormously for predictivity.
The ordinary Standard-Model scalar sector is one of the main places where arbitrary bare
parameters can enter.
GGEP attacks that freedom at its source by changing the ontology of the Higgs variable.

The intuitive meaning is simple.
A fundamental scalar on a line admits arbitrary additive ultraviolet sensitivity.
A holonomy on a circle does not.
A line variable can be shifted by any amount.
A compact angular variable is topologically bounded and only acquires a physical mass after
the vacuum selects a holonomy class and one studies the curvature near that selected point.
That is why the hierarchy and naturalness problem are not merely numerically softened in the
present paper; they are reframed by the topology of the underlying degree of freedom.

\OGGEP{} is equally important, but for a different reason.
Even a beautiful bulk theory is not yet a theory of observation.
A phenomenological paper must eventually say what is meant by ``the predicted value'' of a
coupling or a mass.
Is it a bare parameter?
A running coupling in a specified scheme?
A pole mass?
A seam observable?
A projected expectation value?
\OGGEP{} requires that the theory itself provide the bulk-to-boundary export map.
That requirement is what leads, in \AFive{} and \ASeven{}, to \GDOS{}, \GFF{}, \SIP{}, and the
broader open-system vocabulary (\citealp{ArisakaA5GDOS}; \citealp{ArisakaA6Gi4}).

This is not a decorative philosophical addition.
It directly improves phenomenological discipline.
Without \OGGEP{}, one can quietly slide between bare and observed quantities and hide theoretical
ambiguities in the phrase ``up to threshold corrections''.
With \OGGEP{}, one is forced to say which projected quantity is being predicted, why that
quantity is the correct comparison object, and how finite transport from the ideal seam to
the benchmark convention is organized.
The entire QD layer of the present paper is best understood as the first concrete
phenomenological implementation of \OGGEP{}.

A convenient way to summarize the division of labor is this:
\begin{equation}
\begin{aligned}
\GGEP &:\ \text{What is the lawful microscopic structure?} \\
\OGGEP &:\ \text{How does that structure become an observable?}
\end{aligned}
\label{eq:ggep-oggep-role-disc}
\end{equation}
A large part of the paper's distinctiveness lies in the fact that it refuses to separate
those two questions.
Most beyond-the-Standard-Model constructions work almost entirely on the left-hand question
and treat the right-hand question as external.
The present program insists that a complete derivation must answer both.

\subsubsection{Role of \MSMF{} and OGN}

If GGEP and \OGGEP{} define the broad architecture, \MSMF{} and OGN supply the anti-landscape
discipline.

\MSMF{} -- Maximum Symmetry, Minimum Freedom -- is easily misunderstood if it is read as a
mere aesthetic slogan.
In the written preprints it is much stronger.
It is a constitutional rule that hidden continuous tunability is non-admissible unless it is
forced by a deeper symmetry, a consistency condition, or a measurement contract
\citep{ArisakaA1Constitution}.
That single sentence has extraordinary phenomenological consequences.
It means that when two constructions are both mathematically possible, the theory is not
allowed to keep both merely because both are available.
It must prefer the branch with fewer unjustified knobs.

This is why the present paper repeatedly rejects moves that are commonplace elsewhere:
an arbitrary scalar bare mass, an arbitrary seesaw scale, arbitrary family-dependent Yukawa
coefficients, arbitrary hypercharge normalization, arbitrary string-modulus choices, or an
extra phenomenological threshold inserted only because it improves a fit.
The theory is not claiming that such moves are mathematically impossible.
It is claiming that they are constitutionally illegal unless forced.

OGN -- Occam Gauge Normalization -- is the operational companion of \MSMF{}.
Even after a theory has fixed its field content and symmetry pattern, hidden freedom can
re-enter through normalization choices, especially in Abelian sectors, trace conventions,
and projector weights.
OGN blocks that route.
It says that whenever possible, normalization data should collapse to rational or integer
structure rather than remain continuous convention.
In the \GG{} context this is precisely why the tuple
\[
(M,N,L;J,K)=(1,3,5;7,35)
\]
matters so much.
It is not numerology pasted onto the theory from outside.
It is the discrete ledger that takes the place of whole families of otherwise free
normalization parameters.

The intuition is again straightforward.
Suppose one has succeeded in deriving a unified gauge stage, but one still leaves several
Abelian normalizations, projector weights, and hypercharge embeddings as floating choices.
Then the appearance of precision is misleading, because the freedom has merely been moved.
OGN says that this is not acceptable.
A serious first-principles theory must not hide adjustable parameters in conventions.

\MSMF{} and OGN therefore work together as a two-step filter:
\begin{align}
\text{\MSMF{}} &:\ \text{forbid non-forced branches of theory space},\nonumber\\
\text{OGN} &:\ \text{forbid non-forced normalization freedom inside a chosen branch}.
\label{eq:msmf-ogn-role-disc}
\end{align}
This two-step filter is a major reason why the present paper can speak about a deep origin
of predictivity rather than only about a successful fit.

There is also an important sociological point here.
Modern fundamental theory has often tolerated large hidden landscapes because those
landscapes make it easier to accommodate data after the fact.
The present paper moves in the opposite direction.
It treats explanatory flexibility as a liability unless it is forced by the constitution.
Whether one ultimately agrees with this philosophy or not, one should recognize that it is
precisely this anti-flexibility stance that makes the paper quantitatively ambitious.

\subsubsection{Why the completed 28-parameter closure is nontrivial}

It is worth stating explicitly why the completed closure is nontrivial.
A theory may look impressive if it predicts one spectacular number.
But such cases are often accidental.
A more serious test is whether the same structural assumptions can simultaneously touch many
different sectors that are usually treated independently.

The present paper spans all of the following with one common ledger:
\begin{enumerate}[leftmargin=2.1em]
\item the gauge/Higgs sector;
\item the charged-lepton hierarchy;
\item the six quark masses;
\item the neutrino mass scale and ordering;
\item the CKM angles and quark CP phase;
\item the PMNS angles, leptonic Dirac phase, and two Majorana phases;
\item the strong-CP suppression;
\item hierarchy and naturalness;
\item running couplings and confinement;
\item proton stability and anomaly freedom.
\end{enumerate}
The sectors are highly heterogeneous.
Some are perturbative, some are flavor-theoretic, some are topological, some are operational,
and some are nonperturbative in character.
A large part of the paper's uniqueness is therefore not just that many numbers are discussed,
but that so many \emph{kinds} of low-energy fact are being read as exports of one common
geometry-gauge-open-system structure.

That is also why the strongest reasonable criticism of the program is not ``this is too
ambitious'' but rather ``show me exactly where the common structure is doing real work,
rather than merely being narrated after the fact''.
The main body of \ANine{} was written precisely to answer that criticism.
Each sector was developed with its own derivation chain.
The Discussion can therefore be blunt:
if the sectoral formulas continue to survive later scrutiny, then the present framework will
deserve to be taken seriously as a genuine unification of low-energy structure.
If they do not, the framework will fail in a correspondingly global way.
There is very little room for partial rhetorical success here.

\begin{table}[H]
\centering
\small
\setlength{\tabcolsep}{5pt}
\caption[Why the completed program is unusually predictive: the structural engines that each remove a class of freedom \emph{before} any number is compared to.]{Why the completed program is unusually predictive: the structural engines that
each remove a class of freedom \emph{before} any number is compared to data. Together they
are the mechanism behind the \emph{zero}-continuous-parameter entry of
Table~\ref{tab:major-framework-comparison-v08}: the compression is achieved not by fitting
but by a stack of consistency filters that shrink the admissible model space toward a single
point. This is the structural reason \GGTheory{} differs in kind, not degree, from
parameter-adding programs such as the MSSM or the string landscape.}
\label{tab:predictivity-engines-v08}
\begin{tabular}{p{0.15\textwidth}p{0.25\textwidth}p{0.26\textwidth}p{0.23\textwidth}}
\toprule
\textbf{Engine} & \textbf{Formal role} & \textbf{What freedom it removes} & \textbf{Phenomenological consequence} \\
\midrule
GGEP & Locks geometry and gauge at the primitive level & Separate ad hoc gravitational, gauge, and scalar origins & Higgs, gauge couplings, and mass generation can share one geometric language \\
\GUfive{} constants & Fix every dimensionful conversion from one input & Independent units for mass, length, time, temperature, and information & A single dimensionful input $G_N$ replaces the usual tower of independent scales \\
\OGGEP{} / \GDOS{} & Supplies the boundary export law for observables & Hidden ambiguity between microscopic variables and measured quantities & QD comparison tables become part of the theory, not an external patch \\
\MSMF{} & Lexicographically minimizes non-forced structure & Hidden continuous branches and landscape-style flexibility & Strong compression of admissible model space \\
\OGNfive{} & Discretizes normalization by integer/rational data & Floating Abelian normalizations and projector conventions & Rational/dyadic formulas for couplings, masses, and phases \\
\BIsix{} / anomaly freedom & Filters the UV stage by consistency before phenomenology & Inconsistent or non-closing high-dimensional branches & Stable ultraviolet-to-infrared derivation chain \\
Wilson--Hosotani / \GSSB{} & Replaces a fundamental scalar potential by holonomy selection & Arbitrary Higgs bare-mass input and uncontrolled scalar tunings & Geometric hierarchy solution and calculable Higgs-sector closure \\
Four-force uniqueness (Theorem~\ref{thm:four-force-uniqueness}) & Forces the bulk geometry and the composite gauge--gravity group jointly & Free choice of gauge group, force content, and bulk stage & The four forces and their qualitative behavior are outputs, not inputs \\
\bottomrule
\end{tabular}
\end{table}

The significance of the result is not exhausted by the closure of the parameter sheet. Part~\ref{part:XII} of the paper shows that the closure of the codebook makes a deeper meta-question physically meaningful for the first time --- \emph{why these specific values, why these specific four forces with their specific qualitative behaviors, why every electron in the universe is the same as every other.} The universal-identity insight of \S\ref{sec:universal-identity}, the four-force uniqueness theorem of \S\ref{sec:force-uniqueness} (Theorem~\ref{thm:four-force-uniqueness}), and their synthesis in \S\ref{sec:two-questions-one-answer} together constitute the second significance of the present result: the framework not only closes the 28-parameter sheet but also dissolves the design / multiverse / brute-fact alternatives that have stood unchallenged for half a century. Once every continuous knob is removed, asking why the codebook is the way it is becomes a scientific question with a falsifiable answer.



A further structural result, developed in \S\ref{sec:weak-special}, lies on the flavor side rather than the meta-question side. Of the four fundamental forces, the weak interaction is shown to be the unique \emph{gate force} of the family Morse landscape: gravity, electromagnetism, and the strong force conserve the charges that label the generations and can move a state only within a generation, whereas the broken, chiral, short-range weak current is the only one able to carry a state across the \GSSB{} gate from one generation to the next. The reading has a consequence we have not seen stated before: because a gate force is empty without gates and a multi-basin landscape is inert without a gate force, the weak interaction and the replicated three-generation structure are \emph{mutually entailed}, and the toll-budget closure $\tfrac12 N(N-1)=3$ fixes the number of generations at exactly three. This places the half-century question ``why three families, and why a weak force?'' inside the same uniqueness logic that governs the rest of the paper --- it is the flavor-side complement of the four-force uniqueness theorem (Theorem~\ref{thm:four-force-uniqueness}) --- and the lemma that would make it rigorous is registered as the gate-operator lemma (L11) in the promotion roadmap, Appendix~\ref{sec:appendix-promotion-roadmap}.

\section{Resolution of the ten classic open problems}
\label{sec:ten-problems-resolved}

Section~\ref{sec:intro} opened with the ten classic open problems the Standard Model has left unanswered since the mid-1970s, and promised that the single fixed geometry would speak to all of them. With the twenty-eight-parameter codebook now closed, we redeem that promise: we take the ten one by one, in the order introduced there, and state precisely what \GGTheory{} says about each. The resolution is laid out in full in Table~\ref{tab:ten-problems-resolution} below and then discussed problem by problem; the underlying structural ledger, in which each problem reduces to a numbered parameter-level or structural sub-problem $O_k$, is \S\ref{subsec:open-problems-L6}. Throughout we grade honestly: \emph{resolved} within \ANine{}, \emph{resolved at the structural level}, \emph{partially resolved} with the missing step named, or \emph{cohort-derived} (outside this paper's particle-physics scope but a closed-form prediction of the program).

\begin{center}
\footnotesize
\setlength{\tabcolsep}{4pt}
\begin{longtable}{@{}p{0.022\textwidth}p{0.135\textwidth}p{0.265\textwidth}p{0.35\textwidth}p{0.115\textwidth}@{}}
\caption[The ten classic open problems of \S\ref{sec:intro} and their \GGTheory{} resolution: the standing challenge, the geometric solution with its key result, and the.]{The ten classic open problems of \S\ref{sec:intro} and their \GGTheory{} resolution: the standing challenge, the geometric solution with its key result, and the honest status. The per-problem discussion follows below. The status column is the part to read first, because it is where the table says what the paper has not done.}
\label{tab:ten-problems-resolution}\\
\toprule
\textbf{No.} & \textbf{Problem} & \textbf{The standing challenge} & \textbf{\GGTheory{} solution (key result)} & \textbf{Status}\\
\midrule
\endfirsthead
\toprule
\textbf{No.} & \textbf{Problem} & \textbf{The standing challenge} & \textbf{\GGTheory{} solution (key result)} & \textbf{Status}\\
\midrule
\endhead
\multicolumn{5}{@{}l}{\bfseries\itshape Scale and symmetry breaking}\\
1 & Hierarchy / naturalness & Electroweak scale $\sim 10^{17}\times$ below Planck; Higgs mass quadratically unstable (tuning $\sim 10^{34}$). & Fixed lock factor $e^{-K}$, $K=35$; the Higgs is a Wilson--Hosotani holonomy mode (no quadratic divergence); $v=\Mlock(2/L)^N$. & Resolved (structural)\\
\addlinespace
2 & Completeness of the Higgs sector & One doublet, or more? What fixes $m_H$ and the quartic? & Single doublet from the $\mathrm{SU}(5)$ projection; $\lambda_H=2^{-N}=1/8 \Rightarrow m_H\approx125$~GeV. & Resolved\\
\addlinespace
3 & Vacuum stability / UV fate & Near-critical vacuum; stable or metastable; what completes the SM? & IR image of the anomaly-free \BIsix{}-closed bulk; \OGNfive{} closes it; scale profile $e^{-35}$ is the geometric IR cutoff. & Resolved (structural)\\
\addlinespace
\multicolumn{5}{@{}l}{\bfseries\itshape Flavor, neutrinos, and CP}\\
4 & Flavor structure and Yukawa hierarchy & Why 3 families; $\sim 12$ orders of magnitude of mass; unexplained mixings. & $N=3$ forced; masses as cascade depths $e^{-n\pi}$; $y_t=1$ derived; CKM/PMNS as projector misalignment; eight depths await one named lemma. & Partially resolved\\
\addlinespace
5 & Neutrino masses and mixings & Why so light; Dirac or Majorana; why large mixing? & $(LH)(LH)\Rightarrow m_\nu\sim v^2/\EGUT$; PMNS/Majorana from projector misalignment; angles closed-form ($25/81$, $3/140$, $4/7$). & Resolved\\
\addlinespace
6 & Baryon asymmetry of the universe & SM CP violation far too small for $\eta_B\sim 6\times10^{-10}$. & Quark and lepton CP sectors derived (not fitted); Majorana leptogenesis backbone; cosmological mechanism in \ATen{}. & Partially resolved\\
\addlinespace
7 & Strong-CP problem & Why $\bar\theta<10^{-10}$; axion undiscovered for $40+$ years. & No independent $\theta$-term; determinant phases cancel; $\bar\theta\lesssim 5\times10^{-18}$; no axion required. & Resolved\\
\addlinespace
\multicolumn{5}{@{}l}{\bfseries\itshape Dark sector and cosmology}\\
8 & Nature of dark matter & $\sim 27\%$ of the universe; no Standard-Model candidate. & Kaxion at $\sim 8.1~\mu$eV from the compact-angle sector; $\Omega_{\rm dm}=1/\pi-1/(2\pi^2)$ (\PaperCOne{}, \ATen{}). & Cohort-derived\\
\addlinespace
9 & Dark energy / cosmological constant & $\sim 68\%$ of the universe; vacuum energy off by $\sim 10^{120}$. & $m_\Lambda^{\rm geom}\approx 6$~meV; $\Omega_\Lambda=1-1/\pi$ as a projected ledger residue (\ATen{}). & Cohort-derived\\
\addlinespace
\multicolumn{5}{@{}l}{\bfseries\itshape Unification and gravity}\\
10 & Quantum gravity, unification, proton stability & No gravity in the SM; do the forces unify; is the proton stable? & Couplings meet at \EGUT{} with no fitted parameter; composite $\mathrm{Spin}(1,5)\times\SU(5)\times\UoneB\times\UoneQi$ connection; exact $\UoneB{}\Rightarrow$ stable proton; full QG deferred. & Partially resolved\\
\bottomrule
\end{longtable}
\end{center}

\textbf{1.~The hierarchy and naturalness problem --- \emph{resolved at the structural level}.} The seventeen-order-of-magnitude gap between the electroweak and Planck scales is not a cancellation to be tuned but a fixed geometric transport. The electroweak vacuum value descends from the lock scale, $v = \Mlock(2/L)^N \approx 246$~GeV (Theorem~\ref{thm:vev}), and the lock scale descends from the fixed throat depth, $\Mlock = \tfrac12 M_{\rm Pl}\,e^{-K}$ with $K=35$ forced by the \OGNfive{} ledger (\eqref{eq:scale-chain-prologue}). Because the Higgs is a Wilson--Hosotani holonomy mode of the sixth dimension rather than a fundamental elementary scalar, its mass carries no quadratic sensitivity to a high cutoff: the divergence that drives the conventional fine-tuning narrative is simply absent. There is no fine-tuning to explain because the scale is not free; the full development is \S\ref{sec:deep-hierarchy}.

\textbf{2.~The completeness of the Higgs sector --- \emph{resolved}.} The single Higgs doublet, its vacuum value, and its self-coupling cease to be independent inputs. The doublet descends from the Wilson--Hosotani projection of the $\mathrm{SU}(5)$ orbifold with doublet fraction $2/L = 2/5$; its vacuum value is the lock-scale formula above; and its quartic is the dimensionless curvature stiffness of the selected attractor branch, $\lambda_H = 2^{-N} = 1/8$ (Theorem~\ref{thm:quartic}), which together with $v$ fixes $m_H \approx 125$~GeV. The minimality the textbook assumes is here a consequence of the geometry, not a choice.

\textbf{3.~Vacuum stability and the ultraviolet fate of the Standard Model --- \emph{resolved at the structural level}.} The near-criticality of the measured electroweak vacuum loses its mystery once the Standard Model is read as the infrared boundary image of the anomaly-free \GG{} bulk. The bulk closes under the \BIsix{} Green--Schwarz identity \citep{ArisakaA3BI6}, the unique \OGNfive{} ledger $(1,3,5;7,35)$ fixes its content, and the scale profile $e^{-35}$ supplies a finite geometric infrared cutoff. The ultraviolet completion is therefore not a Landau pole or a tunneling instability but the six-dimensional geometry itself; the apparent boundary metastability is the shadow of a manifestly ultraviolet-complete bulk.

\textbf{4.~The flavor structure and the Yukawa hierarchy --- \emph{partially resolved}.} The number of families is no longer accidental: $N=3$ is forced by the three-leg theorem (\S\ref{sec:N3-theorem}; \citealp{ArisakaP2N3}). The twelve-orders-of-magnitude mass spread is organized structurally --- charged leptons by dyadic depth, quarks by angular winding with branch factors, neutrinos by the quadratic scale $v^2/\EGUT$ --- with each Yukawa carrying the cascade-depth decoration $e^{-n\pi}$ of the \GFF{} Morse cascade (\S\ref{sec:lepton-cascade-depth}, \S\ref{sec:quark-cascade-depth}), and the CKM block following as projector misalignment (\S\ref{sec:CKM}). The cascade-top coupling $y_t = 1$ is a confirmed parameter-free prediction. What is not yet closed is the per-fermion depth: the eight non-top charged-fermion cascade depths await the equivariant-cohomology calculation on $\mathbb{C}P^2$ doubled by $\SU(2)_L$ (item $O_{21}$ of \S\ref{subsec:open-problems-L6}) and remain \textbf{Asserted} pending it. The hierarchy is thus given a geometric skeleton, with a single named lemma standing between that skeleton and a full derivation.

\textbf{5.~The origin of neutrino masses and mixings --- \emph{resolved}.} The Majorana operator $(LH)(LH)$ fixes the neutrino mass scale at $m_\nu \sim v^2/\EGUT$, explaining the extreme lightness with no new tunable coupling, and the misalignment of the dyadic charged-lepton basis with the constitutional Majorana basis fixes the whole PMNS/Majorana block (\S\ref{sec:neutrino-cascade-depth}, \S\ref{sec:PMNS}). The three mixing angles emerge as closed forms ($\sin^2\theta_{12} = 25/81$, $\sin^2\theta_{13} = 3/140$, $\sin^2\theta_{23} = 4/7$), the leptonic Dirac phase as $\delta_{\rm CP} = \pi + 2\arcsin(1/7) = 196^\circ$, and the Majorana phases as the discrete data $\alpha_{31} = \pi$, $\alpha_{21} = 0$ --- the large, near-maximal leptonic angles and the small quark angles now reading as two projector misalignments rather than two unrelated accidents. The forward test is the DUNE / Hyper-Kamiokande measurement of $\delta_{\rm CP}$ over 2027--2032.

\textbf{6.~The baryon asymmetry of the universe --- \emph{conditionally resolved; the mechanism is carried in this paper}.} The Standard Model fails the Sakharov program twice over: for want of enough \CP{} violation, and for want of a first-order electroweak transition at the measured Higgs mass. \ANine{} changes the starting point by \emph{deriving}, rather than fitting, both the quark and lepton \CP{} sectors, and by supplying a Majorana neutrino block whose selection rule leaves leptogenesis as the only admissible mechanism. It then carries the mechanism itself, in Part~\ref{part:lepto}: the $\Delta L = 2$ channel at the derived lock scale, the flavored asymmetry driven by the derived Majorana parity, the sphaleron conversion at $28/79$, and an assembled baryon-to-photon ratio. The grading must be stated with the result: the particle-physics inputs are structural, the mechanism is developed here rather than deferred, and its closure rests on one open finite-temperature transport condition whose home is the cosmology companion \citep{ArisakaA10Cosmos}. \emph{Conditionally} resolved, therefore --- and the condition is named wherever the chain is used.

\textbf{7.~The strong-CP problem --- \emph{resolved}.} The strong-CP angle is suppressed structurally, with no axion and no Peccei--Quinn relaxation. The first-order \GG{} seam admits no independent $\theta$-term; the quark-determinant phases are relative holonomy phases that cancel, so $\arg\det(M_uM_d)=0$ at first order; and the natural quantum-defect topological leakage is pushed to $\bar\theta \lesssim 5\times10^{-18}$ (\S\ref{sec:strong-CP-derivation}), eight orders of magnitude below the neutron-electric-dipole-moment bound. The smallness the axion was invented to explain is here a protected topological zero.

\textbf{8.~The nature of dark matter --- \emph{cohort-derived (out of \ANine{} scope)}.} Dark matter is not one of the Standard Model's twenty-eight parameters, so it is not an internal task of \ANine{}; but it is not left open by the program. The compact-angle / holonomy sector of the same geometry supplies a concrete candidate --- the Kaxion, with mass set by the lock-length integer $K=35$ at $\sim 8.1~\mu$eV ($1.95$~GHz) \citep{ArisakaC1Kaxion} --- and the relic abundance $\Omega_{\rm dm} = 1/\pi - 1/(2\pi^2)$ follows in \citep{ArisakaA10Cosmos}. The prediction is sharply falsifiable: a haloscope signal away from the predicted fractional frequency would break the ledger.

\textbf{9.~Dark energy and the cosmological constant --- \emph{cohort-derived (out of \ANine{} scope)}.} The $120$-order-of-magnitude vacuum-energy discrepancy is recast as a projected ledger residue rather than a free constant. The geometric dark-energy scale $m_\Lambda^{\rm geom} = M_{\rm lock}^2/\overline{M}_{\rm Pl} \approx 6$~meV and the boundary inventory $\Omega_\Lambda = 1 - 1/\pi$ are derived in the cosmology companion \citep{ArisakaA10Cosmos}; like dark matter, this lies outside the particle-physics scope of \ANine{} but is a closed-form prediction of the cohort, not an open question.

\textbf{10.~Quantum gravity, unification, and proton stability --- \emph{partially resolved}.} The three threads of the deepest problem separate cleanly. \emph{Unification:} the three gauge couplings meet at $\EGUT$ with no continuous fitted parameter intervening between $M_Z$ and the unification scale (Figure~\ref{fig:gg-unification}), and gravity and the gauge interactions descend together from one composite $\mathrm{Spin}(1,5)\times\SU(5)\times\UoneB\times\UoneQi$ connection on the bulk \citep{ArisakaA2GG6}. \emph{Proton stability:} the gauged $\UoneB{}$ renders baryon number exact, so the proton is stable at tree level, with violation admissible only in units of $N=3$; this is a conditional boundary theorem (\AFive{} T-GDOS-9.8; Theorem~\ref{thm:proton-stability-a9}), resting on the (H2) descent step that realizes the bulk $\UoneB{}$ as baryon number; a single proton-decay event would falsify it. \emph{Quantum gravity:} the composite connection is the first layer of a quantum-gravitational description; its full dynamics --- graviton scattering at the lock scale, infrared cutoffs --- remain genuinely open and are the natural subject of a successor program.

\medskip
\noindent Counting honestly: within \ANine{}, five of the ten are resolved (three fully --- the Higgs sector, the neutrino sector, and strong-CP --- and two at the structural level --- the hierarchy and the ultraviolet fate); three are partially resolved with the missing step named explicitly (the flavor cascade-depth lemma, the cosmological baryogenesis mechanism, and full quantum gravity); and two --- dark matter and dark energy --- are derived in the cohort, outside the particle-physics scope of this paper. No item of the ten is left without a concrete account from the single geometry --- which is exactly the claim Section~\ref{sec:intro} set out to earn.

\section{Comparison with other theories and models}
\label{sec:disc-comparison}

A professional discussion section should compare the present program not only to the
Standard Model itself but also to the main beyond-the-Standard-Model directions that have
framed high-energy thinking for decades. The goal here is not to dismiss those programs.
On the contrary, each exists for a reason and each solved problems that the field regarded as
important. The purpose of comparison is to clarify exactly what \GG{}/\GDOS{} is and what it is
not.

\subsubsection{The Standard Model as benchmark and foil}

The Standard Model remains the unmatched benchmark of precision particle physics.
Its gauge principle, renormalizability, anomaly cancellation by representation theory,
radiative control, and broad experimental success are extraordinary
.
Any unification program that fails to preserve that empirical success is not interesting.

The present paper does \emph{not} position itself as a rejection of the Standard Model.
Its stance is more subtle.
The Standard Model is treated as the correct low-energy effective seam theory, but not as the
deepest constitutional layer.
In that sense the Standard Model plays two roles simultaneously.

First, it is the empirical gold standard.
All comparison tables in \ANine{} are judged against the mature Standard-Model benchmark.

Second, it is the object to be reinterpreted.
The present paper asks whether the Standard Model can be read as a theorem-level low-energy
image of a geometry-gauge-locked six-dimensional bulk plus an explicit open-system projection.
That is a much stronger claim than ordinary model-building.
It says that the Standard Model may be not the starting constitution, but the boundary export
of a deeper constitution.

From a phenomenological standpoint the key difference is parameter philosophy.
The Standard Model accepts the 28 parameters as empirical input.
\ANine{} asks whether those 28 numbers can instead be generated from a far smaller
constitutional ledger.
In this limited but very important sense, the Standard Model is the benchmark that \GG{} must
respect and the explanatory target that \GG{} tries to transcend.

\subsubsection{Comparison with the MSSM}

The MSSM and related supersymmetric frameworks were developed because they address several
serious issues in an elegant way: gauge-coupling unification improves, radiative electroweak
symmetry breaking becomes natural, fermion masses can be embedded in richer high-scale
structures, and stable weakly interacting dark-matter candidates arise almost automatically in
many realizations.

Those are real virtues.
It would be intellectually careless to deny them.
Supersymmetric model building also trained the field to think carefully about RG
running, threshold effects, flavor danger, proton decay operators, and precision matching.
In that historical sense, the MSSM program is one of the great schools of modern
phenomenology.

The present framework differs from the MSSM in a very direct way.
It does not try to solve the hierarchy problem by pairing each Standard-Model field with a
superpartner whose loop contribution cancels the scalar sensitivity.
Instead it changes the status of the Higgs variable itself.
The Higgs is not fundamental in the same way, because it is not an elementary scalar on a line
with an arbitrary bare mass.
It is a Wilson-line holonomy of the higher-dimensional connection.
Thus the present paper's answer to hierarchy/naturalness is not ``cancel the dangerous
corrections by new partner states'' but ``redefine the underlying variable so that the
dangerous correction logic is no longer the primitive one''.

This difference has several consequences.

First, the particle-content burden is much smaller.
No superpartner tower is introduced in order to make the parameter derivation work.

Second, the source of predictivity is different.
In the MSSM or its descendants, many new soft-breaking parameters must be organized or
constrained.
In \GG{}/\GDOS{}, the goal is to avoid reopening that freedom in the first place.

Third, proton stability has a different status.
In supersymmetric unification one typically needs to work hard to control dangerous
dimension-4 and dimension-5 baryon-violating operators.
In the present program the local \(U(1)_B\) sector is meant to render proton stability
structural rather than accidental.

A balanced assessment is therefore this.
The MSSM offers a powerful and historically important answer to hierarchy and unification.
\GG{} offers a more radical answer:
it removes the ordinary scalar problem by geometrizing the Higgs variable and it refuses to pay
for predictivity by enlarging the continuous-parameter sector.
Whether nature prefers the supersymmetric route or the geometric-holonomy route is not a
matter for rhetoric.
It is exactly the kind of question that collider and low-energy precision tests can decide.

\subsubsection{Comparison with perturbative superstring theory and the landscape problem}

Perturbative superstring theory made two kinds of contribution that are highly relevant to the
present paper.
One is technical, and one is conceptual.

The technical contribution is the Green--Schwarz anomaly-cancellation mechanism \citep{GreenSchwarz1984}, which is the historical ancestor of the \BIsix{} structure used in \GG{} and was the first concrete realization of the geometry--gauge equivalence later axiomatized as AX-2 \citep{ArisakaA3BI6}.
The conceptual contribution is the modern bulk-boundary way of thinking, culminating in
bulk--boundary correspondence, which helps motivate the idea that geometry can encode quantum gauge
dynamics and that a bulk theory should include an explicit export map to boundary data
.

The present paper therefore does \emph{not} present itself as a theory developed in ignorance
of string theory.
On the contrary, it openly borrows two of string theory's great lessons:
anomaly inflow and bulk-boundary reasoning.

What it does \emph{not} borrow is the full ontology of fundamental strings, branes, ten- or
eleven-dimensional compactification landscapes, large moduli sectors, or vacuum multiplicity
as a basic explanatory framework.
That is the decisive distinction.

The present program sets the ultraviolet endpoint at \(E_{\rm GUT}\), not at the Planck scale,
and treats the primitive dynamical objects as those of an ordinary field-theoretic
gauge-gravity-tensor system in six dimensions.
This is an intentionally austere choice.
Its advantage is predictivity.
Its danger is that it may be wrong if nature genuinely requires stringy degrees of freedom at a
deeper level.
But the paper's methodological stance is clear:
one should not introduce an enormous ultraviolet ontology unless the constitution or the data
force it.

That stance is tied directly to the landscape issue.
In the broad literature, landscape reasoning is often taken to mean that there may exist a vast
set of metastable or phenomenologically distinct low-energy vacua, only some of which resemble
our world.
The price of such flexibility is weakened predictivity.
\MSMF{} and OGN were designed precisely as anti-landscape clauses.
They do not claim that many mathematical vacua are impossible.
They claim that a constitutional theory should not treat non-forced vacuum multiplicity as an
acceptable endpoint of explanation.

In that sense \GG{} is neither anti-string polemic nor string imitation.
It is better understood as a selective inheritance:
keep the anomaly and renormalization-group lessons, discard the large ontological overhead unless and
until it becomes unavoidable.

\subsubsection{Comparison with other quantum-gravity programs}

The phrase ``other quantum-gravity theories'' covers a wide family of efforts:
canonical loop quantum gravity, spin-foam variants, asymptotic safety, causal dynamical
triangulations, causal sets, group field theory, and other nonperturbative programs.
These approaches are diverse, and it would be unfair to treat them as one homogeneous target.
Still, they do share a common emphasis:
they ask how spacetime geometry itself should be quantized or completed in the ultraviolet
.

That emphasis is valuable.
Indeed, one reason such programs are respected is that they take the gravitational question
seriously rather than postponing it indefinitely.

The present program differs from them by inverting the order of priority.
Instead of starting from ``how do we quantize geometry?'', it begins from
``how do geometry, gauge structure, and observation become one admissible constitutional
system?''
The distinction matters.
A program may have a beautiful ultraviolet geometry yet still remain silent about the Standard
Model parameter problem, proton stability, measurement, or the explicit derivation of
low-energy observables.
\GG{}/\GDOS{} tries to close that gap by making observation and phenomenology part of the primary
design target.

This does \emph{not} mean that other quantum-gravity programs are somehow scientifically
inferior.
It means that the questions being optimized are different.
Loop quantum gravity, for example, is an exceptionally ambitious program for background
independence and quantum geometry.
Asymptotic safety is an exceptionally ambitious program for ultraviolet completeness by RG fixed
points.
But neither is usually organized around the simultaneous derivation of the full Standard-Model
parameter set \emph{and} an operational boundary dictionary for measurement.
\GG{}/\GDOS{} is organized around exactly that conjunction.

A fair comparison is therefore not ``which framework solves everything?''
No current framework does.
A fairer comparison is ``which freedoms are taken as primitive, which are prohibited, and which
phenomenological burdens are shouldered directly?''
On that criterion, the uniqueness of \GG{} becomes easy to state:
\begin{quote}
\emph{it is a no-SUSY, no-string, anomaly-locked, measurement-aware, parameter-compressing
unification program whose core claim is that the Standard Model is a boundary theorem rather
than an empirical starting axiom.}
\end{quote}
That sentence is not the proof.
The proof attempt is the whole paper.
But as a summary of architectural distinctiveness it is accurate.

\begin{table}[H]
\centering
\small
\caption[Architectural comparison of the major unification programs, sharpened to show what is distinctive about \GG{}/\GDOS{}.]{Architectural comparison of the major unification programs, sharpened to
show what is distinctive about \GG{}/\GDOS{}. Reading down the final column: it is the
only program with \emph{zero} continuous free parameters, a \emph{derived}
dark-matter mass, a \emph{closed} cosmological inventory, and a \emph{dated}
falsification timeline. Where the MSSM enlarges the continuous-parameter sector and the
string landscape renders the low-energy parameters environmental, \GG{} forces them as
theorems of one six-dimensional geometry. (That the geometry is \emph{uniquely} forced is
itself a conjecture, not yet a theorem, so the final column is a contrast of posture ---
one vacuum asserted versus $\sim10^{500}$ constructed --- not a proof that \GG{} wins.)
Entries are necessarily schematic.}
\label{tab:major-framework-comparison-v08}
\begin{tabular}{p{0.145\textwidth}p{0.155\textwidth}p{0.175\textwidth}p{0.185\textwidth}p{0.205\textwidth}}
\toprule
\textbf{Question} & \textbf{Standard Model} & \textbf{MSSM / SUSY GUTs} & \textbf{Superstring / landscape} & \textbf{\GG{}/\GDOS{}} \\
\midrule
Primitive ontology & 4D gauge fields and matter & SM $+$ superpartners and soft sector & strings/branes $+$ compactification data & 6D geometry--gauge--tensor bulk $+$ boundary export law \\
Continuous free parameters & $\sim 28$ empirical inputs & $\sim 105$ added soft-breaking parameters & $\sim 10^{500}$ vacua / large moduli space & \textbf{0} (one dimensionful input $G_N$; integer ledger $(1,3,5;7,35)$) \\
Higgs nature & elementary scalar, bare mass & elementary scalar $+$ partners & model-dependent modulus/brane field & Wilson-line holonomy of the 6D connection \\
Hierarchy strategy & accept empirical weak scale & cancellation by partner states & model-dependent, UV-completed at $M_{\mathrm{string}}$ & multiplicative lock, $\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-K}$ \\
Measurement in foundations & external & external & external to the fundamental ontology & \OGGEP{}/\GDOS{} makes measurement part of the theory \\
Proton stability & accidental at renormalizable level & \emph{predicts} decay, $10^{34}$--$10^{36}$~yr & model-dependent & \emph{exact} ($\tau=\infty$; $\Delta B=1,2$ forbidden) \\
Dark matter & none & neutralino LSP (not yet seen) & model-dependent & Kaxion, $8.1~\mu$eV \emph{derived} (\PaperCOne{}) \\
Cosmological inventory & none intrinsic & none intrinsic & anthropic / environmental & $\Omega_m=1/\pi$, $\Omega_\Lambda=1-1/\pi$, switch $z_*=e^7$ (\ATen{}) \\
Predictivity / falsifiability & precise, but parameters are input & many parameters dilute predictions & landscape weakens low-energy predictivity & sharp, dated falsifiers (\S\ref{subsec:falsification-timeline}) \\
Status vs.\ current data & confirmed & no superpartners to $\sim 2$~TeV; naturalness strained & not directly tested & falsifiable by $\sim 2040$ \\
Primary ultraviolet scale & 4D EFT & SUSY/GUT scales & string/Planck scale & GUT-seam completion with geometry--gauge locking \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Why the contrast with the MSSM is decisive.}
The supersymmetric program answers the hierarchy problem by \emph{adding} structure:
a superpartner for every Standard-Model field and, with it, of order one hundred new
soft-breaking parameters. That enlargement was the price of naturalness, and for two
decades it was a price worth paying. But the Large Hadron Collider has now excluded the
colored superpartners --- squarks and gluinos --- up to roughly $2$~TeV, pushing the
natural region into tension and leaving the soft sector underdetermined. \GG{} takes the
opposite route on the same problem: it does not pair the Higgs with a partner, it
\emph{reclassifies} the Higgs as a Wilson-line holonomy, so the elementary-scalar
sensitivity that motivated supersymmetry is no longer the primitive logic. The parameter
ledgers run in opposite directions --- where the MSSM grows the continuous sector from
$\sim 28$ to $\sim 130$, \GG{} collapses it to zero. This is the sharpest single contrast
in Table~\ref{tab:major-framework-comparison-v08}, and it is decided not by taste but by
the superpartner searches already under way.

\paragraph{Why the contrast with superstring theory is decisive.}
\GG{} inherits string theory's two greatest lessons --- Green--Schwarz anomaly inflow
(the ancestor of \BIsix{}) and bulk--boundary reasoning --- and \emph{declines} its
central ontological commitment. In the landscape reading, the roughly $10^{500}$
metastable vacua make the low-energy parameters \emph{environmental}: the Standard Model
is selected, not derived, and predictivity is weakened accordingly. \GG{} sets the
ultraviolet endpoint at $\EGUT{}$ rather than the string scale, treats the primitive
objects as an ordinary six-dimensional gauge--gravity--tensor system, and forbids
non-forced vacuum multiplicity through the \MSMF{}/\OGN{} clauses. The result is the
mirror image of the landscape: the same parameters that string theory leaves to
environmental selection, \GG{} derives as theorems of a single geometry --- though that the geometry is \emph{uniquely}
forced is, at this stage, a conjecture rather than a proven theorem. The wager is
austere and therefore falsifiable --- precisely the property the landscape sacrifices.

The architectural comparison above can be deepened by noting what each framework does \emph{not} do. Supersymmetric grand unification, the string-theoretic landscape, asymptotic safety, and emergent-gravity proposals all leave the deeper meta-question of \S\ref{sec:two-questions-one-answer} intact: each explains some Standard-Model parameters by introducing others (supersymmetric partners, landscape moduli, fixed-point ratios, emergent thresholds), and none has produced a uniqueness theorem at the bulk-geometry level. GG-Theory is the first framework that closes both the parameter sheet and the meta-question. Theorem~\ref{thm:four-force-uniqueness} forces the bulk geometry, the composite gauge--gravity group, and the four-force qualitative architecture jointly, with zero continuous freedom remaining.



\subsubsection{Comparison with ordinary $\mathrm{SU}(5)$ grand unification}

Ordinary $\mathrm{SU}(5)$ unification explains why the observed gauge factors may descend from a single block, and in that historical sense it is an ancestor of the present work. But $\mathrm{SU}(5)$ by itself does not \emph{close} the low-energy parameter sheet: it supplies a high-scale unification relation and then leans on running, thresholds, and symmetry-breaking details to reach the $Z$-pole. The \GG{} program is more rigid in one decisive way --- the same theorem packet that fixes the admissible bulk stage also fixes the \OGNfive{} integers and the Wilson--Hosotani branch logic --- so that the weak angle, the Higgs vev, and their descendants are direct boundary readouts rather than high-scale relics transported downward.

\subsubsection{Comparison with generic gauge--Higgs unification}
\label{ssec:compare-ghu}

Generic gauge--Higgs unification already teaches that a Higgs-like scalar can descend from an extra-dimensional gauge component, and the present framework accepts that lesson --- with the debt named. \citet{Manton1979} first obtained a Higgs-like scalar from the extra-dimensional components of a gauge field; \citet{Hosotani1983} showed that on a multiply connected space the Wilson-line holonomy is itself the order parameter, and \citet{Hosotani1989} developed the resulting dynamics; \citet{HatanakaInamiLim1998} established the property on which \S\ref{sec:deep-hierarchy} leans, that the holonomy's effective potential is finite because no local counterterm for a Wilson-line phase exists. Everything this paper says about \emph{why} the electroweak scale is stable is inherited from that line of work and is not claimed as new. What is new is what the ledger adds to it: generic gauge--Higgs unification leaves the compactification radius, the matter content, and hence the minimum of the holonomy potential as inputs to be scanned, so the mechanism explains the \emph{stability} of the electroweak scale without fixing its \emph{value}. Here the \OGNfive{} integers fix both, and $\theta_H^\star$ is computed rather than scanned. What changes is the surrounding logic: the Wilson--Hosotani mechanism is not an isolated model-building device but a link in a constitutional chain, $\text{\OGGEP{}} \Rightarrow \text{\GG{} stage} \Rightarrow \text{\BIsix{} legality} \Rightarrow \text{\GDOS{} runtime} \Rightarrow \text{electroweak branch readout}$, which is what licenses connecting branch selection to the larger \DGI{}/\GFF{}/\GSSB{}/\GA{} attractor story. The single discipline that makes the comparison meaningful is the refusal to re-introduce freedom upstairs: a familiar failure mode of ambitious unification is to trade twenty-eight low-energy parameters for an even larger set of moduli, thresholds, or compactification choices. The \MSMF{}/\OGN{} insistence guards against exactly that, and the paper's central claim is that for the entire gauge--Higgs sheet no new continuous coefficient is admitted --- which is what makes the closure nontrivial rather than a mere relocation of the same freedom.

\section{The Ten Structural Innovations of the \GGTheory{} Standard Model}
\label{sec:ten-innovations}

The contribution of \ANine{} is to derive the Standard Model's parameter set from
an admissible input ledger that contains no free continuous parameter at all. We
collect here the ten structural innovations that derivation delivers, each
carrying its cohort-global identifier (\S\ref{sec:naming-convention}).

\paragraph{Innovation 1 (I-SM-1): The 28-to-zero compression.} \ANine{} recasts
the 28 Standard-Model parameters as outputs of a framework with \emph{no}
adjustable continuous input --- explaining the Standard Model by removing
freedom, not adding it. The grand-unified tradition compresses 28 to a smaller
but still positive parameter count; the present reading is, in the strict sense
of \citet{ArisakaA4Euclid}, 28-to-zero.

\paragraph{Innovation 2 (I-SM-2): A parameter-free admissible input ledger.} Every
entry of $\mathfrak I_{\rm adm}=\{c,G_N,\hbar;k_B,Q_B\}\cup\{\pi,e\}\cup\OGNfive{}$
is \emph{forced}: the \GUfive{} constants are unit-conversion factors (T-GU5 of
A1), $\pi$ and $e$ are topological constants of the bulk, and the
$(1,3,5;7,35)$ ledger is proved (Theorem T6 of \ATwo{}, $N=3$ closed in
\citet{ArisakaP2N3}). None can be moved continuously without breaking a clause.

\paragraph{Innovation 3 (I-SM-3): Predictivity as theory-space pruning to a single
point.} The deductive arrow $\mathfrak I_{\rm adm}\Rightarrow\mathfrak
O_{\rm SM}^{(28)}$ runs with no continuous freedom at any step, so the admissible
theory space contracts to a single point before any fit begins --- the structural
origin of the unusually large phenomenological predictivity.

\paragraph{Innovation 4 (I-SM-4): The electroweak sector as boundary attractor
coordinates.} The gauge couplings, the weak angle $\sin^2\theta_W=25/108$, the
fine-structure value $\alpha^{-1}(0)=137$, and the strong coupling emerge as
derived rationals from the boundary attractor structure (T-SM-26.1, T-SM-27.1,
T-SM-28.1, T-SM-29.1, T-SM-34.1), not as fitted inputs.

\paragraph{Innovation 5 (I-SM-5): The Higgs sector from Wilson--Hosotani geometry.}
The electroweak vev $v\approx 246$~GeV (T-SM-30.1) and the Higgs quartic
$\lambda_H=1/8$ (T-SM-31.1) are computed from the scale-overlap and
gauge--Higgs structure rather than tuned.

\paragraph{Innovation 6 (I-SM-6): The electroweak masses as theorems.} The Fermi
constant and the $W$, $Z$, and Higgs masses follow as theorems of the derived
gauge--Higgs package (T-SM-33.1 through T-SM-33.4, T-SM-35.1), closing the
gauge/Higgs block.

\paragraph{Innovation 7 (I-SM-7): Fermion masses and mixings as a geometric
codebook.} The twelve fermion masses are scale-overlap codebook entries and the
CKM and PMNS matrices (with Majorana phases) arise as projector misalignments,
turning the flavor sector from a table of inputs into a computed structure.

\paragraph{Innovation 8 (I-SM-8): Proton stability as a conditional boundary
theorem.} Gauged $\UoneB{}$ makes the proton absolutely stable
($\tau_p=\infty$), stated as a boundary theorem (T-SM-60.1) --- conditional on the
descent step (H2) that realizes the bulk $\UoneB{}$ as component-resolved baryon
number, the one step still to be discharged (see the proton-stability appendix)
--- rather than imposed by hand. It remains a sharp, binary falsifier: a single
proton-decay event would refute it.

\paragraph{Innovation 9 (I-SM-9): Four-force uniqueness and the no-extra-constant
principle.} The canonical four-force architecture is shown unique (T-SM-62.1) and
the \MSMF{}/\OGN{} selection principle (T-SM-67.1) forbids importing any further
transcendental constant into a seam-code expression --- the clause that keeps the
input ledger closed.

\paragraph{Innovation 10 (I-SM-10): Rigidity as a falsifiable virtue.} The same
single-point pruning that yields the predictivity makes the construction
maximally fragile: a handful of benchmark failures would collapse it. The
predictivity is therefore presented only together with the sharp falsifiers
(F-SM-1\,\ldots\,F-SM-7 below).

\section{Predictions and falsification}
\label{sec:predictions-falsification}

A framework that derives twenty-eight Standard-Model parameters from the
nine-element ledger $(M,N,L;J,K)=(1,3,5;7,35)$ and the five \GUfive{} constants
$\{c,G_N,\hbar;k_B,Q_B\}$ is not protected by that economy --- it is made
\emph{more} exposed by it. Every derived number is a place the framework can be
killed. Two qualitatively different classes of falsifier are now in play, and the
second class has changed character since the first version of this manuscript: the
dedicated phenomenology papers \PaperPOne{} and \PaperCOne{} are complete, and the
cosmological companion \ATen{} is complete, so the downstream predictions are now
\emph{quantitative} rather than programmatic. We organize the discussion as
(i)~the parameter-free precision post-dictions of already-measured quantities, and
(ii)~the seven dedicated falsification channels F-SM-1--F-SM-7 --- the
neutrino package itself resolved into five separately-killable
sub-channels F-SM-3a--3e --- each now carried
to a definite number by a completed companion.

\subsection{Parameter-free precision post-dictions}
\label{subsec:intrinsic-falsifiers}

Before any new particle is found, the codebook of this paper is already falsifiable
against existing precision data. Each entry of Table~\ref{tab:intrinsic-falsifiers}
is a closed form over the \OGNfive{} ledger with \emph{no} continuous dial; a
single sustained violation at the quoted level falsifies the corresponding clause
of the derivation. These are the cheapest and earliest tests, requiring no new
facility. They are \emph{post-dictions} --- each quantity was measured long ago --- but
they are parameter-free: no continuous dial was turned to reach them. The genuinely
\emph{blind} predictions of unmeasured quantities (the neutrino package, the $3.85$~TeV
lock cluster, the $8.1~\mu$eV Kaxion) follow in F-SM-1--F-SM-3 below.

\begin{table}[H]
\centering
\caption[Parameter-free precision post-dictions: zero-parameter closed forms for already-measured quantities, testable against existing or near-term precision.]{Parameter-free precision post-dictions: zero-parameter closed forms for
already-measured quantities, testable against existing or near-term precision data. Each
is a closed form over the \OGNfive{} ledger (chosen parameter-free, not blind-predicted);
the right-hand column states what observation would falsify it.}
\label{tab:intrinsic-falsifiers}
\small
\begin{tabular}{p{0.215\textwidth}p{0.165\textwidth}p{0.165\textwidth}p{0.345\textwidth}}
\toprule
\textbf{Observable} & \textbf{GG closed form} & \textbf{Value} & \textbf{Falsified if} \\
\midrule
Weak mixing angle & $\sin^2\theta_W = \dfrac{L^2}{4N^3} = \dfrac{25}{108}$ & $0.23148$ & precision electroweak fits settle $> 0.5\%$ away from $25/108$ at the matching scale \\[4pt]
IR fine structure & $\alpha^{-1}(0) = 2^J + N^2$ & $137$ & $\alpha^{-1}(0)$ moves off $137$ (it will not, but the \emph{structural reading} fails if the integer split is wrong) \\[2pt]
Strong coupling & $\alpha_s^{-1}(M_Z) = \pi e$ & $8.539$ & world-average $\alpha_s(M_Z)$ settles $>1\%$ (several $\sigma$) from $1/(\pi e)$ \\[2pt]
Higgs quartic & $\lambda_H = 2^{-N} = 1/8$ & $0.125$ & a measured $\lambda_H$ (HL-LHC/FCC) incompatible with $1/8$ \\[2pt]
$\tau/\mu$ mass ratio & cascade rational & $16.817\pm0.5\%$ & PDG ratio leaves the $\pm0.5\%$ band \\[2pt]
Family number & $N=3$ (closed, \PaperPTwo{}) & exactly $3$ & a fourth chiral generation, or a light sterile state forced into the active sector \\
\bottomrule
\end{tabular}
\end{table}

\subsection{\texorpdfstring{F-SM-1}{F-SM-1}: the 3.85\,TeV lock cluster (\texorpdfstring{\PaperPOne{}}{P1-KK})}
\label{subsec:fsm1}

The scale chain $\EPl \to \EGUT \to \Mlock$ terminates at a number fixed with no
continuous freedom,
\[
\Mlock = \tfrac12\,M_{\mathrm{Pl}}\,e^{-K} = \tfrac12\,M_{\mathrm{Pl}}\,e^{-35}
\simeq 3.849~\mathrm{TeV},
\]
set by the single \OGNfive{} integer $K=35$ and the \GUfive{} constants. The
dedicated signal paper \PaperPOne{} \citep{ArisakaP1KK} now shows that the
prediction is not a single bump but a \emph{multi-state cluster}: a spin-2
graviton anchored at \Mlock{}, accompanied by a spin-1 pair --- the color-octet
Kaluza--Klein gluon $g^{*}$ and the leptophobic baryonic vector $Z'_B$ of the
$\UoneB{}$ sector derived in the present paper. The falsifier is the
\emph{pattern} (the mass coincidence, the spin assignments, and the coupling
hierarchy), not an isolated excess.

This corridor sits precisely at the present experimental frontier. Run-2 dijet
searches at the LHC already exclude generic leptophobic $Z'$ states up to
$\sim 3.9$--$4.0$~TeV (width-dependent) \citep{CMSDijet2020}, color-octet scalars to $\sim 3.7$~TeV,
and Randall--Sundrum-type spin-2 dijet resonances to $\sim 2.6$~TeV.

\paragraph{Why the strongest-looking bound does not apply, stated before a
reader raises it.} Searches for $t\bar t$ resonances quote exclusions on
Randall--Sundrum Kaluza--Klein \emph{gluons} reaching about $4.55$~TeV
\citep{CMSttbar2019} --- above the lock scale, and therefore apparently
fatal to a color-octet member at $3.85$~TeV. It is not, and the reason is
that such limits are set on $\sigma\times\mathrm{BR}$ under a specific
template: the RS KK gluon is a \emph{broad} state, with $\Gamma/M$ of order
$15$--$30$ per cent and a branching ratio to $t\bar t$ near unity, and its
predicted cross section is correspondingly large. The state predicted here
is narrow and its $t\bar t$ branching is far smaller, so the correct
comparison is not the benchmark mass reach but the cross section, which is
given immediately below. This is stated explicitly because a reader who
knows the $4.55$~TeV number and not the width assumption behind it would
reasonably conclude that the cluster is already excluded. The GG cluster
at $3.85$~TeV therefore lives in the window that the High-Luminosity LHC and any
future hadron collider will decisively probe. \emph{Failure:} exclusion of the
spin-1 pair --- the joint mass and coupling pattern of $g^{*}$ and $Z'_B$ ---
across the $3.8$--$4.0$~TeV region at the derived couplings; the spin-2
confirmation belongs to Tier~2 below. The production cross sections and branching ratios
are computed explicitly in \PaperPOne{} (its operational-confrontation section,
at the model-level grade stated there): the un-suppressible color-octet member
delivers $\sigma\times\mathrm{BR}(t\bar t)=8.7\,[7.0$--$12]$~fb at
$3.85$~TeV --- a factor $\sim\!2.9$ below the current broad-template limit, and
crossed by the projected HL-LHC exclusion between roughly $0.5$ and
$2~\mathrm{ab}^{-1}$ --- while the free-field $2.46$~TeV placement of the
spin-1 pair (Scenario~A of \PaperPOne{}'s T-KK-2 dichotomy) is already a
factor $\sim\!3.6$ above the observed limit and is provisionally closed,
resolving the dichotomy toward Scenario~B --- the degenerate
cluster at \Mlock{}.  A corollary of the same free-field machinery
(\PaperPOne{}, Corollary T-KK-2c) now also \emph{derives} the graviton's
gluon coupling, $\kappa_{gg}=0.47/K=0.0136$: the spin-2 anchor contributes
only $0.045$~fb in $t\bar t$ at the LHC (LO$\times$K) --- invisible at any LHC luminosity, with
the published diphoton null retro-dicted by a factor $\sim\!3\times10^{3}$
--- and becomes a direct FCC-hh target ($\sigma\approx205$~fb at 100~TeV).
This falsifier is therefore two-tiered: \emph{Tier~1 (HL-LHC, decisive)} ---
the un-suppressible octet line, crossed within $\sim\!0.5$--$2$~ab$^{-1}$;
\emph{Tier~2 (FCC-hh, confirmatory)} --- the spin-2 anchor at the same mass
with $\Gamma/m=6.3\%$ and the derived infrared branching pattern.  Finding
either without the other falsifies the cluster reading.

\subsection{\texorpdfstring{F-SM-2}{F-SM-2}: Kaxion dark matter at 8.1\,\texorpdfstring{$\mu$}{micro}eV (\texorpdfstring{\PaperCOne{}}{C1-Kaxion})}
\label{subsec:fsm2}

The compact $S^1_\theta$ direction and the additional Abelian structure
$\UoneB{}\times\UoneQi{}$ carry a protected compact-angle pseudoscalar --- the
Kaxion --- whose mass is the canonical cohort value
\[
m_a = 8.1~\mu\mathrm{eV} \;\Longleftrightarrow\; \nu = m_a c^2/h = 1.95~\mathrm{GHz}.
\]
The benchmark $1.95$~GHz is the center of a single narrow predicted window, $1.59$--$2.13$~GHz, whose width is a residual (conjecture-set) cap computation not yet completed in \PaperCOne{}; the parameter-free \emph{fingerprints} --- the $1/35$ twin-peak split and the $5{:}2$ portal lock --- do not depend on that width.
This is a \emph{two-handle} falsifier, and the two handles are locked together
across the cohort. The laboratory handle: a microwave-cavity haloscope tuned to
$1.95$~GHz at the predicted photon coupling, computed in \PaperCOne{}
\citep{ArisakaC1Kaxion}. The cosmological handle: the same Kaxion must
\emph{saturate} the relic abundance derived independently in \ATen{},
$\Omega_{\mathrm{dm}} = 1/\pi - 1/(2\pi^2) \simeq 0.267$. Crucially, the Kaxion sits \emph{exactly} on the DFSZ line --- the faintest, hardest coupling depth the haloscope community targets --- so present-generation cavities do \emph{not} yet reach it. It is the next-generation DFSZ-sensitive haloscopes (the ADMX-EFR upgrade and the IBS-CAPP program \citep{ADMX2025}), being built to scan this band at full DFSZ depth across roughly $2027$--$2030$, that will provide the decisive test. \emph{Failure:} exclusion of the
$1.95$~GHz line at the derived coupling, \emph{or} a measured dark-matter relic
incompatible with $\Omega_{\mathrm{dm}}=1/\pi-1/(2\pi^2)$. The cross-paper lock
\,\ANine{} (sector) $\to$ \PaperCOne{} (mass, coupling) $\to$ \ATen{} (abundance)\,
means no single experiment can confirm the Kaxion in isolation, and no single
exclusion is cheap to evade.

\subsection{\texorpdfstring{F-SM-3}{F-SM-3}: the neutrino package, in five separately-killable channels}
\label{subsec:fsm3}

The lepton sector is the least hadronically shielded exposure of the framework.
The present paper fixes, with no further freedom: normal ordering; a light but
nonzero QD-lifted $m_1$; a mass sum $\sum m_\nu \simeq 59$~meV; a PMNS Dirac phase
$\delta_{\mathrm{PMNS}} \simeq 196^\circ$ (now \emph{Derived (conditional)} under
the leptonic Dirac-phase assignment lemma, consistent with the $6/14/8$ triage);
an atmospheric angle in the upper octant; protected Majorana phases $(0,\pi)$; and
a neutrinoless-double-beta scale $m_{\beta\beta} \simeq 1.8$~meV. The current data
are already pressing on this package, and it is no longer useful to treat
it as one channel: the six statements above are tested by different
experiments on different timetables, and two of them are already moving in
opposite directions. The package is therefore split here into five
separately-killable sub-channels, each with a criterion declared in
advance, a facility, and a date. \textbf{A clean failure of any one of
them refutes the package}; the sub-channels are not a hedge. In brief:
\textbf{F-SM-3a} the mass ordering, \textbf{F-SM-3b} the Majorana floor,
\textbf{F-SM-3c} the mass sum, \textbf{F-SM-3d} the Dirac phase, and
\textbf{F-SM-3e} the atmospheric octant.

One result that is \emph{not} a kill channel belongs first, because it is
the strongest evidence the block has. JUNO's first data \citep{JUNO2026}
give $\sin^2\theta_{12} = 0.3092\pm0.0087$, against the derived closed form
$L^2/3N^3 = 25/81$, QD-completed to $0.30857$: a pull of
\textbf{$0.07\sigma$}, the closest agreement anywhere in the lepton block,
on a measurement that did not exist when the derivation was written. A
framework can survive its kill tests and still be uninteresting; this is
the row that says the block is worth killing.

\paragraph{F-SM-3a --- the mass ordering.} \emph{Prediction:} normal
ordering, forced by the cascade-depth assignment that places the \GGA{} at
the bottom of the Majorana stratum. \emph{Status:} the post-JUNO global
analysis \citep{EstebanJUNO2026} gives
$\Delta\chi^2(\mathrm{IO}-\mathrm{NO}) = 4.6$ without atmospheric data and
$9.4$ with Super-Kamiokande and IceCube --- normal ordering at roughly
$2.2$--$2.3\sigma$, with JUNO having about doubled the discrimination.
Preferred is not established. \emph{Kill:} inverted ordering established at
$5\sigma$. \emph{Facility and date:} JUNO reaches $3\sigma$ standalone
around 2032 and earlier in combination; DUNE and Hyper-Kamiokande add
atmospheric sensitivity through the 2030s. \emph{What dies:} the
cascade-depth assignment of \S\ref{sec:neutrino-cascade-depth}, and with it
the light spectrum. There is no fallback ordering.

\paragraph{F-SM-3b --- the Majorana floor.} \emph{Prediction:} Majorana
neutrinos with protected parities $(\alpha_{21},\alpha_{31}) = (0,\pi)$ and
$m_{\beta\beta} \simeq 1.8$~meV. \emph{Status:} KamLAND-Zen
\citep{KamLANDZen2024} gives $T^{0\nu}_{1/2} > 3.8\times10^{26}$~yr on
$^{136}$Xe, i.e.\ $m_{\beta\beta} \lesssim 28$--$122$~meV; the combined
LEGEND-200, GERDA and \textsc{Majorana Demonstrator} $^{76}$Ge result
\citep{LEGEND2025} gives $> 1.9\times10^{26}$~yr; CUORE \citep{CUORE2025}
gives $> 3.5\times10^{25}$~yr on $^{130}$Te. \emph{Kill:} a confirmed
$0\nu\beta\beta$ line anywhere in the tonne-scale window of $9$--$21$~meV,
or an established Dirac nature. \emph{Facility and date:} CUPID at
$9$--$15$~meV from the mid-2030s, LEGEND-1000 at $9$--$21$~meV.
\emph{What dies:} the dial-free door as the matter portal, and with it the
whole of Part~\ref{part:lepto}.

\emph{This channel is one-sided, and the paper says so twice.} A signal
kills the prediction; a null result at tonne scale proves nothing about it,
because the predicted $1.8$~meV sits an order of magnitude below the
reach. \S\ref{sec:PMNS} shows that the margin is larger than the spectrum
alone requires --- the derived parity places $m_{\beta\beta}$ in the lowest
quarter of the range accessible to a generic normal-ordering spectrum. That
is a cost of the parity being derived rather than chosen, and it is not to
be presented as a virtue.

\paragraph{F-SM-3c --- the mass sum. \emph{This channel is firing.}}
\emph{Prediction:} $\sum m_\nu = 59.17$~meV. \emph{Status, and the reason
it cannot take the comfortable bound:} this paper's own channel F-SM-6
pins $w = -1$ exactly and two-sidedly, so a framework-consistent reader is
not free to score against the $w_0w_a$CDM row. DESI DR2 with CMB
\citep{DESI2025} gives $\sum m_\nu < 64.2$~meV ($95\%$, $\Lambda$CDM) ---
eight per cent above the prediction --- and the Feldman--Cousins variant of
the \emph{same} analysis gives $< 53$~meV, which the prediction does not
survive. For comparison the same data under $w_0w_a$CDM give $< 163$~meV,
ACT DR6 \citep{ACTDR62025} gives $< 89$~meV, and an SPT-3G-based CMB choice
gives $< 110$~meV. \emph{Kill:} a robust $\sum m_\nu < 58$~meV at high
significance under a stable $\Lambda$CDM treatment. \emph{Facility and
date:} no new facility is required; DESI DR3 with CMB, in the
$\sim$2026--2028 window. \emph{What dies:} the neutrino mass block, and
therefore the seam coefficient $\pi/\EGUT$ that fixes it.

The honest statement is not that this channel will fire soon but that
\emph{it is firing now, at the one-sided edge, under the framework's own
dark-energy prior}. The DESI analysis itself reports a $3\sigma$ tension
between the cosmological constraint and the oscillation floor, and the
recurring preference for a negative effective $\sum m_\nu$ in the
literature is a symptom of the same squeeze. Because it needs no new
facility, this remains the channel most likely to be decided first.

\paragraph{F-SM-3d --- the Dirac phase.} \emph{Prediction:}
$\delta_{\mathrm{PMNS}} = \pi + 2\arcsin(1/7)$, QD-completed to
$196.06^\circ$. \emph{Status:} the T2K--NOvA joint analysis
\citep{T2KNOvA2025} gives, for normal ordering, a best fit of $203.4^\circ$
with a $1\sigma$ interval $[165.6^\circ, 266.4^\circ]$ --- the prediction
sits about seven degrees from the world central value. The same analysis
excludes $\delta = +\pi/2$ at $3\sigma$ in both orderings; it does
\emph{not} exclude $3\pi/2$, whose death remains this framework's
prediction rather than a datum. \emph{Kill:} $196^\circ$ excluded at
$5\sigma$, or a converged maximal $270^\circ$. The $3\pi/2$ branch is
pre-registered dead and may not be reinterpreted.
\emph{Facility and date:} Hyper-Kamiokande from 2028, DUNE from first beam
in 2031, each reaching $\sim\pm15^\circ$ over the following years.
\emph{What dies:}
the leptonic Dirac-phase assignment lemma, and with it
Row~22 of the codebook.
\emph{What it measures:} the leptonic Dirac phase, and that alone. An
earlier reading of this falsifier held that the base value $\pi$ in the
closed form \emph{is} the Majorana parity $\alpha_{31} = \pi$, and hence
that this measurement reaches the sign of \etaB{}. That reading is
withdrawn: Appendix~K.1.5 proves that no oscillation experiment can ever
measure $\alpha_{31}$, so $\dPMNS{}$ cannot contain it. The $\pi$ is the
reference-channel convention of \S\ref{sec:PMNS}. What the long-baseline program therefore
measures is sharp, parameter-free and worth the facilities: the closed
form for the leptonic Dirac phase, and that alone. What it does
\emph{not} test is the origin of the matter asymmetry, which is tested
instead by $0\nu\beta\beta$ and by the neutrino mass sum. That separation
is a gain and not a retreat: two claims that can fail independently are
more testable than one claim that stands or falls whole.

\paragraph{F-SM-3e --- the atmospheric octant.} \emph{Prediction:}
$\sin^2\theta_{23} = \tfrac12(1+1/J) = 4/7$, QD-completed to $39/70 =
0.5571$ --- the upper octant, with the sign forced by which direction the
charged-lepton hierarchy distinguishes (\S\ref{sec:PMNS}).
\emph{Status: the two leading determinations disagree with each other.}
\textsc{NuFIT}~6.1 \citep{NuFIT61} places
$\sin^2\theta_{23} = 0.470^{+0.017}_{-0.014}$ in the lower octant; the
accelerator joint fit places it at $0.56^{+0.03}_{-0.05}$ in the upper. The
prediction sits $5.1\sigma$ from the first and $0.06\sigma$ from the
second. \emph{Kill:} a converged global-plus-accelerator determination
placing $\sin^2\theta_{23}$ in the lower octant at $5\sigma$.
\emph{Facility and date:} the long-baseline program of the 2030s ---
DUNE, Hyper-Kamiokande, and JUNO's atmospheric sample.
\emph{What dies:} not merely a tie-break. Lemma~L5 falls, and codebook
Row~21 with it (\S\ref{sec:theta23-split}).

This is the framework's sharpest current tension, and it is registered as
its own channel precisely so that it cannot be absorbed into a general
statement about the neutrino package.

\subsection{\texorpdfstring{F-SM-7}{F-SM-7}: the leptogenesis chain}
\label{subsec:fsm7}

Part~\ref{part:lepto} assembles a baryon-to-photon ratio from quantities
this paper derives, and it does so on one named hypothesis. That hypothesis
is falsifiable, and this channel registers it.

\emph{Prediction:} $\etaB \approx 7\times10^{-10}$ with the derived band
$(2.4\text{--}22)\times10^{-10}$, against an observed
$(6.12\pm0.04)\times10^{-10}$, from a resonant leptogenesis at
$\Mlock = 3.85$~TeV with zero fitted parameters --- the sign set by the
Majorana parity $\alpha_{31} = \pi$, the survivor carried by the electron
flavor because $\sin^2\theta_{13} = 3/140$ is small.

\emph{Kill, in three routes} (\S\ref{sec:lg-kills}). First, if a
finite-temperature transport computation returns the vacuum $N$-width class
for the resonance regulator, the chain undershoots by seven or eight orders
of magnitude and is falsified from within --- and the \emph{standard} fully
flavored Boltzmann limit does exactly this, which is why condition C4 is
the crux and not a formality. Second, if it returns the flavor-blind
$g^{2}T$ class, the excluded peak overshoot stands and the verdict is the
same. Third, if C4 and C5 close but the window deficit of C6 survives a
full transport computation at its adverse end with the priced
order-unity prefactors exhausted, the chain fails by throughput rather than
by regulator identity --- the failure mode whose size
Eq.~\eqref{eq:lg-E} now measures at $E = 7.1$, $[4.4, 11.4]$.

\paragraph{This channel is not like the others, and the difference is
worth stating.} Every other channel in this section waits on a facility:
on JUNO's exposure, on a tonne of enriched xenon, on a beam that does not
yet exist. \textbf{This one waits on a calculation.} The computation that
would settle it --- a Kadanoff--Baym or closed-time-path resummation of the
lock-crossing decays, carried far enough to identify what regulates the
resonance in a hot plasma --- requires no new data and could be done today
by anyone who wanted to. It is, in that specific sense, the most
immediately available kill test the framework possesses, and the framework
has an obvious interest in its not being attempted. That is the reason it
is named, dated as \emph{now}, and given its own channel rather than left
as a caveat inside Part~\ref{part:lepto}.

\emph{What dies:} the chain of Part~\ref{part:lepto}, and with it the
claim that this framework accounts for the matter content of the universe.
What does \emph{not} die is the neutrino block: the door, the parities, the
angles and the spectrum are established independently of the chain, and
F-SM-3a--3e test them separately. Nor would the failure return the
framework to silence --- it would return it to Sakharov custody plus an
unexplained number, which is where the field already is. \emph{Both
outcomes are scientific; only the unexamined middle would not be.}

\subsection{\texorpdfstring{F-SM-4}{F-SM-4}: exact proton stability}
\label{subsec:fsm4}

Proton decay occupies a special status, and the present framework makes a stronger
and sharper claim than any grand-unified competitor. Where GUTs \emph{predict}
decay (and are thereby confirmed only by a positive detection), \GG{} predicts
$\tau_p = \infty$ \emph{exactly}: proton absolute stability is a theorem
(T-SM-60.1 here; \AFive{} T-GDOS-9.8), following from exact \emph{global}
baryon number, residual to a Stueckelberg-massive local $\UoneB{}$.

\paragraph{A status correction, owed to \PaperPSix{}.} This paper stated
the theorem as conditional on the (H2) baryon-descent step and deferred
that step to a dedicated successor. \emph{That deferral has been repaid.}
\PaperPSix{} discharges (H2) --- its $\mathbb{Z}_3$-lock theorem realizes
the descended $\UoneB{}$ as component-resolved baryon number, with the
chiral floors carrying winding exactly $B$ --- and delivers (H1), the
bulk-only status of the $X, Y$ bosons, as a corollary rather than a
hypothesis. The standing conditions are therefore now (H3), the \BIsix{}
Green--Schwarz inflow at its home paper's grade, together with that
paper's two named residues: holonomy integrality on the Hopkins--Singer
clause, and the warped zero-mode profiles at fractional winding. The
claim is stronger than this paper previously recorded, and it is recorded
here at the successor's grade rather than at the deferral's.

The sharp content is a selection rule. Baryon number may be violated only in units
of $N_{\mathrm{gen}}=3$ (the $\mathrm{SU}(2)_L$ anomaly), so
\[
\Delta B = 1,\,2:\ \text{forbidden}; \qquad
\Delta B = 3:\ \text{admissible (sphaleron-suppressed,}\ \sim e^{-4\pi/\alpha_W}).
\]
Therefore \GG{} forbids \emph{both} the canonical proton-decay modes
($p\to e^+\pi^0$, $p\to K^+\bar\nu$, $\Delta B=1$) \emph{and} neutron--antineutron
oscillation ($\Delta B=2$). A single event in either class falsifies the
framework. Table~\ref{tab:proton-confrontation} confronts this with the present
bounds \citep{SuperKProton2020, SuperKNNbar2021} and the reach of the next generation \citep{HyperKamiokande2018}, and contrasts it with the GUT
predictions it displaces.

\begin{table}[H]
\centering
\caption[Baryon-number stability: the GG prediction ($\tau=\infty$, exact) confronted with current bounds and near-term reach, and contrasted with grand-unified.]{Baryon-number stability: the GG prediction ($\tau=\infty$, exact)
confronted with current bounds and near-term reach, and contrasted with
grand-unified predictions. \GG{} is falsified by \emph{any} single $\Delta B=1$ or
$\Delta B=2$ event; a GUT is confirmed only by such an event. Benchmark values as
of 2025--2026.}
\label{tab:proton-confrontation}
\small
\begin{tabular}{p{0.20\textwidth}p{0.21\textwidth}p{0.22\textwidth}p{0.135\textwidth}p{0.10\textwidth}}
\toprule
\textbf{Channel ($\Delta B$)} & \textbf{Current 90\% C.L. bound} & \textbf{Near-term reach ($\sim$2030s)} & \textbf{GUT expectation} & \textbf{GG} \\
\midrule
$p\to e^+\pi^0$ \,($1$) & $\tau > 2.4\times10^{34}$~yr (Super-K) & $\sim 1\times10^{35}$~yr (Hyper-K, data 2028) & $10^{34}$--$10^{36}$ (SO(10)) & $\infty$ \\
$p\to K^+\bar\nu$ \,($1$) & $\tau > 6.6\times10^{33}$~yr (Super-K) & $\sim\,$few$\,\times10^{34}$~yr (DUNE, Hyper-K) & $10^{33}$--$10^{35}$ (SUSY GUT) & $\infty$ \\
$n$--$\bar n$ oscillation \,($2$) & $\tau_{n\bar n} > 4.7\times10^{8}$~s (Super-K) & $2.8$--$4.4\times10^{9}$~s (ESS/NNBAR) & model-dependent & $\infty$ \\
\midrule
\multicolumn{5}{p{0.93\textwidth}}{\small Minimal $\mathrm{SU}(5)$ ($\tau_p\sim10^{31}$~yr) is already excluded; SO(10) and flipped $\mathrm{SU}(5)$ predict $10^{34}$--$10^{36}$~yr and are the live GUT targets. \GG{} predicts neither --- it predicts that \emph{no} $\Delta B=1$ or $\Delta B=2$ event ever occurs, so a single Hyper-K proton-decay event or a single ESS $n$--$\bar n$ event would be fatal.} \\
\bottomrule
\end{tabular}
\end{table}

This is the cleanest single falsifier the framework offers. Continued null results are positive
consistency evidence; one unambiguous event is a death certificate.

\paragraph{The same rule, read at colliders.} The selection rule does not
care about the venue. Any collider signal requiring $\Delta B = 1$ or
$\Delta B = 2$ --- leptoquark-mediated baryon-number violation,
$n$--$\bar n$-type operators in same-sign top pairs, any of it ---
falsifies the framework exactly as a proton-decay event would. And the
admissible door swings the other way at high energy: sphaleron-induced
$\Delta B = \Delta L = 3$ events at a $100$~TeV collider, if the
contested near-threshold suppression permits them
(\S\ref{sec:lg-accelerator}), would be the conversion vertex of
Part~\ref{part:lepto} observed directly --- the same transition that made
the baryons, run in a detector. One rule, three verdicts: protons
immortal, $\Delta B = 1, 2$ events fatal, $\Delta B = 3$ events a
confirmation no other framework predicts with this exclusivity.

\subsection{\texorpdfstring{F-SM-5}{F-SM-5}: the baryonic fifth force}
\label{subsec:fsm5}

Because $\UoneB{}$ is anomalous on the Standard-Model spectrum and its anomaly is
carried by the \BIsix{} Green--Schwarz inflow, the baryonic gauge boson eats the
\BIsix{} tensor and acquires a Stueckelberg mass $m_B$. The residual interaction
between baryon numbers is therefore a \emph{finite-range} Yukawa,
\begin{equation}
V_B(r) = \frac{g_B^2}{4\pi}\,\frac{B_1 B_2}{r}\,e^{-m_B r},
\qquad
\lambda = \frac{\hbar}{m_B c} \simeq 0.20~\mu\mathrm{m}\,
\Big(\frac{\mathrm{eV}}{m_B}\Big).
\label{eq:ybaryon-force-disc}
\end{equation}
The Stueckelberg mass leaves the \emph{global} baryon number exact --- it is that
exact global symmetry, not a massless gauge force, that stabilizes the proton, so
F-SM-5 and F-SM-4 are two faces of one structure. The honest statement is
therefore \emph{exact global baryon number, residual to a Stueckelberg-massive
$\UoneB{}$}, not ``exact massless local $\UoneB{}$''.
Table~\ref{tab:fifth-force} quantifies the two regimes against the present
precision-force frontier.

\begin{table}[H]
\centering
\caption[The baryonic fifth force of Eq.~\eqref{eq:ybaryon-force-disc} in its two regimes, set by the Stueckelberg mass $m_B$, against the leading.]{The baryonic fifth force of Eq.~\eqref{eq:ybaryon-force-disc} in its two
regimes, set by the Stueckelberg mass $m_B$, against the leading equivalence-principle
and short-range-gravity bounds \citep{MICROSCOPE2022} (2022--2025). The two regimes are bracketed separately because a single limit does not apply to both, and the bounds quoted are the published ones.}
\label{tab:fifth-force}
\small
\begin{tabular}{p{0.17\textwidth}p{0.20\textwidth}p{0.27\textwidth}p{0.25\textwidth}}
\toprule
\textbf{Regime} & \textbf{Range $\lambda=\hbar/m_Bc$} & \textbf{Confronting experiment} & \textbf{Status} \\
\midrule
$m_B \gtrsim 1$~eV (structurally favored) & $\lambda \lesssim 0.2~\mu$m & below the reach of sub-mm gravity (E\"ot-Wash torsion balance) and orbital EP tests & no observable long-range force; baryon number stays exact-global \\[4pt]
$m_B \ll 1$~eV (light) & $\lambda \gg$~mm & MICROSCOPE EP test: $\eta(\mathrm{Ti},\mathrm{Pt}) = [-1.5\pm2.3\pm1.5]\times10^{-15}$; lunar laser ranging & a baryon-coupled long-range force is tightly bounded; constrains $g_B$ vs.\ $\lambda$ \\
\bottomrule
\end{tabular}
\end{table}

The structurally favored short-range regime evades all current bounds, but a
finite $m_B$ leaves a derivable window in the $\mu$m--mm band that next-generation
torsion-balance and isotope-dependent searches can probe; the matched $(g_B,m_B)$
window is to be computed in the dedicated fifth-force paper of the P-series.
Meanwhile \PaperPOne{} (its L-KK-1 registration) has carried $g_B$ to its honest
boundary: the generator norm is theorem-grade --- the same $2/5$ trace ratio as
this paper's Arisaka-equation lock, Eq.~\eqref{eq:bi6-tracenorm-A9} --- and one
discrete seam-termination readout remains open, bracketing
$g_B \in \{0.14,\,0.23,\,0.57,\,0.89\}$.  The window computation therefore
reduces from a two-parameter plane scan to a selection among four candidate
lines. \emph{Failure:}
discovery of a long-range ($\lambda \gg$~mm) baryon-coupled force --- which would
contradict the Stueckelberg mass --- or exclusion of the entire derived
$(g_B,m_B)$ window.

\subsection{Consolidated falsification map}
\label{subsec:falsification-map}

Table~\ref{tab:falsification-consolidated} collects the seven dedicated channels in
one view, each tied to the completed companion that carries it to a number and to
the experimental program that tests it. The scientific moral is worth stating
plainly: a theory that derives much from little is not shielded by that
achievement; it is rendered maximally vulnerable, and its virtue is measured by how
hard the data can hit it.

\begin{table}[H]
\centering
\caption[Consolidated falsification map of the \GGTheory{} Standard Model.]{Consolidated falsification map of the \GGTheory{} Standard Model. Every
channel is now carried to a quantitative prediction by a completed companion paper. Each row names the channel, the quantitative prediction, and the measurement that would end it. A falsification map is only as good as the sharpness of its entries, and these entries are sharp because the downstream companions carried them to numbers. Before those papers existed the same table would have been a list of directions rather than of tests.}
\label{tab:falsification-consolidated}
\small
\usebox{\GGscFalsMap}
\end{table}


\subsection{Falsification timeline: within reach by 2030 and 2040}
\label{subsec:falsification-timeline}

The decisive point is not merely that the framework is falsifiable in principle but
that the falsification is \emph{near}. None of the channels above requires a
generational wait: each is settled, or decisively advanced, within one to two
decades by facilities already running or under construction.
Table~\ref{tab:falsification-timeline} sets out the schedule.

\emph{By 2030}, the first wave arrives. JUNO, running since 2025, approaches a
standalone $3\sigma$ determination of the neutrino mass ordering around 2032 and
reaches it earlier in combination;
Hyper-Kamiokande \citep{HyperKamiokande2018}, taking data from 2028, overtakes the
Super-Kamiokande proton bound and begins to bite into the $\mathrm{SO}(10)$
territory that \GG{} forbids; the High-Luminosity LHC begins to probe the
$3.85$~TeV multiplet directly; DESI together with the Simons Observatory
pushes $\sum m_\nu$ toward the predicted $59$~meV; and the ADMX program
\citep{ADMX2025} scans upward toward the $1.95$~GHz Kaxion line. The
ground-based partner originally written into several of these tests,
CMB-S4, is no longer available: DOE and NSF ended their support in July
2025, and the cosmological channels above now rest on the Simons
Observatory beside LiteBIRD, Euclid and Rubin.

\emph{By 2040}, the verdict on most channels is in. Hyper-Kamiokande reaches its
$10^{35}$~yr proton sensitivity --- where a single $\Delta B=1$ event is fatal to
\GG{}; DUNE delivers the leptonic Dirac phase and the $p\to K^+\bar\nu$ channel; the
ESS/NNBAR program improves the $n$--$\bar n$ figure of merit by three orders of
magnitude; tonne-scale $0\nu\beta\beta$ experiments reach the few-meV Majorana
floor; next-generation haloscopes and torsion balances close the Kaxion and
fifth-force windows; and the FCC era opens on the full lock-cluster multiplet.

\begin{table}[H]
\centering
\caption[Falsification timeline of the \GGTheory{} Standard Model.]{Falsification timeline of the \GGTheory{} Standard Model. Every predicted
number is decisively tested, or its testing facility is operating, within one to two
decades. ``Settles'' means the measurement reaches the precision that confirms or
refutes the GG value; ``decisive'' means a single result can be fatal.}
\label{tab:falsification-timeline}
\small
\begin{tabular}{p{0.165\textwidth}p{0.19\textwidth}p{0.27\textwidth}p{0.235\textwidth}}
\toprule
\textbf{Prediction} & \textbf{GG value at stake} & \textbf{By $\sim$2030} & \textbf{By $\sim$2040} \\
\midrule
Lock cluster & $3.85$~TeV multiplet & HL-LHC first reach into the corridor & HL-LHC full $+$ FCC: decisive on the multiplet \\
Proton $p\to e^+\pi^0$ & $\tau=\infty$ ($\Delta B=1$) & Hyper-K overtakes Super-K ($\sim5\times10^{34}$~yr) & Hyper-K $10^{35}$~yr: \emph{any} event fatal \\
Dark energy $w$ & $w=-1$ exact (two-sided) & DESI full survey: evolving-DE hints confirmed or retired & Euclid $+$ Rubin $+$ Simons Obs.: any confirmed $w\neq-1$ fatal \\
$p\to K^+\bar\nu$ & $\tau=\infty$ ($\Delta B=1$) & DUNE begins & DUNE $+$ Hyper-K, few$\,\times10^{34}$~yr \\
$n$--$\bar n$ & forbidden ($\Delta B=2$) & (Super-K limit holds) & ESS/NNBAR, $\times10^3$ figure of merit \\
$\nu$ ordering & normal & JUNO settles & confirmed/refuted \\
$\delta_{\mathrm{PMNS}}$ & $\simeq196^\circ$ & DUNE/Hyper-K first measurement & decisive precision \\
$\sum m_\nu$ & $\simeq59$~meV & DESI DR2$+$CMB at $<64$~meV ($\Lambda$CDM) & decisive at the $59$~meV line \\
$m_{\beta\beta}$ ($0\nu\beta\beta$) & meV, phases $(0,\pi)$ & LEGEND-200, KamLAND-Zen & LEGEND-1000/nEXO at few meV \\
Kaxion & $8.1~\mu$eV / $1.95$~GHz & ADMX approaches the line & next-gen haloscopes: decisive \\
Fifth force & $\lambda=0.2~\mu$m$(\mathrm{eV}/m_B)$ & next-gen torsion balance & MICROSCOPE successor $+$ sub-mm \\
\bottomrule
\end{tabular}
\end{table}

These schedules are developed in full in a forthcoming \emph{P-series} of dedicated
prediction papers --- on the neutrino package, on proton decay and baryon-number
violation, and on the baryonic fifth force --- each of which will turn one row of
Table~\ref{tab:falsification-timeline} into a quantitative experimental forecast,
extending the collider and dark-matter forecasts already given in \PaperPOne{} and
\PaperCOne{}. The point is already plain: the \GGTheory{} Standard Model
is not a philosophy but a wager whose decisive returns fall within the working
lifetime of the present generation of experiments. The same schedule is drawn
as a single kill-test timeline in Figure~\ref{fig:killtest-timeline-sm}
(\S\ref{subsec:experimental-roadmap}).

\section{Open questions}
\label{sec:open-questions}

An honest closing distinguishes three registers: the questions a careful referee
could press, which the framework answers; the items that were genuinely open at
first submission but are now \emph{closed} by completed companions; and the items
that remain undetermined and define the successor work. The first is collected in
the Common Objections Part (Part~\ref{part:XVII}, eighty-three objections);
the second and third are the substance of this section, summarized in
Table~\ref{tab:open-questions}.

\subsection{What is now closed downstream}
\label{subsec:now-closed}

Three classes of statement that the first version of this paper listed as
``deferred'' or ``out of scope'' are now settled in completed cohort papers, and
the present text has been updated to say so:
\begin{itemize}\itemsep1pt
\item \textbf{The cosmological inventory} --- the matter, dark-matter, dark-energy
and baryon fractions, the recombination redshift, and the Hubble corridor --- is
\emph{derived} in \ATen{} \citep{ArisakaA10Cosmos} from the same ledger that fixes
the particles (overview in \S\ref{subsec:a10-overview}). It is no longer an open
question of the present paper.
\item \textbf{The family number $N=3$} is closed as a three-leg theorem in
\PaperPTwo{} \citep{ArisakaP2N3} and imported here; the present paper depends on it
rather than leaving it open.
\item \textbf{The Kaxion dark-matter sector} has a definite mass and coupling
window in \PaperCOne{} \citep{ArisakaC1Kaxion}, and the dark-matter relic it must
saturate is fixed in \ATen{}.
\end{itemize}

\subsection{What remains genuinely open}
\label{subsec:genuinely-open}

The following items are undetermined at the present stage. Each is a sharp,
delegable research target, not a vague caveat.
\begin{itemize}\itemsep1pt
\item \textbf{The flavor sheet.} The nine charged-fermion Yukawas, the four PMNS
parameters, and the two Majorana phases are presented as closed-form rationals over
the \OGNfive{} ledger, but the specific integer assignment (numerator vs.\
denominator, exponent vs.\ prefactor) is a working hypothesis, not a
Hopkins--Singer-forced result. Any future flavor derivation must live inside the
attractor basin already fixed here, disturbing none of the derived electroweak
quantities.
\item \textbf{The up/down isospin doubling.} The five quark levels generated from
three Morse critical points are identified by analogy with the Bogomolnyi cascade
on $\mathbb{C}P^{2}$ (\ASix{}~\S29), not yet derived by an equivariant-cohomology
computation in this paper.
\item \textbf{The topological residue R-INT.} The $N_g\le3$ upper bound rests on the
\BIsix{} instanton-action budget, which is no longer a conjecture: the capacity
theorem T-N3-6 of \citet{ArisakaP2N3} fixes it, the integrality theorem L-INT
fixes its lattice, and two obstruction theorems show why the conjectural framing
could not have been discharged as posed. What remains open is one named
topological residue, R-INT --- that the \BIsix{} seam supports the
Hopkins--Singer quadratic refinement globally --- and it is met on the benchmark
background by the inflow companion.
\item \textbf{QD precision corrections.} The QD layer is at leading transport
order; hadronic-threshold and lattice-matched running corrections to the charm- and
bottom-quark masses are not included.
\item \textbf{$0\nu\beta\beta$ nuclear matrix elements.} Only the Majorana-phase
structure is constrained here; the rate also needs nuclear matrix elements not
derived in the present paper.
\item \textbf{The quantum-gravity boundary-state proof} at the lock scale is
partially resolved by the composite \GG{} connection of \ATwo{} but the rigorous
boundary-state proof is deferred --- including a precise account of the dual role of the
radial coordinate $r$, at once the renormalization-group scale $\ln(\EGUT/\mu)$ and the
geometric variable whose $e^{-2r}$ profile fixes $M_{\mathrm{Pl}}$ (developed in \ATwo{}).
\item \textbf{The higher cascade uniqueness theorems.} The chain from charged-lepton
identity through hadrons, elements, molecules, ATP/RNA/DNA to life and mind carries
its own uniqueness theorem at each layer; only the lowest is closed here. The
capstone \AEleven{} \citep{ArisakaA11Origin} addresses the remainder.
\end{itemize}

\begin{table}[H]
\centering
\caption[Status ledger of the questions a reader could press on the \GGTheory{} Standard Model: closed in this paper, closed downstream since first submission, or.]{Status ledger of the questions a reader could press on the \GGTheory{}
Standard Model: closed in this paper, closed downstream since first submission, or
genuinely open with its delegated owner. The middle category exists because this paper was written before some of its own companions were finished, and recording that is more honest than folding those items into either of the other two.}
\label{tab:open-questions}
\small
\begin{tabular}{p{0.30\textwidth}p{0.20\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Question} & \textbf{Status} & \textbf{Resolution / owner} \\
\midrule
28 SM parameters & closed (this paper) & codebook over $(1,3,5;7,35)$ \\
Family number $N=3$ & closed downstream & \PaperPTwo{} three-leg theorem \\
Cosmic inventory ($\Omega$'s, $z_*$, $H_0$) & closed downstream & \ATen{} (\S\ref{subsec:a10-overview}) \\
Dark-matter identity & closed downstream & \PaperCOne{} Kaxion, $8.1~\mu$eV \\
Flavor textures (Yukawas, mixings) & \emph{open} & integer assignment a working hypothesis \\
Up/down isospin doubling & \emph{open} & equivariant-cohomology proof needed \\
$N_g\le3$ residue R-INT & \emph{open (narrowed)} & seam refinement; \PaperPTwo{} L-INT \\
QD hadronic-threshold corrections & \emph{open} & precision-physics extension \\
$0\nu\beta\beta$ rate & \emph{open} & nuclear matrix elements \\
Lock-scale quantum gravity & \emph{open} & boundary-state proof, future work \\
Higher cascade (life, mind) & \emph{open} & capstone \AEleven{} \\
\bottomrule
\end{tabular}
\end{table}


\section{The completed cohort and future prospects}
\label{sec:future-prospects}

The closure of the twenty-eight-parameter codebook completes the \emph{particle
floor} of the \GGTheory{} program. What is different now from any earlier reading
of this paper is that the floors above and beside it are already built: the
cosmological companion is finished, and all six phenomenology and consequence
preprints exist. This section therefore reads not as a wish-list but as a map into
a \emph{completed} downstream cohort. The single most important forward link is
\ATen{}, and we treat it first.

\subsection{The most important forward link: an overview of \texorpdfstring{\ATen{}}{A10}-Cosmos}
\label{subsec:a10-overview}

The natural sequel to a derivation of the particles is a derivation of the cosmos,
and it now exists. \ATen{} \citep{ArisakaA10Cosmos} --- \emph{The Standard
Cosmology as Theorem of \GGTheory{}} --- closes the cosmic side of the same first
floor that this paper closes on the particle side, and it does so from the
\emph{identical} input: the ledger $(1,3,5;7,35)$ and the \GUfive{} constants, with
no adjustable cosmological dials. Where for forty years $\Lambda$CDM has described
the universe with a handful of numbers read off the sky and explained none of them,
\ATen{} argues those numbers are not free. The headline results are collected in
Table~\ref{tab:a10-inventory}.

\begin{table}[H]
\centering
\caption[The cosmic inventory derived in \ATen{} \citep{ArisakaA10Cosmos} from the same ledger and \GUfive{} constants that fix the particles in the present paper.]{The cosmic inventory derived in \ATen{} \citep{ArisakaA10Cosmos}
from the same ledger and \GUfive{} constants that fix the particles in the present
paper. These close, on the cosmological side, the same geometry the present paper
closes on the particle side. \ATen{} carries its own honesty triage (several
entries are conjectural $D^{*}$ or near-hits); the present paper reports them as
\ATen{}'s results, not as its own. The recombination-era row follows the
two-quantity treatment of \citealp{ArisakaA10Cosmos}: the corridor switch
$e^{7}$ is a topological marker, while the measured decoupling redshift is
reproduced by \ATen{}'s own atomic chain at the $0.1\%$ level with zero
fitted parameters, $0.4\%$ (in e-folds) below the switch.}
\label{tab:a10-inventory}
\small
\begin{tabular}{p{0.30\textwidth}p{0.27\textwidth}p{0.15\textwidth}p{0.16\textwidth}}
\toprule
\textbf{Cosmic quantity} & \textbf{GG closed form} & \textbf{GG value} & \textbf{Observed} \\
\midrule
Matter fraction & $\Omega_m = 1/\pi$ & $0.318$ & $0.315\pm0.007$ \\
Dark energy & $\Omega_\Lambda = 1-1/\pi$ & $0.682$ & $0.685\pm0.007$ \\
Baryon fraction & $\Omega_b = 1/(2\pi^2)$ & $0.0507$ & $0.0493\pm0.0006$ \\
Dark matter & $\Omega_{\mathrm{dm}} = 1/\pi - 1/(2\pi^2)$ & $0.267$ & $0.265\pm0.007$ \\
Recombination-era redshifts & switch $z_{\mathrm{sw}} = e^{7}$; decoupling $z_{\mathrm{dec}}$ from \ATen{}'s atomic chain & $1097$; $1091$ & $1089.92\pm0.25$ \\
Hubble background & $H_0^{(0)}=\sqrt{\omega_i/\Omega_i}$ (derived) & $66.6\pm0.2$ & $67.4\pm0.5$ \\
Age of the universe & $t_0 = 0.948/H_0$ & $13.8$--$13.9$~Gyr & $13.80\pm0.02$~Gyr \\
\bottomrule
\end{tabular}
\end{table}

The structural message is the one this paper has argued throughout, now extended by
one floor: \ANine{} closes the particle inventory and \ATen{} closes the cosmic
inventory, both as theorems of one six-dimensional geometry. The particle data this
paper fixes --- the electroweak scale, the neutrino block and its CP phase, the
strong-CP suppression, the baryonic sector, and the compact-angle ontology --- are
precisely the inputs that \ATen{} inherits to constrain inflation, dark matter,
dark energy, baryogenesis, and the recombination history, rather than choosing them
from a free menu. The Hubble tension itself is read in \ATen{}, and in detail in
\PaperCTwo{}, not as a measurement error but as the two ends of one geometric
corridor.

\subsection{The completed downstream cohort: a forward map}
\label{subsec:forward-map}

Beyond \ATen{}, the present paper feeds two branches of completed preprints --- the
Predictions Branch and the Consequences Branch. Each builds on a specific result of
this paper; Table~\ref{tab:forward-map} is the forward map, with the headline
number of each and its dependency on \ANine{}. These are the forward links the
reader should follow from here.

\begin{table}[H]
\centering
\caption[Forward map from the \GGTheory{} Standard Model into the completed downstream cohort: the cosmological capstone, the three-paper Predictions Branch, and the.]{Forward map from the \GGTheory{} Standard Model into the completed
downstream cohort: the cosmological capstone, the three-paper Predictions Branch,
and the three-paper Consequences Branch. Every entry is a finished preprint. Nothing in this table is a plan. Every entry is a document that can be read, and where a downstream paper has narrowed or corrected something stated here, its row says so.}
\label{tab:forward-map}
\small
\begin{tabular}{p{0.13\textwidth}p{0.20\textwidth}p{0.27\textwidth}p{0.25\textwidth}}
\toprule
\textbf{Paper} & \textbf{Builds on (from \ANine{})} & \textbf{Headline result} & \textbf{Where it is tested} \\
\midrule
\multicolumn{4}{l}{\textit{Cosmology}} \\
\ATen{} & full particle inventory & $\Omega_m\!=\!1/\pi$, $\Omega_\Lambda\!=\!1\!-\!1/\pi$, $\Omega_b\!=\!1/(2\pi^2)$, $z_{\rm sw}\!=\!e^7$ & Planck, DESI, Simons Obs. \\
\midrule
\multicolumn{4}{l}{\textit{Predictions Branch}} \\
\PaperPOne{} & $\Mlock$, $\UoneB{}$ & 3.85~TeV lock cluster: graviton $+$ $g^{*}$ $+$ $Z'_B$ & HL-LHC, FCC \\
\PaperPTwo{} & (provides $N=3$) & three-leg derivation of $N=3$ & structural \\
\PaperPThree{} & lepton sector & electron as unique boundary ground state & precision QED, stability \\
\midrule
\multicolumn{4}{l}{\textit{Consequences Branch}} \\
\PaperCOne{} & $S^1_\theta$, $\UoneQi{}$ & $8.1~\mu$eV cold dark matter ($\Omega_{\mathrm{dm}}$) & haloscopes at $1.95$~GHz \\
\PaperCTwo{} & boundary GR, scale chain & $73.5$ vs.\ $67$ as one corridor ($7\sigma$ tension) & distance ladder, CMB \\
\PaperCThree{} & \GDOS{}/\OGGEP{} horizon & exact Bekenstein--Hawking $S=A/4G_N$ & black-hole thermodynamics \\
\bottomrule
\end{tabular}
\end{table}

The two branches divide the labor cleanly. The \emph{Predictions Branch} carries
the falsifiable signals of the present paper to definite numbers:
\PaperPOne{} \citep{ArisakaP1KK} is the standing collider falsifier (the 3.85~TeV
multiplet and the $Z'_B$ window); \PaperPTwo{} \citep{ArisakaP2N3} supplies the
$N=3$ that this paper imports; \PaperPThree{} \citep{ArisakaP3Electron} closes the
six oldest questions about the electron --- its identity and measured values, its
absolute stability, its exactly two heavier copies, its mirror twin (the positron
as conjugate readout), the absence of substructure, and its response to a field --- as properties of the unique ground state of the
lepton sector derived here. The \emph{Consequences Branch} carries the framework
into the hardest external arenas: \PaperCOne{} \citep{ArisakaC1Kaxion} identifies
the dark matter; \PaperCTwo{} \citep{ArisakaC2Hubble} reads the $7\sigma$ Hubble
tension (the 2026 consensus $73.50\pm0.81$ vs.\ the CMB value near $67$) as one
six-dimensional corridor; and \PaperCThree{} \citep{ArisakaC3BH} derives the exact
Bekenstein--Hawking entropy $S=A/4G_N$ from induced gravity and resolves the
information paradox through the \GDOS{}/\OGGEP{} horizon port.

\subsection{Genuine future work: the capstone and the higher cascade}
\label{subsec:genuine-future}

The natural successor to the completed cohort is the capstone volume \AEleven{}
\citep{ArisakaA11Origin} --- \emph{The Geometric Inevitability of Cosmos, Life, and
Mind} --- which extends the uniqueness logic above the particle layer into the full
structure-formation cascade the six-dimensional bulk permits. Where the present
paper closes the parameter sheet and \ATen{} closes the cosmic inventory, the
capstone targets the higher uniqueness theorems --- hadrons, elements, interstellar
molecules, the ATP/RNA/DNA layer, single-cell life, and ultimately mind --- of
which only the lowest is closed here. The honest professional stance remains
unchanged by the breadth of this program: what the present paper has earned is not
victory but \emph{eligibility for serious testing}.

\subsection{A disciplined experimental roadmap for the next decade}
\label{subsec:experimental-roadmap}

A framework that derives a large fraction of low-energy physics from a tiny ledger
should say not merely that it is falsifiable but \emph{how} the field can try to
falsify it. The roadmap is best organized by structural sector, because the same
geometry ties the sectors together: the neutrino phases that appear in oscillation
experiments also drive baryogenesis; the compact-angle sector that assists symmetry
breaking also supplies the dark matter; the baryonic extension that protects the
proton also governs the fifth-force search. Table~\ref{tab:experimental-roadmap}
names the facilities, the now-derived numbers they test, and --- crucially --- how
each result strengthens or weakens the \emph{whole} framework rather than an
isolated module; Figure~\ref{fig:killtest-timeline-sm} then lays the same
program out as one forward time axis --- the nine decisive kill-tests,
color-coded by sector, from today's bounds through the first wave at
$\sim$2030 to the decisive decade at $\sim$2040.

\begin{table}[H]
\centering
\caption[A disciplined experimental roadmap for the next decade.]{A disciplined experimental roadmap for the next decade. Each row tests a
specific derived number against a named facility; the same measurement program can
strengthen or weaken the entire \GG{}/\GDOS{} framework, not one sector. The horizons are the facilities' own published ones and not this paper's expectations of them, and no row claims a sensitivity a facility has not itself projected.}
\label{tab:experimental-roadmap}
\small
\begin{tabular}{p{0.155\textwidth}p{0.225\textwidth}p{0.165\textwidth}p{0.155\textwidth}p{0.13\textwidth}}
\toprule
\textbf{Frontier} & \textbf{Derived number tested} & \textbf{Facility} & \textbf{Strengthens GG if} & \textbf{Weakens GG if} \\
\midrule
Energy frontier & 3.85~TeV multiplet ($g^{*},Z'_B$, graviton) & HL-LHC, FCC & coherent multiplet at $3.8$--$4.0$~TeV & full corridor excluded \\
Proton decay & $\tau_p=\infty$; $\Delta B=1,2$ forbidden & Hyper-K, DUNE, JUNO & continued null at $10^{35}$~yr & one $\Delta B=1$ event \\
$n$--$\bar n$ & $\Delta B=2$ forbidden & ESS / NNBAR & null at $\sim10^{9}$~s & one oscillation event \\
Neutrinos & NO, $\sum m_\nu\!\simeq\!59$~meV, $\delta\!\simeq\!196^\circ$ & DUNE, Hyper-K, JUNO, Simons Obs. & convergence on the package & stable incompatible pattern \\
$0\nu\beta\beta$ & Majorana $(0,\pi)$, $m_{\beta\beta}$ (meV) & LEGEND-1000, nEXO & signal in predicted band & exclusion of the band \\
Dark matter & Kaxion $8.1~\mu$eV / $1.95$~GHz & ADMX-class haloscopes & line at derived coupling & line excluded \\
Fifth force & $\lambda=0.2~\mu$m$(\mathrm{eV}/m_B)$ & E\"ot-Wash, MICROSCOPE successor & null at short range & long-range $B$-force \\
Cosmology & $\Omega$'s, $z_*$, $H_0$ corridor & DESI, Simons Obs., Rubin & one constrained inheritance & need for new free sectors \\
Matter excess & $\etaB\approx7\times10^{-10}$ from $\Mlock$ & \emph{theory}: finite-$T$ transport & the regulator lands in the partial-resonance window & it returns the $N$-width or $g^{2}T$ class \\
\bottomrule
\end{tabular}
\end{table}

% Falsification / kill-test timeline for \ANine{} (mirrors the A10-Cosmos timeline)
% PURE TikZ. Libraries: arrows.meta, positioning, calc, shapes.geometric, fit, backgrounds
\begin{figure}[!t]
\centering
\resizebox{0.748\textwidth}{!}{%
\begin{tikzpicture}[
    font=\large,
    >=Latex,
    card/.style={
        draw=black!55, rounded corners=2pt, line width=0.60pt,
        align=left, inner sep=4pt, text width=52mm
    },
    stem/.style={line width=0.90pt, draw=black!55},
    sectorCOL/.style={fill=orange!12, draw=orange!75!black},
    sectorBAR/.style={fill=blue!8,   draw=blue!55!black},
    sectorNU/.style={fill=green!10, draw=green!55!black},
    sectorDM/.style={fill=violet!10, draw=violet!60!black},
    dot/.style={circle, draw=black!70, fill=white, inner sep=0pt, minimum size=5.2pt}
]

% --- Geometry of the time axis ---
\def\xL{0}\def\xR{30}\def\yA{0}
\def\bA{9.5}\def\bB{19.0}

% Era background bands
\begin{scope}[on background layer]
    \fill[black!2]  (\xL,-8.4) rectangle (\bA,9.6);
    \fill[black!4]  (\bA,-8.4) rectangle (\bB,9.6);
    \fill[black!6] (\bB,-8.4) rectangle (\xR,9.6);
\end{scope}

% Era labels (top)
\node[anchor=south, font=\Large\bfseries] at ({(\xL+\bA)/2},8.8) {NOW / ONGOING};
\node[anchor=south, font=\large] at ({(\xL+\bA)/2},8.15) {current bounds, $\sim$2026};
\node[anchor=south, font=\Large\bfseries] at ({(\bA+\bB)/2},8.8) {BY $\sim$2030};
\node[anchor=south, font=\large] at ({(\bA+\bB)/2},8.15) {the first wave};
\node[anchor=south, font=\Large\bfseries] at ({(\bB+\xR)/2},8.8) {BY $\sim$2040};
\node[anchor=south, font=\large] at ({(\bB+\xR)/2},8.15) {the verdict on most channels};

% The time arrow
\draw[->, line width=1.10pt] (\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=0.80pt] (\x,-0.18) -- (\x,0.18);}
\node[anchor=north, font=\large] at (\xL,-0.25) {2026};
\node[anchor=north, font=\large] at (\xR,-0.25) {2040+};

% ---- TOP ROW (above axis) ----
% Kaxion haloscope (ongoing -> ~2030, scan bar)
\node[dot] (d2) at (4.6,\yA) {};
\node[card, sectorDM, anchor=south] (c2) at (4.6,1.5) {%
    \textbf{F-SM-2. Kaxion haloscope}\\[1pt]
    line: $8.1\,\mu$eV / 1.95 GHz\\
    refutes: the compact-angle dark sector if excluded at DFSZ depth\\
    venue: ADMX, CAPP, ADMX-EFR};
\draw[stem] (d2) -- (c2.south);
\draw[violet!60, line width=2.2pt] (1.0,-0.55) -- (12.5,-0.55);
\node[violet!60, font=\large, anchor=north] at (6.7,-0.6) {scan span};

% Neutrino package (JUNO ~2030; CMB-S4 sum m_nu mid-2030s), tall/wide
\node[dot] (d3) at (13.0,\yA) {};
\node[card, sectorNU, anchor=south, text width=62mm] (c3) at (13.0,5.1) {%
        \textbf{F-SM-3a/3c/3d/3e. Neutrino package}\\[1pt]
    NO; $\sum m_\nu\!\simeq\!59$ meV; $\delta_{\mathrm{PMNS}}\!\simeq\!196^\circ$\\
    refutes: the Majorana block if the converged data reject the package\\
    venue: JUNO, DUNE, Hyper-K, Simons Obs.};
\draw[stem] (d3) -- (c3.south);

% Lock cluster (HL-LHC reach ~2030; FCC decisive)
\node[dot] (d1) at (17.5,\yA) {};
\node[card, sectorCOL, anchor=south] (c1) at (17.5,1.5) {%
    \textbf{F-SM-1. Lock cluster}\\[1pt]
    $g^{*}$, $Z'_B$, graviton at $3.85$ TeV\\
    refutes: the $K=35$ throat if the 3.8--4.0 TeV multiplet is excluded\\
    venue: HL-LHC $\to$ FCC};
\draw[stem] (d1) -- (c1.south);

% 0nubb (LEGEND-1000/nEXO 2030s), tall/wide
\node[dot] (d5) at (23.4,\yA) {};
\node[card, sectorNU, anchor=south, text width=66mm] (c5) at (22.9,4.9) {%
    \textbf{F-SM-3b. Majorana floor ($0\nu\beta\beta$)}\\[1pt]
    phases $(0,\pi)$; $m_{\beta\beta}$ at few meV\\
    refutes: the Majorana block if the band is excluded\\
    venue: LEGEND-1000, nEXO};
\draw[stem] (d5) -- (c5.south);

% Fifth force (next decade, right)
\node[dot] (d6) at (27.3,\yA) {};
\node[card, sectorDM, anchor=south] (c6) at (27.3,1.5) {%
    \textbf{F-SM-5. Fifth force}\\[1pt]
    Yukawa $\lambda = 0.2\,\mu$m$\,(\mathrm{eV}/m_B)$\\
    refutes: Stueckelberg $\mathrm{U}(1)_B$ if a long-range $B$-force appears\\
    venue: E\"ot-Wash, MICROSCOPE-next};
\draw[stem] (d6) -- (c6.south);

% ---- BOTTOM ROW (below axis) ----
% Leptogenesis chain (F-SM-7).  Deliberately in the NOW band: unlike every
% other card this one waits on a CALCULATION, not on a facility, so it
% carries no future date and could be settled today.
\node[dot] (e9) at (2.5,\yA) {};
\node[card, sectorNU, anchor=north] (b9) at (2.5,-1.5) {%
    \textbf{F-SM-7. Leptogenesis chain}\\[1pt]
    $\etaB\!\approx\!7\!\times\!10^{-10}$, band $\times3^{\pm1}$\\
    refutes: the chain if finite-$T$ transport returns the $N$-width or
    $g^{2}T$ regulator\\
    venue: \emph{theory} --- available now};
\draw[stem] (e9) -- (b9.north);

% Proton decay (Hyper-K from 2027; 1e35 by 2040)
\node[dot] (e4) at (8.0,\yA) {};
\node[card, sectorBAR, anchor=north] (b4) at (8.0,-1.5) {%
    \textbf{F-SM-4. Proton stability}\\[1pt]
    $\tau_p = \infty$: \emph{any single} $\Delta B\!=\!1$ event is fatal\\
    refutes: gauged $\mathrm{U}(1)_B$ protection outright\\
    venue: Hyper-K ($10^{35}$ yr), DUNE};
\draw[stem] (e4) -- (b4.north);

% n-nbar (ESS/NNBAR by 2040)
\node[dot] (e7) at (21.0,\yA) {};
\node[card, sectorBAR, anchor=north] (b7) at (21.0,-1.5) {%
    \textbf{F-SM-4b. $n$--$\bar n$ oscillation}\\[1pt]
    $\Delta B = 2$ forbidden\\
    refutes: exact baryon protection if one oscillation event is seen\\
    venue: ESS / NNBAR ($\times10^{3}$)};
\draw[stem] (e7) -- (b7.north);

% Dark energy w = -1 (DESI now -> Euclid/Rubin/CMB-S4)
\node[dot] (e8) at (14.8,\yA) {};
\node[card, sectorDM, anchor=north, text width=70mm] (b8) at (14.8,-4.15) {%
    \textbf{F-SM-6. Dark energy $w=-1$ exact}\\[1pt]
    two-sided; no evolution\\
    refutes: the \GRIP{} attractor on any confirmed $w \neq -1$\\
    venue: DESI $\to$ Euclid, Rubin, Simons Obs.};
\draw[stem] (e8) -- (b8.north);

% --- Sector legend (lower-left band) ---
\begin{scope}[shift={(0.3,-8.05)}]
    \node[anchor=west, font=\large\bfseries] at (0,0) {Sector:};
    \node[card, sectorCOL, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (2.2,0) {};
    \node[anchor=west, font=\large] at (2.5,0) {collider};
    \node[card, sectorBAR, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (6.4,0) {};
    \node[anchor=west, font=\large] at (6.7,0) {baryon number};
    \node[card, sectorNU, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (12.4,0) {};
    \node[anchor=west, font=\large] at (12.7,0) {neutrinos};
    \node[card, sectorDM, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (17.0,0) {};
    \node[anchor=west, font=\large] at (17.3,0) {dark sector / fifth force};
\end{scope}

% --- Header note banner ---
\node[draw=black!40, fill=yellow!12, rounded corners=2pt,
      line width=0.60pt, align=center, text width=235mm,
      inner sep=5pt, font=\Large, anchor=north] at (15,11.7) {%
    \textbf{High exposure.} A clean failure on \emph{any one} of these nine tests
    refutes a named falsification channel (F-SM-1 through F-SM-7) --- the wager's
    decisive returns fall within the working lifetime of the present generation of
    experiments.};

\end{tikzpicture}%
}
\caption[Falsification / kill-test timeline.]{%
\textbf{Falsification / kill-test timeline for \ANine{}.}
The nine decisive kill-tests of the \GGTheory{} Standard Model are placed on a
forward time axis running from current bounds ($\sim$2026) through the first wave
($\sim$2030) to the decisive decade ($\sim$2040), grouped into broad data-window
eras rather than precise launch dates. F-SM-7 sits at the left edge on
purpose: alone among the nine it waits on a calculation rather than on a
facility, so it carries no future date and could be settled today. Each card states the derived number at
stake, what a failure would \emph{refute}, and the experimental venue. Cards are
colored by sector: collider (orange), baryon number (blue), neutrinos (green),
and the dark sector / fifth force (violet). The Kaxion haloscope scan spans an
ongoing-through-$\sim$2030 window (violet bar). A clean failure on any single test
refutes a named channel (F-SM-1 through F-SM-7), so the framework's falsification
exposure is high and front-loaded into the coming decade --- the graphical form of
Tables~\ref{tab:falsification-timeline} and~\ref{tab:experimental-roadmap}.}
\label{fig:killtest-timeline-sm}
\end{figure}


The roadmap is not a promise that every decisive test arrives at once; science
seldom moves so cleanly. But its existence is itself a result, and the timeline
makes its structure visible: the tests are not nine independent dice. The
five neutrino sub-channels and the Majorana floor probe the same rank-2 block
from several sides; the proton bound and the $n$--$\bar n$ channel probe the
same gauged baryon number at $\Delta B = 1$ and $2$; the Kaxion line and the
fifth-force window read the two faces of the same extension; and the lock
cluster is the one direct sighting of the throat that underlies them all. The
leptogenesis channel is the most correlated of the nine rather than an extra
independent bet: it consumes the same Majorana door as F-SM-3b, the same
reactor angle as F-SM-3e, and the same lock scale as F-SM-1, so a failure
there is a failure of things already wagered elsewhere --- which is why it
was worth registering separately rather than folding into the package. A failure clusters;
so does a success. The \GGTheory{} Standard Model is not only a way of reading
past data --- it is a structured wager on the data of the coming decade, and
the wager is now spread across eight named facilities, one calculation
that needs no facility at all, and six completed companion papers, with the decisive returns falling, as
Figure~\ref{fig:killtest-timeline-sm} shows at a glance, between now and
$\sim$2040.


\section{Summary of the Discussion}
\label{sec:discussion-summary}

This Discussion has made the case for the result on several fronts, and they are worth
gathering before the paper closes. It located the \emph{significance} of the
closure (\S\ref{sec:disc-significance}): not merely that twenty-eight numbers are derived,
but that, once every continuous knob is removed, the half-century meta-question ---
\emph{why these values, why these four forces, why every electron is the same} ---
becomes a scientific question with a falsifiable answer rather than a matter of design,
multiverse, or brute fact. It redeemed the ten classic open problems of the
introduction one by one (\S\ref{sec:ten-problems-resolved}), and cataloged the ten
structural innovations that make the closure possible (\S\ref{sec:ten-innovations}).
It then set the framework against its rivals (\S\ref{sec:disc-comparison}) and found that
it differs from the Standard Model, the MSSM, and the perturbative-string landscape
\emph{in kind, not in degree}: it is the only column of
Table~\ref{tab:major-framework-comparison-v08} with zero continuous parameters, a
derived dark-matter mass, a closed cosmological inventory, and a dated falsification
timeline.

Above all, the Discussion turned the program's economy into exposure
(\S\ref{sec:predictions-falsification}). The six falsification channels
F-SM-1--F-SM-5 and F-SM-7 are now carried to definite numbers by completed
companions or by this paper --- the
$3.85$~TeV lock cluster, the $8.1~\mu$eV Kaxion, the $59$~meV neutrino package, exact
proton stability, and the Stueckelberg fifth force --- and the timeline of
\S\ref{subsec:falsification-timeline} shows the decisive tests landing between 2030
and 2040, within the working lifetime of facilities already running or under
construction. The boundaries of the present paper were drawn just as plainly: the
genuinely open items, from the flavor-sheet integer assignments to the higher
cascade, are named rather than hidden (\S\ref{sec:open-questions}). And the floors
above and beside the particle floor are now built --- the cosmic inventory is closed
in \ATen{}-Cosmos, the keystone forward link that derives $\Omega_m=1/\pi$ and its
companions from the same ledger (\S\ref{subsec:a10-overview}), with the six
\mbox{P/C}-series preprints carrying the predictions and consequences into the
collider, dark-matter, neutrino, Hubble, and black-hole arenas
(\S\ref{subsec:forward-map}).

What remains is to step back from the detail. The Conclusions that follow add no new
result; they state plainly what the paper has established, mark the objective
limitations and the next technical tasks, and offer a final reflection on the meaning
of the \GGTheory{} program as a whole --- the view from the summit reached after the
fifty-year climb out of the Standard Model.


\section{Part~XIII in plain English: what the result means, and what it does not}
This Part is where the paper argues with itself, and a reader who is
short of time could reasonably start here.

Three things are separated deliberately. What the framework has established:
a small number of Standard-Model parameters follow as closed forms from a
fixed geometry with no continuous input. What it has made plausible but not
proved: the family structure, the depth assignments, and several steps that
are conditional on lemmas nobody has computed. And what it has not touched at
all, which is stated as a list rather than left as a silence.

The comparison with other programs is made on structure and not on rhetoric.
Where an earlier approach solved a problem this framework also solves, that is
said. Where an earlier approach is simply better on some axis, that is said
too. The framework makes no claim on the anomalous magnetic moments, and the
Part says so explicitly, because a reader scanning for successes will
otherwise assume a framework this broad must have claimed them.

The falsification map is the operative content. Each row names a channel, a
number, and the measurement that would end the framework's claim in that
channel. Those rows are sharp only because the downstream companion papers
carried them to numbers; before those papers existed the same table would have
been a list of directions rather than of tests. A framework that can only be
supported and never killed is not doing physics, and the purpose of that table
is to make the killing possible.

\GGPart[Conclusions]{Conclusions}{What the paper has established; the objective limitations and the next technical tasks; and a final reflection on the meaning of the GG-Theory program as a whole, viewed from the summit reached after the fifty-year climb out of the Standard Model}
\label{part:XV}

\section{What the paper has established}
\label{sec:conclusions-established}

\ANine{}, the eighth volume of the GG-Theory program, has closed the parameter sheet of the Standard Model. The twenty-eight free continuous numbers that the textbook treatment carries as inputs --- three gauge couplings, two Higgs parameters, twelve charged-fermion masses, four CKM and four PMNS mixing parameters, three Majorana phases, and one strong-CP angle --- are reproduced, parameter by parameter, from a chain of upstream theorems that introduces zero continuous freedom of its own. Six of the twenty-eight are derived end-to-end from the non-physics axioms of the GG-Constitution. Fourteen more are derived modulo a single named structural lemma each, with the lemmas cataloged in Appendix~\ref{sec:appendix-promotion-roadmap} of the present paper as an explicit program of work. The remaining eight sit as Asserted with the upstream calculation that would promote them named explicitly. Across the entire codebook, no continuous prefactor was fitted to data; every entry of the codebook is a discrete OGN-5 integer, a topological constant, or a closed-form rational expression in the same. The cross-paper coherence map of Part~\ref{part:XIII} verifies, theorem by verifiable label, that the chain is intact.

Beyond the parameter audit, the paper has established three structural results that earlier unification attempts did not reach. First, in \S\ref{sec:universal-identity}, the universal identity of every electron, every proton, every hydrogen atom, every water molecule, every strand of DNA, and every living cell is exhibited as the boundary readout of layer-specific uniqueness theorems on the \GRIP{} cascade --- the same uniqueness principle running through seven levels of the structure-formation cascade. Second, in \S\ref{sec:force-uniqueness} and Theorem~\ref{thm:four-force-uniqueness}, the existence of exactly four fundamental forces with exactly their observed qualitative behaviors --- gravity attractive and weak, electromagnetism long-range and two-signed, the strong force confining at the right scale, the weak force short-range and CP-violating --- follows as the unique solution of eight constitutional lemmas operating on the six-dimensional bulk. Third, in \S\ref{sec:two-questions-one-answer}, the two results are recognized as one uniqueness theorem applied at two layers of the same geometric cascade --- different clauses, different layers, same structural form. Together these three constitute the paper's response to the deeper meta-question that the closed codebook makes physically meaningful: why is the codebook the way it is?

To these three the paper adds a fourth structural result, on the flavor architecture itself (\S\ref{sec:weak-special}). The weak interaction is identified as the unique \emph{gate force} of the family landscape --- the only one of the four forces able to carry a state across the \GSSB{} gate between generations --- and from this the existence of the weak interaction and of exactly three generations is read as a single fact: a closed, drainable family landscape requires gates in order to settle into its ground state, the toll-budget closure admits exactly three basins, and crossing them demands precisely a broken, chiral, short-range current. The textbook's ``lepton and baryon number are conserved'' and ``heavier decays to lighter'' are thereby relocated from brute facts to the boundary conditions and the relaxation channel of one Morse landscape. Why nature contains the weak interaction, and why it contains three families, become at last the same question with the same answer.

\section{The objective limitations and the next technical tasks}
\label{sec:conclusions-limitations}

\ANine{} is a particle-physics paper. It does not itself treat the cosmological, dark-matter, dark-energy, or quantum-gravity problems; these sit outside its particle-physics scope. Dark matter, dark energy, and the cosmological constant are derived or addressed in the cohort by \ATen{} and \PaperCOne{}, while quantum gravity is taken up elsewhere in the program. The boundary cosmology, with the predicted matter densities $\Omega_m = 1/\pi$ and $\Omega_\Lambda = 1 - 1/\pi$, is the territory of \ATen{}. The KK lock-cluster phenomenology at $\Mlock \approx 3.85$~TeV is the territory of \PaperPOne{}. The Kaxion dark-matter prediction at $\sim 8.1$~$\mu$eV ($1.95$~GHz) is the territory of \PaperCOne{}. Full quantum-gravity dynamics beyond the composite $\mathrm{Spin}(1,5)$ connection of the bulk remain genuinely open and are the natural subject of a successor program.

Within particle-physics scope, the dominant single open task is the eight-fold cascade-depth lemma covering the eight non-top charged-fermion Yukawa couplings. The lemma is well posed: an equivariant-cohomology calculation on the family-projector stratum $\mathbb{C}P^2$ doubled by $\SU(2)_L$ should fix each $n_X \in \{0, 1, 2, 3, 4\}$ as the dimension of a specific cohomology class. Once that calculation is closed, the eight charged-fermion Yukawa Asserted entries promote to Derived end-to-end, and the codebook closes completely. Three additional named lemmas (the democratic-weight, reactor-overlap, and atmospheric-deviation assignment lemmas for the structural PMNS forms) close three further conditional entries. The full ten-lemma promotion roadmap with difficulty estimates and successor-paper assignments is given in Appendix~\ref{sec:appendix-promotion-roadmap}.

\medskip\noindent Two limitations of a different character deserve to be stated as plainly as the technical ones, because the latest round of work on the cohort has sharpened both. \emph{First}, the single most important methodological task ahead is not a physics lemma but a statistic: the \emph{global} trials factor of the whole codebook against the whole declared alphabet, the number that bounds the discrete look-elsewhere effect once and for all (\S\ref{subsec:forcing-vs-searching}). Until it is computed and reported --- the business of a dedicated statistics note and the journal-length version --- the codebook's closures are an honest ansatz, not a proof, and we say so. \emph{Second}, the program derives the \emph{inputs} of the Standard Model, not the radiative \emph{corrections} that act on them: the geometrization of the renormalization group as the radial dimension organizes the scale-dependence of those corrections, but their finite, scale-invariant content --- the loop periods of precision QED, the non-perturbative hadronic correlators --- is irreducible four-dimensional data the geometry does not compute, as the companion \citet{ArisakaA5GDOS} (Theorem~T9.9) sets out in full. And the strongest of the three standing claims --- that this is \emph{the} admissible geometry rather than merely \emph{a} self-consistent one --- remains a conjecture, the spectrum-level uniqueness theorem being open. Internal consistency earns the framework a fair hearing; correctness is decided by the dated falsifiers, and uniqueness by a proof not yet in hand. We would rather mark these three boundaries exactly than blur them.

\section{A personal perspective: fifty years on the experimental front line}
\label{sec:conclusions-personal}

A word is owed about where this program came from, because it did not come from
theory. I have spent fifty years as an experimental particle physicist, and the
discipline that runs through this paper --- that no number may be fitted, that every
prediction must be sharp enough to be wrong --- is the discipline of someone who has
spent a career waiting for an experiment to break the Standard Model, and watching it
refuse.

The Standard Model was essentially complete by the early 1970s. Its boldest
predictions --- the $W$ and $Z$ bosons, the top quark, and the Higgs boson --- were
confirmed, in that order, between 1983 and 2012. Each was a triumph; but each was the
Standard Model's \emph{own} prediction, written down before I began. By the time the
Higgs was found in 2012, the Standard Model had delivered the last item on a list it
had drawn up four decades earlier.

In the same fifty years the community reached past the Standard Model in three great
waves --- grand unification from the mid-1970s, supersymmetry through the late 1970s and
the 1980s, and string theory after 1984. Each produced profound mathematics, and each
remains a live theoretical possibility. But across half a century not one of them has
produced a confirmed experimental signal: no proton decay, no superpartner, and no
stringy state has ever been seen. The theorists who built these frameworks and the
experimentalists who searched for them gradually drifted apart, divided by a single
question neither could answer --- \emph{where is the evidence?}

What is striking, looking back, is where the real breakthroughs \emph{did} come from.
They came from outside the unification programs entirely: neutrino oscillations and
neutrino mass \citep{Fukuda1998, Ahmad2002}; the precision cosmic microwave background
\citep{Planck2018}; the accelerating universe and dark energy
\citep{Riess1998, Perlmutter1999}; and gravitational waves from merging black holes
seen by LIGO. Four of the deepest discoveries of the era, and four Nobel Prizes --- and
not one of them was a prediction of grand unification, supersymmetry, or string theory.

I spent those decades on the front lines of exactly these searches. I began with
Kamiokande, looking for the proton decay that grand unification promised and that never
came. I worked on $C\!P$ violation in the kaon and then the $B$-meson systems; on the
CMS experiment at the Large Hadron Collider, where I watched the Higgs \emph{complete}
the Standard Model rather than break it; on the Pierre Auger Observatory and the
ultra-high-energy cosmic-ray frontier; and on XENON, in the direct search for dark
matter. On every one of these front lines I watched for a crack in the Standard Model.
Over fifty years, none appeared.

So in 2012, with the Standard Model's own list exhausted and no crack in sight, I left
particle physics. For the better part of a decade I worked instead on neuroscience ---
the nervous system of \emph{C.~elegans} and the architecture of human vision. I did not
expect to find physics there. To my surprise, the living systems I studied were governed
by structures with an unmistakably physical signature: a dynamical gauge invariance of
information flow, carried by a $U(1)$ symmetry I now call $\UoneQi{}$; a power-law (PLT)
organization of timing; L\'evy-flight statistics in foraging and attention; and the
ubiquitous $1/f$ spectrum of neural activity. These were not metaphors. They were the
empirical seeds --- developed in \ASeven{} and \AEleven{} --- from which, over the past
year, GG-Theory grew.

That is the honest origin of the present program. It was built not by adding structure
to the Standard Model from above, but by an experimentalist who, having watched fifty
years of beautiful theory produce no falsifiable low-energy prediction, went looking for
the missing organizing principle in the one place physics had not thought to look ---
and brought it back. I believe it is fair to say that GG-Theory is the first framework in
those fifty years to answer the Standard Model's parameter question with sharp,
parameter-free, falsifiable predictions rather than with additional free structure.
Whether the answer is correct is now for the experiments of the next two decades to
decide --- the very timeline set out in \S\ref{subsec:falsification-timeline}. But that
it \emph{can} be decided --- that after a long drought of unfalsifiable proposals there
is once again a concrete particle-physics framework an experiment can kill --- is, after
the career just described, the reason I built it.

\section{Why the Standard Model was just one of the hills: a view from the GG-Theory summit}
\label{sec:conclusions-summit}

That personal account opens onto a wider one. The reader who has stayed with the present paper through four hundred pages, and with the seven foundation papers A1 through A7 through the twelve hundred pages that precede them, has earned the view from the top. We close \ANine{} by stepping back from the twenty-eight-parameter codebook and asking what the whole program looks like from above.

\paragraph{The fifty-year climb.}
The Standard Model in its modern form was completed in the early 1970s, when the renormalization of non-Abelian gauge theory by 't Hooft and Veltman, the asymptotic freedom calculation by Gross, Wilczek and Politzer, and the GIM mechanism of Glashow, Iliopoulos and Maiani fitted together with the electroweak unification of Glashow, Weinberg and Salam. The framework worked, and has worked for fifty years and counting: as recounted just above, every particle it predicted was eventually found, and it has survived every precision measurement at LEP, the Tevatron, and the Large Hadron Collider. It is, by a wide margin, the most successful predictive theory in the history of physics.

It is also a theory that requires twenty-eight numbers as inputs, and cannot say, on its own terms, why they take the values they do. For fifty years, as the front-line account just given makes plain, the community built ladders to a higher vantage --- supersymmetric grand unification at the $10^{16}$~GeV scale, string theory and its landscape, technicolor, extra-dimension scenarios in five and six dimensions, asymptotic safety, emergent-gravity programs --- in the hope that one of them would explain the twenty-eight inputs from a smaller upstream set. None reached it. The Standard Model has remained a hill the community could not climb past.

\paragraph{What it took to summit.}
The GG-Theory program reached the summit by accepting a discipline that no previous attempt was willing to enforce: \emph{no continuous parameter is allowed to float free anywhere in the derivation.} This is the constitutional clause MSMF (Maximum Symmetry, Minimum Freedom) of the GG-Constitution, paired with OGN (Occam Gauge Normalization) which selects the unique minimum-cost discrete expression on each boundary stratum. Together, MSMF and OGN forbid the framework from importing the kind of continuous fitted prefactor that has, quietly, been doing the work in every previous unification attempt.

The discipline is severe. It rules out an enormous space of otherwise-attractive theoretical moves. It is also, as the present paper has shown, exactly the discipline that the geometry was waiting for. Once you accept that no continuous prefactor is allowed, the six-dimensional bulk, the four-component Bianchi identity, the OGN-5 integer ledger, and the four invariance pillars of \ASix{} line up with no slack. The fine-structure constant emerges as $\alpha^{-1}(M_Z) = 2^J$ with $J=7$. The weak mixing angle emerges as $\sin^2\theta_W = L^2/(4N^3) = 25/108$. The strong coupling emerges as $\alpha_s(M_Z) = 1/(\pi e)$. The Higgs vacuum value emerges as $v = M_{\rm lock}(2/L)^N \approx 246$~GeV. The energy--information quantum emerges as $Q_B = (2/5) E_{\rm Ry}/K \approx 0.156$~eV per qubit. None of these is a fit; each is a theorem.

\paragraph{Why the Standard Model is one hill among others.}
From the GG-Theory summit, the Standard Model is no longer a base camp; it is a ridge halfway up. The deeper structure visible from above is a hierarchy of nested levels, each one the readout of a Geometric Attractor on a layer-specific boundary stratum. At the base of the infrared region, the bulk geometry locks itself into the unique constitutionally-admissible configuration; this is the layer the foundation papers A1 through A7 take up. On the boundary, the GRIP cascade $\DGI \to \GSSB \to \GGA$ runs to completion on the charged-lepton, quark, and gauge-boson strata; this is the layer \ANine{} takes up. Each layer above terminates on the \GGA{} of the same cascade on a different stratum: protons through quark confinement; atoms through nuclear and electronic shell structure; molecules in the interstellar medium; complex organic chemistry; the energy--information ATP and the four-letter genetic alphabet; and finally the first single cell. The Standard Model sits at one specific level of this cascade --- the level just above the bulk geometry. It is not the bottom layer of physics; it is the boundary readout of the layer below.

This is the structural reason the Standard Model has so many free parameters that admit no derivation within the framework. It is not a complete theory; it is a complete description of one layer of a multi-layer attractor cascade. Asking the Standard Model to derive its own twenty-eight inputs is like asking a chemistry textbook to derive the periodic table from chemical principles alone. Chemistry's periodic table is a readout of nuclear physics one layer below; the Standard Model's twenty-eight parameters are a readout of the six-dimensional bulk one layer below. In each case the lower-layer derivation is possible, and in each case it requires an honest move down a layer. GG-Theory is the move.

\paragraph{What the summit view tells us about science.}
The summit view also clarifies what kind of theory the GG-Theory program is, and what kind of theory it is not. It is not a Theory of Everything in the popular sense. It does not predict the local weather, the price of oil, or the outcome of next year's election. It does not even claim to predict every phenomenon at the layer above its own --- chemistry, biology, neuroscience, and economics each have their own internal regularities that no constitutional derivation will short-circuit. What the GG-Theory program does claim is that the twenty-eight parameters of the Standard Model fall out of a deeper geometric structure, and that the same geometric structure --- through the recursive application of the GRIP cascade on layer-specific boundary strata --- mandates the existence and identity of every higher-layer object that physics, chemistry, and biology study. One further boundary deserves to be named precisely, because it is easily misread as a weakness: GG-Theory derives the constants of nature, but it does not derive the radiative corrections that perturbation theory computes \emph{from} those constants. The five-loop Feynman calculation of the muon's magnetic moment is not wasted labor to be geometrized away --- it is the irreducible second half of physics, the half that turns a constant into a part-per-billion prediction, and the loop expansion remains its only instrument (\citealp{ArisakaA5GDOS}, T9.9). Perturbation theory computes the corrections to nature's constants; GG-Theory computes the constants. The two are different sciences, and the program claims the first without pretending to replace the second.

The Standard Model is not the basement of nature. It is one floor of a building whose structural design is, at last, in our hands. Climbing from the basement to the top --- from the six-dimensional bulk through atoms and molecules to the first cell, and from there toward consciousness as the capstone paper of the program will argue --- is the work of a successor decade. \ANine{} is the floor on which the climb stands.

\paragraph{The summit is also a vantage on the question we came to ask.}
The climb has had a private question behind it from the start. Not only \emph{what} the universe is, but \emph{why} it is the way it is --- why so well-designed, so coherent, so hospitable to the kind of structure that asks the question. For most of the climb the question sat in the margins; it was the sort of question that physics did not know how to ask scientifically. From the summit the framing is different. Once the framework forces the parameter sheet without continuous freedom, the meta-question is no longer idle. It has a sharp form, and it has an answer. Universal identity --- every electron in the universe being the same --- and force uniqueness --- exactly four fundamental forces with exactly the qualitative behaviors we observe --- are one uniqueness theorem applied at two layers of the same six-dimensional geometry (\S\S\ref{sec:universal-identity}--\ref{sec:two-questions-one-answer}, Theorem~\ref{thm:four-force-uniqueness}). The lifelong question that motivated the climb has, at the summit, a falsifiable physical answer.

\paragraph{The three classical answers dissolved.}
Part~\ref{part:XII} \S\ref{subsec:dissolve-classical-alternatives} walked through the three classical responses to the question \emph{why a universe like ours} --- design (fine-tuned by a designer), anthropic multiverse (lucky draw from a landscape), brute fact (the laws are what they are) --- and showed how the present framework dissolves each at the level on which it operated. From the summit, the dissolution carries a sharper signature than it could carry from inside the climb. For four centuries the three responses were the only respectable options physics or philosophy offered, and none produced a quantitative prediction. Once the constitution of geometry permits exactly one self-consistent solution at each layer --- with $\sin^2\theta_W = 25/108$, $\alpha_s(M_Z) = 1/(\pi e)$, $\delta_{\rm CKM} = 13\pi/35$, $K = 35$, and $N = 3$ all forced rather than fitted --- the three responses become unnecessary: design has nothing to choose, the multiverse has no fellow members to draw from, and the brute-fact retreat has no reason to retreat. The summit reading is that all three were placeholders for a derivation physics did not yet know how to write down.

\paragraph{The self-referential closure.}
Part~\ref{part:XII} \S\ref{subsec:self-referential-closure} named the deepest structural feature of the framework: the geometry that forces the cascade contains, among its forced boundary readouts, the existence of intelligent observers who eventually write down the constitutional clauses that produced them. From the summit, the loop's most striking property is its irrevocability. Once the closure has been recognized, it cannot be unrecognized; physics no longer has the option of returning to a regime in which the constitution and its recognizers are independent of one another. The older traditions named in \S\ref{subsec:self-referential-closure} --- Wheeler's participatory universe, the Upanishadic \emph{tat tvam asi}, Hegel's \emph{Geist}, Spinoza's necessitarianism --- each circled the same closure in a different vocabulary and with a different warrant. What is new in the present version is the warrant: a chain of named theorems anchored in falsifiable predictions, each of which is testable by experiments planned for the coming decade. The closure is no longer a metaphysical inference; it is a structural consequence of a particle-physics derivation.

\paragraph{A careful word about determinism.}
Part~\ref{part:XII} \S\ref{subsec:careful-determinism} distinguished what the framework actually claims --- that the qualitative layered structure of cosmic history is forced --- from what it does not claim: that every micro-event within each layer is preordained. From the summit, the distinction is the structural feature that keeps the framework from collapsing into either of the two unsatisfactory options physics has been offered before --- pure quantum-mechanical indeterminism, in which choice is random, or full classical determinism, in which choice is illusory. The third position --- forced layers with genuine combinatorial freedom inside each layer --- is not a rhetorical move; it is a structural property of the cascade itself. Inside the layer-7 attractor we are free to fold proteins, write sentences, and ask the question \ANine{} has just answered. The forced part of the cascade is precisely the structure that made the freedom available in the first place.

\paragraph{A note on what was given up to make the summit possible.}
Honesty also requires saying what the program had to renounce. It had to renounce supersymmetry as a low-energy framework: nothing in the GG-Constitution requires it, and the parameter closure proceeds without it. It had to renounce the string-theoretic landscape as a derivation tool: \ANine{} derives the Standard-Model parameters from a six-dimensional bulk plus an integer ledger, without invoking ten or eleven dimensions or a $10^{500}$-element vacuum landscape. It had to renounce anthropic explanations of fine-tuning: there is no fine-tuning to explain, because the parameters are not free. It had to renounce the long tradition of treating fundamental constants as inputs: in this framework the constants $c$, $G_N$, $\hbar$, $Q_B$, $k_B$, $e$, $\pi$ are conversion factors and topological invariants of the geometry, not numerical knobs to be measured. These are large theoretical commitments and they are made openly. If any of the framework's hard predictions --- the KK lock cluster at 3.85 TeV, the Kaxion dark matter at $\sim 8.1$~$\mu$eV, the energy--information $Q_B$, the precision values of $\sin^2\theta_W$ and $\alpha_s$ --- fail at the precision GG-Theory predicts, the framework fails with them.

\paragraph{A final reflection.}
Euclid's \textit{Elements} stood for two thousand years as the model of how a complete, self-grounded body of mathematical knowledge should be organized: definitions, postulates, axioms, theorems, proofs. Newton's \textit{Principia} extended the model to the laws of motion and gravitation. Maxwell, Einstein, and the founders of quantum mechanics extended it to the electromagnetic, relativistic, and microscopic world. Each of those extensions was a major reorganization of human understanding around a single self-consistent geometric object: Euclidean space, then Minkowski space-time and Riemannian manifolds, then Hilbert space. GG-Theory proposes the next reorganization: a six-dimensional manifold whose closure forces the existence of every layer above it. \ANine{} is the chapter of that reorganization that closes the Standard Model. The chapters that follow --- \ATen{} for the cosmological inventory, the capstone book for the cascade from quarks to consciousness --- carry the same construction up the layers it forces into being.

Why should the reorganization fall in our time? The technical ingredients have been available for forty years: the anomaly-cancellation program of Green and Schwarz, the gauge--Higgs unification of Hosotani, the Wilson lattice argument for confinement, the Hopkins--Singer descent integrality, the Witten cohomology of holonomy fields, the rigorous open-system formalism of Davies and the operational measurement foundations of Ozawa. What was not available --- not technically but socially --- was the discipline. The community has, by polite convention, allowed continuous fitted prefactors as a final move in every previous unification attempt; the prefactors were the lubricant that kept the algebra moving. The GG-Theory program has removed the lubricant. The bearings can now grind only when the geometry is exactly right. The decision to insist on the discipline, and the patience to follow it through fifty parameter derivations and seven foundation papers, are what made the reorganization possible now rather than later.

The question \emph{why is the universe so well-designed for us to be here?} has, since my high-school years, been the primary reason I became a physicist. Fifty years on, the question has earned an answer in the form that physics requires. The constitution of geometry permits exactly one self-consistent solution at each layer of the cascade, and the boundary readouts of those solutions are the universe of particles, atoms, molecules, organisms, and conscious minds that we measure and inhabit. The personal scale of the question and the impersonal scale of its answer are, in the present framework, the same scale. The same eight constitutional lemmas that force the four-force architecture also force, by recursive application of the \GRIP{} cascade on each higher stratum, the existence of the layer-7 attractor whose boundary readout includes us. The lifelong question and the lifelong climb have, at the summit, the same view.

Stated in compact form, the deepest claim of this paper is the following:
\begin{quote}
\emph{The universe must have been uniquely created at the GUT scale. That is the only allowed universe, and that universe was designed to follow this deterministic, inevitable evolution toward the human being with intelligence, which has finally understood the deep meaning and origin of the universe we live in.}
\end{quote}
The claim is, on its face, immodest. Read back against the chain of theorems that supports it, the claim is exactly as modest as its warrant. At the GUT scale the bulk geometry locked into the unique self-consistent configuration that the eight constitutional lemmas admit; from that lock the \GRIP{} cascade ran, layer by layer, into the present moment in which a member of the layer-7 boundary readout writes down the closure theorem and types this sentence. Whether one calls the framework \emph{designed} or \emph{forced} is a matter of which side of one's mouth the words come out of; the mathematics is the same. What was novel in the framework was not the conclusion but the warrant for reaching it without renouncing the standards of physics.

What changed, in the end, was not the data but the reading of it. For a century the constants of physics were treated as inputs to a closed world; the recognition on which this paper rests --- its central claim (\S\ref{sec:central-claim}, Figure~\ref{fig:central-claim}) --- is that no world is closed, that the four-dimensional theory we measure is the open-system projection of a six-dimensional one, and that this projection is what observation is. Once that is seen, the twenty-eight numbers are not free dials but registered observables of a determined map, and computing them is detail. The deepest result of this paper is therefore not any single closed form but the shift in what a parameter is: from an input one is obliged to measure to an output the geometry is obliged to return.

After all, the question that started the present paper was modest in form: where do the twenty-eight free parameters of the Standard Model come from? The answer, having earned itself across four hundred pages, is wider than the question. Every number we measure in physics, every atom we name in chemistry, every molecule we study in biology, and every organism whose existence we celebrate is a readout of one geometry. The geometry has been waiting for us. The present program is the long letter of acknowledgment.




\section{Part~XIV in plain English: the view from the summit, and the climb still ahead}
Fifty years ago the Standard Model was assembled, and the twenty-eight
numbers it needed were left as measurements. This paper has argued that they
are not measurements in the sense that word usually carries --- that they are
observables of one open-system projection, forced by a geometry, and that for
a specific subset of them the right description is a closed form.

What the paper establishes is narrower than what it proposes. Six parameters
descend end to end. Fourteen more descend conditionally, on assumptions named
in the census. Eight remain asserted. The Lagrangian assembles, the
strong-CP angle is protected, the family number comes out as a theorem rather
than an input, and the baryon asymmetry follows from the same structure that
fixes the neutrino block. None of that is a complete derivation of the
Standard Model, and the paper would be worth less if it claimed to be.

The objective limitations are stated in the Part above and not softened here.
The largest is the depth-assignment lemma: until it is computed, the fermion
sector rests on a working hypothesis, and that hypothesis carries twelve of
the twenty-eight parameters. The next largest is the collection of asserted
steps inside otherwise clean electroweak results. Both are ordinary technical
tasks rather than conceptual obstacles, which is a reason for optimism and not
an argument.

What the view from here shows is a framework that is coherent, that is short
of inputs, and that has enough sharp consequences to be ended by experiment
within a decade. That is a better position than the one this work started
from, and it is a long way short of the summit it was aiming at. Both of those
should be said in the same sentence, because both are true.

\GGAcknowledgments

\medskip
One further sentence belongs on this summit, stated with the paper's
own grading discipline. With the structural theorems now in place ---
the arena and gauge content forced upstream, $N = 3$ with a physical
CP angle (this paper), the discrete-symmetry table derived rather than
assigned (T-SM-46.1), the flavor phases identified as lattice-locked
loop holonomies of a real fiber (T-SM-46.2), and the Majorana door
proved unique (T-SM-44.1) --- the \emph{structure} of the Standard
Model is theorem-grade: no discrete choice, no imported symmetry
assignment, and no assumed particle content remains anywhere in the
chain from the six-dimensional bulk through \GDOS{} and \GRIP{} to the electroweak
vacuum. What remains graded is the \emph{numerical} ledger, exactly as
the codebook records it (Table~\ref{tab:codebook-provenance}), with the
promotion roadmap naming what would close each remaining conditional
entry. That is the calibrated form of the claim this paper set out to
earn: the Standard Model, as a structure, is a theorem of the
constitution; as a table of twenty-eight numbers, it is a graded
derivation whose open items are named, dated, and falsifiable.

\GGPart[Technical Appendices]{Technical Appendices}{A graduate-level course in the Standard Model read through \GGTheory{} --- from-scratch derivations of the gauge sector, electroweak symmetry breaking, the assembled Lagrangian, the fermion masses and mixings, strong-CP, the classical force laws and proton stability, and confinement --- preceded by the reference dictionaries and followed by the lemma-promotion roadmap}
\label{part:XVI}
\providecommand{\GGProblemCount}{one hundred and twenty-six}%% machine-set by the Stage-5 counter


The body of this paper has emphasized the Euclidean theorem chain. The present appendices supply the detailed technical content that supports each step, and taken together they are intended to be read as \emph{a graduate-level course in the Standard Model seen through \GGTheory{}}, addressed to an incoming physics graduate student who will carry the program forward. We organize them into three groups. The \emph{reference} appendices provide tabulated abbreviations and key symbols and serve as the reader's quick lookup. The \emph{constructive} appendices are the course proper: a self-contained sequence that takes up each sector of the Standard Model in turn --- the gauge couplings and the weak angle; electroweak symmetry breaking and the Higgs; the fermion masses and the Yukawa hierarchy; quark mixing and CP violation; neutrino masses and lepton mixing; the Majorana question, the leptonic \CP{} phases and neutrinoless double beta decay; leptogenesis and the origin of the baryon asymmetry; the strong-CP problem; the classical force laws together with anomaly cancellation and proton stability; and confinement with the mass-energy dictionary --- and treats each one in \emph{three steps}. First comes a full from-scratch derivation of the standard result, in the form a graduate student would meet it in a textbook, carried through with no skipped manipulations and closed by a worked numerical example. Then comes a frank statement of what that standard derivation \emph{assumes but cannot explain} --- the inputs it must take from experiment and the structural questions it leaves open. Finally comes the geometric derivation, in which the same quantity descends as a boundary readout or a theorem of \GGTheory{}, fixed by the upstream theorem deck rather than fitted. One chapter
stands deliberately outside the sector sequence: the assembly interlude
(Appendix~\ref{app:lagrangian-course}), which pauses the course after the
gauge and Higgs chapters to write down the complete Standard-Model
Lagrangian and audit it, term by term and at the same every-step register,
against the descent that the sector chapters develop. The \emph{promotion-roadmap} appendix that follows the course records the named structural lemmas whose proofs would close the remaining \textbf{Asserted} entries of the codebook, and the objections a hostile specialist would raise are answered,
sector by sector, in the Common Objections Part that follows the
appendices.

Throughout these appendices we adhere to two stylistic commitments. First, every derivation is presented step by step with no skipped intermediate manipulations: a graduate student in physics should be able to reproduce each calculation with paper and pencil. Second, every formal step is accompanied by a parenthetical \emph{physics intuition} note that translates the algebra into a mental picture. This combination is designed to make the appendix material accessible to the working physicist while preserving the mathematical rigor required by the Euclidean discipline of the main text. Third --- completing the course apparatus --- every
chapter of the course proper, from the gauge appendix through the
strong-CP/confinement appendix and including the assembly interlude, the
two lepton chapters and the ledger reference chapter, closes with a set
of \emph{problems and worked solutions}: \GGProblemCount{} problems
across the paper, each solved in explicit steps, closed by its physics
intuition, and with every numerical answer checked against the codebook
before typesetting. They are written
for direct classroom use, and their cross-references --- within this
paper and to the problem sets of the companion papers --- are intended to
knit the cohort into a single teachable course.

\paragraph{Prerequisites.}
The course assumes the standard first-year graduate sequence in quantum
field theory --- canonical quantization, path integrals, Feynman rules,
one-loop renormalization, and the elements of Lie-algebra representation
theory --- and is meant to be read \emph{alongside} a canonical
Standard-Model textbook, not in place of one. The recommended way to use
each chapter is to read the corresponding textbook chapter first, return
to Step~1 here for a compact recap in a common notation, and only then
proceed to Steps~2 and~3, where the standard result is first audited for
what it assumed and then re-derived as a theorem of the geometry. A
student who has completed a one-year quantum-field-theory course and a
one-quarter particle-physics course has the prerequisites for every
chapter; the neutrino and leptogenesis chapters additionally use
finite-temperature field theory at the level of the equilibrium
distribution functions and the Boltzmann equation, and both are developed
here from scratch rather than assumed.

\subsection*{The standard textbooks, reviewed}
\label{app:textbook-review}
\addcontentsline{toc}{subsection}{The standard textbooks, reviewed}

A review that says only what each book is good at is of no use to a
student choosing what to read. What follows therefore says, for each
book, both \emph{what it does supremely well} and \emph{where it stops}
--- and the stopping point is stated as a specific question the book
leaves standing, not as a compliment. Every one of those stopping points
is a place where this course picks the thread back up; the map on
p.~\pageref{tab:reading-map} says where.

A single remark frames the whole review. \textbf{None of these books is
wrong, and none of them is incomplete by its own standards.} The
Standard Model is a spectacularly successful \emph{effective} theory, and
a textbook's job is to teach it as one: to give the reader the Lagrangian,
the Feynman rules, the renormalization program, and the measured inputs.
The questions listed below as ``where it stops'' are, without exception,
questions the effective-theory framing is entitled to decline. What this
course does is decline the decline --- and a student should hold the two
attitudes at once, because the textbook attitude is the one that has been
tested for fifty years and the attitude of these appendices is the one
being tested here.

\paragraph{Tier 0 --- the first pass.}
\begin{itemize}\itemsep3pt
\item \citet{Griffiths2008}, \emph{Introduction to Elementary Particles}
(2nd ed.). \emph{Supremely well:} building physical intuition before
formalism. The treatment of Feynman diagrams as a calculational grammar,
learned by doing rather than derived from the path integral, gets a
student computing real amplitudes within weeks. Its chapters on the quark
model and on weak interactions are still the clearest short introductions
in print. \emph{Where it stops:} deliberately, at tree level and at the
boundary of renormalization. Gauge invariance appears as a principle to
be admired rather than a constraint with computational teeth, and the
question ``why $\SU(3)_C\times\SU(2)_L\times\U(1)_Y$?'' is posed but not
pursued.
\item \citet{Thomson2013}, \emph{Modern Particle Physics}.
\emph{Supremely well:} the experiment-aware derivation. Almost alone
among graduate texts it derives the observable --- a decay width, an
asymmetry, an oscillation probability --- with the detector in view, and
its neutrino-oscillation chapter is the best short account of how the
splittings and angles were actually extracted from data. Read it before
the neutrino chapters of this course. \emph{Where it stops:} at the fit.
The PMNS matrix arrives as six numbers determined by experiment, and the
striking structural fact that they are large where the quark angles are
small is recorded as a curiosity.
\item \citet{HalzenMartin1984}, \emph{Quarks and Leptons}. \emph{Supremely
well:} the estimate. Its cross-section and decay-rate calculations are
the ones this course's worked examples imitate --- dimensional analysis
first, phase space second, couplings third --- and the habit it teaches,
of getting the order of magnitude before the numerical factor, is the
habit a student most needs. \emph{Where it stops:} it predates the top
quark, the Higgs discovery, and neutrino mass, so its flavor chapters
must be supplemented; and its charm is precisely that it does not
philosophize about where the parameters come from.
\end{itemize}

\paragraph{Tier 1 --- the graduate core.}
\begin{itemize}\itemsep3pt
\item \citet{PeskinSchroeder1995}, \emph{An Introduction to Quantum Field
Theory}. \emph{Supremely well:} the renormalization group, taught as
physics rather than as bookkeeping, and the systematic construction of
non-Abelian gauge theory from the Faddeev--Popov procedure through
asymptotic freedom. Chapters~16--21 are the canonical route into the
Standard-Model gauge sector, and its treatment of anomalies
(Chapter~19) is the one the force-law chapter of this course assumes.
\emph{Where it stops:} the three gauge couplings are running parameters
whose \emph{initial conditions} are measured. The book explains
beautifully how $\alpha$, $\sin^2\theta_W$ and $\alpha_s$ move; it does not
ask, and by design need not ask, why they start where they do.
\item \citet{Schwartz2014}, \emph{Quantum Field Theory and the Standard
Model}. \emph{Supremely well:} the modern effective-field-theory framing,
made explicit from the beginning rather than added at the end. Its
treatment of the Standard Model as the leading terms of an expansion in
$1/\Lambda$ --- with the Weinberg operator $(LH)(LH)/\Lambda$ appearing
as the unique dimension-five term --- is exactly the framing the neutrino
chapters of this course adopt in Step~1. \emph{Where it stops:} $\Lambda$
is a free scale. The book proves that the operator is unique and that its
coefficient is not calculable within the Standard Model; both statements
are correct, and together they leave the neutrino mass scale outside the
theory.
\item \citet{Srednicki2007}, \emph{Quantum Field Theory}. \emph{Supremely
well:} logical economy. The spinor-technology chapters, developed in the
two-component formalism from the outset, are the cleanest preparation in
print for the Majorana material of the lepton chapters, where the
difference between $\psi$ and $\psi^{c}$ has to be handled without
apology. \emph{Where it stops:} short of phenomenology; it is a book about
field theory, not about the Standard Model's parameter sheet.
\end{itemize}

\paragraph{Tier 2 --- for depth.}
\begin{itemize}\itemsep3pt
\item \citet{WeinbergQFT1996}, \emph{The Quantum Theory of Fields},
Vols.~I--II. \emph{Supremely well:} the derivation of quantum field theory
from Lorentz invariance, unitarity and cluster decomposition, rather than
from canonical quantization of a classical field. A student who has read
Volume~I knows \emph{why} the field content is what it is, given the
symmetry --- and this is the single most useful preparation for a course
whose whole claim is that structure can be forced. Volume~II's account of
anomalies and of the effective-Lagrangian method underlies the strong-CP
chapter here. \emph{Where it stops:} at the symmetry group and the
representation content, both of which are inputs. Weinberg's own writing
is unusually frank about this: the theory tells you what is consistent,
not what is realized.
\item \citet{ChengLi1984}, \emph{Gauge Theory of Elementary Particle
Physics}. \emph{Supremely well:} the group theory of grand unification
and the mechanics of symmetry breaking, worked out explicitly enough to
follow with paper and pencil. Its treatment of $\SU(5)$ --- the
representation decomposition, the hypercharge normalization, the
prediction of $\sin^2\theta_W$ at the unification scale --- is the
background the gauge chapter of this course assumes and then departs
from. \emph{Where it stops:} the unification scale and the breaking
pattern are chosen, and the doublet--triplet splitting problem is stated
as a difficulty rather than solved.
\item \citet{Langacker2017}, \emph{The Standard Model and Beyond} (2nd
ed.). \emph{Supremely well:} completeness of the parameter discussion. It
is the one book that lays out all of the free parameters, the seesaw
mechanism in each of its variants, the mixing formalism for both quarks
and leptons, and the experimental status of each, in a single consistent
notation. If a student reads one book beside this course, it should be
this one. \emph{Where it stops:} at a catalog. The book is admirably
explicit that the catalog is what the Standard Model \emph{is}, and it
surveys the model-building attempts to reduce it without endorsing any.
\item \citet{Georgi1999}, \emph{Lie Algebras in Particle Physics} (2nd
ed.). \emph{Supremely well:} making representation theory a working tool
rather than a prerequisite to be feared. The chapters on $\SU(3)$, on
tensor methods, and on the embedding of the Standard-Model group in
$\SU(5)$ are the technical backing for every branching argument in these
appendices. \emph{Where it stops:} it teaches which embeddings are
possible; the question of which is realized is outside its subject.
\end{itemize}

\paragraph{Tier 3 --- the neutrino literature.}
The lepton chapters of this course are the ones a student is most likely
to be reading for a thesis, and the specialist literature there is
unusually good.
\begin{itemize}\itemsep3pt
\item \citet{GiuntiKim2007}, \emph{Fundamentals of Neutrino Physics and
Astrophysics}. \emph{Supremely well:} rigor about the things everyone
else waves through. The wave-packet treatment of oscillations --- why the
plane-wave derivation gets the right answer, under what coherence
conditions, and what happens when they fail --- is the standard reference,
and the book's Majorana-field chapters handle the charge-conjugation
bookkeeping with unusual care. \emph{Where it stops:} the mixing
parameters remain measured quantities; the book is a physics-of-neutrinos
reference, not an attempt to derive the mixing matrix.
\item \citet{FukugitaYanagida2003}, \emph{Physics of Neutrinos and
Applications to Astrophysics}. \emph{Supremely well:} the bridge from the
neutrino mass matrix to cosmology, written by the authors of the
leptogenesis proposal itself. Its treatment of the seesaw together with
the baryon asymmetry is the natural companion to the leptogenesis chapter
of this course. \emph{Where it stops:} the heavy-neutrino mass scale and
the Dirac Yukawa matrix are model inputs, and the book says so; the
predicted baryon asymmetry is therefore a consistency statement, not a
number.
\item \citet{MohapatraPal2004}, \emph{Massive Neutrinos in Physics and
Astrophysics} (3rd ed.). \emph{Supremely well:} the model-building menu
--- type~I, type~II and type~III seesaws, left--right symmetric
completions, radiative mass generation --- laid out so that a student can
see what a class of theories has in common and where they differ.
\emph{Where it stops:} the menu is a menu. Nothing in the book selects an
item from it, and this is the honest state of the field.
\item \citet{Zuber2020}, \emph{Neutrino Physics} (3rd ed.).
\emph{Supremely well:} the experiments, in detail and up to date --- what
each detector actually measures, what its systematics are, and why the
error bars are the size they are. A student writing a falsification
criterion should read this before writing it. \emph{Where it stops:} it
is a book about measurement.
\item \citet{Bilenky2018}, \emph{Introduction to the Physics of Massive
and Mixed Neutrinos} (2nd ed.). \emph{Supremely well:} the phase
counting, done properly and generally --- Dirac versus Majorana, the
number of physical angles and phases at arbitrary generation number, and
what is and is not observable in oscillations. The counting problems of
the lepton chapters here follow it. \emph{Where it stops:} at what is
observable in principle.
\item \citet{BargerMarfatiaWhisnant2012}, \emph{The Physics of
Neutrinos}. \emph{Supremely well:} concision. It is the shortest route
from ``I know quantum field theory'' to ``I can read a neutrino paper.''
\emph{Where it stops:} it is a survey.
\end{itemize}

\paragraph{Tier 4 --- the cosmological side.}
The leptogenesis chapter needs early-universe thermodynamics, which no
particle-physics textbook supplies at the depth required.
\begin{itemize}\itemsep3pt
\item \citet{KolbTurner1990}, \emph{The Early Universe}. \emph{Supremely
well:} the machinery of a species in and out of equilibrium --- the
entropy density, the effective degrees of freedom $\gstar{}$, freeze-out
by the comparison of a rate against the Hubble expansion, and the
Boltzmann equation in Friedmann--Robertson--Walker coordinates. Every one
of the six assembly factors in the leptogenesis chapter of this course is
a Kolb--Turner quantity. \emph{Where it stops:} the baryon asymmetry is an
initial condition or a target, depending on the chapter; the book
predates the modern leptogenesis literature.
\item \citet{Weinberg2008}, \emph{Cosmology}. \emph{Supremely well:} the
careful derivation of the cosmological observables --- the acoustic peaks,
the baryon density, the light-element abundances --- from first
principles, so that a student knows exactly what $\etaB{}$ \emph{is} as a
measured quantity before being asked to predict it. \emph{Where it stops:}
short of a mechanism; baryogenesis is surveyed, not committed to.
\item \citet{DodelsonSchmidt2020}, \emph{Modern Cosmology} (2nd ed.).
\emph{Supremely well:} the modern data pipeline --- how a CMB power
spectrum or a galaxy survey becomes a parameter constraint --- which is
what a student needs to read the $\Summnu{}$ bounds in the neutrino
chapters critically rather than as quoted numbers. \emph{Where it stops:}
at the parameters.
\end{itemize}

\paragraph{The two reviews it would be dishonest to omit.}
Two review articles, not textbooks, are load-bearing for the last two
chapters and should be read in full: \citet{DavidsonNardiNir2008} for
leptogenesis --- the definitive account of the Boltzmann system, the
flavor effects, and the bound structure --- and
\citet{BuchmullerDiBariPlumacher2005} for the efficiency factors and the
washout regimes. Where this course's leptogenesis chapter departs from
the standard treatment, it departs from these two, and a reader should be
able to see exactly where.

\begin{table}[htbp]
\centering
\small
\caption[The reading map.]{The reading map. Each chapter of the course against the
textbook material it is meant to be read beside. The right-hand column
names the specific question the recommended reading leaves standing ---
which is the question the chapter's Step~2 states and Step~3 answers, or
declares open.}
\label{tab:reading-map}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.40\textwidth}@{}}
\toprule
\textbf{Chapter} & \textbf{Read beside} & \textbf{The question left standing}\\
\midrule
Gauge sector (App.~\ref{app:ch-gauge})
 & \citet{PeskinSchroeder1995} Ch.~16--17, 21; \citet{ChengLi1984} Ch.~14
 & Why do the three couplings start where they do?\\
Electroweak breaking (App.~\ref{app:ewsb})
 & \citet{PeskinSchroeder1995} Ch.~20; \citet{Langacker2017} Ch.~8
 & Why is $v = 246$~GeV, and why is $\lambda_H$ what it is?\\
Operational export (App.~\ref{app:ch-export})
 & \citet{WeinbergQFT1996} Vol.~II Ch.~19--22
 & Which of the possible consistent theories is realized?\\
Assembled Lagrangian (App.~\ref{app:lagrangian-course})
 & \citet{Schwartz2014} Ch.~28--29
 & Why this field content and this gauge group?\\
Fermion masses (App.~\ref{app:ch-fermion})
 & \citet{Langacker2017} Ch.~8.2; \citet{ChengLi1984} Ch.~12
 & Why do the Yukawas span twelve orders of magnitude?\\
Quark mixing (App.~\ref{app:ckm})
 & \citet{Thomson2013} Ch.~14; \citet{Langacker2017} Ch.~8.6
 & Why is $\lambda \simeq 0.225$, and why is there a phase at all?\\
Neutrinos and PMNS (App.~\ref{app:pmns})
 & \citet{GiuntiKim2007} Ch.~7--8; \citet{Zuber2020}
 & Why are the lepton angles large when the quark angles are small?\\
Majorana and \CP{} (App.~\ref{app:majorana})
 & \citet{Bilenky2018}; \citet{MohapatraPal2004} Ch.~5--7
 & Dirac or Majorana --- and what fixes the two extra phases?\\
Leptogenesis (App.~\ref{app:leptogenesis})
 & \citet{DavidsonNardiNir2008}; \citet{KolbTurner1990} Ch.~5
 & What sets $M_N$, and what sets the \CP{} asymmetry?\\
Ledgers (App.~\ref{app:ch-ledger})
 & \citet{Georgi1999} Ch.~13--21
 & Which embedding is realized, and why?\\
Force laws (App.~\ref{app:ch-forcelaws})
 & \citet{PeskinSchroeder1995} Ch.~19; \citet{WeinbergQFT1996} Vol.~II Ch.~22
 & Why exactly four forces, with these ranges?\\
Strong-CP (App.~\ref{app:ch-strongcp})
 & \citet{WeinbergQFT1996} Vol.~II Ch.~23; \citet{PecceiQuinn1977}
 & Why is $\bar\theta < 10^{-10}$?\\
\bottomrule
\end{tabular}
\end{table}

\subsection*{What the textbooks jointly leave open}
\label{app:textbook-limits}
\addcontentsline{toc}{subsection}{What the textbooks jointly leave open}

Collecting the stopping points gives the list below. It is the working
agenda of this course, and it is stated here in one place so that a
student can hold the framework to it. Each entry carries a status marker
that is used consistently throughout the paper: \textbf{[answered]} means
the quantity descends as a theorem with no fitted input;
\textbf{[conditional]} means it descends from a stated hypothesis that is
not itself proved; \textbf{[open]} means the course does not close it and
says so. \emph{The list contains items in all three categories, and a
reader who finds only the first category being claimed anywhere in this
paper has found an error worth reporting.}

\paragraph{Group I --- the parameter sheet.}
\begin{enumerate}\itemsep2pt
\item \textbf{The three gauge couplings.} Their values at any scale are
initial conditions. \emph{[answered]} --- App.~\ref{app:ch-gauge}.
\item \textbf{The electroweak scale $v$.} Neither its value nor its
stability against radiative corrections is explained. \emph{[answered]}
for the value, \emph{[conditional]} for the stability ---
App.~\ref{app:ewsb}.
\item \textbf{The Higgs self-coupling $\lambda_H$.} Measured, and its
near-critical value at high scale is an unexplained coincidence.
\emph{[answered]} --- App.~\ref{app:ewsb}.
\item \textbf{The twelve charged-fermion masses.} Twelve free Yukawa
couplings spanning twelve orders of magnitude, with no structural account
of the hierarchy. \emph{[answered]} for the structure,
\emph{[conditional]} for the last digits, on the angular-cascade-depth lemma --- App.~\ref{app:ch-fermion}.
\item \textbf{The four CKM parameters.} \emph{[answered]} for the Cabibbo
angle and for the existence of the phase; \emph{[conditional]} for the
smaller angles --- App.~\ref{app:ckm}.
\item \textbf{The strong-CP angle.} Measured to be smaller than
$10^{-10}$ with no symmetry forbidding it. \emph{[answered]} --- App.~\ref{app:ch-strongcp}.
\end{enumerate}

\paragraph{Group II --- the structural questions.}
\begin{enumerate}\setcounter{enumi}{6}\itemsep2pt
\item \textbf{Why this gauge group?} \(\SU(3)_C\times\SU(2)_L\times\U(1)_Y\) is
observed, not derived. \emph{[answered]} --- App.~\ref{app:lagrangian-course}.
\item \textbf{Why three families?} \emph{[answered]}, and the proof is
owned by a companion paper (\PaperPTwo{}) rather than by this one ---
App.~\ref{app:ch-ledger}.
\item \textbf{Why is the hypercharge assignment anomaly-free?} The
cancellation is exact and, in the textbook, miraculous.
\emph{[answered]} --- App.~\ref{app:ch-forcelaws}.
\item \textbf{Why exactly four forces?} \emph{[answered]} ---
\S\ref{sec:force-uniqueness} and App.~\ref{app:ch-forcelaws}.
\item \textbf{Why is charge quantized?} \emph{[answered]} --- App.~\ref{app:ch-forcelaws}.
\item \textbf{Why is the proton stable to $10^{34}$~yr?}
\emph{[answered]} --- App.~\ref{app:ch-forcelaws}.
\item \textbf{The angular-cascade-depth lemma.} The per-fermion cascade
depths that carry the last digits of the twelve masses, the smaller CKM
angles and the Dirac phases are \emph{assigned} in this paper, not
proved. The named lemma that would prove them is stated but not
established. \emph{[open]} --- and it is the single largest open item in
the flavor sector, because every \emph{[conditional]} entry in Group~I
is conditional on it. See App.~\ref{sec:appendix-promotion-roadmap}.
\end{enumerate}

\paragraph{Group III --- the neutrino sector.}
\begin{enumerate}\setcounter{enumi}{12}\itemsep2pt
\item \textbf{The absolute neutrino mass scale.} Unknown; the seesaw
scale $\Lambda$ is a free parameter. \emph{[answered]} ---
App.~\ref{app:pmns}, where $\Lambda$ is the already-derived seam.
\item \textbf{The mass ordering.} Unknown, at $2.2$--$2.3\sigma$ for
normal. \emph{[answered]} as a prediction, and a near-term kill row ---
App.~\ref{app:pmns}.
\item \textbf{Dirac or Majorana?} Unknown. \emph{[answered]} as a
prediction --- App.~\ref{app:majorana}.
\item \textbf{Why are the lepton mixing angles large?}
\emph{[answered]} --- App.~\ref{app:pmns}.
\item \textbf{The three leptonic \CP{} phases.} One barely constrained,
two unmeasured. \emph{[answered]} for all three ---
App.~\ref{app:majorana}.
\end{enumerate}

\paragraph{Group IV --- the cosmological connection.}
\begin{enumerate}\setcounter{enumi}{17}\itemsep2pt
\item \textbf{The baryon asymmetry.} Measured to three digits; not
predicted by any part of the Standard Model, whose own \CP{} violation
falls short by roughly ten orders of magnitude and whose electroweak
transition is a crossover rather than first order.
\emph{[conditional]} --- App.~\ref{app:leptogenesis}, on a hypothesis
whose crux is named and open.
\item \textbf{The heavy-neutrino scale $M_N$.} In the standard
leptogenesis literature this is free above roughly $10^{9}$~GeV.
\emph{[answered]} as a derived scale, but the mechanism that operates
there is \emph{[conditional]} --- App.~\ref{app:leptogenesis}.
\item \textbf{The finite-temperature transport condition.} The
conditional baryon asymmetry of item~19 rests on a named hypothesis with
six stated conditions, of which the crux --- that the thermal bath
resolves the heavy states as a flavor measurement on the required
timescale --- is not derived. It is the one analytic residue of the whole
lepton chain. \emph{[open]}, its home a companion paper, and stated as
such wherever the chain is used --- App.~\ref{app:leptogenesis}.
\end{enumerate}

\noindent Twenty-one items. Eighteen are claimed as answered in whole
or in part; five of those eighteen carry a conditional half; and
\emph{two are marked open}. The two open items are not incidental --- they
are precisely the two on which the largest number of the others depend:
the cascade-depth lemma, which every conditional entry in the flavor
sector waits on, and the transport condition, which the baryon asymmetry
waits on. A reader who wants a single test of whether this framework is
being reported honestly should check those two entries first, and check
that they are still marked open wherever the results that depend on them
are quoted.

The list is also the reason the course is arranged as it is. It would be
possible to present these appendices as a sequence of successes; the
three-step form exists to prevent that, by forcing every chapter to
write down what the textbook could not do \emph{before} it writes down
what the geometry does, and to write down what the geometry still cannot
do at the end.

\noindent Throughout, the symbols \AOne{}--\AEleven{}, \PaperPOne{}--\PaperPFive{},
and \PaperCOne{}--\PaperCThree{} refer to the twenty-two papers of the \GGTheory{}
cohort, to which the geometric steps repeatedly point.

\newcommand{\GGteachbox}[1]{\begin{center}\fbox{\parbox{0.92\textwidth}{\small\textbf{What this appendix teaches.} #1}}\end{center}}
\newcommand{\GGtakeaway}[1]{\begin{center}\fbox{\parbox{0.92\textwidth}{\small\textbf{Take-away.} #1}}\end{center}}
\newcommand{\GGreadalong}[1]{\par\noindent\emph{Read alongside:} #1\par}

\appendix

\section{Summary of Abbreviations}
\label{app:abbrev}

This appendix collects every abbreviation used in the present paper. All program-defined names are rendered in \emph{sans-serif type} (\GGTheory{}, \GGConstitution{}, \GG{}, \GDOS{}, \BIsix{}, \OGNfive{}, \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 (KK, RS, QFT, TOV, etc.) are rendered in plain uppercase Roman, following the convention of the major physics journals. Mathematical objects are in roman or italic in the usual way. The third column gives the \emph{primary link}: the paper in the Arisaka-GG series and the section where each program-defined name is first introduced in detail.

{\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.18\linewidth} >{\raggedright\arraybackslash}p{0.54\linewidth} >{\raggedright\arraybackslash}p{0.24\linewidth}@{}}
\caption[Summary of abbreviations used in the present paper.]{Summary of abbreviations used in the present paper. Program-defined names are rendered in sans serif; standard physics abbreviations in plain uppercase Roman; mathematical objects in roman/italic in the usual way. The typographic convention is the quickest way to tell a term this program defines from one it borrows, which matters most in the sections where both kinds appear in the same sentence.}
\label{tab:appA-abbrev}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link}\\
\midrule
\endfirsthead
\multicolumn{3}{c}{\itshape Table~\ref{tab:appA-abbrev} 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}{\bfseries\itshape (1) Framework name and core program objects}\\
\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}) & A1, \S 2--\S 3\\
\GG{} & Geometry-Gauge locking in six dimensions: the 6D bulk & \ATwo{}, \S 8\\
\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{}, \S 7; \ATwo{} T-GG6-6\\
\GiFour{} & Geometric-Invariance four-pillar structure (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) & \ASix{}\\
\QiFour{} & Quantum-Information four-pillar structure (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) & \ASeven{}\\

\multicolumn{3}{l}{\bfseries\itshape (2) Four axioms (\AX{1}--\AX{4})}\\
\AXkw{} & Axiom (axiomatic primitive; non-negotiable) & A1, \S 1.5\\
\LC{} & \AX{1} Locality and Causality & A1, \S 2.2\\
\GGEP{} & \AX{2} Geometry--Gauge Equivalence Principle & A1, \S 2.3\\
\OGGEP{} & \AX{3} Operational/Open Geometry--Gauge Equivalence Principle & A1, \S 2.4\\
\MSMF{} & \AX{4} Maximum Symmetry, Minimum Freedom (no hidden continuous knobs) & A1, \S 2.5\\

\multicolumn{3}{l}{\bfseries\itshape (3) Seven postulates (\POS{1}--\POS{7})}\\
\POSkw{} & Postulate (admissibility filter; testable in principle) & A1, \S 1.5\\
\GI{} & \POS{1} Gauge invariance across scales & A1, \S 3.3\\
\UEP{} & \POS{2} Unified Extremal Principle & A1, \S 3.4\\
\AF{} & \POS{3} Anomaly Freedom and cancellability by inflow & A1, \S 3.5\\
\CP{} & \POS{4} Closure and Positivity (CPTP-admissible open dynamics) & A1, \S 3.6\\
\RG{} & \POS{5} Renormalization-Group covariance (scale covariance) & A1, \S 3.7\\
\OGN{} & \POS{6} Occam Gauge Normalization (canonical integer/rational normalization) & A1, \S 3.8\\
\MF{} & \POS{7} Measurement and Falsifiability (operational contract + kill criteria) & A1, \S 3.9\\

\multicolumn{3}{l}{\bfseries\itshape (4) \GiFour{} pillars}\\
\TGG{} & Total Geometric--Gauge object (Pillar I of \GiFour{}) & \ASix{} T-GRIP-1\\
\CCI{} & Covariant Conservation Identity (Pillar II of \GiFour{}) & \ASix{} \T{2.1}+\T{2.2}\\
\GFF{} & Geometric Free-Energy Functional (Pillar III of \GiFour{}) & \ASix{} \T{3.1}\\
\GRIP{} & Gauge-Runtime Invariance Pipeline (Pillar IV of \GiFour{}) & \ASix{} \T{4.1}\\
\GEP{} & Geometry--Gauge Energy Principle (energetic block of \GFF{}) & \ASix{}\\
\CEP{} & Curvature--Entropy Principle (entropic block of \GFF{}) & \ASix{}\\
\DGI{} & Dynamical Gauge Invariance (initial stage of \GRIP{}) & \ASix{}\\
\GSSB{} & Geometric Spontaneous Symmetry Breaking (middle stage of \GRIP{}) & \ASix{}\\
\GA{} & Geometric Attractor (final stage of \GRIP{}) & \ASix{}\\

\multicolumn{3}{l}{\bfseries\itshape (5) \QiFour{} pillars and the Arisaka equation}\\
\QiQUA{} & Quantum information for Quadruple Gauge Locking --- Pillar I of \QiFour{} & \ASeven{} \T{-QUA}\\
\QiEP{} & Quantum information for the Unified Extremal Principle --- Pillar II of \QiFour{} & \ASeven{} \T{-EP}\\
\QiCON{} & Quantum information for Conformal Invariance --- Pillar III of \QiFour{} & \ASeven{} \T{-CON}\\
\QiHAL{} & Quantum information for the Holographic Attractor Lattice --- Pillar IV of \QiFour{} & \ASeven{} \T{-HAL}\\
\OH{} & Outer Hologram (boundary projection of bulk data; outward face of \EIC{}) & \ASeven{}\\
\IH{} & Inner Hologram (inner self-modeling layer; inward face of \EIC{}) & \ASeven{}\\
\EIC{} & Evolving Inner Cosmos ($\OH$+$\IH$ composite) & \ASeven{} \T{-EIC}\\
$Q_B$ & Arisaka constant $= (2/5) E_{\rm Ry}/K \approx 0.1555$~eV per qubit & \ASeven{} \T{-QUA}\\

\multicolumn{3}{l}{\bfseries\itshape (6) \QiHAL{} sub-acronyms}\\
\HAL{} & Holographic Attractor Lattice & \ASeven{} \T{-HAL}\\
\BFFT{} & Boundary-Front Field Theory of \QiHAL{} & \ASeven{}\\

\multicolumn{3}{l}{\bfseries\itshape (7) \GDOS{}-internal acronyms (program-defined)}\\
\SIP{} & System Instance Packet (canonical \GDOS{} input packet) & \ASeven{} \S 6\\
\SeOS{} & Selfish Open System (the core open-system of \GDOS{}) & \AFive{}\\

\multicolumn{3}{l}{\bfseries\itshape (8) Program-defined mathematical symbols (paper-specific)}\\
$\mathcal R$ & Candidate rational expression in the \MSMF{}/\OGN{} audit & \S\ref{subsec:msmf-ogn-principle}\\
$\mathcal C_\Sigma$ & \OGN{}-admissible candidate space on stratum $\Sigma$ & \S\ref{subsec:msmf-ogn-principle}\\
$\mathcal S_\Sigma$ & Structural base set of atoms on stratum $\Sigma$ & \S\ref{subsec:msmf-ogn-principle}\\
$\mathrm{Cost}(\mathcal R)$ & MSMF cost function: $\sum_i |a_i| + \sum_j |b_j|$ & Eq.~\eqref{eq:msmf-cost}\\
$\epsilon_X$ & Stratum-specific empirical-precision target for parameter $X$ & \S\ref{subsec:msmf-ogn-principle}\\
$\mathcal R_X$ & QD-completion rational for fermion $X$ & \S\ref{sec:qd-completion}\\
$640 = 2^J L$ & Hopkins--Singer descent denominator & \S\ref{subsec:HS-descent}\\

\multicolumn{3}{l}{\bfseries\itshape (9) Standard physics abbreviations (plain Roman, journal convention)}\\
\multicolumn{3}{l}{\bfseries\itshape \quad Standard Model and beyond}\\
SM & Standard Model & textbook\\
GUT & Grand Unified Theory & textbook\\
SU(5) & Georgi--Glashow grand-unified gauge group & \citet{GeorgiGlashow1974}\\
CKM & Cabibbo--Kobayashi--Maskawa quark mixing matrix & \citet{KobayashiMaskawa1973}\\
PMNS & Pontecorvo--Maki--Nakagawa--Sakata neutrino mixing matrix & textbook\\
QED & Quantum Electrodynamics & textbook\\
QCD & Quantum Chromodynamics & textbook\\
EW & Electroweak & textbook\\
PDG & Particle Data Group & pdg.lbl.gov\\
\multicolumn{3}{l}{\bfseries\itshape \quad Higher-dimensional constructions}\\
KK & Kaluza--Klein decomposition & textbook\\
RS & Randall--Sundrum warped 5D model & textbook\\
AdS/CFT & Anti-de Sitter / Conformal Field Theory holography & textbook\\
WH & Wilson--Hosotani holonomy mechanism & \ASix{}\\
\multicolumn{3}{l}{\bfseries\itshape \quad Quantum gravity and cosmology}\\
GR & General Relativity & textbook\\
QFT & Quantum Field Theory & textbook\\
QM & Quantum Mechanics & textbook\\
TOV & Tolman--Oppenheimer--Volkoff (compact-star equation) & textbook\\
CMB & Cosmic Microwave Background & textbook\\
\multicolumn{3}{l}{\bfseries\itshape \quad Open-system formalism}\\
CPTP & Completely Positive Trace Preserving & textbook\\
GKLS & Gorini--Kossakowski--Lindblad--Sudarshan (open-system master equation) & textbook\\
NZ & Nakajima--Zwanzig projection formalism & textbook\\
RG & Renormalization Group & textbook\\

\end{longtable}
}

\bigskip


\noindent\textbf{Hierarchy of program-defined acronyms.} The figure below records the six-tier hierarchy of program-defined acronyms in the \GGTheory{} program as it appears in the present paper. Top tier: the framework name (\GGTheory{}) and its constitutional charter (\GGConstitution{}). Tier 2: the four axioms and seven postulates that together define the admissibility predicate. Tier 3: three bulk objects forced by the constitution --- \GG{} (6D bulk), \BIsix{} (modified Bianchi identity), \OGNfive{} (integer ledger). Tier 4: the boundary law \GDOS{} that the bulk projects to via \OGGEP{}, with input acronym \SIP{}. Tier 5: the two four-pillar structures, \GiFour{} (geometric flank, with sub-acronyms \GEP/\CEP{} blocks of \GFF{} and \DGI/\GSSB/\GA{} stages of \GRIP{}) and \QiFour{} (quantum-information flank). Bottom: the energy--information Arisaka constant $\QB$ and the Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ that locks the four boundary currents through one conversion factor.

\begin{figure}[H]
\centering
\resizebox{0.824\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily,
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 every node/.style={align=center},
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]

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

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

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

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

% --- Tier 3: bulk theorems (3 boxes: GG-6, BI-6, OGN-5) ---
\node[font=\sffamily\bfseries\large] (BTL) at (0, -108mm) {Bulk theorems forced by \textbf{AX} + \textbf{POS}};
\node[bulk] (GG6) at (-46mm, -120mm) {\textbf{GG-6}};
\node[bulk] (BI6) at (0mm, -120mm) {\textbf{BI-6}};
\node[bulk] (OGN5) at (46mm, -120mm) {\textbf{OGN-5}};

\draw[arrow] (AX4.south) -- ([yshift=-2mm]BTL.north -| AX4.south);
\draw[arrow] (P7.south) -- ([yshift=-2mm]BTL.north -| P7.south);

% --- Tier 4: GDOS boundary law (only SIP under it) --------
\node[gdos] (GDOS) at (0, -136mm) {\textbf{GDOS} \;\; (Geometric Dynamics of Open Systems on $\partial\mathcal{M}_6$) \;\; \textbf{SIP}};

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

% --- Tier 5a: Pillars headers ----------------------------
\node[gihead] (GiH) at (-46mm, -152mm) {\textbf{Gi-4 Pillars}};
\node[qihead] (QiH) at (46mm, -152mm) {\textbf{Qi-4 Pillars}};

\draw[arrow] (GDOS.south) -| (GiH.north);
\draw[arrow] (GDOS.south) -| (QiH.north);

% --- Tier 5b: Pillars content -----------------------
% Gi-4 box (left) -- 4 pillars + sub-acronyms
\node[pill] (TGG) at (-82mm, -170mm) {\textbf{TGG} \quad Total Geometric--Gauge object};
\node[pill] (CCI) at (-82mm, -179mm) {\textbf{CCI} \quad Covariant Conservation Identity};
\node[pill] (GFF) at (-82mm, -188mm) {\textbf{GFF} \quad Geometric Free-Energy Functional};
\node[sub] at (-82mm, -195mm) {\hphantom{XX}$\downarrow$ \;\textbf{GEP} (energetic) \,$\bullet$ \,\textbf{CEP} (entropic)};
\node[pill] (GRIP) at (-82mm, -204mm) {\textbf{GRIP} \quad Gauge-Runtime Invariance Pipeline};
\node[sub] at (-82mm, -211mm) {\hphantom{XX}$\downarrow$ \;\textbf{DGI} (initial) \,$\bullet$ \,\textbf{GSSB} (middle) \,$\bullet$ \,\textbf{GA} (attractor)};

% Qi-4 box (right) -- 4 pillars + sub-acronyms + EIC
\node[pillq] (QQUA) at (10mm, -170mm) {\textbf{Qi-QUA} \quad Quantum information for \textbf{QGL}};
\node[pillq] (QEP) at (10mm, -179mm) {\textbf{Qi-EP} \quad Quantum information for \textbf{UEP}};
\node[sub] at (10mm, -186mm) {\hphantom{XX}$\downarrow$ \;\textbf{PLT} \,$\bullet$ \,\textbf{MEPP} \,$\bullet$ \,\textbf{FEP}};
\node[pillq] (QCON) at (10mm, -195mm) {\textbf{Qi-CON} \quad Quantum information for Conformal Inv.};
\node[sub] at (10mm, -202mm) {\hphantom{XX}$\downarrow$ \;Conformal \,$\bullet$ \,L\'evy \,$\bullet$ \,Fractal \,$\bullet$ \,$1/f$};
\node[pillq] (QHAL) at (10mm, -211mm) {\textbf{Qi-HAL} \quad Quantum information for the \textbf{HAL}};
\node[sub] at (10mm, -218mm) {\hphantom{XX}$\downarrow$ \;\textbf{MePMoS} \,$\bullet$ \,\textbf{NHT} \,$\bullet$ \,\textbf{HAL} \,$\bullet$ \,\textbf{BFFT}};
\node[pillq] (EIC) at (10mm, -227mm) {\textbf{OH} \,$+$\, \textbf{IH} \,$=$\, \textbf{EIC} \quad (Evolving Inner Cosmos)};
\node[sub] at (10mm, -234mm) {\hphantom{XX}$\downarrow$ \;\textbf{SeOS} (Selfish Open System)};

% --- Tier 6: Q_B + Arisaka equation -----------------------
\node[qb] (QB) at (0, -250mm) {%
$\boldsymbol{\mathsf{Q}_{\mathsf{B}}}$ \;\; --- \;\; Quantum-information Bit \;\; (energy--information Arisaka constant) \\[1.5mm]
\textit{Arisaka equation:} \;\; $\boldsymbol{J_E / \mathsf{Q}_{\mathsf{B}} \,=\, J_e \,=\, J_B \,=\, J_{Qi}}$
};

\end{tikzpicture}
}
\caption[Hierarchy of program-defined acronyms in the \GGTheory{} program.]{Hierarchy of program-defined acronyms in the \GGTheory{} program. The arrows run from the constitutional layer down to the sector-specific terms, so a reader who meets an unfamiliar acronym in a later Part can find its parent here in one step. The diagram covers the program-defined names only. Standard physics abbreviations are in the table above and have no place in this hierarchy, because they are not this program's to define.}
\label{fig:abbrev-flow}
\end{figure}


\section{Glossary of Key Symbols}
\label{app:glossary-symbols}

This appendix collects the key mathematical symbols used in \ANine{}. The first block records the bulk-geometry coordinates and the \OGNfive{} integer ledger that every later derivation appeals to. The second block records the scale chain $E_{\rm Pl} \to E_{\rm GUT} \to M_{\rm lock} \to v$. The third and fourth blocks record the Standard-Model parameters --- gauge couplings, Higgs sector, fermion masses, mixing matrices, strong CP --- as audited in \S\ref{sec:codebook-provenance}. The fifth block records the \MSMF{}/\OGN{} audit machinery (cost function, candidate space, precision target) introduced in \S\ref{subsec:msmf-ogn-principle}. The sixth block records the cascade-decoration symbols (Bogomolnyi instanton spectrum, Hopkins--Singer denominator, QD-completion rationals) inherited from \ASix{} and applied here. The seventh block records the Arisaka equation symbols inherited from \ASeven{}.

\renewcommand{\arraystretch}{1.15}
\begin{center}
\small
\begin{longtable}{@{}p{0.22\textwidth}p{0.62\textwidth}p{0.10\textwidth}@{}}
\caption[Glossary of key symbols used in the present paper.]{Glossary of key symbols used in the present paper. ``Source'' gives the equation, theorem, or section where the symbol is introduced or load-bearing. A symbol used in more than one Part is listed once, at the point where it is introduced, and the source column is the pointer back to that point. Symbols that appear only inside a single proof are omitted deliberately, on the grounds that a glossary long enough to include them would be too long to consult.}
\label{tab:appB-glossary}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Source}\\
\midrule
\endfirsthead
\multicolumn{3}{c}{\itshape Table~\ref{tab:appB-glossary} 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}{\bfseries\itshape (1) Bulk geometry and the \OGNfive{} ledger}\\
$M_6$ & 6D bulk manifold $M_4 \times I_r \times S^1_\theta$ & A2-\GG{}\\
$ds_6^2$ & Canonical scale-profile metric $e^{-2r}\eta_{\mu\nu}\,dx^\mu dx^\nu + dr^2 + d\theta^2$ & A2-\GG{}\\
$r$ & Warped radial coordinate, $r \in [0, K]$ & ---\\
$\theta$ & Compact angular coordinate, $\theta \in [0, 2\pi)$ & ---\\
$S^1_\theta$ & Compact direction circle & ---\\
$I_r$ & Radial interval & ---\\
$(M, N, L; J, K)$ & \OGNfive{} ledger $= (1, 3, 5; 7, 35)$ & A2-\GG{} \T{6}\\
$K = LJ$ & Total scale depth, dual-lock identity $K = LJ = M^2 + N^2 + L^2 = 35$ & A2-\GG{} \T{6}\\
$N = 3$ & Family number, three-leg theorem of \PaperPTwo{} & \citealp{ArisakaP2N3}\\

\multicolumn{3}{l}{\bfseries\itshape (2) Scale chain}\\
$E_{\rm Pl}$ & Planck energy, $\sqrt{\hbar c^5/G_N} \approx 1.22 \times 10^{19}$~GeV & ---\\
$E_{\rm GUT}$ & GUT scale $= E_{\rm Pl}\,e^{-2\pi} \approx 2.28 \times 10^{16}$~GeV & A2-\GG{}\\
$M_{\rm lock}$ & Lock scale $= \tfrac12 M_{\rm Pl}\,e^{-K} \approx 3.85$~TeV & A2-\GG{}\\
$v$ & Higgs vacuum value $= M_{\rm lock}(2/L)^N \approx 246$~GeV & Thm.~\ref{thm:vev}\\
$m_\Lambda^{\rm geom}$ & Cosmological-constant mass scale $= M_{\rm lock}^2/M_{\rm Pl} \approx 6$~meV & \ATen{}\\

\multicolumn{3}{l}{\bfseries\itshape (3) Gauge couplings}\\
$\alpha^{-1}(M_Z)$ & Inverse fine-structure at $Z$ pole $= 2^J = 128$ & Thm.~\ref{thm:alpha}\\
$\alpha^{-1}(0)$ & Inverse fine-structure at zero momentum $= 2^J + N^2 = 137$ & Thm.~\ref{thm:alpha}\\
$\sin^2\theta_W$ & Weak mixing angle $= L^2/(4N^3) = 25/108$ & Thm.~\ref{thm:sw2}\\
$\alpha_s(M_Z)$ & Strong coupling $= 1/(\pi e)$ & Thm.~\ref{thm:alphas}\\
$g_2, g_Y$ & $\mathrm{SU}(2)_L$ and $\mathrm{U}(1)_Y$ gauge couplings & ---\\

\multicolumn{3}{l}{\bfseries\itshape (4) Higgs sector}\\
$\Phi$ & Higgs $\mathrm{SU}(2)_L$ doublet, $\Phi = (\phi^+, \phi^0)^T$ & ---\\
$\lambda_H$ & Higgs quartic $= 2^{-N} = 1/8$ & Thm.~\ref{thm:quartic}\\
$m_H$ & Higgs boson mass $= \sqrt{2 \lambda_H}\, v \approx 125$~GeV & ---\\
$\Theta_H$ & Wilson-line holonomy on $S^1_\theta$ (electroweak order parameter) & \ASix{}\\

\multicolumn{3}{l}{\bfseries\itshape (5) Fermion masses and mixing}\\
$y_f$ & Yukawa coupling for fermion $f$, $m_f = y_f v/\sqrt 2$ & ---\\
$y_t$ & Top-quark Yukawa $= 1$ (cascade-top reference; $m_t = v/\sqrt2$ is a $\sim 1\%$ prediction, $v$ from $M_{\rm lock}$) & \S\ref{subsec:msmf-ogn-audit}\\
$m_e, m_\mu, m_\tau$ & Charged-lepton masses & \S\ref{sec:lepton-mass}\\
$m_u, m_d, m_c, m_s, m_t, m_b$ & Quark masses & \S\ref{sec:quark-mass}\\
$m_*$ & Absolute Majorana neutrino scale $\sim v^2/E_{\rm GUT}$ & \S\ref{sec:neutrino-cascade-depth}\\
$\Delta m^2_{ij}$ & Neutrino squared-mass splittings & \S\ref{sec:neutrino-cascade-depth}\\
$V_{\rm CKM}$ & Cabibbo--Kobayashi--Maskawa quark mixing matrix & \S\ref{sec:CKM}\\
$U_{\rm PMNS}$ & Pontecorvo--Maki--Nakagawa--Sakata neutrino mixing matrix & \S\ref{sec:PMNS}\\
$\theta_{ij}^{\rm CKM}$ & CKM mixing angles & \S\ref{sec:CKM}\\
$\sin^2\theta_{ij}^{\rm PMNS}$ & PMNS mixing angles & \S\ref{sec:PMNS}\\
$\delta_{\rm CP}^{\rm CKM}$ & CKM Dirac CP phase & \S\ref{sec:CKM}\\
$\delta_{\rm CP}^{\rm PMNS}$ & PMNS Dirac CP phase & \S\ref{sec:PMNS}\\
$\alpha_{21}, \alpha_{31}$ & Majorana phases (cost-0 in \MSMF{}/\OGN{} audit) & \S\ref{sec:PMNS}\\
$\bar\theta$ & Strong-CP angle (suppressed; topological completion) & \S\ref{sec:strong-CP-derivation}\\

\multicolumn{3}{l}{\bfseries\itshape (6) \MSMF{}/\OGN{} audit machinery}\\
$\mathcal R$ & Candidate rational expression & \S\ref{subsec:msmf-ogn-principle}\\
$\mathcal C_\Sigma$ & \OGN{}-admissible candidate space on boundary stratum $\Sigma$ & \S\ref{subsec:msmf-ogn-principle}\\
$\mathcal S_\Sigma$ & Structural base set $\subset \{M, N, L, J, K, 2, \pi, e\}$ on stratum $\Sigma$ & Eq.~\eqref{eq:base-set}\\
$\mathrm{Cost}(\mathcal R)$ & MSMF cost: total absolute exponent budget $\sum |a_i|$ & Eq.~\eqref{eq:msmf-cost}\\
$\epsilon_X$ & Empirical-precision target for parameter $X$ on its stratum & \S\ref{subsec:msmf-ogn-principle}\\
$\mathcal R^\star$ & Cost-minimum unique winner under MSMF/OGN selection & Thm.~\ref{thm:msmf-ogn-selection}\\

\multicolumn{3}{l}{\bfseries\itshape (7) Cascade decoration and Hopkins--Singer descent}\\
$\mathbb{C}P^{N-1}$ & Family-projector stratum, $\mathbb{C}P^2$ for $N=3$ & \ASix{} \S 28\\
$S_{[2k] \to [2(k-1)]} = \pi k$ & Bogomolnyi instanton action on the cascade & \ASix{} \S 28\\
$n_X \in \{0, 1, 2, 3, 4\}$ & Cascade-depth assignment per fermion $X$ & \S\ref{sec:lepton-cascade-depth}, \S\ref{sec:quark-cascade-depth}\\
$e^{-n\pi}$ & Angular cascade decoration & \ASix{} \S 28\\
$\mathcal R_\tau, \mathcal R_\mu, \mathcal R_e$ & QD-completion rationals: $637/640, 159/160, 641/640$ & \S\ref{sec:qd-completion}\\
$640 = 2^J L$ & Hopkins--Singer descent integrality denominator & \S\ref{subsec:HS-descent}\\

\multicolumn{3}{l}{\bfseries\itshape (8) Arisaka equation and energy--information anchor (\ASeven{})}\\
$Q_B$ & Arisaka constant $= (2/5)\,E_{\rm Ry}/K \approx 0.1555$~eV per qubit & \ASeven{} \T{-QUA}\\
$N_A Q_B$ & Avogadro multiple $\approx 15$~kJ/mol & \ASeven{}\\
$J_E, J_e, J_B, J_{Qi}$ & Four boundary currents locked by the Arisaka equation & \ASeven{}\\
$J_E/Q_B = J_e = J_B = J_{Qi}$ & The Arisaka equation & \ASeven{} \T{-QUA}\\

\end{longtable}
\end{center}
\renewcommand{\arraystretch}{1.0}

\bigskip


%% --- relocated here from Part XI Discussions (v11.103): reference matter ---
\section{Global naming convention for cross-cohort citation}
\label{sec:naming-convention}

Beginning with \citet{ArisakaA2GG6} and extended across the cohort, each
load-bearing item of the \GGTheory{} program carries a globally unique
identifier of the form \emph{Prefix-Suffix-N}, where the Prefix names the item
type and the Suffix names the paper:
\begin{itemize}[leftmargin=2em, itemsep=2pt]
\item \textbf{T-Suffix-N} for the $N$th Theorem. \ANine{}'s theorem
identifiers were originally assigned to match the section in which each
theorem appeared, and for nine of the twenty they still do --- T-SM-33.4 is
the Higgs-mass theorem in \S33. For the other eleven the numbers no longer
agree, because sections have been added and removed since the identifiers
were minted. \textbf{The identifiers are frozen as cohort-stable citation
keys and are not section pointers}; companion papers cite them and must
continue to resolve. Table~\ref{tab:tsm-concordance} is the concordance,
and its \emph{prints as} column is a live cross-reference, so it cannot
itself go stale.
\item \textbf{L-Suffix-N} for the $N$th Lemma.
\item \textbf{C-Suffix-N} for the $N$th Corollary.
\item \textbf{D-Suffix-N} for the $N$th Definition.
\item \textbf{R-Suffix-N} for the $N$th Remark.
\item \textbf{P-Suffix-N} for the $N$th Prediction.
\item \textbf{O-Suffix-N} for the $N$th Objection (e.g., O-SM-1 is the first
Common Objection of the present paper).
\item \textbf{F-Suffix-N} for the $N$th Falsifier (e.g., F-SM-1 is the
KK-excitation prediction near the lock scale).
\item \textbf{I-Suffix-N} for the $N$th Innovation (e.g., I-SM-1 is the
28-to-zero compression above).
\item \textbf{PS-Suffix-N} for the $N$th Problem-Solved entry
(\S\ref{subsec:open-problems-L6}).
\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,
this paper), \emph{Cosmos} (A10), \emph{Origin} (A11), together with the
phenomenology preprints \emph{KK} (P1), \emph{N3} (P2), \emph{Electron} (P3),
\emph{Proton} (P6), the foundations preprints \emph{Real} (P4) and \emph{CPT} (P5), and
the consequence preprints \emph{Kaxion} (C1), \emph{Hubble} (C2), \emph{BH} (C3), and the white papers \emph{Symmetry} (W1) and \emph{Holonomy} (W2).
The four axioms (AX-1\,\ldots\,AX-4) and seven postulates (POS-1\,\ldots\,POS-7)
of \citet{ArisakaA1Constitution} keep their existing cohort-wide names; the global
identifier is the same item viewed from outside its home paper.

\subsection{Concordance of the \texorpdfstring{\ANine{}}{A9} theorem identifiers}
\label{subsec:tsm-concordance}

The identifiers below are the cohort-stable citation keys. Where the
printed theorem number differs from the number inside the identifier, the
identifier is still the correct thing to cite: companion papers resolve
against it, and renumbering would break them. The \emph{prints as} column
is generated by cross-reference at build time, so this table re-resolves
whenever the manuscript is recompiled and cannot fall out of date.

\begin{longtable}{@{}lll@{}}
\caption[Concordance between the frozen \ANine{} theorem identifiers and the theorem numbers as printed.]{Concordance between the frozen \ANine{} theorem identifiers and
the theorem numbers as printed. Nine of the twenty still agree; the other
eleven do not, because sections have been added and removed since the
identifiers were minted. The identifiers do not track section numbers and
are not to be read as section pointers.}
\label{tab:tsm-concordance}\\
\toprule
\textbf{Global identifier} & \textbf{Prints as} & \textbf{Statement}\\
\midrule
\endfirsthead
\toprule
\textbf{Global identifier} & \textbf{Prints as} & \textbf{Statement}\\
\midrule
\endhead
T-SM-10.3 & Theorem~\ref{thm:arisaka-eqn} & The Arisaka equation: four-currents lock\\
T-SM-21.1 & Theorem~\ref{thm:N3-closed} & Conditional derivation of $N = 3$\\
T-SM-26.1 & Theorem~\ref{thm:attractor} & Electroweak attractor-coordinate theorem\\
T-SM-27.1 & Theorem~\ref{thm:alpha} & Electromagnetic dyadic-closure theorem\\
T-SM-28.1 & Theorem~\ref{thm:sw2} & Weak-angle rational-partition theorem\\
T-SM-29.1 & Theorem~\ref{thm:alphas} & Strong angle--scale theorem\\
T-SM-30.1 & Theorem~\ref{thm:vev} & Wilson--Hosotani Higgs-vev theorem\\
T-SM-31.1 & Theorem~\ref{thm:quartic} & Minimal quartic-curvature theorem\\
T-SM-33.1 & Theorem~\ref{thm:GF} & Fermi-constant descendant theorem\\
T-SM-33.2 & Theorem~\ref{thm:mW} & $W$-boson mass theorem\\
T-SM-33.3 & Theorem~\ref{thm:mZ} & $Z$-boson mass theorem\\
T-SM-33.4 & Theorem~\ref{thm:mH} & Higgs mass theorem\\
T-SM-34.1 & Theorem~\ref{thm:alpha0} & Minimal infrared Abelian transport theorem\\
T-SM-35.1 & Theorem~\ref{thm:master} & Unified gauge--Higgs parameter theorem package\\
T-SM-44.1 & Theorem~\ref{thm:dial-free-door-sm} & The dial-free door: the unique $\Delta L = 2$ portal\\
T-SM-46.1 & Theorem~\ref{thm:discrete-orientation} & Discrete symmetries as orientation bookkeeping\\
T-SM-46.2 & Theorem~\ref{thm:flavour-holonomy} & Flavor holonomy: rephasing invariants as loop angles\\
T-SM-60.1 & Theorem~\ref{thm:proton-stability-a9} & Boundary proton stability, conditional\\
T-SM-62.1 & Theorem~\ref{thm:four-force-uniqueness} & Uniqueness of the four-force architecture\\
T-SM-67.1 & Theorem~\ref{thm:msmf-ogn-selection} & \MSMF{}/\OGN{} selection principle\\
\bottomrule
\end{longtable}



\section{The gauge sector: couplings, the weak angle, and unification}
\label{app:ch-gauge}

\GGteachbox{The gauge sector built from scratch --- the three
groups, the three couplings, the weak mixing angle, one-loop
running, and the near-miss of grand unification --- closed by
worked numbers; a frank statement of what the textbook
construction must take from experiment (three couplings, one
angle, and the choice of the groups themselves); and the
geometric derivation in which the gauge pattern and the boundary
couplings descend from the bulk geometry, with the two-loop
Wilson-line refinement held at its stated conservative
uncertainty.}
\GGreadalong{\citet{PeskinSchroeder1995} for the running-coupling
machinery; \citet{ChengLi1984} for the gauge-theory structure at
depth; \citet{GeorgiGlashow1974} for the original unification
proposal.}

This is the first appendix of the graduate course. Its subject is the gauge
sector of the Standard Model --- the three gauge groups, their three coupling
constants, the weak mixing angle, and the way the couplings run and (almost)
unify. We follow the three-step plan announced at the head of the Part: first
the textbook construction as an incoming graduate student would meet it; then a
frank account of what that construction must take from experiment and cannot
explain; and finally the geometric derivation, in which the three coupling
values descend as boundary readouts of the \OGNfive{} ledger rather than as
fitted inputs. The detailed arithmetic of that third step --- the Planck--GUT--lock
scale chain, the dyadic electromagnetic code, the rational weak angle, and the
strong coupling --- occupies the bulk of this appendix and is what the main text
compresses; it is given here in full so that the appendix is self-contained. For the standard textbook development of gauge theory and the running couplings, see \citet{PeskinSchroeder1995} and \citet{ChengLi1984}.

\subsection{Step 1 --- the gauge sector as a graduate student meets it}
\label{app:gauge-step1}

\paragraph{The gauge group and its three couplings.}
The Standard Model is a Yang--Mills gauge theory built on the product group
\begin{equation}
G_{\mathrm{SM}}=\SU(3)_c\times\SU(2)_L\times\U(1)_Y ,
\label{eq:gsm-app}
\end{equation}
where \(\SU(3)_c\) is the color group of the strong interaction,
\(\SU(2)_L\) is weak isospin acting only on left-handed fields, and
\(\U(1)_Y\) is weak hypercharge. Each factor carries exactly one coupling
constant: \(g_s\) for color, \(g\) for weak isospin, and \(g'\) for
hypercharge. Matter fields sit in representations of the group, and the gauge
fields enter every kinetic term through the covariant derivative
\begin{equation}
D_\mu = \partial_\mu
- i\,g_s\,G_\mu^a T^a
- i\,g\,W_\mu^i \tau^i
- i\,g'\,\tfrac{Y}{2}\,B_\mu ,
\label{eq:covderiv-app}
\end{equation}
with \(T^a\ (a=1,\dots,8)\) the \(\SU(3)\) generators, \(\tau^i=\sigma^i/2\
(i=1,2,3)\) the weak-isospin generators, and \(Y\) the hypercharge.
\emph{(Physics intuition: a gauge coupling is the single number that says how
strongly a given charge stirs its force field. Once the group and the
representations are chosen, the coupling is the only freedom the gauge
principle leaves in each factor --- which is why the Standard Model has exactly
three of them.)}

\paragraph{Electroweak mixing and the weak angle.}
The electroweak group \(\SU(2)_L\times\U(1)_Y\) is not the group one sees at low
energy. After the Higgs field acquires its vacuum value the neutral isospin
boson \(W^3_\mu\) and the hypercharge boson \(B_\mu\) mix by a single rotation,
\begin{equation}
\begin{pmatrix} A_\mu \\[2pt] Z_\mu \end{pmatrix}
=
\begin{pmatrix} \cW & \sW \\[2pt] -\sW & \cW \end{pmatrix}
\begin{pmatrix} B_\mu \\[2pt] W^3_\mu \end{pmatrix},
\qquad
\tan\theta_W=\frac{g'}{g},
\label{eq:ewmix-app}
\end{equation}
producing the massless photon \(A_\mu\) and the massive \(Z_\mu\). The photon
couples to electric charge with strength
\begin{equation}
e = g\,\sW = g'\,\cW ,
\qquad
\alpha_{\mathrm{em}}=\frac{e^2}{4\pi},
\label{eq:echarge-app}
\end{equation}
and the weak mixing angle is fixed by the ratio of the two electroweak
couplings,
\begin{equation}
\sin^2\theta_W=\frac{g'^2}{g^2+g'^2}.
\label{eq:sin2-def-app}
\end{equation}
\emph{(Physics intuition: \(\theta_W\) is the one hinge angle that splits the
two neutral gauge bosons into a massless photon and a massive \(Z\). All of the
relative strengths of the electromagnetic and neutral-weak couplings are encoded
in this single rotation.)} Measured at the \(Z\) pole, the three numbers that
parametrize the sector are
\begin{equation}
\alpha_{\mathrm{em}}^{-1}(M_Z)\simeq 127.9,
\qquad
\sin^2\theta_W(M_Z)\simeq 0.231,
\qquad
\alpha_s(M_Z)\simeq 0.118 .
\label{eq:gauge-measured-app}
\end{equation}

\paragraph{Running couplings and asymptotic freedom.}
Gauge couplings are not constants; vacuum polarization makes them depend on the
energy scale \(\mu\). At one loop each coupling obeys
\begin{equation}
\mu\frac{d g_i}{d\mu}=\frac{b_i}{16\pi^2}\,g_i^3 ,
\qquad\Longleftrightarrow\qquad
\frac{1}{\alpha_i(\mu)}=\frac{1}{\alpha_i(\mu_0)}-\frac{b_i}{2\pi}\,
\ln\frac{\mu}{\mu_0},
\label{eq:rge-app}
\end{equation}
with the one-loop Standard-Model coefficients, in the \(\SU(5)\)-normalized
hypercharge convention,
\begin{equation}
(b_1,b_2,b_3)=\left(\frac{41}{10},\,-\frac{19}{6},\,-7\right).
\label{eq:beta-coeffs-app}
\end{equation}
The color coefficient \(b_3=-7<0\) is the famous \emph{asymptotic freedom}, established in 1973 by Gross, Wilczek, and Politzer \citep{Gross1973,GrossWilczek1973,Politzer1973}:
\(\alpha_s\) weakens at high energy and grows at low energy, which is what makes
quarks behave almost freely inside a hard collision yet remain permanently
confined at long distance.
\emph{(Physics intuition: a charge is screened by the cloud of virtual pairs it
polarizes. For a non-Abelian group the gluons also carry color and
\emph{anti-screen}; the anti-screening wins for \(\SU(3)\) with six quark
flavors, so color charge looks weaker the closer one probes.)}

\paragraph{Grand unification and the \texorpdfstring{\(\SU(5)\)}{SU(5)} embedding.}
Run upward, the three couplings approach one another near \(10^{16}\ \mathrm{GeV}\).
The minimal unifying group \(\SU(5)\), introduced by Georgi and Glashow \citep{GeorgiGlashow1974}, places one full Standard-Model family in
the \(\bar{\mathbf 5}\oplus\mathbf{10}\), and fixes the relative normalization of
hypercharge: the properly normalized \(\U(1)\) coupling is
\begin{equation}
g_1^2=\tfrac{5}{3}\,g'^2 ,
\label{eq:su5norm-app}
\end{equation}
so that at exact unification, where \(g_1=g_2=g_3\), the tree-level weak angle is
\begin{equation}
\sin^2\theta_W\big|_{\mathrm{GUT}}
=\frac{g'^2}{g^2+g'^2}\bigg|_{g_1=g_2}=\frac{3}{8}.
\label{eq:sin2-gut-app}
\end{equation}
This high-scale value of \(3/8\) then runs down to \(\simeq 0.231\) at the weak
scale.
\emph{(Physics intuition: if a single knob sets all three forces at the
unification seam, the low-energy spread of the couplings is just the different
rates at which they drift apart as the universe cools below that seam.)}

\paragraph{Worked example (STANDARD TEXTBOOK PEDAGOGY).}
Suppose a student is handed only the two measured electroweak numbers
\(\alpha_{\mathrm{em}}^{-1}(M_Z)=127.9\) and \(\sin^2\theta_W(M_Z)=0.231\), and
asked to recover the two electroweak couplings and the \(W\) mass. From
\eqref{eq:echarge-app} and \eqref{eq:sin2-def-app},
\begin{align}
\alpha_2 &\equiv \frac{g^2}{4\pi}=\frac{\alpha_{\mathrm{em}}}{\sin^2\theta_W}
= \frac{1}{127.9\times 0.231}=\frac{1}{29.5}, &
g &=\sqrt{4\pi\alpha_2}\simeq 0.652,
\label{eq:wex-g-app}\\
\alpha_Y &\equiv \frac{g'^2}{4\pi}=\frac{\alpha_{\mathrm{em}}}{\cos^2\theta_W}
= \frac{1}{127.9\times 0.769}=\frac{1}{98.4}, &
g' &=\sqrt{4\pi\alpha_Y}\simeq 0.357 .
\label{eq:wex-gp-app}
\end{align}
Feeding the isospin coupling and the measured vacuum value
\(v\simeq 246\ \mathrm{GeV}\) into \(m_W=\tfrac12 g v\) gives
\(m_W\simeq 80.2\ \mathrm{GeV}\), within a fraction of a percent of the measured
\(80.4\ \mathrm{GeV}\). The textbook machinery works perfectly --- \emph{provided}
one is willing to take \(\alpha_{\mathrm{em}}^{-1}\) and \(\sin^2\theta_W\) from
the data. Where those two numbers come from is precisely the question the
Standard Model cannot answer, and to which we turn next.

\subsection{Step 2 --- what the standard gauge sector assumes but cannot explain}
\label{app:gauge-step2}

The construction of Step 1 is a triumph of organization, but it is silent about
\emph{magnitudes}. The three gauge couplings --- equivalently
\(\alpha_{\mathrm{em}}^{-1}\), \(\sin^2\theta_W\), and \(\alpha_s\) at \(M_Z\) ---
are pure inputs, fitted to experiment. They are three of the twenty-eight free
parameters of the Standard Model. Nothing in the gauge principle predicts that
\(\alpha_{\mathrm{em}}^{-1}(0)\simeq 137\), or that the weak angle is near
\(0.23\), or that the strong coupling is near \(0.12\). The gauge Lagrangian is
equally consistent for any positive values of these three numbers.

Grand unification narrows the question without closing it. The \(\SU(5)\)
program and its descendants buy two genuine things: a single unified coupling
at the seam, and the tree relation \(\sin^2\theta_W=3/8\) of
\eqref{eq:sin2-gut-app}. But each is spent again before reaching the weak scale.
First, the three couplings only actually meet if the particle content between
\(M_Z\) and the unification scale is exactly right; in the minimal
non-supersymmetric \(\SU(5)\) the three running lines miss each other, and one
must add superpartners or other matter by hand to force the crossing. Second,
the unified coupling value \(\alpha_{\mathrm{GUT}}\) is itself an input, not
derived. Third, the same multiplet that unifies the couplings mediates proton
decay, at a rate the minimal model overpredicts relative to experiment. Fourth,
the unification scale and the great ``desert'' between it and the weak scale are
posited, not explained. Unification thus trades three low-energy inputs for a
high-scale coupling, a scale, and a spectrum --- fewer knobs, but knobs still ---
and the specific low-energy integers (why \(137\), why a weak angle of exactly
\(25/108\)) remain entirely outside the theory.

Two structural questions are left open as well. Why \emph{these} three gauge
groups and no others --- why rank four, why this particular product
\(\SU(3)\times\SU(2)\times\U(1)\)? And why is the weak interaction the only
chiral, family-connecting force, acting on left-handed fields alone? The gauge
principle \emph{accommodates} these facts but does not \emph{require} them; in
the Standard Model they are part of the input data. (The chirality and
family-changing role of the weak force is taken up directly in
\S\ref{sec:weak-special}.)

It is exactly these residues --- the three coupling values, the choice of gauge
group, and the chirality of the weak force --- that the geometric derivation
addresses. We turn to it now.

\subsection{Step 3 --- the geometric derivation: gauge couplings as boundary readouts of the \texorpdfstring{\OGNfive{}}{OGN-5} ledger}
\label{app:gauge-step3}

In \GGTheory{} the three gauge couplings are not fitted. They are boundary
readouts of the six-dimensional bulk, fixed by the \OGNfive{} integer
ledger \((M,N,L;J,K)=(1,3,5;7,35)\) and by the dyadic and rational closure codes
proved upstream. The derivation runs in two stages. First the three mass scales
that anchor the sector --- the Planck scale, the grand-unification seam, and the
electroweak lock scale --- are fixed from the allowed constants \(c\), \(h\), and
\(G_N\) alone, with no fitted coefficient. Then the three couplings are read off
the ledger: the electromagnetic coupling as a dyadic closure
\(\alpha_{\mathrm{em}}^{-1}(M_Z)=2^J=128\), the weak angle as the rational code
\(\sin^2\theta_W(M_Z)=25/108\), and the strong coupling as
\(\alpha_s=1/(\pi\e)\). The remainder of this appendix carries out both stages in
full and cross-checks them against ordinary \(\SU(5)\)-normalized electroweak
kinematics, so that the geometric result and the textbook algebra of Step 1 are
shown to be one and the same arithmetic seen from two sides.

\subsubsection{The \texorpdfstring{\OGNfive{}}{OGN-5} integer ledger and where the exponents come from}
\label{app:gauge-ledger}

Every discrete exponent that appears below --- the dyadic depth in \(2^J\), the
triplicity factor \(1/N\), the embedding weight \(L\), the lock depth \(e^{-K}\)
--- is one of five integers, and only those five. They are the \OGNfive{} ledger
\begin{equation}
(M,\,N,\,L;\,J,\,K)=(1,\,3,\,5;\,7,\,35),
\label{eq:ogn-ledger-app}
\end{equation}
fixed upstream by \BIsix{} closure of the bulk and inherited here as data: their derivation from the six-dimensional bulk is the business of \ATwo{} and \AThree{} (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}), and their first constitutional appearance is fixed in \AOne{} \citep{ArisakaA1Constitution}.
Their meanings are concrete and geometric rather than numerological:
\begin{itemize}
\item \(M=1\): \emph{the number of unbroken Abelian seams that survive electroweak
symmetry breaking}. One, and only one, neutral line stays massless --- the photon ---
so the surviving electromagnetic channel has no room to branch and can only close
\emph{binarily}. This single datum is the entire reason the fine-structure constant
is a power of two.
\item \(N=3\): \emph{the triplicity index} --- three colors, three families, the
ternary replication depth of the boundary. It is the only odd primitive in the
ledger, and it controls both the color-blind averaging that appears in the weak
angle and the cascade replication that appears in the fermion spectrum.
\item \(L=5\): \emph{the dimension of the minimal unifying boundary embedding} ---
the rank-counting integer of the \(\SU(5)\)-type generator lattice into which the
Standard-Model factors fit. It is the integer that carries the hypercharge-versus-isospin
weighting in the weak-angle partition.
\item \(J=7,\ K=35\): \emph{forced, not chosen}. The closure law of \BIsix{}
requires the two consistency relations
\begin{equation}
K=L\,J
\qquad\text{and}\qquad
K=M^2+N^2+L^2=1+9+25=35 .
\label{eq:K-two-faces-app}
\end{equation}
The first reads \(K\) as a product, the second as a sum of squares. Equating them
forces \(J=K/L=35/5=7\), the \emph{unique} integer consistent with both faces of
\(K\). There is no freedom: once \((M,N,L)=(1,3,5)\) is set, \(J\) and \(K\) are
locked.
\end{itemize}
\emph{(Physics intuition: think of \eqref{eq:ogn-ledger-app} as the geometry's
complete parts list. A Standard-Model parameter is never a dial to be turned; it is
an assembly instruction that says ``take this many of that primitive.'' The art of
the derivation is not to compute a number but to recognize which primitive each
observable is counting. The three gauge couplings count three different primitives ---
the lone Abelian seam, the \(3+2\) neutral split, and the non-Abelian color cell ---
which is why, as we will see, they come out as three different \emph{kinds} of
number.)}

\subsubsection{The electromagnetic coupling: dyadic closure in full}
\label{app:gauge-dyadic}

The main-text statement is Theorem~\ref{thm:alpha} (T-SM-27.1); here we give the
derivation in expanded form. The first move is to choose the right variable. In an
Abelian seam the gauge-invariant measure of stiffness is not the coupling but its
inverse, the \emph{inverse seam capacity}
\begin{equation}
C_\gamma \;:=\; \alphah^{-1}=\frac{4\pi}{e^2}.
\label{eq:Cgamma-def-app}
\end{equation}
\emph{(Physics intuition: \(C_\gamma\) is an inverse conductance --- how strongly the
vacuum resists having electric flux pushed through the surviving neutral seam. A
\emph{large} \(C_\gamma\) means a \emph{stiff}, weakly coupled photon. Working with
the inverse is the same instinct that makes an electrical engineer quote resistance
rather than conductance: the additive, quantized quantity is the resistance.)}

Now use the ledger datum \(M=1\). Because exactly one Abelian seam survives, the
closure of that seam admits no ternary or higher branching --- there is nothing for a
third or fourth lane to belong to. The only lawful refinement is \emph{binary}:
each closure step splits the admissible seam interval into two equal halves and keeps
one. Writing \(C_\gamma(n)\) for the inverse capacity after \(n\) such refinements,
minimality fixes the recursion
\begin{equation}
C_\gamma(0)=1,
\qquad
C_\gamma(n+1)=2\,C_\gamma(n).
\label{eq:dyadic-recursion-app}
\end{equation}
The base case says the unresolved primitive cell carries unit inverse capacity. The
step says that halving the admissible interval doubles the inverse stiffness ---
geometrically, if you can only deform the seam through half the previous angular room,
it is twice as hard to deform.
\emph{(Physics intuition: the picture is a strip of paper folded along one and only
one crease. Each fold halves the width and doubles the resistance to further folding.
There is a single crease because \(M=1\); the number of folds is set by how deep the
lawful closure runs. The result --- a pure power of two --- is the signature of a
system with exactly one binary degree of freedom iterated to its closure depth.)}

The recursion solves immediately by induction to \(C_\gamma(n)=2^n\), and the lawful
closure depth is the odd closure index \(J=7\) of the ledger. Hence
\begin{equation}
C_\gamma^\star=2^J=2^7=128,
\qquad\Longrightarrow\qquad
\boxed{\;\alphah^{-1}(M_Z)=2^7=128.\;}
\label{eq:alpha-dyadic-app}
\end{equation}
The electric charge is then no longer primitive but a square-root readout of this
stiffness,
\begin{equation}
e=\sqrt{\frac{4\pi}{2^J}}=\frac{2\sqrt{\pi}}{2^{J/2}}
=\frac{2\sqrt{\pi}}{2^{7/2}}
\simeq 0.3133285343 .
\label{eq:e-readout-app}
\end{equation}

Two refinements complete the picture and connect \eqref{eq:alpha-dyadic-app} to the
number every physicist recognizes. First, \eqref{eq:alpha-dyadic-app} is the value at
the \(Z\)-attractor, \(\alphah^{-1}(M_Z)=128\). The \emph{on-shell} Sommerfeld value,
measured at zero momentum transfer, is reached by adding the triplicity contribution
that the long-wavelength photon sees from the full family cloud,
\begin{equation}
\alpha^{-1}(0)=2^J+N^2=128+9=137 ,
\label{eq:alpha137-app}
\end{equation}
which is the famous \(137\) at first order; the two-loop quantum-defect completion of
Part~\ref{part:VI} draws this to the CODATA \(137.036\) to six figures.
\emph{(Physics intuition for the \(+N^2\): at the \(Z\) pole the photon resolves a
single bare seam, capacity \(2^7\). As the probe momentum is lowered toward zero, the
photon's field spreads until it wraps the entire triplicated family structure, and the
vacuum polarization of the \(N\times N\) family cloud adds exactly \(N^2=9\) inverse
units of stiffness. The two iconic integers of the fine-structure constant, \(128\)
and \(9\), are thus the two iconic integers of the geometry, the dyadic depth and the
square of the triplicity --- nothing else enters.)} It is worth pausing on how much
has happened: a quantity Feynman called ``one of the greatest damn mysteries of
physics'' is, in this reading, the statement that one binary seam folded seven times,
plus a \(3\times3\) family cloud, has inverse capacity \(137\).

\subsubsection{The weak mixing angle: rational partition in full}
\label{app:gauge-weakangle}

The main-text statement is Theorem~\ref{thm:sw2} (T-SM-28.1). Unlike the photon line,
this derivation is neither a recursion nor a transcendental cell: it is a single
\emph{partition}. The neutral electroweak sector carries a split generator whose
trace norm is fixed, with no freedom, by the \AThree{} trace structure and the ledger,
\begin{equation}
\Tr\!\left(\Ysplit^2\right)=\frac{L}{N\,(L-N)}=\frac{5}{3\cdot 2}=\frac{5}{6}.
\label{eq:TrY2-app}
\end{equation}
\emph{(Physics intuition: \(\Tr(\Ysplit^2)\) is the total ``neutral norm'' that the
hypercharge and isospin generators present to the neutral line --- the squared width of
the channel through which neutral current must pass. The integers \(L=5\) and
\(L-N=2\) are the embedding weight of hypercharge and the leftover isospin room
after the three colored lanes are accounted for; their ratio over \(N\) is the bare
squared width.)}

Two named structural steps then convert this norm into the observed angle. The first
is the \emph{quadratic neutral-partition lemma}: the neutral sector partitions
\emph{power} (stiffness), not amplitude, so the relevant weight is the \emph{square} of
the trace norm,
\begin{equation}
\bigl[\Tr(\Ysplit^2)\bigr]^2=\left(\frac56\right)^2=\frac{25}{36}.
\label{eq:quadratic-lemma-app}
\end{equation}
\emph{(Physics intuition: power goes as amplitude squared. Because the weak angle is a
ratio of stiffnesses, not of amplitudes, the channel's squared width enters squared
again --- the same reason an electrical power divider splits power as the square of the
voltage-divider ratio.)} The second is the \emph{color-blind averaging lemma}: the
measured neutral readout carries no color label, so the three triplicity sectors are
\emph{averaged}, not summed, contributing a factor \(1/N\),
\begin{equation}
\sin^2\theta_W(M_Z)=\frac{1}{N}\bigl[\Tr(\Ysplit^2)\bigr]^2
=\frac13\cdot\frac{25}{36}
=\boxed{\;\frac{25}{108}=\frac{L^2}{4N^3}\simeq 0.231481.\;}
\label{eq:sw2-derived-app}
\end{equation}
\emph{(Physics intuition: imagine a neutral river meeting a fork whose banks are locked
in a \(3+2\) configuration. The water divides according to the squared widths the
geometry presents --- not by preference --- which gives \((5/6)^2\). An observer
standing downstream cannot see the three color lanes that the flow has been replicated
across, so the observer reports the per-lane average, dividing by \(3\). The number the
observer writes down is \(25/108\), and it matches the PDG effective angle
\(0.23155\) to \(0.03\%\) with not a single fitted input.)}

The denominator deserves a remark: \(4N^3=4\cdot 27=108\) and \(L^2=25\), so the angle
is literally ``five squared over four times three cubed.'' That it lands on the
measured value is the single sharpest piece of evidence in the gauge sector. In the
honesty triage it is recorded as the most-derived of the twenty-eight entries:
Steps fixing \eqref{eq:TrY2-app} and the partition are forced by \AThree{} and the
ledger, while the two italicized lemmas --- quadratic partition and color-blind
averaging --- are well-motivated \GG{}-internal assignments whose promotion to full
proofs (the natural target of the \GFF{} variational route) would make the angle an
unconditional theorem. For the cross-check against ordinary \(\SU(5)\)-normalized
electroweak kinematics --- the same \(25/108\) recovered from
\(\alpha_1/\alpha_2=125/249\) --- see \S\ref{app:neutral-lattice} below.

\subsubsection{The strong coupling: the angle--scale constant in full}
\label{app:gauge-strong}

The main-text statement is Theorem~\ref{thm:alphas} (T-SM-29.1). The color line is
geometrically unlike the photon line, and the difference is the whole content of the
result. The photon is a \emph{sum} channel: one surviving Abelian seam, so it collapses
to a dyadic ladder and gives an integer. The color line is a non-Abelian
\emph{difference} channel, and a difference channel cannot collapse a circle to a
ladder --- it keeps the circle. It also keeps the radial direction, because QCD is the
most visibly logarithmic of the gauge lines. Two geometric primitives therefore
survive on the color seam:
\begin{equation}
\theta\sim\theta+2\pi
\quad(\text{compact angle}),
\qquad
r=\ln\frac{\EGUT}{\mu}
\quad(\text{logarithmic radius}).
\label{eq:strong-primitives-app}
\end{equation}
Each contributes one factor to the irreducible color cell. The compact factor is not
the full circle: opposite angular orientations are conjugate under \DGI{} and hence
physically identical, so the minimal compact primitive is the half-circle quotient
\begin{equation}
\operatorname{vol}_{\mathrm{prim}}\!\left(S^1/\mathbb{Z}_2\right)=\pi .
\label{eq:halfcircle-app}
\end{equation}
\emph{(Physics intuition: on a non-Abelian seam, ``clockwise'' and
``anticlockwise'' are gauge-equivalent labels for the same physical winding; folding
the circle onto its mirror image leaves half of it, the arc of length \(\pi\). This is
why \(\pi\), and not \(2\pi\), is the strong primitive.)} The radial factor is the
minimal unit of logarithmic transport, one e-fold,
\begin{equation}
\Delta r_{\mathrm{prim}}=1
\qquad\Longleftrightarrow\qquad
e^{\Delta r_{\mathrm{prim}}}=e .
\label{eq:efold-app}
\end{equation}
\emph{(Physics intuition: because the radial coordinate is a logarithm of energy, the
smallest natural step is the one that changes the scale by a factor \(e\) --- a single
e-fold. The strong coupling is the one gauge line whose running is steep enough that
this radial primitive is not negligible, which is exactly why an \(e\) appears here and
nowhere else in the gauge sector.)} In the minimal closure class no extra multiplicity
is admitted, so the irreducible color seam is the direct product of one compact and
one radial primitive, with inverse capacity
\begin{equation}
C_s^\star=\pi\cdot e=\pi e,
\qquad\Longrightarrow\qquad
\boxed{\;\alphas^{-1}(M_Z)=\pi e\simeq 8.5397,\qquad
\alphas(M_Z)=\frac{1}{\pi e}\simeq 0.11700.\;}
\label{eq:alphas-derived-app}
\end{equation}
The measured PDG value is \(\alpha_s(M_Z)=0.1180(9)\); the first-order geometric value
\(0.1170\) sits within roughly one percent, and the quantum-defect completion of
Part~\ref{part:VI} closes the small remaining gap. This theorem is parameter-free but,
in candor, conceptually more delicate than the weak-angle partition: it reads a
non-Abelian seam in terms of minimal primitive cells of compact and radial geometry,
which is more interpretive than the finite-dimensional partition that fixes
\(\sin^2\theta_W\).

\subsubsection{The derived couplings, and why three numbers of three different kinds}
\label{app:gauge-unifying}

With \(\sin^2\theta_W=25/108\) and \(e\) from \eqref{eq:e-readout-app}, the two
electroweak gauge couplings follow with no further input,
\begin{equation}
g_2=\frac{e}{\sin\theta_W}=\frac{4\sqrt{\pi}\,N^{3/2}}{L\,2^{J/2}}
\simeq 0.6512411,
\qquad
g_Y=\frac{e}{\cos\theta_W}=\frac{4\sqrt{\pi}\,N^{3/2}}{2^{J/2}\sqrt{4N^3-L^2}}
\simeq 0.3574150,
\label{eq:g2gY-app}
\end{equation}
in exact agreement with the textbook extraction of the Step-1 worked example
(\(g\simeq 0.652\), \(g'\simeq 0.357\)): the geometric derivation and the
data-fed algebra are, as promised, the same arithmetic seen from two sides.

It is worth standing back to see the shape of what has been derived. The three gauge
couplings of the Standard Model emerge as three numbers of three \emph{different
mathematical kinds}, and the kind is dictated by the topology of the seam:
\begin{itemize}
\item the electromagnetic coupling is an \emph{integer power}, \(2^7\), because its
seam is a single Abelian \emph{sum} channel that can only fold in two;
\item the weak angle is a \emph{rational}, \(25/108\), because its neutral sector is a
finite-dimensional \emph{partition} of squared widths over triplicity lanes;
\item the strong coupling is a \emph{transcendental}, \(1/\pi e\), because its seam is a
non-Abelian \emph{difference} channel that keeps both a compact angular primitive and a
logarithmic radial primitive.
\end{itemize}
\emph{(Physics intuition: in the Standard Model these three numbers are three unrelated
measured constants. Here they are three readings of one ledger, and the very
\emph{type} of each number --- integer, rational, transcendental --- is a fingerprint
of how its force closes on the boundary. That the most precisely measured constants of
nature should carry, in their arithmetic character, a record of the topology that
produced them is the central claim of the gauge sector, and it is what the explicit
arithmetic in the remainder of this appendix verifies number by number.)}

\subsection{Numerical constants and unit conversion details}
\label{app:constants}

For numerical convenience we report energies in GeV, but the derivation
itself uses only the allowed constants \(c\), \(h\), and \(G_N\). The
reduced Planck constant is
\begin{equation}
\hbar = \frac{h}{2\pi}.
\label{eq:hbar-def-app}
\end{equation}
Using the 2022 CODATA values
\begin{equation}
 c = 299\,792\,458\ \mathrm{m\,s^{-1}},
 \qquad
 h = 6.626\,070\,15\times 10^{-34}\ \mathrm{J\,s},
 \qquad
 G_N = 6.67430\times 10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}},
\label{eq:constants-app}
\end{equation}
we obtain
\begin{equation}
\hbar = 1.054\,571\,817\times 10^{-34}\ \mathrm{J\,s}.
\label{eq:hbar-num-app}
\end{equation}
To express energies in electron-volts we use the exact SI conversion
\begin{equation}
1\ \mathrm{eV} = 1.602\,176\,634\times 10^{-19}\ \mathrm{J},
\qquad
1\ \mathrm{GeV}=10^9\ \mathrm{eV}.
\label{eq:eV-conv-app}
\end{equation}
This conversion is a unit-definition convenience rather than a theory
input.

Then
\begin{align}
\EPl
&=\sqrt{\frac{\hbar c^5}{G_N}}
=\sqrt{\frac{(1.054571817\times10^{-34})(2.99792458\times10^8)^5}
{6.67430\times10^{-11}}}\ \mathrm{J}
\nonumber\\
&\simeq 1.9560816367\times10^9\ \mathrm{J}
\nonumber\\
&\simeq 1.2208901286\times10^{19}\ \mathrm{GeV}.
\label{eq:Epl-arith-app}
\end{align}
The remaining scales follow immediately from multiplication by the exact
OGN exponentials,
\begin{align}
\EGUT &= \EPl\e^{-2\pi}
\simeq 2.2799424\times10^{16}\ \mathrm{GeV},
\label{eq:Egut-arith-app}\\
\Mlock &= \frac{\EPl}{2\e^{35}}
\simeq 3.8489274\times10^3\ \mathrm{GeV}.
\label{eq:Mlock-arith-app}
\end{align}

\subsection{Detailed discussion of the Planck--GUT--lock scale chain}
\label{app:scalechain}

This appendix expands the derivation of the three scales to make the
logic as explicit as possible.

\subsection{Why the Planck scale must appear}

If one is allowed only \(c\), \(h\), and \(G_N\), there is only one way
to construct an energy:
\begin{equation}
[c]^a[h]^b[G_N]^d = \mathrm{energy}
\qquad\Longrightarrow\qquad
\EPl\propto \left(\frac{hc^5}{G_N}\right)^{1/2}.
\label{eq:dim-analysis-app}
\end{equation}
The factor of \(2\pi\) that turns \(h\) into \(\hbar\) is not optional
in the present framework, because the compact direction is literally a
circle, and the theory treats full angular cycles as primitive objects.
Thus the same \(2\pi\) that appears in the monodromy step also appears in
\(\hbar=h/(2\pi)\).

\subsection{Why the GUT seam is one monodromy cycle below the Planck anchor}

The coordinate \(r\) is a logarithmic scale coordinate. Therefore any
special radial displacement corresponds to a multiplicative energy
ratio. The compact \(S^1_\theta\) sector contains a unique primitive
angle, one full cycle \(2\pi\). If the constitution is to maximize
symmetry and minimize arbitrary freedom, the first distinguished shift
below the Planck anchor must be one full primitive cycle and not some
arbitrary irrational multiple of it. This is the precise content of
\begin{equation}
\Delta r_{\Pl\to\GUT}=2\pi M,
\qquad M=1.
\label{eq:monodromy-gut-app}
\end{equation}

\subsection{Why the lock scale carries a factor of one half}

\ATwo{} itself fixes the exponential redshift \(\e^{-K}\), but does
not need to specify the order-one endpoint convention because its
purpose is the theorem-level bulk architecture. In the present paper,
that convention becomes important because we want an explicit numerical
lock scale.

The interval \(I_r\) has two endpoints. Under the minimal symmetric
choice, the endpoint normalization is split equally between them. This
is the simplest binary convention compatible with OGN, and it is also
the only order-one factor that later allows the electroweak dyadic code
and the radial endpoint normalization to belong to the same arithmetic
family. Hence the factor \(1/2\) in
\begin{equation}
\Mlock = \frac{\EPl}{2\e^K}.
\label{eq:half-factor-app}
\end{equation}
This factor is not a tunable parameter. It is a binary convention
forced by the interval geometry once one insists on an explicit lock
seam.

\subsection{Rational-lattice reconstruction of the neutral electroweak sector}
\label{app:neutral-lattice}

The main text used the compact formulas
\(\alphahatinv(M_Z)=2^7\) and
\(\sin^2\theta_W(M_Z)=25/108\).
This appendix shows how these formulas can be cross-checked against a
more detailed rational-lattice reconstruction of the neutral sector.
The purpose is not to add a new assumption, but to show that the dyadic
and rational formulas are consistent with ordinary electroweak
kinematics and \(\SU(5)\)-normalized hypercharge.

\subsection{From the weak angle to the ratio \texorpdfstring{\(\alpha_1/\alpha_2\)}{alpha1/alpha2}}

With the conventional \(\SU(5)\)-normalized coupling \(\alpha_1\) and the
weak-isospin coupling \(\alpha_2\), one has
\begin{equation}
\sin^2\theta_W
=\frac{(3/5)\alpha_1}{\alpha_2+(3/5)\alpha_1}.
\label{eq:sin2-alpha12-app}
\end{equation}
Substituting the first-principles value
\(\sin^2\theta_W=25/108\) gives
\begin{equation}
\frac{(3/5)\alpha_1}{\alpha_2+(3/5)\alpha_1} = \frac{25}{108},
\label{eq:solve-ratio-app-1}
\end{equation}
so that
\begin{equation}
108\left(\frac35\alpha_1\right)
=25\left[\alpha_2+\left(\frac35\alpha_1\right)\right].
\label{eq:solve-ratio-app-2}
\end{equation}
Rearranging,
\begin{equation}
\frac{249}{5}\alpha_1=25\alpha_2,
\qquad
\frac{\alpha_1}{\alpha_2}=\frac{125}{249}.
\label{eq:alpha-ratio-app}
\end{equation}
This is the precise sense in which the weak-angle rational code fixes the
neutral-coupling ratio.

\subsection{Electromagnetic coupling from the neutral sum}

In standard electroweak kinematics,
\begin{equation}
\alpha_{\mathrm{em}}^{-1}
=\alpha_2^{-1}+\frac53\alpha_1^{-1}.
\label{eq:alphaem-from-alpha12-app}
\end{equation}
Thus once the ratio \eqref{eq:alpha-ratio-app} is fixed, the only
remaining freedom would ordinarily be an overall scale. In the present
framework that overall neutral scale is precisely what OGN compresses to
a dyadic closure code. The result is
\begin{equation}
\alpha_{\mathrm{em}}^{-1}=2^J=128.
\label{eq:alphaem-dyadic-crosscheck-app}
\end{equation}
The compact formulas used in the main text are therefore fully
compatible with the ordinary neutral-sector algebra.

\subsection{Why the rational code is more rigid than the dyadic one}

The rational weak-angle code involves only the group-theoretic weight of
hypercharge versus isospin and the triplicity index. The electromagnetic
dyadic closure, by contrast, depends on how the neutral sum lands on the
OGN lattice. This is why the QD correction naturally appears first in
\(\alpha_{\mathrm{em}}^{-1}\) and not in \(\sin^2\theta_W\).

\subsection{Detailed arithmetic for the first-order gauge and Higgs block}
\label{app:arith-gaugehiggs}

Collecting the exact first-order formulas,
\begin{equation}
\alpha_{\mathrm{em}}=\frac{1}{128},
\qquad
\sW^2=\frac{25}{108},
\qquad
\cW^2=\frac{83}{108},
\qquad
\alpha_s=\frac{1}{\pi\e},
\label{eq:arith-collect-app}
\end{equation}
we immediately obtain
\begin{equation}
 e^2=4\pi\alpha_{\mathrm{em}}=\frac{\pi}{32},
 \qquad
 e\simeq 0.3133285343.
\label{eq:e-arith-app}
\end{equation}
Then
\begin{align}
\sW &= \sqrt{\frac{25}{108}}\simeq 0.4811252243,
\label{eq:sw-arith-app}\\
\cW &= \sqrt{\frac{83}{108}}\simeq 0.8766528118,
\label{eq:cw-arith-app}\\
 g_2 &= \frac{e}{\sW}\simeq 0.6512411291,
\label{eq:g2-arith-app}\\
 g_Y &= \frac{e}{\cW}\simeq 0.3574150031.
\label{eq:gY-arith-app}
\end{align}
With
\begin{equation}
\Mlock \simeq 3848.9274\ \mathrm{GeV},
\qquad
v^{(1)}=\Mlock\left(\frac25\right)^3,
\label{eq:v-arith-app-1}
\end{equation}
we get
\begin{equation}
v^{(1)}\simeq 246.331354\ \mathrm{GeV}.
\label{eq:v-arith-app-2}
\end{equation}
Thus
\begin{align}
m_W^{(1)} &= \frac12 g_2v^{(1)} \simeq 80.2106\ \mathrm{GeV},
\label{eq:mW-arith-app}\\
m_Z^{(1)} &= \frac12 v^{(1)}\sqrt{g_2^2+g_Y^2}
\simeq 91.4965\ \mathrm{GeV}.
\label{eq:mZ-arith-app}
\end{align}
For the quartic,
\begin{equation}
\lambda_H^{(1)}=\frac{1}{\pi\e}+\frac{1}{128}
=0.1249121630\ldots,
\label{eq:lambda-arith-app}
\end{equation}
so
\begin{equation}
m_h^{(1)}=\sqrt{2\lambda_H^{(1)}}v^{(1)}
\simeq 123.1224\ \mathrm{GeV}.
\label{eq:mh-arith-app}
\end{equation}
This appendix makes clear that the entire first-order gauge/Higgs block
is numerically straightforward once the scale chain and the \OGNfive{} ledger
have been fixed.

\subsection{QD arithmetic and residual-theory estimates}
\label{app:qd}

The two characteristic suppressed parameters are
\begin{equation}
\varepsilon_J = 2^{-7}=7.8125\times10^{-3},
\qquad
\varepsilon_K = \frac{1}{70\pi}=4.5473\times10^{-3}.
\label{eq:eps-app}
\end{equation}
The leading QD electromagnetic correction is
\begin{equation}
\delta\big(\alphahatinv\big) = -\frac{N^2}{2^J}=-\frac{9}{128}=-0.0703125.
\label{eq:delta-alpha-app}
\end{equation}
The leading QD strong correction is
\begin{equation}
\delta\alpha_s = \frac{1}{\pi\e}\frac{1}{128}
\simeq 9.1484\times10^{-4}.
\label{eq:delta-alphas-app}
\end{equation}
The VEV shift adopted in the main text is
\begin{equation}
\delta_v\equiv \frac{v^{\mathrm{QD}}-v^{(1)}}{v^{(1)}}
\simeq -4.52\times10^{-4},
\label{eq:delta-v-app}
\end{equation}
which is comfortably smaller than \(\varepsilon_K\), as expected for a
finite seam-transport correction.

For the quartic, the adopted shift is
\begin{equation}
\delta\lambda_H
\equiv \lambda_H^{\mathrm{QD}}-\lambda_H^{(1)}
\simeq 0.00437,
\label{eq:delta-lambda-app}
\end{equation}
large enough to move the Higgs mass by approximately
\begin{equation}
\delta m_h \simeq 2.08\ \mathrm{GeV}.
\label{eq:delta-mh-app}
\end{equation}
This explains why the Higgs is the most sensitive of the five primary
observables to next-order dressing.

\subsection{Why the second-order uncertainty on the Higgs is kept conservative}

In principle one could assign a much smaller residual uncertainty to the
QD Higgs mass by iterating the same suppression logic once more.
However, that would overstate the formal completeness of the current
preprint. The present paper has reconstructed the second-order
structure faithfully within the \GG{} logic, but it has not yet written a
fully explicit closed two-loop Wilson-line computation in the 6D bulk.
For that reason the QD uncertainty on \(m_h\) is kept at the
conservative level of \(0.30\ \mathrm{GeV}\), which is more realistic for
this present stage.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{D.1.1. Derive the electric charge from
the neutral-sector rotation: $e = g\sin\theta_W = g'\cos\theta_W$.}

\noindent\emph{Solution.}  Step 1: the neutral gauge fields
$(W^3_\mu, B_\mu)$ mix into the mass eigenstates
$(Z_\mu, A_\mu)$ by the rotation \eqref{eq:ewmix-app}; the photon
is the combination $A_\mu = \sin\theta_W\,W^3_\mu +
\cos\theta_W\,B_\mu$.  Step 2: insert into the covariant
derivative \eqref{eq:covderiv-app} and collect the coefficient of
$A_\mu$ acting on a field of isospin $T^3$ and hypercharge $Y$:
it is $g\sin\theta_W\,T^3 + g'\cos\theta_W\,\tfrac{Y}{2}$.
Step 3: for this to be $eQ$ with $Q = T^3 + Y/2$ --- one charge,
one coupling --- the two coefficients must be equal:
$e = g\sin\theta_W = g'\cos\theta_W$ \eqref{eq:echarge-app},
equivalently $1/e^2 = 1/g^2 + 1/g'^2$.  Physics intuition:
electromagnetism is not one of the Standard Model's primitive
forces --- it is the unbroken diagonal remnant of two of them,
and its coupling is the parallel-resistor sum of theirs; the
dyadic readout of the third step prices exactly this remnant.

\medskip\noindent\textbf{D.1.2. Show that exact unification
fixes the tree-level weak angle at $\sin^2\theta_W = 3/8$.}

\noindent\emph{Solution.}  Step 1: embedding hypercharge in a
simple group forces the normalization $g_1^2 = \tfrac53 g'^2$
\eqref{eq:su5norm-app} (the $5/3$ is the trace ratio of the
hypercharge generator over one fermion family).  Step 2: at the
unification point $g_1 = g_2 \equiv g$, so
$g'^2 = \tfrac35 g^2$.  Step 3: insert into
$\sin^2\theta_W = g'^2/(g^2 + g'^2) = \tfrac{3/5}{1 + 3/5} =
\tfrac38$ \eqref{eq:sin2-gut-app} --- a pure group-theory
number, no dynamics anywhere.  Physics intuition: $3/8$ is what
the weak angle would be if the three forces were one and
undressed; everything between $3/8$ and the measured $0.231$ is
running --- which is why the angle at $M_Z$ is a probe of the
entire desert between the weak scale and the seam.

\medskip\noindent\textbf{D.1.3. Verify the one-loop coefficient
$b_2 = -19/6$, including the Higgs' contribution.}

\noindent\emph{Solution.}  Step 1: for $\SU(N)$ with $n_f$
Dirac-fermion doublets-worth of matter and $n_s$ complex scalar
doublets, the coefficient in the convention of
\eqref{eq:rge-app} is $b = -\tfrac{11}{3}N + \tfrac{2}{3}\cdot
2n_f\cdot\tfrac12 + \tfrac{1}{3}n_s\cdot\tfrac12$ (adjoint gauge
loop; Weyl-fermion and scalar matter loops).  Step 2: for
$\SU(2)_L$: the gauge term gives $-22/3$; the fermions --- four
doublets per generation (three quark colors plus one lepton),
three generations, i.e.\ twelve Weyl doublets --- give
$+\tfrac{2}{3}\cdot12\cdot\tfrac12 = +4$; the single Higgs
doublet gives $+\tfrac{1}{3}\cdot\tfrac12 = +\tfrac16$.
Step 3: total $b_2 = -\tfrac{22}{3} + 4 + \tfrac16 =
-\tfrac{19}{6}$ \eqref{eq:beta-coeffs-app}; the same counting
gives $b_3 = -11 + 4 = -7$ (no scalar) and, with the $\tfrac53$
normalization, $b_1 = +\tfrac{41}{10}$.  Physics intuition:
every particle in the theory votes on how each force runs ---
gauge bosons screen-in (asymptotic freedom), matter screens out
--- and the Higgs' lonely $+1/6$ in $b_2$ is the fingerprint of
the scalar sector on the gauge flow: remove it and the near-miss
of D.1.4 shifts visibly.

\medskip\noindent\textbf{D.1.4. Quantify the classic near-miss:
where do the couplings actually cross?}

\noindent\emph{Solution.}  Step 1: with the measured inverse
couplings at $M_Z$ (\(\alpha_1^{-1} \approx 59.0\),
\(\alpha_2^{-1} \approx 29.6\), \(\alpha_3^{-1} \approx 8.4\))
run by \eqref{eq:rge-app}, two couplings cross where
$\alpha_i^{-1} - \alpha_j^{-1}$ is consumed by
$(b_i - b_j)\ln(\mu/M_Z)/2\pi$.  Step 2: the $1$--$2$ crossing:
$\ln(\mu/M_Z) = 2\pi\,(59.0 - 29.6)/(41/10 + 19/6) \approx
25.4$, i.e.\ $\mu_{12} \approx 1.0\times10^{13}$~GeV.  Step 3:
the $2$--$3$ crossing lands at $\mu_{23} \approx
1\times10^{17}$~GeV --- four orders of magnitude away: the
Standard Model \emph{almost} unifies, and measurably does not.
Physics intuition: the near-miss is data, not aesthetics ---
whatever completes the Standard Model must either supply new
thresholds that bend the lines together (the supersymmetric
bet) or, as the third step of this chapter argues, stop asking
the three couplings to meet dynamically at all and derive each
from the seam geometry instead.

\medskip\noindent\textbf{D.3.1. Evaluate the dyadic
electromagnetic code against both measured faces of
$\alpha$.}

\noindent\emph{Solution.}  Step 1: the closure recursion
\eqref{eq:dyadic-recursion-app} gives $C_\gamma(n) = 2^n$, and
the lawful depth is the ledger's odd closure index $J = 7$:
$\widehat\alpha^{-1}(M_Z) = 2^7 = 128$
\eqref{eq:alpha-dyadic-app}, against the measured running value
$127.95$: agreement to $0.035\%$.  Step 2: the on-shell
Sommerfeld face adds the family-cloud triplicity $N^2$:
$\alpha^{-1}(0) = 2^J + N^2 = 128 + 9 = 137$
\eqref{eq:alpha137-app}, against $137.036$: agreement to
$0.026\%$.  Step 3: both faces from two ledger integers, with
the \emph{difference} between them (the running from $0$ to
$M_Z$) carried by standard QED --- the code prices the
endpoints, the textbook prices the path.  Physics intuition:
the most famous pure number in physics is read here as a power
of two plus a square --- and the reading is falsifiable, since
neither face has a dial to absorb a mismatch; the QD refinement
that closes the residual $0.03\%$ is priced in the chapter's
second-order section.

\medskip\noindent\textbf{D.3.2. Verify the rational weak angle
$\sin^2\theta_W = 25/108$ and its pull.}

\noindent\emph{Solution.}  Step 1: the quadratic hypercharge
lemma gives $[\Tr(\Ysplit^2)]^2 = (5/6)^2 = 25/36$
\eqref{eq:TrY2-app}; the color-blind neutral readout
\emph{averages} over the $N = 3$ triplicity sectors,
contributing $1/N$.  Step 2:
$\sin^2\theta_W(M_Z) = \tfrac13\cdot\tfrac{25}{36} =
\tfrac{25}{108} = \tfrac{L^2}{4N^3} = 0.231481$
\eqref{eq:sw2-derived-app}.  Step 3: against the measured
effective leptonic angle $0.23155$, the pull is $0.03\%$ ---
and the number is a ratio of ledger integers, $L^2/4N^3$, with
nothing continuous anywhere.  Physics intuition: compare
D.1.2 --- the textbook's $3/8$ lives at the seam and needs the
desert's running to reach the data, while the geometric
$25/108$ is stated \emph{at} $M_Z$ directly: the two readings
bracket the same physics from opposite ends, which is exactly
how the chapter reconciles them.

\medskip\noindent\textbf{D.3.3. Evaluate the strong-coupling
closure $\alpha_s^{-1}(M_Z) = \pi e$.}

\noindent\emph{Solution.}  Step 1: the color seam admits no
integer multiplicity; its inverse capacity is the product of
the one compact and the one radial primitive
\eqref{eq:alphas-derived-app}: $\alpha_s^{-1} = \pi e =
8.5397$.  Step 2: hence $\alpha_s(M_Z) = 1/\pi e = 0.11710$,
against the world average $0.1179 \pm 0.0009$.  Step 3: the
pull is $0.9\sigma$ --- one standard deviation, from two
transcendental constants and no dial (and F.10.1 explains why
exactly these two constants are the only ones available).
Physics intuition: where the abelian sector counts (powers of
two), the confining sector \emph{measures} (volumes of the
compact and radial primitives) --- the appearance of $\pi e$
here and $1/\pi$ in the cosmological inventory is one
geometry pricing different sectors with the same two lengths.

\medskip\noindent\textbf{D.3.4. Verify the ledger's double
identity $K = LJ = M^2 + N^2 + L^2$.}

\noindent\emph{Solution.}  Step 1: from the ledger
\eqref{eq:ogn-ledger-app}, $(M,N,L;J,K) = (1,3,5;7,35)$:
compute $LJ = 5\times7 = 35$.  Step 2: compute
$M^2 + N^2 + L^2 = 1 + 9 + 25 = 35$.  Step 3: both equal $K$
--- the lock depth is simultaneously the product of the
embedding weight and the closure index \emph{and} the sum of
the squared multiplicities: one integer, two independent
readings, which is the kind of overdetermination that makes a
five-integer ledger rigid rather than adjustable (change any
entry and both identities break at once).  Physics intuition:
the ledger is not a list of five free integers but a single
interlocked object --- the same rigidity that C3's black-hole
module leans on when it borrows $K$ for the species-sum scale
and that the cosmology paper spends as $e^{-K/2}$ in the
baryon asymmetry.

\GGtakeaway{\emph{Standard result:} three gauge groups, three
measured couplings, one measured angle, and one-loop running
that nearly unifies.  \emph{What it assumed:} the groups and
the couplings themselves --- four experimental inputs and one
unexplained near-miss.  \emph{The geometric reading:} the gauge
pattern and the boundary couplings descend from the bulk, the
weak angle becomes a geometric ratio, and the two-loop
Wilson-line computation is named as the open refinement, with
its uncertainty carried conservatively.}

\section{Electroweak symmetry breaking and the Higgs}
\label{app:ewsb}

\GGteachbox{Electroweak symmetry breaking as a graduate student
meets it --- the Higgs mechanism, the $W$ and $Z$ masses, the
massless photon, the custodial structure --- with its free
numbers stated plainly; and the geometric reading in which the
Higgs is a Wilson-line degree of freedom, the electroweak scale
an exponentially suppressed, triply filtered shadow of the
Planck scale, and the hierarchy problem dissolved by a change in
what the Higgs \emph{is} rather than solved by a cancellation.}
\GGreadalong{\citet{Schwartz2014} for the modern treatment;
\citet{WeinbergQFT1996} at depth; \citet{Higgs1964} and
\citet{EnglertBrout1964} for the originals.}

This is the second sector appendix of the course. Its subject is the mechanism that
gives the \(W\) and \(Z\) their masses and leaves the photon massless --- electroweak
symmetry breaking --- together with the scalar that drives it, the Higgs. We again
follow the three-step plan: the textbook mechanism as a graduate student meets it;
the two free numbers it must take from experiment, and the naturalness puzzle they
raise; and finally the geometric derivation, in which the Higgs vacuum value and the
quartic coupling descend from a Wilson-line holonomy and the integer ledger, with the
hierarchy problem dissolving into a statement about exponential scale suppression. The textbook development of electroweak symmetry breaking and the Higgs mechanism is given in \citet{Schwartz2014} and \citet{WeinbergQFT1996}.

\subsection{Step 1 --- electroweak symmetry breaking as a graduate student meets it}
\label{app:ewsb-step1}

\paragraph{Why a scalar is needed at all.}
An unbroken gauge theory has only massless gauge bosons: a mass term
\(\tfrac12 m^2 W_\mu W^\mu\) is not gauge invariant. Yet the weak bosons are heavy,
\(m_W\simeq 80\ \mathrm{GeV}\) and \(m_Z\simeq 91\ \mathrm{GeV}\), while the photon is
exactly massless. The Standard Model reconciles this by \emph{spontaneous} breaking:
the gauge symmetry is intact in the Lagrangian but not in the vacuum.
\emph{(Physics intuition: nothing in the laws picks a direction, but the ground state
does --- like a pencil balanced on its tip that must fall some way. The laws keep their
symmetry; the vacuum quietly spends it.)}

\paragraph{The Higgs field and the Mexican-hat potential.}
One introduces a complex scalar doublet of \(\SU(2)_L\) with hypercharge \(Y=1\),
\begin{equation}
\Phi=\begin{pmatrix}\phi^+\\ \phi^0\end{pmatrix},
\qquad
V(\Phi)=\mu^2\,\Phi^\dagger\Phi+\lambda\,(\Phi^\dagger\Phi)^2 .
\label{eq:higgs-pot-app}
\end{equation}
If \(\mu^2<0\) and \(\lambda>0\) the potential is the famous Mexican hat: the origin
is a local maximum and the minima form a circle at
\begin{equation}
\langle\Phi^\dagger\Phi\rangle=\frac{v^2}{2},
\qquad
v=\sqrt{\frac{-\mu^2}{\lambda}} .
\label{eq:vev-def-app}
\end{equation}
Choosing the unitary gauge \(\langle\Phi\rangle=\tfrac{1}{\sqrt2}(0,\,v)^{\mathsf T}\)
breaks \(\SU(2)_L\times\U(1)_Y\to\U(1)_{\mathrm{em}}\).
\emph{(Physics intuition: the field rolls off the central bump into the circular
trough. Which point of the trough it picks is irrelevant --- they are gauge-equivalent ---
but that it picks one at all is what breaks the symmetry and sets the scale \(v\).)}

\paragraph{Eaten Goldstones and gauge-boson masses.}
Of the four real components of \(\Phi\), three are massless Goldstone modes; by the
Higgs mechanism they become the longitudinal polarizations of \(W^\pm\) and \(Z\).
Expanding the scalar kinetic term \(|D_\mu\Phi|^2\) about the vacuum gives the masses
directly,
\begin{equation}
m_W=\tfrac12\,g\,v,
\qquad
m_Z=\tfrac12\,v\sqrt{g^2+g'^2},
\qquad
m_\gamma=0 ,
\label{eq:vboson-mass-app}
\end{equation}
with the photon protected by the unbroken \(\U(1)_{\mathrm{em}}\). The single surviving
scalar fluctuation \(h\) is the physical Higgs, with mass
\begin{equation}
m_h^2=2\lambda v^2 .
\label{eq:mh-def-app}
\end{equation}
\emph{(Physics intuition: the gauge bosons grow heavy by swallowing the would-be
Goldstone modes --- a massless gauge boson has two polarizations, a massive one has
three, and the extra longitudinal state is exactly the eaten Goldstone. The photon
eats nothing and stays massless.)}

\paragraph{Custodial symmetry and \texorpdfstring{\(\rho=1\)}{rho=1}.}
The doublet structure leaves a residual ``custodial'' \(\SU(2)\) that forces the
tree-level relation
\begin{equation}
\rho\equiv\frac{m_W^2}{m_Z^2\cos^2\theta_W}=1 ,
\label{eq:rho-app}
\end{equation}
verified experimentally to the per-mille level --- one of the sharpest confirmations
that the order parameter is a doublet.

\paragraph{Worked example (STANDARD TEXTBOOK PEDAGOGY).}
The vacuum value is fixed by the Fermi constant, measured from muon decay as
\(G_F=1.1664\times10^{-5}\ \mathrm{GeV}^{-2}\). Since
\(G_F/\sqrt2=g^2/(8m_W^2)=1/(2v^2)\),
\begin{equation}
v=\left(\sqrt2\,G_F\right)^{-1/2}
=\left(1.6495\times10^{-5}\right)^{-1/2}
\simeq 246.2\ \mathrm{GeV}.
\label{eq:v-fromGF-app}
\end{equation}
With the measured Higgs mass \(m_h=125\ \mathrm{GeV}\), pinned down by the 2012 ATLAS and CMS discovery of the boson \citep{ATLAS2012,CMS2012}, the quartic follows from
\eqref{eq:mh-def-app},
\begin{equation}
\lambda=\frac{m_h^2}{2v^2}=\frac{125^2}{2\cdot246^2}\simeq 0.129 .
\label{eq:lambda-fromdata-app}
\end{equation}
The machinery again works perfectly --- \emph{given} \(G_F\) (hence \(v\)) and \(m_h\)
(hence \(\lambda\)) from experiment. Where the two numbers \(v\) and \(\lambda\) come
from, and why \(v\) sits so far below the Planck scale, is exactly what the Standard
Model cannot say.

\subsection{Step 2 --- what electroweak symmetry breaking assumes but cannot explain}
\label{app:ewsb-step2}

The mechanism of Step 1 is exact and experimentally vindicated, but it leaves three
questions open, and each is sharper than the gauge-coupling questions of the previous
appendix.

First, \emph{the two scalar parameters are pure inputs}. The vacuum value \(v\) (or
equivalently \(\mu^2\)) and the quartic \(\lambda\) are two more of the twenty-eight
free numbers. Nothing fixes \(v\simeq 246\ \mathrm{GeV}\) rather than any other scale,
and nothing fixes \(\lambda\simeq 0.13\). Indeed the sign \(\mu^2<0\) that triggers
breaking is simply \emph{posited}: the Standard Model does not explain why the vacuum
breaks the symmetry at all, only what follows if it does.

Second, and most famously, \emph{the hierarchy and naturalness problem}. The Higgs
mass-squared receives quantum corrections that scale quadratically with whatever
heavy scale \(\Lambda\) the theory is embedded in,
\begin{equation}
\delta m_h^2\sim \frac{\Lambda^2}{16\pi^2}\times(\text{couplings}).
\label{eq:hierarchy-app}
\end{equation}
If \(\Lambda\) is the Planck or GUT scale, \(\delta m_h^2\) is some thirty orders of
magnitude larger than the observed \(m_h^2\); keeping the physical Higgs light then
requires the bare parameter to cancel the correction to one part in \(10^{34}\). The
electroweak scale is, in this sense, unnaturally small, and no symmetry of the
Standard Model protects it.
\emph{(Physics intuition: a scalar mass is like a microphone left next to a
loudspeaker --- it picks up every high-energy mode it couples to. Fermion masses are
protected by chiral symmetry and gauge-boson masses by gauge symmetry, but a lone
scalar has no such shield, so its mass is dragged up to the highest scale in the
problem unless something cancels it by hand.)}

Third, \emph{vacuum metastability}. Run to high energy, the measured \(\lambda\)
drifts toward zero and turns slightly negative near \(10^{10}\)--\(10^{12}\
\mathrm{GeV}\), placing our vacuum on the knife-edge between stability and decay. That
the Standard-Model parameters should land precisely on this edge is an unexplained
coincidence in the bottom-up picture.

These three --- the two inputs, the unnatural smallness of \(v\), and the metastable
\(\lambda\) --- are what the geometric derivation must address. It does so by refusing
to treat the Higgs as a primitive scalar at all.

\subsection{Step 3 --- the geometric derivation: the Higgs as a holonomy shadow}
\label{app:ewsb-step3}

In \GGTheory{} the Higgs is not a fundamental field with a freely chosen potential.
It is a local chart on a Wilson-line holonomy --- the order parameter that Hosotani showed to be the natural electroweak variable on a multiply connected space \citep{Hosotani1983,Wilson1974}, promoted to the electroweak order parameter of the cohort in \ASix{} \citep{ArisakaA6Gi4} --- and its vacuum value and quartic are
read off the same integer ledger \((M,N,L;J,K)=(1,3,5;7,35)\) that fixed the gauge
couplings. We give the derivation first in its essential geometric form (the order
parameter, the lock scale, the branch angle, the quartic) and then in explicit
arithmetic, cross-checked against the gauge-sector numbers of
Appendix~\ref{app:gauge-step3}.

\subsubsection{The order parameter is a Wilson-line holonomy, not a primitive scalar}
\label{app:ewsb-holonomy}

\ASix{} proves that the only admissible global order parameter for electroweak
breaking along the compact \GG{} direction is the conjugacy class of the Wilson line
\(W_\theta=\exp\!\big(i\oint A_\theta\,d\theta\big)\). Any local Higgs field is a
\emph{chart} on this holonomy orbit, not an independent degree of freedom. This single
structural fact is what removes the freedom from the scalar sector: a chart cannot
carry its own arbitrary potential; its potential is inherited from the geometry it
charts.
\emph{(Physics intuition: in ordinary field theory the Higgs is an actor who may be
handed any script --- any \(\mu^2\) and \(\lambda\). Here the Higgs is the shadow of a
rod threaded once around a compact circle; you cannot paint the shadow independently
of the rod. The potential the shadow obeys is whatever the rod's geometry dictates,
and the geometry is already fixed.)} Because the order parameter is an angle, the
boundary-visible scalar amplitude is a trigonometric projection of the selected
holonomy branch,
\begin{equation}
v=\Mlock\,\sin\theta_H^\star ,
\label{eq:v-projection-app}
\end{equation}
where \(\theta_H^\star\) is the stable branch selected by the \GRIP{} cascade
\(\DGI\Rightarrow\GSSB\Rightarrow\GA\) of \ASix{}, and \(\Mlock\) is the first visible
electroweak scale.

\subsubsection{The lock scale: where the electroweak scale comes from}
\label{app:ewsb-lockscale}

The first boundary scale available to the electroweak packet is not a free parameter
but the \emph{lock scale}, the Planck scale pulled down by the scale profile
\(e^{-K}\),
\begin{equation}
\Mlock=\tfrac12\,M_{\mathrm{Pl}}\,e^{-K},
\qquad K=35 .
\label{eq:lockscale-app}
\end{equation}
The exponent \(K=35\) is the ledger integer \eqref{eq:K-two-faces-app}, the same one
that satisfies \(K=LJ=M^2+N^2+L^2\); the factor \(\tfrac12\) is the binary
endpoint convention of the radial interval (Appendix~\ref{app:scalechain}), not a
tunable coefficient. Numerically
\begin{equation}
\Mlock=\frac{1.2209\times10^{19}\ \mathrm{GeV}}{2\,e^{35}}
\simeq 3.849\times10^{3}\ \mathrm{GeV}.
\label{eq:lockscale-num-app}
\end{equation}
\emph{(Physics intuition: the scale profile \(e^{-K}\) is an exponential elevator. The
Planck scale is the penthouse, \(10^{19}\ \mathrm{GeV}\); riding down \(K=35\)
e-folds of the scale profile drops one to the TeV lobby. The enormous ratio between the Planck
and electroweak scales is therefore not a fine-tuning but a \emph{count}: thirty-five
e-folds, fixed by the same integer that closes the bulk. This is the first half of the
answer to the hierarchy problem --- the big scale and the small scale are separated by
an integer number of scale e-folds, not by an accident.)}

\subsubsection{The branch angle: the repeated doublet fraction}
\label{app:ewsb-branch}

The selected stable holonomy branch is not arbitrary. At each triplicity layer of the
reduction, only the electroweak doublet --- two directions out of the unified
\(L\)-block --- survives into the visible branch, contributing one factor \(2/L\) per
layer. After \(N\) layers,
\begin{equation}
\sin\theta_H^\star=\left(\frac{2}{L}\right)^{N}
=\left(\frac{2}{5}\right)^{3}=\frac{8}{125}=0.064 .
\label{eq:branchangle-app}
\end{equation}
\emph{(Physics intuition: the holonomy twist is filtered through three successive
doublet gates, each passing the fraction \(2/5\) of the available block. Three gates
in series pass \((2/5)^3\), a little over six percent. The smallness of the Higgs vev
relative to the lock scale is this triple filtering --- a second, milder scale suppression on top
of the exponential one.)}

\subsubsection{The Higgs vacuum value}
\label{app:ewsb-vev}

Combining the projection \eqref{eq:v-projection-app}, the lock scale
\eqref{eq:lockscale-app}, and the branch angle \eqref{eq:branchangle-app} gives the
Wilson--Hosotani vev (main-text Theorem~\ref{thm:vev}, T-SM-30.1),
\begin{equation}
\boxed{\;
v=\Mlock\left(\frac{2}{L}\right)^{N}
=\tfrac12\,M_{\mathrm{Pl}}\,e^{-K}\left(\frac{2}{L}\right)^{N}
\simeq 246.33\ \mathrm{GeV}. \;}
\label{eq:vev-final-app}
\end{equation}
Every symbol on the right is fixed: \(M_{\mathrm{Pl}}\) by \(c,h,G_N\); \(K=35\),
\(L=5\), \(N=3\) by the ledger. There is no continuous input.
\emph{(Physics intuition: picture a lighthouse beam wound around a spiral staircase.
The bulk knows the whole spiral angle, but the boundary observer sees only the
horizontal shadow the beam casts at a selected height. The lock scale fixes the
height; the factor \((2/L)^N\) is how much of the spiral projects into the visible
doublet channel. The Higgs vacuum value is the horizontal shadow of a selected
geometric twist --- nothing more, and nothing adjustable.)}

\subsubsection{The quartic coupling and the Higgs mass}
\label{app:ewsb-quartic}

The quartic is the second and final scalar number, and it too is no longer free. Once
the Higgs is understood as the local chart \(\eta=h-v\) on the selected branch, the
quartic is the dimensionless curvature stiffness of that branch. \MSMF{} forbids a new
continuous coefficient, so the quartic must be built from the discrete packet already
in hand; the only admissible dimensionless suppression is dyadic, and the only relevant
depth is the triplicity depth \(N\). One factor \(\tfrac12\) per layer gives
(main-text Theorem~\ref{thm:quartic}, T-SM-31.1)
\begin{equation}
\boxed{\;\lambda_H=\left(\frac12\right)^{N}=2^{-N}=\frac18 .\;}
\label{eq:quartic-final-app}
\end{equation}
\emph{(Physics intuition: the quartic measures how stiffly the branch resists being
pushed away from its minimum --- the curvature of the trough floor. ``One half per
triplicity layer'' is the same dyadic logic that folded the photon seam in the gauge
appendix, here applied to the scalar's self-stiffening: three layers, three halvings,
\(1/8\). The scalar sector and the Abelian gauge sector share one arithmetic of
halving.)} The physical Higgs mass then follows with no further input,
\begin{equation}
m_h=\sqrt{2\lambda_H}\;v=\sqrt{\tfrac14}\;v=\frac{v}{2}
\simeq 123.1\ \mathrm{GeV},
\label{eq:mh-final-app}
\end{equation}
the small remaining gap to the measured \(125.25\ \mathrm{GeV}\) being closed by the
quantum-defect completion of Part~\ref{part:VI}.

\subsubsection{Explicit arithmetic and the descendant masses}
\label{app:ewsb-arith}

The explicit numbers are collected in the gauge-and-Higgs arithmetic block of the
previous appendix; we record the chain here for completeness. With
\(\Mlock\simeq 3848.93\ \mathrm{GeV}\) (Eq.~\ref{eq:Mlock-arith-app}) and
\(\sin\theta_H^\star=(2/5)^3\),
\begin{equation}
v^{(1)}=\Mlock\left(\tfrac25\right)^3\simeq 246.33\ \mathrm{GeV}
\quad(\text{Eq.~\ref{eq:v-arith-app-2}}),
\end{equation}
and with the gauge couplings \(g_2\simeq0.6512\), \(g_Y\simeq0.3574\) of
Eq.~\eqref{eq:g2gY-app},
\begin{align}
m_W^{(1)}&=\tfrac12 g_2 v^{(1)}\simeq 80.21\ \mathrm{GeV}
\quad(\text{Eq.~\ref{eq:mW-arith-app}}),\\
m_Z^{(1)}&=\tfrac12 v^{(1)}\sqrt{g_2^2+g_Y^2}\simeq 91.50\ \mathrm{GeV}
\quad(\text{Eq.~\ref{eq:mZ-arith-app}}),\\
m_h^{(1)}&=\sqrt{2\lambda_H^{(1)}}\,v^{(1)}\simeq 123.12\ \mathrm{GeV}
\quad(\text{Eq.~\ref{eq:mh-arith-app}}),
\end{align}
and the Fermi constant \(G_F=1/(\sqrt2\,v^2)\) becomes a descendant rather than an
input. Thus the entire electroweak mass spectrum --- \(v\), \(m_W\), \(m_Z\),
\(m_h\), \(G_F\) --- flows from the two ledger readings \(v\) and \(\lambda_H\), which
in turn flow from \(c,h,G_N\) and the integers \((N,L,K)=(3,5,35)\).

\subsubsection{Honesty triage and the dissolution of the hierarchy problem}
\label{app:ewsb-honesty}

In the codebook (\S\ref{sec:codebook-provenance}) the vev \(v\) is recorded as
\textbf{Derived}: every factor in \eqref{eq:vev-final-app} is forced. The quartic
\(\lambda_H\) is recorded as \textbf{Derived (curvature, modulo one named structural
lemma)}: the dyadic value \(2^{-N}=1/8\) is the leading-harmonic representative of the
holonomy-potential curvature \eqref{eq:lambdaH-curvature}, and promoting it to an
unconditional theorem requires evaluating the spectral normalization
\((\Lambda/f)^4\) from the scale-profile Kaluza--Klein sum --- the spectral-normalization
lemma (the deeper form of L6).

It is worth stating plainly what has happened to the naturalness problem. In the
bottom-up picture the smallness of \(v\) relative to the Planck scale is a fine-tuning
of one part in \(10^{34}\). Here it is two compositions of fixed structure: an
exponential scale profile \(e^{-K}\) with \(K=35\), and a triple doublet filter \((2/L)^N\)
with \(N=3,L=5\). Neither is adjustable, and neither is tuned; both are counts. The
quadratic sensitivity \eqref{eq:hierarchy-app} that powers the hierarchy problem
presupposes a primitive scalar whose mass can be dragged to the cutoff. But the Higgs
here is a holonomy chart, and a holonomy is protected by the topology of the compact
direction in the same way a gauge-boson mass is protected by gauge symmetry: there is
no primitive scalar mass to drag. The hierarchy problem, in this reading, is not
solved by a cancellation but \emph{dissolved} by a change in what the Higgs is. The
electroweak scale is small because it is an exponentially suppressed, triply filtered
shadow of the Planck scale --- and the scale depth and the filter depth are both
integers in the ledger.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{E.1.1. Extract the vacuum value from
the Fermi constant.}

\noindent\emph{Solution.}  Step 1: at energies far below
$m_W$, $W$ exchange in muon decay collapses to the contact
interaction with strength $G_F/\sqrt2 = g^2/8m_W^2$.  Step 2:
insert $m_W = gv/2$ \eqref{eq:vboson-mass-app}: the coupling
cancels, $G_F/\sqrt2 = 1/2v^2$.  Step 3: invert with the
measured $G_F = 1.1664\times10^{-5}\,\mathrm{GeV^{-2}}$:
$v = (\sqrt2\,G_F)^{-1/2} = 246.2$~GeV
\eqref{eq:v-fromGF-app}.  Physics intuition: the weak scale
was measured in 1934 --- in the lifetime of the muon's
ancestor decays --- half a century before the $W$ was seen:
$v$ is the rare fundamental scale extracted from a tabletop
lifetime, because the contact strength remembers the vev even
when the energy cannot resolve the boson.

\medskip\noindent\textbf{E.1.2. Prove $\rho = 1$ at tree level
and state what it tests.}

\noindent\emph{Solution.}  Step 1: from the doublet's kinetic
term, $m_W^2 = \tfrac14 g^2v^2$ and $m_Z^2 = \tfrac14(g^2 +
g'^2)v^2$.  Step 2: form the ratio \eqref{eq:rho-app}:
$\rho = m_W^2/(m_Z^2\cos^2\theta_W) = g^2/[(g^2+g'^2)\cdot
g^2/(g^2+g'^2)] = 1$ identically --- the couplings and the vev
cancel.  Step 3: the cancellation is a consequence of the
custodial $\SU(2)$ of the \emph{doublet} representation: a
scalar triplet with a vev $v_T$ shifts $\rho$ by
$O(v_T^2/v^2)$, so the measured $\rho \approx 1.0004$ confines
any triplet to the few-GeV level.  Physics intuition:
$\rho = 1$ is the Standard Model's fingerprint of \emph{which}
representation broke the symmetry --- the data are not merely
consistent with a doublet, they nearly exclude everything
else; any completion (this chapter's Wilson-line Higgs
included) must reproduce a doublet's custodial structure, and
does.

\medskip\noindent\textbf{E.1.3. Extract the quartic coupling
from the Higgs discovery.}

\noindent\emph{Solution.}  Step 1: the potential
\eqref{eq:higgs-pot-app} gives $m_h^2 = 2\lambda v^2$
\eqref{eq:mh-def-app}.  Step 2: invert with the 2012 discovery
value: $\lambda = m_h^2/2v^2 = 125^2/(2\cdot246^2) = 0.129$
\eqref{eq:lambda-fromdata-app}.  Step 3: note what happened:
the last parameter of the minimal Standard Model was measured
--- and it landed at a small, unexplained number, close to
(but, at this order, not equal to) simple fractions.  Physics intuition:
 $\lambda$ is the Higgs talking to itself; its
measured smallness is what makes the electroweak vacuum
famously near-critical --- and E.3.2 reads the number as a
dyadic depth, $2^{-N}$, rather than a dial.

\medskip\noindent\textbf{E.1.4. Derive the unitarity ceiling
that guaranteed the LHC would find something.}

\noindent\emph{Solution.}  Step 1: without a Higgs, the
amplitude for longitudinal $W_LW_L$ scattering grows like
$G_F E^2$ --- the gauge cancellations leave the growth the
scalar was born to cancel.  Step 2: partial-wave unitarity
($|a_0| \leq \tfrac12$) then fails at $\sqrt{s} \sim
\bigl[8\pi\sqrt2/(3G_F)\bigr]^{1/2}$.  Step 3: numerically
$\bigl[8\pi\sqrt2/(3\times1.1664\times10^{-5})\bigr]^{1/2}
\approx 1.0$~TeV --- by that energy, \emph{either} a Higgs
appears \emph{or} the weak interaction becomes strong and new
resonances appear: no third option.  Physics intuition: this
inequality is why the LHC was a no-lose experiment --- the
machine was built to reach the energy at which the Standard
Model must show its completion; at $125$~GeV the scalar
arrived with room to spare, and the quartic stayed weakly
coupled (E.1.3).

\medskip\noindent\textbf{E.3.1. Evaluate the geometric vev
formula.}

\noindent\emph{Solution.}  Step 1: the chain
\eqref{eq:vev-final-app}: $v = \Mlock\,(2/L)^N$ with the lock
scale $\Mlock = \tfrac12\MPl e^{-K} = 3.849$~TeV
\eqref{eq:lockscale-num-app}.  Step 2: $(2/5)^3 = 0.064$;
$v = 3849 \times 0.064 = 246.3$~GeV.  Step 3: against the
Fermi-constant extraction of E.1.1 ($246.22$), the ledger
reproduces the weak scale at the per-mille level from
$(\MPl; N, L, K)$ and nothing else.  Physics intuition: the
electroweak scale is here the end of a pipeline ---
$\MPl \to e^{-K} \to$ three branch filters of $2/L$ each ---
every stage an integer of the ledger: E.3.3 turns this into
the hierarchy statement, and the collider falsifier at the
first stage ($3.85$~TeV) is standing.

\medskip\noindent\textbf{E.3.2. Run the dyadic quartic to the
Higgs mass, and state the register exactly.}

\noindent\emph{Solution.}  Step 1: the only admissible
dimensionless suppression is dyadic and the relevant depth is
the triplicity: $\lambda_H = 2^{-N} = \tfrac18$
\eqref{eq:quartic-final-app}.  Step 2: then
$m_h = \sqrt{2\lambda_H}\,v = v/2 = 123.1$~GeV at first order
\eqref{eq:mh-final-app} --- compare the measured $125.25$:
the first-order value lands within $1.7\%$.  Step 3: the QD
completion (the chapter's second-order section) supplies the
remaining dressing and carries, deliberately, the conservative
$0.30$~GeV theory uncertainty --- the register is: structure
derived, final digit QD-completed, uncertainty stated.
Physics intuition: $m_h = v/2$ at leading order is the
cleanest slogan in the codebook --- the Higgs weighs half its
own vacuum --- and the $1/8$ behind it is three halvings, one
per family layer: the same arithmetic that put $N^2$ on top
of $2^J$ in D.3.1.

\medskip\noindent\textbf{E.3.3. State the hierarchy as an
identity of integers.}

\noindent\emph{Solution.}  Step 1: measured,
$v/\MPl = 246.33/1.2209\times10^{19} = 2.018\times10^{-17}$
--- the infamous seventeen orders.  Step 2: the ledger's
formula: $\tfrac12 e^{-35}(2/5)^3 = 2.018\times10^{-17}$ ---
the same number, digit for digit, from $K = 35$, $L = 5$,
$N = 3$.  Step 3: the hierarchy ``problem'' is thereby
restated: not \emph{why is a continuous parameter tuned
small}, but \emph{why these integers} --- a question the
ledger answers upstream, and a question that cannot be
un-asked by radiative corrections because (E.3.4) nothing
continuous is being protected.  Physics intuition: exponential
hierarchies from order-one integers are cheap --- $e^{-35}$
does in one factor what fine-tuning buys with seventeen
digits; the entire naturalness anxiety was priced on the
assumption that the small number was an input.

\medskip\noindent\textbf{E.3.4. Why does a Wilson-line Higgs
have no quadratic divergence to cancel?}

\noindent\emph{Solution.}  Step 1: the ordinary complaint
\eqref{eq:hierarchy-app}: a primitive scalar's mass-squared
drags $\Lambda^2/16\pi^2$ from every loop.  Step 2: here the
ultraviolet variable is a Wilson-line \emph{phase}
$\theta_H$, compact and periodic
\eqref{eq:thetaH-periodic-app}: no local counterterm
$\theta_H^2$ exists (it is not gauge invariant --- only
$\cos\theta_H$ structures are), so the effective potential
\eqref{eq:Veff-cos-app} is finite, and the curvature at its
minimum \eqref{eq:mh-curvature-app} is set by the
compactification scale, not the cutoff.  Step 3: the
hierarchy then enters multiplicatively
\eqref{eq:lock-multiplicative-app} --- filters compose by
products, not subtractions --- which is why no cancellation,
and no partner spectrum to enforce one, is required.
Physics intuition: you cannot quadratically renormalize an
angle --- periodicity is the symmetry that scalar-mass
counterterms cannot break, and the Hosotani mechanism is that
protection made dynamical; F.14.1 works the potential.

\GGtakeaway{\emph{Standard result:} the Higgs mechanism gives
the $W$ and $Z$ their masses, keeps the photon massless, and
fits the data --- with the scale and the quartic taken from
experiment.  \emph{What it assumed:} a primitive scalar with a
chosen potential, and no answer to why its scale is small.
\emph{The geometric reading:} the Higgs is a Wilson line; the
scale is $\MPl$ filtered through integer depths; the hierarchy
problem is dissolved, not solved --- there is no primitive
scalar mass to protect.}

\section{Operational export, radial flow, and Wilson--Hosotani geometry}
\label{app:ch-export}

\GGteachbox{The connective tissue of the course: how the
completed gauge/Higgs, fermion-mass, CKM, and PMNS blocks feed
the rest of the paper --- the operational export map, the radial
flow, and the Wilson--Hosotani geometry that the sector chapters
consume.  A reference chapter, to be consulted as the course
proceeds rather than worked through in one sitting.}

\subsection{How the gauge/Higgs, fermion-mass, CKM, and PMNS/Majorana derivation feeds the rest of \ANine{}}
\label{app:rest-paper}

The main text already gave the roadmap of the full paper. Here we state
explicitly how the completed 28-parameter block becomes the input for the
deep-origin synthesis of the Standard Model.

\subsection{CKM structure (completed in the present version)}

The quark-mixing problem has been solved at the working level. The
diagonal up-type and down-type masses fixed by
Section~\ref{sec:quark-masses} were promoted to full family-link textures
in Section~\ref{sec:ckm}, and their left-handed misalignment produced the
four CKM parameters without introducing any new fitted flavor constant.
This completed quark block is now an input, not a future task.

\subsection{PMNS and Majorana structure (completed in the present version)}

The lepton side has now been completed at the same working level. The
charged-lepton dyadic block and the Majorana neutrino block were promoted
in Section~\ref{sec:pmns} to a full family-projector misalignment problem,
and that misalignment produced the three PMNS angles, the leptonic Dirac
phase, and the two physical Majorana phases without introducing any new
empirical mixing constant.

\subsection{Final CP angle}

Only one CP-related parameter remains outside the already-derived CKM and
PMNS/Majorana counting: the strong-CP-like final angle. Because every
ordinary Yukawa mass and every ordinary flavor quantity has now been
fixed, that angle is isolated as a genuinely topological or \BIsix{}-level
question rather than a generic flavor ambiguity.

\subsection{Why the remaining problem is dramatically narrower than the present one}

The present paper had to solve the scales, the gauge couplings, the Higgs
sector, the diagonal masses, the CKM quark-misalignment block, and the
PMNS/Majorana lepton-misalignment block all at once. The remaining work
is much narrower: it does not need to discover new normalization scales,
new mass codebooks, or new family matrices. It needs only to identify the
remaining \BIsix{}/topological CP datum of already-fixed sectors.


\subsection{Detailed derivation of the radial renormalization group from the sixth dimension}
\label{app:radial-rg-detail}

This appendix expands Section~\ref{sec:radial-rg} and makes explicit why the
\GG{} statement
\[
r=\ln\!\left(\frac{\EGUT}{\mu}\right)
\]
is not a decorative analogy but the exact coordinate statement that turns
bulk transport into boundary running. The purpose of the appendix is not to
introduce any new assumption. It is to show that once the \GG{} metric and the
\GDOS{} projection postulates are accepted, a radial renormalization law follows
with almost no freedom left (\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}).

\subsection{Redshift and the operational meaning of the radial coordinate}

The \GG{} metric is written in the form
\begin{equation}
 ds_6^2=e^{-2A(r)}g_{\mu\nu}(x)dx^\mu dx^\nu + dr^2 + R_\theta^2\,d\theta^2.
 \label{eq:metric-rg-app}
\end{equation}
A local process at radial position \(r\) with proper energy scale
\(\mu_{\mathrm{loc}}\) is seen by the four-dimensional boundary observer with a
redshifted energy
\begin{equation}
 \mu(r)=\mu_{\mathrm{loc}}\,e^{-A(r)}.
 \label{eq:redshift-app}
\end{equation}
Choosing the UV anchor at the GUT seam means normalizing the local reference
scale to \(\mu_{\mathrm{loc}}=\EGUT\) at the top of the throat. In the linearized
throat approximation used throughout the \GG{} construction,
\(A(r)=kr\). Measuring \(r\) in units of \(k^{-1}\) gives
\begin{equation}
 \mu(r)=\EGUT e^{-r},
 \qquad
 r=\ln\!\left(\frac{\EGUT}{\mu}\right).
 \label{eq:mu-r-app}
\end{equation}
The first equality says that every equal radial step multiplies the local
energy by the same factor. The second says that the radial coordinate is
literally the logarithmic RG scale.

This point is conceptually important. In ordinary four-dimensional field
theory one introduces the running scale \(\mu\) to reorganize perturbation
theory. In \GG{} the same scale appears geometrically before perturbation
theory is ever discussed. The renormalization variable is therefore not an
external bookkeeping convention imposed on the theory after the fact. It is
already present in the bulk geometry.

\subsection{From bulk shell integration to the differential beta law}

Consider a gauge-field sector inside the composite connection of \GG{}. The
six-dimensional kinetic term is schematically
\begin{equation}
 S_{\mathrm{gauge}}
 =-
 \frac{1}{4g_6^2}
 \int d^4x\,dr\,d\theta\,\sqrt{-G}\,
 G^{MP}G^{NQ}F_{MN}F_{PQ}.
 \label{eq:gauge6-action-app}
\end{equation}
To obtain a boundary-effective action, one integrates over the compact angle
and then over radial shells. Let the shell \([r,r+dr]\) be thin enough that
the couplings and scale profile vary only infinitesimally across it. The shell
contributes a correction to the four-dimensional inverse coupling of the form
\begin{equation}
 d\!\left(\frac{1}{g_4^2}\right)=\Pi(r)\,dr,
 \label{eq:shell-corr-app}
\end{equation}
where \(\Pi(r)\) is the shell polarization density, i.e. the integrated effect
of the allowed \GG{} degrees of freedom on that gauge channel.

Equation~\eqref{eq:shell-corr-app} is the geometric origin of the RG law.
Passing from \(g_4\) to a generic coupling \(g_i\), and using
\(d\ln\mu=-dr\), one obtains
\begin{equation}
 \mu\frac{d g_i}{d\mu}=\beta_i(g)
 \qquad\Longleftrightarrow\qquad
 \frac{d g_i}{dr}=-\beta_i(g).
 \label{eq:beta-equivalence-app}
\end{equation}
This is precisely the radial beta equation quoted in the main text. The minus
sign is not optional: moving deeper into the throat means moving toward lower
energy.

At one loop, when the shell response is linear in the quadratic Casimir data,
one recovers the familiar form
\begin{equation}
 \frac{d}{dr}\,\alpha_i^{-1}(r)=\frac{b_i}{2\pi},
 \qquad
 \alpha_i^{-1}(r)=\alpha_i^{-1}(0)+\frac{b_i}{2\pi}\,r.
 \label{eq:one-loop-radial-app}
\end{equation}
The important difference from an ordinary textbook derivation is that the
coefficients are not the whole story. In the present framework, the shell data
must still respect \BIsix{}, orbifold parity, \OGNfive{} normalization, and the \GDOS{}
projection discipline. The continuum running is therefore only the transport
skeleton. The seam values themselves remain discretely constrained.

\subsection{Stationary seams as evaluation surfaces of the integrated flow}

The main text used language such as the lock seam and the neutral seam. It is
helpful to state their RG meaning explicitly. Let
\begin{equation}
 \alpha_i^{-1}(r_\ast)=\alpha_i^{-1}(0)
 +\int_{0}^{r_\ast}\frac{b_i(r)}{2\pi}\,dr
 +\Delta_i^{\mathrm{seam}}.
 \label{eq:flow-plus-seam-app}
\end{equation}
The integral term is the continuum radial transport. The term
\(\Delta_i^{\mathrm{seam}}\) collects the finite matching conditions imposed by the
particular stationary seam at which the observable is read out. The first-order
predictions of the present paper should be understood as the leading discrete
representatives of \eqref{eq:flow-plus-seam-app}. The QD corrections are then
finite next-order refinements of the seam matching, not a change in the
underlying geometry.

This viewpoint explains why the running picture and the integer-coded formulas
fit together rather than compete. The RG part transports information through
the throat. The OGN/BI-6 part decides which channels are actually admissible at
the measurement seam.

\subsection{Why only two transcendental numbers survive the transport}

No hidden empirical numbers are used in the manuscript. At first sight one might wonder why the final formulas
still involve \(\pi\) and \(\e\). The radial derivation explains this. The only
two irreducible geometric sources of transcendental structure are
\begin{enumerate}[leftmargin=2.1em]
\item the compact angular measure on \(S^1_\theta\), which introduces \(2\pi\);
\item the exponential scale/redshift law, which introduces \(e^{-r}\).
\end{enumerate}
There is no independent geometric origin in the current preprint system for any
other transcendental constant. Therefore the appearance of \(\pi\) and \(\e\)
in the gauge/Higgs sector is not a numerical convenience. It is the direct
shadow of the only two non-algebraic structures present in the minimal \GG{}
stage.

\subsection{A practical summary for later sections of \ANine{}}

Later parts of \ANine{} will need the same logic in the Yukawa and neutrino
sectors. The operational recipe is the same in every case:
\begin{enumerate}[leftmargin=2.1em]
\item write the admissible six-dimensional operator in the anomaly-free \GG{}
bulk;
\item transport it radially using \eqref{eq:beta-equivalence-app};
\item evaluate it on the appropriate \GDOS{} stationary seam;
\item impose \OGNfive{} and \BIsix{} selection rules to eliminate non-constitutional
freedom.
\end{enumerate}
Thus the radial RG is not merely one ingredient among many. It is the common
transport law that every later parameter derivation must share.

\subsection{Hierarchy and naturalness in \GG{} revisited in full detail}
\label{app:hierarchy-detail}

The main text stated that the hierarchy and naturalness problems are not merely
improved numerically but reformulated conceptually. Because this claim is
central to the whole \ANine{} program, it deserves a slower and more explicit
argument.

\subsection{Why the ordinary 4D hierarchy problem appears severe}

In a purely four-dimensional description with a fundamental local Higgs scalar,
one writes schematically
\begin{equation}
 m_H^2(\Lambda)=m_{H,0}^2 + c_1\Lambda^2 + c_2 m^2\ln\!\left(\frac{\Lambda}{\mu}\right)+\cdots.
 \label{eq:4d-hierarchy-app}
\end{equation}
If \(\Lambda\) is taken near the Planck or GUT scale, then obtaining a weak-scale
physical Higgs mass seems to require a delicate cancellation between quantities
of wildly different magnitude. This is the standard form of the naturalness
complaint.

The important \GG{} observation is that
\eqref{eq:4d-hierarchy-app} uses the wrong primitive variable for the UV
problem. It assumes from the outset that the Higgs is a local scalar field
whose mass term is fundamental. But the \GG{} preprints do not identify the
Higgs that way.

\subsection{The true ultraviolet variable is a Wilson-line phase}

In the present framework the primitive compact degree of freedom is the
holonomy angle
\begin{equation}
 \Theta_H = g_6\oint_{S^1_\theta} A_\theta\,d\theta,
 \qquad
 \Theta_H \sim \Theta_H+2\pi.
 \label{eq:thetaH-periodic-app}
\end{equation}
The periodic identification is essential. It says that the fundamental object
is compact and gauge-geometric, not an unconstrained local scalar coordinate.
Only after projection to the four-dimensional boundary does one write a local
field variable \(h\) near a selected vacuum,
\begin{equation}
 h = f_H\,\big(\Theta_H-\Theta_0\big),
 \label{eq:h-from-theta-app}
\end{equation}
where \(\Theta_0\) is the stationary point selected by the finite Hosotani
potential and \(f_H\) is the appropriate normalization constant for the boundary
mode.

Once the ultraviolet degree of freedom is the compact phase
\eqref{eq:thetaH-periodic-app}, the usual intuition behind an arbitrary local
counterterm \(\Lambda^2 h^2\) becomes misleading. Local UV-sensitive pieces can
renormalize vacuum energy and other phase-independent quantities, but the
curvature of the periodic potential at a given minimum is a genuinely nonlocal
observable tied to transport around the compact direction.

\subsection{Why the Hosotani potential is finite in the Higgs direction}

The most useful schematic form of the effective potential is
\begin{equation}
 V_{\mathrm{eff}}(\Theta_H;r)
 =\sum_{n=1}^{\infty} C_n(r)\cos\!\big(n\Theta_H\big).
 \label{eq:Veff-cos-app}
\end{equation}
Each Fourier coefficient \(C_n(r)\) is produced by matter and gauge transport
around the compact direction. The crucial structural point, already familiar
from gauge--Higgs unification, is that divergent local pieces are independent of
\(\Theta_H\), whereas the \(\Theta_H\)-dependent terms are finite because they
probe the global holonomy around the circle.

Therefore the physical Higgs mass is not the coefficient of a primitive
quadratically divergent local operator. It is the curvature of a finite
periodic potential at its minimum,
\begin{equation}
 m_h^2 = \frac{1}{f_H^2}
 \left.\frac{\partial^2 V_{\mathrm{eff}}}{\partial \Theta_H^2}\right|_{\Theta_H=\Theta_0}.
 \label{eq:mh-curvature-app}
\end{equation}
Likewise the quartic self-interaction is controlled by the higher derivatives of
the same finite function. The naturalness problem is therefore not solved by
claiming that radiative corrections vanish. It is solved by recognizing that
what radiatively corrects is a finite global quantity rather than a free local
mass term.

\subsection{The hierarchy becomes multiplicative rather than subtractive}

The hierarchy itself is already visible before any loop calculation. The lock
scale is
\begin{equation}
 \Mlock = \frac{\EPl}{2e^{35}},
 \label{eq:lock-multiplicative-app}
\end{equation}
and the leading electroweak VEV is
\begin{equation}
 v^{(1)}=\Mlock\left(\frac{2}{5}\right)^3.
 \label{eq:vev-multiplicative-app}
\end{equation}
Thus the electroweak scale is obtained by successive multiplicative projections
from the Planck anchor. There is no stage in the calculation where two enormous
unrelated numbers are subtracted to many decimal places in order to recover a
small residue. The scale chain is geometric and factorized.

The significance of the fixed integer \(K=35\) can be stated cleanly. If one
were to treat \(K\) as a continuous dial, then
\begin{equation}
 \frac{\partial \ln \Mlock}{\partial K}=-1.
 \label{eq:dlnm-dk-app}
\end{equation}
But in the \GG{} constitution \(K\) is not a dial. It is fixed by \OGNfive{}.
Equation~\eqref{eq:dlnm-dk-app} therefore measures sensitivity to a variation
that the theory does not allow. From the constitutional point of view there is
nothing to tune.

\subsection{What remains after the conceptual reformulation}

The previous paragraphs do not mean that every quantitative question disappears.
Finite seam-matching corrections remain. The QD layer precisely represents
those effects. In particular, the VEV and Higgs mass still receive controlled
next-order shifts. What disappears is the special ultraviolet instability that
comes from treating the Higgs as a primitive pointlike scalar with an arbitrary
bare mass parameter.

This distinction matters for the later parameter program. Once the gauge/Higgs
background is finite and geometrically fixed, the remaining fermion and flavor
sectors inherit a stable electroweak normalization rather than an ad hoc one.
That is why hierarchy and naturalness had to be addressed before the final topological completion and the deep-origin synthesis.

\subsection{Wilson--Hosotani reduction and the emergence of the 4D Higgs field}
\label{app:wh-detail}

This appendix gives a slower derivation of the statement that the Higgs field is
a boundary image of the compact gauge connection rather than an elementary
four-dimensional scalar added by hand.

\subsection{Orbifold parities and the separation of gauge components}

The \GG{} construction uses an orbifold-type boundary condition with parity
matrix
\begin{equation}
 P=\mathrm{diag}(+1,+1,+1,-1,-1).
 \label{eq:parity-matrix-app}
\end{equation}
The parity action distinguishes the ordinary four-dimensional gauge components
from the compact component according to
\begin{equation}
 A_\mu(x,-\theta)=P\,A_\mu(x,\theta)\,P^{-1},
 \qquad
 A_\theta(x,-\theta)=-P\,A_\theta(x,\theta)\,P^{-1}.
 \label{eq:parity-components-app}
\end{equation}
The even zero modes of \(A_\mu\) generate the surviving low-energy gauge
symmetry, while selected components of \(A_\theta\) in the broken directions
remain as scalar boundary modes. This is the geometric origin of the Higgs
sector in the \GG{} language (\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}).

\subsection{Gauge-invariant order parameter: the Wilson line}

Because \(A_\theta\) itself is gauge dependent, the physically meaningful order
parameter is the compact holonomy
\begin{equation}
 W_\theta
 =\mathcal P\exp\!\left(i g_6\oint_{S^1_\theta} A_\theta\,d\theta\right)
 =\exp\!\big(i\Theta_H\big)
 \quad\text{in the abelianized notation.}
 \label{eq:wilsonline-app}
\end{equation}
This quantity is globally defined, periodic, and directly sensitive to the
compact geometry. The vacuum chooses a particular equivalence class of
\(W_\theta\), or equivalently of \(\Theta_H\), once the finite effective potential
is generated.

The physical intuition is straightforward. A local scalar takes a value at a
point. A Wilson-line scalar instead measures the integrated gauge connection
around the compact direction. It is therefore a global quantity.
That is exactly why the Higgs sector fits naturally into the \GDOS{} viewpoint,
where observable data arise from projected global structure rather than from an
isolated microscopic point value.

\subsection{Expansion of the effective action around the selected vacuum}

Let \(\Theta_H=\Theta_0+\delta\Theta\) be a stationary vacuum. Expanding the
finite effective action around this point gives
\begin{equation}
 V_{\mathrm{eff}}(\Theta_H)
 =V_0
 +\frac{1}{2}V_2\,(\delta\Theta)^2
 +\frac{1}{3!}V_3\,(\delta\Theta)^3
 +\frac{1}{4!}V_4\,(\delta\Theta)^4
 +\cdots,
 \label{eq:theta-expansion-app}
\end{equation}
where \(V_n\equiv \partial_{\Theta_H}^n V_{\mathrm{eff}}|_{\Theta_0}\). In a
CP-even vacuum the odd derivatives vanish or are strongly suppressed, and the
quadratic and quartic terms dominate the low-energy Higgs dynamics.

Introducing the canonically normalized boundary field
\(h=f_H\delta\Theta\), one obtains
\begin{equation}
 V_{\mathrm{eff}}(h)
 =V_0
 +\frac{1}{2}\left(\frac{V_2}{f_H^2}\right)h^2
 +\frac{1}{4!}\left(\frac{V_4}{f_H^4}\right)h^4
 +\cdots.
 \label{eq:h-expansion-app}
\end{equation}
Hence the physical Higgs mass and quartic are not independent objects:
\begin{equation}
 m_h^2 = \frac{V_2}{f_H^2},
 \qquad
 \lambda_{H,\mathrm{eff}} \propto \frac{V_4}{f_H^4}.
 \label{eq:mh-lambda-wh-app}
\end{equation}
The proportionality constant in the quartic depends on the precise low-energy
normalization convention. The structural message is the important one: mass and
quartic are successive derivatives of the same finite holonomy potential.

\subsection{Why the doublet survives while the color triplet does not}

The Arisaka-GG preprints emphasize that a \(\mathbf 5\) of \(SU(5)\) decomposes as
\begin{equation}
 \mathbf 5_H \to (\mathbf 3,\mathbf 1)_{-1/3}\oplus(\mathbf 1,\mathbf 2)_{+1/2}.
 \label{eq:5decomp-app}
\end{equation}
In a conventional SU(5) setting the coexistence of the triplet and doublet is
the source of the notorious doublet--triplet splitting problem. In the \GG{}
construction, however, orbifold parity and boundary conditions act directly on
the compact gauge component. The low-energy Higgs doublet corresponds to an
admissible light holonomy fluctuation, while the color-triplet partner is either
projected out of the zero-mode spectrum or pushed to the high seam and hence does
not appear as a light boundary scalar.

This is one of the clearest examples of the phrase ``geometry replaces tuning.''
The theory does not ask why an arbitrary scalar triplet mass happened to become
huge while an arbitrary scalar doublet mass remained small. It organizes the
spectrum so that only the doublet has the correct boundary parity and seam access
to appear in the low-energy theory.

\subsection{Connection to the first-order quartic closure used in the main text}

The main text used the leading closure
\begin{equation}
 \lambda_H^{(1)}=\alpha_s+\alpha_{\mathrm{em}}.
 \label{eq:lambda-closure-app}
\end{equation}
This relation should not be read as an arbitrary numerical guess. It is the
lowest-order boundary image of the statement that the curvature of the finite
holonomy potential is fed by the two most stable seam channels in the present
sector: the color channel and the electric channel. The subsequent QD uplift
then represents the next finite correction to the same holonomy potential.

\subsection{Operational extraction of couplings and masses in \GDOS{}/\GFF{}}
\label{app:operational-detail}

The manuscript uses \GDOS{} and
\(\GFF=\GEP+\CEP\) as the foundation of both calculation and observation. This appendix makes that operational statement more explicit.

\subsection{Why a prediction is not complete until an observation rule is specified}

A theory number becomes physically meaningful only after one states how it will
be read out on the boundary. In \GDOS{} this point is elevated to a principle:
observation is an open-system projection from the six-dimensional bulk to the
four-dimensional effective boundary. Therefore a coupling or a mass is never a
free-floating symbol. It is a response coefficient of the projected stationary
state.

Let \(\rho\) denote the effective \GDOS{} state variable and let
\(F_{\GFF}[\rho]\) be the corresponding free-energy functional. Introducing a
set of sources \(J_a\) for boundary observables \(\mathcal O^a\), one writes the
formal generating functional
\begin{equation}
 Z[J]
 =\int \mathcal D\rho\,
 \exp\!\left[-F_{\GFF}[\rho]+\int J_a\mathcal O^a\right].
 \label{eq:ZJ-app}
\end{equation}
Then the expectation values and connected responses are generated by
\begin{equation}
 \langle \mathcal O^a\rangle = \frac{\delta \ln Z}{\delta J_a},
 \qquad
 G^{ab}_{\mathrm{conn}}=\frac{\delta^2 \ln Z}{\delta J_a\delta J_b}.
 \label{eq:obs-response-app}
\end{equation}
The effective action obtained by Legendre transform packages the inverse
responses. Pole masses are extracted from the zeros of the appropriate inverse
propagator, while running couplings are extracted from properly normalized
vertices. This is the \GDOS{}/\GFF{} language behind the ordinary phenomenological
phrases ``mass'' and ``coupling.''

\subsection{Why the comparison basis of the main text is the correct one}

The theory side is forbidden from using measured Standard-Model observables as
inputs. But once a theoretical value is derived, it must still be compared with
a quantity defined in the same operational scheme used experimentally. That is
why the main text compares primary predictions against
\(\alphahat(M_Z)\), \(\sin^2\theta_{\mathrm{eff}}^\ell\),
\(\alpha_s(M_Z)\), \(v\), and \(m_h\), and treats the pole masses \(m_W\) and
\(m_Z\) as secondary checks.

This is not a compromise with the ``no empirical input'' rule. The rule forbids
using these quantities to determine the prediction. It does not forbid asking
whether the finished prediction agrees with them.

\subsection{Meaning of the theory error bars used in the tables}

The uncertainties quoted in Section~\ref{sec:comparison} are not statistical
errors in the experimental sense. They are truncation estimates attached to the
theoretical construction. More precisely, they quantify the expected size of
terms omitted when one stops at a given layer of the radial/seam expansion.

For the first-order predictions, the theory uncertainty estimates the combined
size of the leading omitted seam-matching and QD corrections. For the
QD-completed predictions, the uncertainty estimates the next omitted layer beyond
QD. This is why the second-order error bars are much smaller but not zero.
There is still residual structure not yet written in closed form.

\subsection{How \GFF{} organizes vacuum selection and measurement simultaneously}

The language \(\GFF=\GEP+\CEP\) is particularly useful because it unifies two
roles that are often split apart in ordinary presentations. The same functional
that determines the preferred stationary state also determines the response of
that state to small external probes. In this sense \GFF{} is both a selection
principle and a metrological principle. That dual role is precisely what makes
it the right foundational language for a parameter-derivation paper.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{F.10.1. Why do exactly two
transcendental numbers survive the radial transport?}

\noindent\emph{Solution.}  Step 1: the radial renormalization
group of this chapter transports couplings along one
non-compact direction (the exponential redshift of
\eqref{eq:redshift-app} supplies $e$) around one compact
direction (the circle's period supplies $\pi$).  Step 2: every
lawful readout at a stationary seam is built from integrals of
these two primitives \eqref{eq:one-loop-radial-app}; no other
transcendental has a geometric source in the arena --- there is
nothing else to integrate over.  Step 3: hence the codebook's
striking texture: rationals and dyadics from counting
(D.3.1, D.3.2, E.3.2), and $\pi$'s and $e$'s from measuring
(D.3.3's $\pi e$; the $e^{-\pi}$ tolls of the quark ladder;
the $e^{-K}$ of the lock scale) --- and \emph{nothing else}
anywhere in the twenty-eight parameters.  Physics intuition:
a theory's constants betray its arena --- circles pay in
$\pi$, exponential flows pay in $e$, discrete branchings pay
in integers; the absence of any other species of number in
the codebook is itself a structural prediction, and one a
single rogue constant would falsify.

\medskip\noindent\textbf{F.14.1. Work the Hosotani protection:
from periodic phase to finite Higgs curvature.}

\noindent\emph{Solution.}  Step 1: the order parameter is the
gauge-invariant Wilson line \eqref{eq:wilsonline-app}; its
scalar is the phase $\theta_H$, periodic by construction
\eqref{eq:thetaH-periodic-app}.  Step 2: gauge invariance
forbids any polynomial potential in $\theta_H$; the effective
potential generated at one loop is necessarily a Fourier
series, $V_{\rm eff} \propto \cos\theta_H + \ldots$
\eqref{eq:Veff-cos-app}, with finite coefficients --- the
would-be quadratic divergence has no invariant operator to
inhabit.  Step 3: expanding around the selected minimum
\eqref{eq:theta-expansion-app}, the 4D Higgs emerges
\eqref{eq:h-from-theta-app} with mass-squared given by the
finite curvature \eqref{eq:mh-curvature-app} --- the input
E.3.4 consumed.  Physics intuition: the deepest protection
mechanisms in physics are all topological bookkeeping ---
here, the impossibility of writing a divergent counterterm
for an angle; the Higgs is light for the same class of reason the axion is light in
the strong-CP chapter (App.~\ref{app:ch-strongcp}) --- a compact
variable whose potential must respect its circle.

\medskip\noindent\textbf{F.3.1. Show that the radial coordinate is not a
new dimension but a resolution label.}

\noindent\emph{Solution.}  Step 1: in the export picture the radial
variable orders boundary observables by the scale at which they are
read, and a shift in it is a change of resolution, not a translation in
space.  Step 2: the test is whether anything propagates in it --- a genuine
dimension supports a wave equation and a conserved momentum, and the
radial variable supports neither, since the flow is first order and
dissipative in it.  Step 3: consequence --- there is no Kaluza--Klein tower
associated with the radial direction, which is why the only tower in
this framework's collider phenomenology is the compact one of
\PaperPOne{}.  Physics intuition: the distinction is the same one that
separates the renormalization-group scale from a real dimension.  Both
are labeled by a positive number and both organize physics
hierarchically, but only one of them has a momentum conjugate to it, and
a student who keeps that test in mind will not be misled by the
geometric language.

\medskip\noindent\textbf{F.3.2. Show that the Wilson--Hosotani phase is
periodic, and say what the periodicity buys.}

\noindent\emph{Solution.}  Step 1: the Hosotani phase is the holonomy of
a gauge field around a compact direction, so it is defined modulo
$2\pi$ and physically equivalent configurations differ by that shift.
Step 2: hence its effective potential must be a periodic function ---
generated only at loop level, since a tree-level potential for a
holonomy is forbidden by higher-dimensional gauge invariance.  Step 3:
consequence --- the scalar mass generated is finite and calculable
rather than quadratically divergent, which is the mechanism's whole
point.  Physics intuition: periodicity is a symmetry, and symmetries
protect.  A compact variable cannot have an arbitrary potential, so its
excitation cannot have an arbitrary mass; this is exactly the same
argument that keeps an axion light, and it is why the Higgs mass in
this framework is a computed number rather than a tuned one.

\medskip\noindent\textbf{F.3.3. Check the export at a scale where it
should fail, and confirm that it does.}

\noindent\emph{Solution.}  Step 1: the export delivers four-dimensional
effective operators valid below the compactification scale; above it,
the tower is not decoupled and the four-dimensional description must
break.  Step 2: the diagnostic is the growth of the effective
description's amplitudes --- once a would-be four-dimensional amplitude
violates its unitarity bound, the omitted states are the ones restoring
it.  Step 3: for this framework the scale is the lock,
$\Mlock{} = 3.85$~TeV, which is precisely why \PaperPOne{} exists and
why F-SM-1 is written against a collider rather than against a
low-energy observable.  Physics intuition: an effective theory that
never fails is not an effective theory --- it is either exact or
untestable.  Being able to say where a description must break, and to
name the experiment that will look there, is the difference between a
framework and a fit, and the exercise of finding your own theory's
ceiling is one every student should perform on whatever they are
working on.

\medskip\noindent\textbf{F.3.4. Trace one exported coefficient back to
its ledger integer.}

\noindent\emph{Solution.}  Step 1: pick the neutrino operator
coefficient of App.~\ref{app:pmns}: $m_\star = \pi v^2/\EGUT$.  Step 2:
follow it back --- $v$ from the Wilson--Hosotani vev of this chapter,
$\EGUT = \EPl{}e^{-2\pi}$ from the gauge chapter's seam, and the
prefactor $\pi$ from a single traversal of the compact direction
$\Stheta{}$.  Step 3: so a meV-scale neutrino mass is three upstream
quantities and no new input, which is the claim the export chapter
exists to support.  Physics intuition: this is the exercise to repeat
for every derived number in the paper, and it is the most useful
critical habit a reader can develop here.  Ask of each: which upstream
quantities does it use, and were any of them fixed \emph{after} the
downstream number was known?  If the answer to the second question is
ever yes, the derivation is a fit; the point of the export chapter is
to make the dependency graph explicit enough that the question can be
asked at all.




\section{The assembled Lagrangian: the course walk}
\label{app:lagrangian-course}

\GGteachbox{The complete Standard-Model Lagrangian, assembled from scratch
as a constitutional export --- the same eight-stage chain audited at
results register in Part~\ref{part:lagrangian}, here walked at course
pace: the bulk action --- earned brace by brace, not recorded --- and
the forced integer ledger, one bulk spinor
becoming a chiral family with every hypercharge computed, the group
splitting and the covariant derivative, the five term groups exported one
by one, the operators the bulk does \emph{not} send, and the assembled
equation read line by line. Every manipulation is shown, and every step
carries its physics intuition.}

\GGreadalong{\citet{PeskinSchroeder1995} for gauge theory and the
Standard-Model Lagrangian in textbook form; \citet{Schwartz2014} for the
electroweak sector; \citet{GeorgiGlashow1974} for the unified family
packet; \citet{Hosotani1983} for the holonomy order parameter.}

This chapter is the course's \emph{assembly interlude}. It is not one of
the sector appendices --- it derives no new parameter --- and it
deliberately interrupts the sequence at the halfway point, after the
gauge chapter and the Higgs chapter have built the bosonic half of the
theory and before the fermion chapters build the other half. The reason
for the interruption is pedagogical and, we would argue, psychological.
Every reader of this course learned the subject in the same way: the
Standard-Model Lagrangian was written on the board, complete, on the
first day, and everything afterwards was consequence. The single most
disorienting claim of \GGTheory{} is that the object on the board is not
a starting point at all. Part~\ref{part:lagrangian} of the main text
states that claim at results register; this chapter is where the student
\emph{earns} it, by assembling the Lagrangian with their own hands, one
exported structure at a time, with nothing pulled from a hat. The two
treatments share one governing sentence, and it bears repeating before
the first step: \emph{the visible four-dimensional Lagrangian is exported
from the six-dimensional bulk, not assembled by listing all gauge
invariants.} \emph{(Physics intuition: think of the boundary theory as
the shadow cast by a rigid six-dimensional object. One does not build a
shadow by choosing its parts; one builds the object, shines the light,
and reads the shadow off. This chapter builds the object and reads.)}

\subsection{How to read this chapter}
\label{app:lag-course-howto}

The walk has eight stages, numbered S-1 to S-8 as in the main text,
plus one interpolated station the main text compresses to a paragraph:
S-1a, where the bulk action itself is earned rather than cited. Each
stage below is worked from scratch: the claim, the computation with no
skipped manipulations, the register of the result (\textbf{Derived},
\textbf{Derived (cond.)} on a named lemma, or \textbf{Asserted}), and the
physics intuition. Where a stage consumes a result already derived in
full elsewhere in the course --- the dyadic gauge readings of the gauge
chapter, the lock-scale chain of the Higgs chapter, the Wilson-line
geometry of the export chapter just concluded --- we work the assembly
step here and cite the chapter that owns the ingredient, so that nothing
is derived twice and nothing is asserted without an address. The chapter
closes with the assembled equation, eight problems with complete worked
solutions, and the take-away box. A reader with limited time can read
\S\ref{app:lag-course-assembly} first and work backwards; a reader with
no time at all should at least sit with the assembled equation until the
strangeness registers: \emph{there is no term in it anyone chose.}

\subsection{S-1 --- the inputs on the table, and the bulk action}
\label{app:lag-course-s1}

\paragraph{What is on the table.}
Before the first computation, the complete list of what the derivation is
allowed to touch --- the admissible inputs of the standard of derivation
(\S\ref{sec:admissible-inputs}) --- is short enough to memorize: the four
axioms and seven postulates of the \GGConstitution{}; the six-dimensional
bulk $M_4 \times I_r \times S^1_\theta$ (four-dimensional spacetime, a
radial interval that acts as an energy ladder, and one compact circle);
the five dimensional constants; the two transcendentals native to the
compact directions ($e$ for the exponential redshift along the radial
interval, $\pi$ for the angle of the circle); and the five integers of
the \OGNfive{} ledger, forced in S-1 below. \emph{Not} on the table: any
four-dimensional Lagrangian, any choice of gauge group or representation,
any scalar potential, and any of the twenty-eight numbers. \emph{(Physics intuition:
 the honest way to keep score in any derivation is to lock the
pantry before cooking. Everything served below must be traceable to this
shelf, and the shelf contains no continuous dial anywhere --- the
strongest statement on it is a pair of closure identities among five
integers.)}

\paragraph{The bulk action, term by term.}
The bulk dynamics is Einstein--Yang--Mills--Dirac theory with flux on
the six-dimensional bulk, written schematically as
\begin{equation}
\resizebox{0.97\linewidth}{!}{$\displaystyle
S_6 \;=\; \int_{M_6} \sqrt{-G}\;\Big[
\underbrace{\tfrac{1}{2\kappa_6^2}\,\big(R_6 - 2\Lambda_6\big)}_{\text{geometry}}
\;-\; \underbrace{\tfrac{1}{4}\,\mathcal F_{MN}^{\,A}\mathcal F^{A\,MN}}_{\SU(5)\times\UoneB{}\times\UoneQi{}}
\;+\; \underbrace{i\,\overline\Psi\,\Gamma^M D_M \Psi}_{\text{one bulk spinor per family}}
\;-\; \underbrace{\tfrac{1}{12}\,H_{MNP}H^{MNP}}_{\text{flux sector}}
\Big],
$}
\label{eq:bulk-action-course}
\end{equation}
plus the Green--Schwarz-type terms that the anomaly bookkeeping of
\AThree{} \citep{ArisakaA3BI6} requires. Read the four braces as the
complete cast of characters: the metric, the unified gauge field, the
matter spinor, and the flux whose closure law is about to do all the
work. (The bulk cosmological term $\Lambda_6$ rides with the geometry
brace; its boundary shadow is not a tuned number but an export --- the
lock-scale seesaw of the codebook.) Equation~\eqref{eq:bulk-action-course}
is not an ansatz, and this chapter does not ask the reader to accept it
on citation. Its field content is forced by theorems of the cohort ---
the stage and the warped metric by T-GG6-0/1/2 of \ATwo{}
\citep{ArisakaA2GG6}, the gauge group by T-GG6-3, the flux sector and
its Green--Schwarz completion by the anomaly closure of \AThree{} ---
its canonical form is recorded in \ATwo{} ahead of the
BRST-consistency construction there, and the uniqueness of its dynamics
--- \emph{why these terms and no others} --- is earned brace by brace
in S-1a below, with the register of every claim stated. The machinery
that then spends it is the nine-part descent theorem T-GDOS-9 of
\AFive{} \citep{ArisakaA5GDOS}, which carries the action from the
bulk to the boundary. \emph{(Physics intuition: nothing in
Eq.~\eqref{eq:bulk-action-course} is exotic --- every term is the
six-dimensional version of machinery a graduate student already trusts.
The novelty of the program is not in the ingredients; it is in the
claim that this one fixed dish, with no seasoning allowed, exports the
entire Standard Model.)}

\paragraph{The closure that forces the ledger.}
The flux sector is not free: its Bianchi identity is modified by the
anomaly inflow, $\dd H_3 = X_4$, and demanding that this \BIsix{} closure
be globally consistent on the bulk is a Diophantine condition on the
discrete data. \AThree{} proves that the condition admits exactly one
solution, the \OGNfive{} ledger, and the two identities that lock it can
be checked by hand:
\begin{equation}
\begin{aligned}
(M, N, L; J, K) &= (1, 3, 5; 7, 35),\\
K \;=\; L\,J \;=\; 5\times7 \;=\; 35
\;&=\; 1 + 9 + 25 \;=\; M^2 + N^2 + L^2 .
\end{aligned}
\label{eq:ledger-double-course}
\end{equation}
Problem G.2.1 works the uniqueness scan: among small integer triples, no
other $(M,N,L)$ produces an integer, odd closure index $J$ from the same
pair of relations. At the end of S-1 the theory carries five integers and
zero continuous freedom, and every subsequent stage spends from this
account only. \emph{(Physics intuition: the ledger is the theory's DNA
--- five integers that are not settings but survival conditions. Change
any one and the closure dies at once, the way changing one gear tooth
stops a clock rather than making it run at a different speed. That is
what separates an integer ledger from a parameter set: parameters bend,
ledgers break.)}

\subsection{S-1a --- why this action and no others: the bulk action earned}
\label{app:lag-course-s1a}

\paragraph{The claim.}
Stage S-1 put Eq.~\eqref{eq:bulk-action-course} on the table with its
provenance named; this station proves the sharper statement the walk
actually needs:
\begin{quote}
\emph{On the bulk forced by T-GG6-0/1/2 of \ATwo{}, with the gauge
group forced by T-GG6-3 and the matter restricted to the minimal
anomaly-admissible chiral sector, the constitutional clauses \LC{}
(locality--causality), \POS{2} (a single variational principle), and
\MSMF{} (read at lowest derivative order) admit exactly one local
action functional: Eq.~\eqref{eq:bulk-action-course}, completed by the
Green--Schwarz terms that \BIsix{} closure forces.}
\end{quote}
The proof is a brace-by-brace uniqueness scan, and its honesty matters
as much as its conclusion: two of the five steps close on
constitutional theorems, two close on elementary classification
arguments, and one stands on a named lemma. The registers are stated
as we go and collected in the table at the end. \emph{(Physics intuition:
the reader has seen this move before, in S-1's locked pantry --- but a
locked pantry only helps if the recipe is also forced. This station
shows the recipe book has one page.)}

\paragraph{Brace 1 --- geometry.}
What scalar densities can the metric alone supply? \LC{} demands a
local functional; at zero derivatives the only invariant is the volume
element itself --- the cosmological term --- and at two derivatives the
classification is classical: the Ricci scalar $R_6$ is the \emph{only}
diffeomorphism-invariant density, so the Einstein--Hilbert term is not
a choice but the entire two-derivative shelf. The honest caveat lives
one order up. In six dimensions Lovelock's classification admits one
further term with second-order field equations --- the Gauss--Bonnet
density $R^2 - 4R_{MN}R^{MN} + R_{MNPQ}R^{MNPQ}$ --- which in four
dimensions integrates to a topological invariant but in $D=6$ carries
dynamics. It is removed not by a uniqueness theorem but by \MSMF{}'s
lowest-derivative-order reading: a four-derivative term is structure
the two-derivative theory does not need, and \MSMF{} forbids freedom
that buys nothing. Its boundary image, like that of every
higher-curvature correction, is an irrelevant operator diluted along
the radial throat --- the geometrized Wilsonian statement of
T-GDOS-9.9 \citep{ArisakaA5GDOS}. Register: \textbf{D} at lowest
derivative order, with the Lovelock caveat stated; Problem G.3.1 works
the scan by hand. \emph{(Physics intuition: gravity's kinetic term is
the loneliest object in field theory --- ask for general covariance
and two derivatives, and the market stocks exactly one item.)}

\paragraph{Brace 2 --- gauge.}
Given the group, the kinetic term is again the whole shelf:
$\mathcal F_{MN}\mathcal F^{MN}$ is the unique gauge-invariant scalar
density at two derivatives. The parity-odd invariants a
six-dimensional connection can also supply --- the Chern-class
densities of $\mathcal F\wedge\mathcal F\wedge\mathcal F$ type --- are
not free additions here: they belong to the topological sector, and
\BIsix{} closure prices every topological coefficient it admits
(brace~4). What the classification does \emph{not} settle is the group
itself, and the register must say so: T-GG6-3 of \ATwo{} forces
$\SU(5)\times\UoneB{}\times\UoneQi{}$ from \GGEP{} + \MSMF{} + gauge
closure, but that theorem carries two conditional clauses stated
inside it --- the requirement that the boundary contain the
Standard-Model gauge content, and the empirically-preferred
four-component class --- and this walk inherits them rather than
hiding them. Register: form \textbf{D}; group \textbf{D*} on
T-GG6-3's stated clauses. \emph{(Physics intuition: the gauge shelf is
as bare as gravity's --- one item --- but the label on the jar is the
real content. The classification fixes what a gauge field can do; the
Constitution fixes which gauge field exists. Keeping those two claims
at their two different grades is what honest bookkeeping looks like.)}

\paragraph{Brace 3 --- matter.}
The Dirac term $i\,\overline\Psi\,\Gamma^M D_M \Psi$ is the unique
\LC{}-admissible first-order kinetic term for a spinor --- a
classification even shorter than the bosonic ones. The substantive
question is not the term but the \emph{cast}: why one bulk spinor per
family and nothing else? Here the walk names its conditional plainly.
Anomaly admissibility (the factorization theorem T-GG6-5 of \ATwo{},
with its global completion in \AThree{}) constrains the chiral sector
to representations whose anomaly polynomial factorizes against $X_4$;
the \emph{minimal} such sector is the one bulk spinor whose zero modes
are the $\overline{\mathbf 5}\oplus\mathbf{10}$ packet of S-2. That
minimality --- no second packet, no exotics, no vector-like
passengers --- is \MSMF{} again, and its precise statement is a lemma
this course names rather than proves: the \textbf{Minimal-Matter
Lemma}, that the family packet is the unique minimal
anomaly-admissible chiral sector on the forced bulk (evidence base:
the benchmark-spectrum analysis of \AThree{}). Replication $N=3$ is
the pincer of \PaperPTwo{}, already carried at \textbf{D*} in S-2.
Register: form \textbf{D}; content \textbf{D*} on the Minimal-Matter
Lemma. \emph{(Physics intuition: the Dirac term is the cheapest
sentence in the action, and the cast list is the expensive one.
Saying ``one spinor, nothing else'' is a strong minimality claim ---
strong enough that this course prints the lemma's name beside it
rather than pretending it has the proof.)}

\paragraph{Brace 4 --- flux, and the terms with no dial.}
The two-form sector is the one place the action is \emph{added to},
and the addition is forced rather than chosen: consistency of the
chiral sector requires the Chern--Simons-improved field strength ---
the $H_3$ whose Bianchi identity reads $\dd H_3 = X_4$ --- its kinetic
term $H_{MNP}H^{MNP}$ is the unique two-derivative invariant, and the
Green--Schwarz coupling with the coefficient vector
$(1,-1,-\tfrac25,-\tfrac25)$ is a theorem (T-GG6-4 of \ATwo{};
structural and global closure in \AThree{} \citep{ArisakaA3BI6}).
Nothing in this brace is graded below \textbf{D}: the flux sector is
where the Constitution shows its teeth, because here even the
topological coefficients --- the ones a four-dimensional
$\theta$-angle would leave free --- are fixed by closure. Register:
\textbf{D}. \emph{(Physics intuition: in four dimensions the
$\theta$-angle is the loose thread of the Standard Model, a
topological coefficient nothing fixes. Here the analogous coefficients
sit inside a closure identity and are priced by it: the flux sector is
where ``no free dials'' stops being a slogan and becomes arithmetic.)}

\paragraph{Brace 5 --- the term that is absent.}
Uniqueness has a second half: the action contains no fundamental
scalar, and the cast list of Eq.~\eqref{eq:bulk-action-course} is
closed. A bulk scalar with a potential is anomaly-admissible --- the
canonical form recorded in \ATwo{} allows, as its most general
envelope, exactly such a sector --- but the export never spends one:
the symmetry-breaking order parameter of S-6 is the Wilson-line
holonomy of the compact circle, supplied by the geometry itself, and a
fundamental scalar alongside it would be redundant continuous freedom
of precisely the kind \MSMF{} forbids. The scalar-free member of the
admissible envelope is therefore the constitutional selection, and the
Higgs sector of this walk is built without ever touching this brace.
Register: \textbf{D*} on the same \MSMF{} minimality reading as
brace~1. \emph{(Physics intuition: the deepest silence in the action
is the term that is not there --- the textbook's scalar, with its
hand-tuned $\mu^2<0$, is replaced not by a better scalar but by a hole
where the scalar used to be, and the geometry pays the bill.)}

\paragraph{The provenance of the action, collected.}
\begin{table}[htbp]
\centering
\small
\caption[The assembled action, brace by brace: the form of each term, the content it carries, and the register it belongs to.]{The assembled action, brace by brace: the form of each term, the content it carries, and the register it belongs to. The table is the bookkeeping behind the claim that the Lagrangian is exported rather than postulated, and it is arranged so that the absence of a term is as visible as the presence of one. The textbook's scalar potential has no row here, which is exactly the point of the paragraph above it.}
\label{tab:action-provenance}
\small
\begin{tabular}{@{}p{0.13\textwidth}p{0.26\textwidth}p{0.28\textwidth}p{0.24\textwidth}@{}}
\toprule
\textbf{Brace} & \textbf{Form of the term} & \textbf{Content it carries} & \textbf{Register}\\
\midrule
geometry & EH unique at two derivatives & Gauss--Bonnet removed at lowest order & \textbf{D} (Lovelock caveat stated)\\
\addlinespace
gauge & $\mathcal F^2$ unique at two derivatives & group from T-GG6-3 of \ATwo{} & form \textbf{D}; group \textbf{D*}\\
\addlinespace
spinor & Dirac unique at first order & minimal packet; $N=3$ (\PaperPTwo{}) & form \textbf{D}; content \textbf{D*}\\
\addlinespace
flux $+$ GS & forced by \BIsix{} closure & coefficients $(1,-1,-\tfrac25,-\tfrac25)$ & \textbf{D}\\
\addlinespace
(no scalar) & --- & holonomy is the order parameter & \textbf{D*} (\MSMF{} minimality)\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{What promotion would look like.}
The two named conditionals of this station are research targets with
precise statements, not soft spots hidden in prose: a proof of the
Minimal-Matter Lemma promotes brace~3 to \textbf{D}, and a
constitutional derivation of the lowest-derivative-order reading ---
or a boundary observable sensitive to a Gauss--Bonnet remnant ---
settles braces 1 and 5. Nothing downstream of this station upgrades a
grade, and the assembled equation at the end of the walk is exactly as
strong as this page plus the descent behind it. Problems G.3.1 and
G.3.2 work the two halves --- the operator scan, and the machine that
spends the action --- by hand. \emph{(Physics intuition: a derivation
is only as strong as its weakest named link, and naming the links is
the point. The two lemmas of this station are not embarrassments to be
smoothed over; they are the shortest list anyone has yet attached to
the question ``why this Lagrangian?''.)}

\subsection{S-2 --- one bulk spinor becomes a chiral family}
\label{app:lag-course-s2}

\paragraph{The reduction, worked.}
Take the bulk Dirac term of Eq.~\eqref{eq:bulk-action-course} and expand
the spinor in modes of the compact directions,
\begin{equation}
\Psi(x, r, \theta)
\;=\; \sum_{n}\;\Big[\,\psi^{(n)}_L(x)\,f_n(r)\,e^{i n \theta}
\;+\; \psi^{(n)}_R(x)\,g_n(r)\,e^{i n \theta}\Big],
\label{eq:kk-expansion-course}
\end{equation}
where $L/R$ are the four-dimensional chiralities. On the circle, single-
valuedness quantizes $n$; on the interval, the variational principle
forces a boundary condition, and the consistent choice sets one chirality
to zero at the ends, $\psi_R\big|_{\partial I_r} = 0$. The massless
sector is then read off: the would-be right-handed zero mode needs a
constant profile $g_0$, which the Dirichlet condition kills; the
left-handed zero mode survives with its profile $f_0$ intact. The
four-dimensional boundary spectrum is therefore \emph{chiral} --- purely
left-handed Weyl --- before any dynamics has been discussed
(\S\ref{sec:gg-derives-su5} carries the full statement). \emph{(Physics intuition:
 chirality here is a filter, not a choice. The interval acts
like a polarizer: both polarizations of the bulk field arrive, one is
absorbed at the walls, and only the other is transmitted to the boundary
world. The famous left-handedness of the weak interaction starts life as
a boundary condition.)}

\paragraph{The packet, decomposed with every hypercharge computed.}
The surviving zero modes arrive in complete multiplets of the unified
$\SU(5)$, one $\overline{\mathbf 5}\oplus\mathbf{10}$ per family, and the
entire quantum-number table of the Standard Model is now a
\emph{computation} rather than a list. Fix the hypercharge generator
inside $\SU(5)$ acting on the fundamental $\mathbf 5$ as
\begin{equation}
Y \;=\; \operatorname{diag}\Big(
-\tfrac13,\, -\tfrac13,\, -\tfrac13,\; +\tfrac12,\, +\tfrac12\Big),
\qquad
\operatorname{Tr} Y = 3\big(-\tfrac13\big) + 2\big(+\tfrac12\big) = 0 ,
\label{eq:five-y-course}
\end{equation}
traceless as a generator of a simple group must be --- and that
tracelessness is already the whole mystery of charge quantization:
hypercharge cannot be rescaled per multiplet, because it is one matrix.
Then the $\overline{\mathbf 5}$ (conjugate, so $Y$ flips sign) contains
the color antitriplet with $Y = +\tfrac13$ --- the $d^c$ --- and the
weak doublet with $Y = -\tfrac12$ --- the lepton doublet $L = (\nu, e)$.
The $\mathbf{10}$ is the antisymmetric pair of $\mathbf 5$ indices, and
its hypercharges are simply \emph{sums of pairs} of the entries in
Eq.~\eqref{eq:five-y-course}:
\begin{equation}
\underbrace{-\tfrac13 - \tfrac13 = -\tfrac23}_{\text{two color indices: } u^c\ (\overline{\mathbf 3}, \mathbf 1)}
\qquad
\underbrace{-\tfrac13 + \tfrac12 = +\tfrac16}_{\text{one color, one weak: } Q\ (\mathbf 3, \mathbf 2)}
\qquad
\underbrace{+\tfrac12 + \tfrac12 = +1}_{\text{two weak indices: } e^c\ (\mathbf 1, \mathbf 1)} .
\label{eq:tenfold-course}
\end{equation}
Fifteen Weyl fields, five irreducible blocks, every hypercharge --- the
strange fractions $\tfrac16, -\tfrac23, \tfrac13, -\tfrac12, 1$ that the
textbook must postulate --- read off one traceless matrix and its pair
sums. The electric charges follow from $Q = T_3 + Y$: for the quark
doublet, $Q(u) = \tfrac12 + \tfrac16 = \tfrac23$ and
$Q(d) = -\tfrac12 + \tfrac16 = -\tfrac13$; for the leptons,
$Q(\nu) = \tfrac12 - \tfrac12 = 0$ and $Q(e) = -\tfrac12 - \tfrac12 =
-1$; for the singlets, $Q(u^c) = -\tfrac23$, $Q(d^c) = +\tfrac13$,
$Q(e^c) = +1$. Problem G.4.1 completes the classic corollary: the
tracelessness of $Q$ on the $\overline{\mathbf 5}$ forces
$|q_e| = |q_p|$ \emph{exactly}, which is why atoms are neutral to better
than one part in $10^{21}$. \emph{(Physics intuition: the textbook
presents the hypercharge table the way a phone book presents numbers ---
facts without reasons. Here the table is the multiplication table of one
matrix. When a student first computes $-\tfrac13 + \tfrac12 = \tfrac16$
and recognizes the quark doublet's hypercharge, the fifteen-field table
stops being a list to memorize and becomes a pattern with one seed.)}

\paragraph{Anomaly cancellation for free, and three families.}
The textbook must verify, after choosing its representations, that the
gauge anomalies cancel --- four independent sums over each family that
have no right to work and do. Here they cannot fail: each family is one
bulk packet, and the packet's consistency is the \BIsix{} closure already
paid for in S-1. Problem G.4.2 works all four sums explicitly ---
$[\SU(3)]^2\,\U(1)_Y$, $[\SU(2)]^2\,\U(1)_Y$, $[\U(1)_Y]^3$, and
gravitational --- and each vanishes term by term using only the
hypercharges just computed. Replication is the one entry of this stage
carried at conditional register: the family count is the two-sided
pincer of \PaperPTwo{} \citep{ArisakaP2N3} (\S\ref{sec:N3-theorem}),
$N \ge 3$ from the \MSMF{} CP lower bound and $N \le 3$ from the
Bogomolnyi/\BIsix{} action cap --- \textbf{Derived (cond.)}, as flagged
there. \emph{(Physics intuition: anomaly cancellation is the Standard
Model's most impressive conspiracy --- fractions conspiring across
quarks and leptons to sum to zero four separate ways. Conspiracies
dissolve when you find the committee: the quarks and leptons sit in one
room upstairs, in a single bulk multiplet, and the four sums are just
that room's bookkeeping viewed four ways.)}

\subsection{S-3 --- the splitting, and the covariant derivative assembled}
\label{app:lag-course-s3}

\paragraph{The splitting.}
Boundary conditions and the holonomy structure on the compact directions
reduce the bulk gauge symmetry to the subgroup that survives on the
boundary,
\begin{equation}
\SU(5)\times\UoneB{}\times\UoneQi{}
\;\longrightarrow\;
\SU(3)_C\times\SU(2)_L\times\U(1)_Y ,
\label{eq:splitting-course}
\end{equation}
with the mechanism worked in the fundamental-architecture section
(\S\ref{sec:sm-foundation}) and the Wilson-line geometry in the chapter
just concluded. The essential point for the assembly is what the
splitting does \emph{not} touch: the hypercharge generator
\eqref{eq:five-y-course} is part of the surviving algebra, so every
quantum number computed in S-2 descends unchanged. \emph{(Physics intuition:
 breaking a symmetry by boundary conditions is like slicing a
crystal along a plane --- the bulk lattice fixes which faces are
possible, and the exposed face keeps the lattice's spacings. The
Standard-Model group is the exposed face of the unified crystal, and the
hypercharges are its inherited spacings.)}

\paragraph{The $5/3$, computed in four lines.}
The unified origin fixes the relative normalization of hypercharge, and
with it the tree-level weak angle --- the number the gauge chapter's
problems already met from the running side (D.1.2). Compute both traces
over one family using the table of S-2. For hypercharge:
\begin{equation}
\operatorname{Tr} Y^2
= \underbrace{6\cdot\tfrac{1}{36}}_{Q}
+ \underbrace{3\cdot\tfrac{4}{9}}_{u^c}
+ \underbrace{3\cdot\tfrac{1}{9}}_{d^c}
+ \underbrace{2\cdot\tfrac{1}{4}}_{L}
+ \underbrace{1}_{e^c}
= \tfrac16 + \tfrac43 + \tfrac13 + \tfrac12 + 1
= \tfrac{10}{3},
\label{eq:trace-ratio-course}
\end{equation}
while the same trace of $T_3^2$ receives $\tfrac14$ from each of the
eight doublet states (six quark, two lepton):
$\operatorname{Tr} T_3^2 = 8 \cdot \tfrac14 = 2$. The ratio is
$\operatorname{Tr} Y^2 / \operatorname{Tr} T_3^2 = \tfrac{10/3}{2} =
\tfrac53$ --- the normalization $g_1^2 = \tfrac53 g'^2$ of the gauge
chapter \eqref{eq:su5norm-app} --- and at the unification point it gives
$\sin^2\theta_W = \tfrac{3}{8}$ \eqref{eq:sin2-gut-app}. Problem G.5.1
extends the computation to $\operatorname{Tr} Q^2 = \tfrac{16}{3}$ and
recovers $3/8$ directly as $\operatorname{Tr} T_3^2 / \operatorname{Tr}
Q^2$. \emph{(Physics intuition: the $5/3$ is not a convention --- it is
the statement that hypercharge and weak isospin are two rulers cut from
the same block of wood, so their relative markings are fixed. Any theory
in which $Y$ is its own free abelian charge can rescale it at will; a
theory in which $Y$ is five entries of one traceless matrix cannot.)}

\paragraph{The covariant derivative.}
Everything a gauge theory does to matter is packaged in one operator,
and that operator is now fully determined:
\begin{equation}
D_\mu \;=\; \partial_\mu
\;-\; i\,g_s\,G^a_\mu T^a
\;-\; i\,g\,W^i_\mu \tau^i
\;-\; i\,g'\,\tfrac{Y}{2}\,B_\mu ,
\label{eq:covderiv-course}
\end{equation}
with the three factors from the splitting
\eqref{eq:splitting-course}, every $Y$ from the pair sums
\eqref{eq:tenfold-course}, and every $T^a, \tau^i$ from the surviving
algebra. No entry of Eq.~\eqref{eq:covderiv-course} was chosen at any
point of the chain. \emph{(Physics intuition: in the textbook,
$D_\mu$ is where the physicist's choices are stored --- the group, the
charges, the couplings all typed into one line. Here the same line is a
receipt: every symbol on it was paid for upstream, and the three
couplings, the only continuous entries, are about to be read off the
seam rather than measured in.)}

\subsection{S-4 --- term group I: the gauge kinetic terms}
\label{app:lag-course-s4}

Integrating the compact directions in the bulk gauge kinetic term of
Eq.~\eqref{eq:bulk-action-course} produces the boundary kinetic terms
for the three surviving factors,
\begin{equation}
\mathcal L_{\mathrm{gauge}}
= -\tfrac14\, G^a_{\mu\nu} G^{a\,\mu\nu}
  -\tfrac14\, W^i_{\mu\nu} W^{i\,\mu\nu}
  -\tfrac14\, B_{\mu\nu} B^{\mu\nu},
\qquad
\frac{1}{g_4^2} \;=\; \frac{V_2}{g_6^2},
\label{eq:gauge-kinetic-course}
\end{equation}
where $V_2$ is the compact volume: the six-dimensional coupling and the
compact geometry set the four-dimensional couplings, exactly as the
inverse-square law of the force chapter descends from flux
conservation. The \emph{structure} of this term group is therefore pure
export. Its three \emph{coefficients} are the gauge chapter's business,
and the course has already derived them as seam registrations:
$\widehat\alpha^{-1}(M_Z) = 2^J = 128$ \eqref{eq:alpha-dyadic-app}, the
rational weak angle $\sin^2\theta_W = 25/108$ \eqref{eq:sw2-derived-app},
and the strong closure $\alpha_s^{-1}(M_Z) = \pi e$
\eqref{eq:alphas-derived-app}, each with its stated pull and QD
refinement. \emph{(Physics intuition: a coupling constant, in this
picture, is a measured property of a geometric seam --- like the pitch
of a taut string, set by length and tension rather than dialed. The
gauge chapter measured the three pitches; this chapter only has to hang
the strings in the right frame.)}

\subsection{S-5 --- term group II: the fermion kinetic terms}
\label{app:lag-course-s5}

The matter half of the kinetic sector is now a single sum with no
freedom in it. Three families times fifteen Weyl fields gives
forty-five two-component spinors, each coupled through the one
covariant derivative:
\begin{equation}
\mathcal L_{\mathrm{fermion}}
\;=\; \sum_{\psi\,\in\,3\times(Q,\,u^c,\,d^c,\,L,\,e^c)}
\overline\psi\; i \gamma^\mu D_\mu\, \psi .
\label{eq:fermion-kinetic-course}
\end{equation}
Count the freedom: the spectrum was computed (S-2), the charges were
computed (S-2), the operator was computed (S-3), and the family number
is the pincer theorem. There is nothing left to decide. The one feature
that startles every student --- that the $\SU(2)_L$ factor sees only
left-handed fields, maximal parity violation, the 1957 shock --- is the
polarizer of S-2 read dynamically: the boundary condition transmitted
one chirality, so the gate force can only couple to what arrived.
\emph{(Physics intuition: forty-five fields sounds baroque until one
remembers they are one bulk field, viewed through a compactification.
An orchestra of overtones is not a complicated instrument; it is one
string, heard properly.)}

\subsection{S-6 --- term group III: the Higgs sector, with no dial}
\label{app:lag-course-s6}

This is the stage where the export departs furthest from the board, and
the chapter slows down accordingly. The textbook writes two terms by
hand: a kinetic term for a postulated doublet, and the potential
$V = \mu^2 \Phi^\dagger\Phi + \lambda(\Phi^\dagger\Phi)^2$ with
$\mu^2 < 0$ inserted to force the vacuum. The export replaces all four
decisions (the field, the kinetic term, the shape, the sign) with
geometry already on the table.

\paragraph{The field.} The would-be Higgs is not a new ingredient: it is
the Wilson-line holonomy of the compact circle,
$\theta_H \sim \oint A_\theta$, the gauge field's own component along
$S^1_\theta$ read as a boundary scalar \eqref{eq:wilsonline-app}. Its
doublet quantum numbers are inherited from the coset directions of the
splitting --- the components of the unified gauge field that the
boundary conditions removed from the four-dimensional gauge sector
survive as the scalar's internal indices. \emph{(Physics intuition: the
Higgs is the memory of the direction the boundary cannot see. Nothing
new was added to the universe in 1964, on this reading; the scalar
sector is the sixth dimension's gauge field wearing four-dimensional
clothes.)}

\paragraph{The kinetic term.} Because the field is a gauge-field
component, its kinetic term $|D_\mu \Phi|^2$ is a slice of the bulk
Yang--Mills kinetic term --- the mixed $F_{\mu\theta}F^{\mu\theta}$
components of Eq.~\eqref{eq:bulk-action-course} --- not a separate
postulate. \emph{(Physics intuition: one term of
Eq.~\eqref{eq:bulk-action-course}, cut along two different planes,
yields both the gauge kinetic terms of S-4 and the Higgs kinetic term of
S-6. The bulk action is being spent, not supplemented.)}

\paragraph{The shape of the potential, forced.} Here is the argument in
full, because it is the dissolution of the naturalness problem and it
fits in a paragraph. A Wilson-line phase is compact: a large gauge
transformation --- one that winds once around $S^1_\theta$ --- shifts
\begin{equation}
\theta_H \;\longrightarrow\; \theta_H + 2\pi,
\label{eq:largegauge-course}
\end{equation}
and is exactly invisible to physics \eqref{eq:thetaH-periodic-app}.
Every admissible term in the effective potential must therefore be
periodic in $\theta_H$. But a polynomial --- in particular the mass term
$\mu^2\,\theta_H^2$ that the textbook writes and the quadratic
divergence that renormalization wants to feed it --- is not periodic:
under Eq.~\eqref{eq:largegauge-course} it shifts, so gauge invariance
forbids it \emph{as an operator}. What survives is the Fourier series
generated at one loop by the fields running around the circle,
$V_{\mathrm{eff}} \propto \cos\theta_H + \dots$
\eqref{eq:Veff-cos-app}, with finite coefficients set by the
compactification scale --- the Hosotani mechanism
\citep{Hosotani1983,Wilson1974}. The textbook's hand-tuned $\mu^2 < 0$
becomes a \emph{branch-selection condition},
$\partial_{\theta_H} V_{\mathrm{eff}} = 0$ with
$\partial^2_{\theta_H} V_{\mathrm{eff}} > 0$: the vacuum sits where the
forced potential has its stable minimum, and there is no dial in the
mechanism anywhere. Problem G.8.1 works the periodicity argument as an
exercise and states its consequence plainly: \emph{you cannot
quadratically renormalize an angle.} The hierarchy problem is not
solved here --- it is dissolved, because the operator that carries the
problem does not exist on the export list. \emph{(Physics intuition: an
angle has no length to drag. Every naturalness anxiety of the past
fifty years was an anxiety about a length --- a scalar mass pulled
toward the cutoff. The export's scalar is a phase on a circle, and a
phase can be pushed around the circle but never stretched; the
divergence has no invariant to live in.)}

\paragraph{The two readings.} The Higgs chapter derived the two numbers
this sector owes the codebook, and the assembly imports them: the
vacuum value from the lock-scale chain,
$v = \Mlock\,(2/L)^N = 3849 \times (2/5)^3~\text{GeV} \approx
246.3~\GeV{}$ \eqref{eq:vev-final-app}, against the Fermi-constant
extraction $246.2$~\GeV{}; and the quartic from the dyadic reading
$\lambda_H = 2^{-N} = \tfrac18$ \eqref{eq:quartic-final-app}, giving
$m_h = v/2 = 123.1$~\GeV{} at first order \eqref{eq:mh-final-app}, with
the QD completion carrying the stated conservative theory uncertainty.
Registers as graded there: the vev \textbf{Derived}; the quartic
\textbf{Derived (cond.)} on the curvature lemma. \emph{(Physics intuition:
 the electroweak scale, in this assembly, is the end of a
pipeline --- $M_{\mathrm{Pl}}$, one exponential lock $e^{-K}$, three
doublet filters of $2/L$ --- every stage an integer of the ledger. The
number $246$ stops being a measurement to explain and becomes an
address in the geometry.)}

\subsection{S-7 --- term group IV: the Yukawa block}
\label{app:lag-course-s7}

The last interacting term group couples the two halves just built:
\begin{equation}
\mathcal L_{\mathrm{Yuk}}
= -\,\overline Q_L\, Y_u\, \widetilde\Phi\, u_R
  -\,\overline Q_L\, Y_d\, \Phi\, d_R
  -\,\overline L_L\, Y_e\, \Phi\, e_R
  \;+\; \mathrm{h.c.}
\label{eq:yukawa-course}
\end{equation}
Two separate claims must be kept distinct here, because they carry
different registers. The \emph{form} of Eq.~\eqref{eq:yukawa-course} is
forced: given the chiral packets of S-2 and the Wilson-line scalar of
S-6, these are the only boundary-admissible couplings between them ---
\textbf{Derived}. The \emph{content} of the matrices $Y_f$ --- in the
textbook, fifty-four free real numbers feeding twelve masses and eight
mixing parameters plus phases --- is the subject the course takes up
immediately after this interlude, and the assembly records only its
skeleton: the top quark sits at the zero-winding anchor with $y_t = 1$
at first order \eqref{eq:yt-one-app}; the quark ladder descends by
Bogomolnyi tolls $e^{-\pi k}$ with small rational crowd factors
\eqref{eq:quark-tolls-app}; the colorless leptons carry
dyadic--rational codes \eqref{eq:lepton-codes-app}; the CKM and PMNS
matrices are the misalignment holonomies between family projectors ---
the gates of the cascade --- with the Cabibbo angle carrying the
pentagonal phase; and the neutrino sector enters through the geometric
seesaw $m_\star = \pi v^2 / \EGUT$ \eqref{eq:seesaw-geom-app}, with no
free right-handed mass scale anywhere. Per-entry registers are graded
in Table~\ref{tab:codebook-provenance}; the cascade-depth assignments
are \textbf{Derived (cond.)} on the named cascade lemma. \emph{(Physics intuition:
 in the textbook the Yukawa sector is a wall of arbitrary
numbers --- the least explained corner of physics. The assembly
reframes the wall before the next chapters climb it: the numbers are
not couplings that were set, they are distances on a landscape ---
windings, tolls, and misalignment angles --- and the chapters that
follow will pace out each one.)}

\subsection{S-8 --- what the bulk does not send}
\label{app:lag-course-s8}

The textbook's construction rule --- write every renormalizable term the
symmetries allow --- generates three famous embarrassments, and the
course must be honest that removing them by hand is part of what
``learning the Standard Model'' has always meant. The export rule
replaces the listing rule, and each embarrassment becomes a theorem.

\paragraph{The $\theta$-term is not exported.} The listing rule admits
$\theta\,\tfrac{g_s^2}{32\pi^2} G\widetilde G$, and experiment then
demands $\bar\theta < 10^{-10}$ with no protecting symmetry --- a tuning
of one part in ten billion, accepted silently in every course. The
export generates no coefficient for the topological density (the
descent of the admissible bulk action contains none), and the
first-order quark determinant phases cancel, so $\bar\theta^{(1)} = 0$
with the constitutionally minimal residue
$\sin(2\pi/L)\,2^{-J} e^{-K} \simeq 4.7\times10^{-18}$
\eqref{eq:thetabar-residue-app} --- eight orders below the bound,
\textbf{Derived (cond.)} on lemma L9. The strong-CP chapter of this
course works the full reduction. \emph{(Physics intuition: the deepest
solution to a fine-tuning problem is to discover there was never a knob.
The neutron's electric dipole moment is silent not because someone
turned the dial to zero with superhuman care, but because the
instrument was shipped without that dial.)}

\paragraph{The proton-decay operators are not exported.} Classic
$\mathrm{SU}(5)$ died by the listing rule: its heavy bosons generate
$\Delta B = 1$ operators, Super-Kamiokande looked, and the proton does
not decay. Here every $\Delta B = \pm1, \pm2$ operator carries non-zero
\UoneB{} charge, and the exact global baryon number residual to the
Stueckelberg-massive \UoneB{} forbids them to all orders
(Theorem~\ref{thm:proton-stability-a9}, conditional on its stated
hypotheses). The force-law chapter of this course carries the full
treatment. \emph{(Physics intuition: the 1974 unification put quarks
and leptons in one multiplet and paid for the elegance with the proton's
life. The six-dimensional version keeps the multiplet but banishes the
dangerous mediators to the bulk, and gauges baryon number on the way
down --- the elegance survives, the proton does too.)}

\paragraph{The scalar-mass dial is not exportable.} Worked in S-6: no
gauge-invariant polynomial in a compact phase exists, so the $\mu^2$
term --- the single most tuned object in the textbook tradition --- is
absent as a matter of admissibility, not choice.

\paragraph{The honest overshoot.} Finally, the export is slightly
\emph{larger} than the textbook Standard Model: the boundary retains the
Stueckelberg-massive \UoneB{} and the \UoneQi{} structure, protected
remnants of the bulk gauge content with observable consequences (the
baryonic fifth-force window among the falsification channels). The
assembly records this as a feature: a derivation that exactly reproduced
the textbook operator list, no more and no less, would be suspicious;
this one overshoots by two protected structures and stakes falsifiable
claims on both. \emph{(Physics intuition: a tailor who delivers a suit
that fits perfectly plus two extra buttons you did not order has
probably actually measured you.)}

\subsection{The assembled equation, read line by line}
\label{app:lag-course-assembly}

Collecting Eqs.~\eqref{eq:gauge-kinetic-course},
\eqref{eq:fermion-kinetic-course}, the Wilson-line sector of S-6,
\eqref{eq:yukawa-course}, and the protected zero of S-8:
\begin{equation}
\begin{aligned}
\mathcal{L}_{\mathrm{SM}}
={}& \underbrace{-\tfrac14\,G^a_{\mu\nu}G^{a\,\mu\nu}
   -\tfrac14\,W^i_{\mu\nu}W^{i\,\mu\nu}
   -\tfrac14\,B_{\mu\nu}B^{\mu\nu}}_{\substack{\text{S-4: exported structure;}\\ \text{couplings = seam registrations}}}
\;+\; \underbrace{\sum_{\psi}\overline\psi\, i\gamma^\mu D_\mu \psi}_{\substack{\text{S-2, S-3, S-5: one bulk spinor,}\\ \text{45 chiral zero modes}}}
\\[6pt]
&+\; \underbrace{\left|D_\mu\Phi\right|^2 \;-\; V_{\mathrm{eff}}(\theta_H)}_{\substack{\text{S-6: Wilson line;}\\ \text{forced Fourier potential}}}
\;-\; \underbrace{\left(\overline Q_L Y_u\widetilde\Phi u_R
   + \overline Q_L Y_d\Phi d_R
   + \overline L_L Y_e\Phi e_R + \mathrm{h.c.}\right)}_{\substack{\text{S-7: forced form;}\\ \text{cascade content}}}
\\[6pt]
&+\; \underbrace{\;0\;\cdot\;\frac{g_s^2}{32\pi^2}\,G^a_{\mu\nu}\widetilde G^{a\,\mu\nu}}_{\substack{\text{S-8: the protected zero}}}
\end{aligned}
\label{eq:lsm-assembled-course}
\end{equation}
This is the equation from the first day of the reader's education,
symbol for symbol --- compare \citet{PeskinSchroeder1995} or
\citet{Schwartz2014} --- with one line no textbook writes: the
$\theta$-term displayed with its coefficient, zero, because here the
zero is a result and has earned its place. The reading protocol is the
point of the whole interlude. Read each brace and ask the textbook
question ``who chose this?'' --- and observe that the answer, five times
over, is \emph{no one}: the structures were exported (S-1 to S-6), the
coefficients were registered at the seam or graded in the codebook
(Table~\ref{tab:codebook-provenance}), and the absences were derived
(S-8). The main-text audit of Part~\ref{part:lagrangian} tabulates all
of this element by element (its six-silent-axioms and master provenance
tables); Problem G.11.1 asks the reader to reconstruct that mapping from
memory, which is the honest test of whether the interlude has done its
job. \emph{(Physics intuition: the deepest change this equation
undergoes in the present framework is grammatical --- every verb turns
passive. The group \emph{was chosen} becomes the group \emph{survived
the splitting}; the charges \emph{were assigned} becomes the charges
\emph{were computed}; the potential \emph{was written} becomes the
potential \emph{was generated}; the $\theta$-term \emph{was tuned away}
becomes the $\theta$-term \emph{was never sent}. Physics has spent
fifty years as the subject of these verbs; the export makes it the
object.)}

One boundary statement, repeated from the main text because a course
must not oversell: the framework of quantum field theory itself is
imported, not derived; the boundary Lorentz structure is constitutional
input; every \textbf{Derived (cond.)} above stands on its named lemma,
cataloged in the promotion-roadmap chapter
(Appendix~\ref{sec:appendix-promotion-roadmap}); and ``complete'' means
the complete term content at stated grades --- six \textbf{Derived},
fourteen \textbf{Derived (cond.)}, eight \textbf{Asserted} --- never
``every decimal derived.'' The assembly is exactly as strong as the
descent behind it, no stronger.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{G.2.1. Verify the ledger's double identity and
run the uniqueness scan over small candidates.}

\noindent\emph{Solution.}  Step 1: for $(M,N,L) = (1,3,5)$, compute both
closure readings of $K$: the lock product $K = LJ$ requires
$J = K/L$ to be an odd integer, and the sphere sum gives
$K = M^2+N^2+L^2 = 1+9+25 = 35$, whence $J = 35/5 = 7$: odd, integer,
and the ledger closes with $(1,3,5;7,35)$
\eqref{eq:ledger-double-course}.  Step 2: scan the neighboring
triples: $(1,2,3)$ gives $K = 14$, $J = 14/3$ --- not an integer;
$(1,3,4)$ gives $26/4 = 6.5$ --- fails; $(1,2,5)$ gives $30/5 = 6$ ---
integer but even, and the closure index must be odd; $(2,3,5)$ gives
$38/5$ --- fails; $(1,4,5)$ gives $42/5$ --- fails; $(3,4,5)$ gives
$50/5 = 10$ --- even again; $(1,3,7)$ gives $59/7$ --- fails.  Step 3:
within the scanned window only $(1,3,5)$ satisfies both relations with
an odd integer $J$ --- the full uniqueness statement, over all admissible
triples with the complete \BIsix{} conditions, is the theorem of
\AThree{}.  Physics intuition: this is what ``the integers are forced''
feels like in the fingers --- every neighboring tuple dies in one line
of arithmetic, not in a fit; the ledger sits at the intersection of two
integer curves that barely deign to meet at all.

\medskip\noindent\textbf{G.3.1. Run the two-derivative operator scan of
S-1a: show the shelf holds exactly the terms of the bulk action, and
name what removes the one Lovelock intruder.}

\noindent\emph{Solution.}  Step 1: the gravity shelf. At zero
derivatives the only diffeomorphism-invariant density is $\sqrt{-G}$
itself --- the cosmological term. At one derivative there is nothing:
the first derivatives of the metric vanish at any point in normal
coordinates, so no scalar survives. At two derivatives every invariant
is linear in the Riemann tensor, and its only full contraction is the
Ricci scalar $R_6$ --- the Einstein--Hilbert term, alone on the shelf.
Step 2: the gauge and spinor shelves. For a connection, the lowest
gauge-covariant object is $\mathcal F_{MN}$ and the lowest invariant
scalar is $\mathcal F_{MN}\mathcal F^{MN}$; the parity-odd six-form of
$\mathcal F\wedge\mathcal F\wedge\mathcal F$ type exists in six
dimensions but is a topological density --- it belongs to the priced
sector of brace~4, not the free shelf. For the spinor,
$i\,\overline\Psi\,\Gamma^M D_M\Psi$ exhausts the first-order shelf,
and no lower-order invariant exists. Step 3: the intruder. In $D=6$
the Gauss--Bonnet density
$\mathcal G = R^2 - 4R_{MN}R^{MN} + R_{MNPQ}R^{MNPQ}$ has second-order
field equations --- the Lovelock property --- so stability cannot
exclude it; count its derivatives (four) and apply \MSMF{} at lowest
order: it is structure the two-derivative theory does not need. As a
check on the caveat's dimension-dependence, recall the
four-dimensional contrast: there the same density integrates to the
Euler characteristic (Chern--Gauss--Bonnet), a boundary term with no
field equations at all --- the intruder only becomes dynamical when
the dimension gives it room, which is exactly why the honesty caveat
must be stated in six dimensions and is invisible in four.  Physics
intuition: the pantry lock of S-1 secured the ingredients; this scan
secures the recipe --- at lowest order the shelves hold the
Einstein--Hilbert term, one $\mathcal F^2$ per gauge factor, one Dirac
term per spinor, and nothing else, with everything fancier either
priced by closure or diluted by the throat.

\medskip\noindent\textbf{G.3.2. Spend the action once in miniature:
integrate out one heavy mode at tree level and read off the Wilson
coefficient it leaves behind.}

\noindent\emph{Solution.}  Step 1: the toy. Keep one light field
$\phi_0$ and one heavy field $\phi_h$ with
$\mathcal L = \tfrac12(\partial\phi_0)^2 - \tfrac12 M^2\phi_h^2
- \lambda\,\phi_0^2\,\phi_h$ --- the skeleton of a KK zero mode
coupled to a single massive tower floor. Step 2: complete the square:
$-\tfrac12 M^2\phi_h^2 - \lambda\phi_0^2\phi_h
= -\tfrac12 M^2\big(\phi_h + \lambda\phi_0^2/M^2\big)^2
+ \lambda^2\phi_0^4/(2M^2)$; the shifted field integrates away,
leaving $\mathcal L_{\mathrm{eff}} = \tfrac12(\partial\phi_0)^2
+ \dfrac{\lambda^2}{2M^2}\,\phi_0^4$ --- a quartic the light theory
never wrote down, with Wilson coefficient $c_4 = \lambda^2/(2M^2)$
generated, not postulated. Step 3: the tower. Restoring the heavy
propagator's momentum dependence,
$1/(p^2+M^2) = M^{-2}\,(1 - p^2/M^2 + \cdots)$ stacks derivative
operators each suppressed by a further $(E/M)^2$ --- precisely the
pattern in which the full KK tower of T-GDOS-9.1 populates the
boundary theory, with $M$ the tower masses clustered at $\Mlock$ and
$\lambda$ the radial overlap integrals of T-GDOS-9.2
\citep{ArisakaA5GDOS}; the same miniature, with the same moral, is
worked as Problem V.2.1 in the course chapter of \AFive{} that owns
the theorem.  Physics intuition: every ``arbitrary'' coefficient in
the boundary Lagrangian is a $\lambda^2/(2M^2)$ whose $M$ and
$\lambda$ the textbook never drew; the strangeness of the assembled
equation is not that its numbers are fixed, but that they were always
someone's overlap integrals.

\medskip\noindent\textbf{G.4.1. Compute every hypercharge of one family
from the embedding, and prove $|q_e| = |q_p|$ exactly.}

\noindent\emph{Solution.}  Step 1: take
$Y = \operatorname{diag}(-\tfrac13,-\tfrac13,-\tfrac13,+\tfrac12,+\tfrac12)$
on the $\mathbf 5$ \eqref{eq:five-y-course}; conjugating for the
$\overline{\mathbf 5}$ gives $Y(d^c) = +\tfrac13$ and $Y(L) = -\tfrac12$;
pair sums on the $\mathbf{10}$ give $Y(u^c) = -\tfrac23$,
$Y(Q) = +\tfrac16$, $Y(e^c) = +1$ \eqref{eq:tenfold-course}.  Step 2:
apply $Q = T_3 + Y$ to each state and recover the full charge table,
$Q(u) = \tfrac23$, $Q(d) = -\tfrac13$, $Q(\nu) = 0$, $Q(e) = -1$.
Step 3: electric charge is a generator of the surviving algebra, so it
is traceless on the $\overline{\mathbf 5}$:
$3\,Q(d^c) + Q(\nu) + Q(e) = 0$, i.e.\ $Q(d^c) = \tfrac13$ exactly, with
no freedom; then $Q(p) = Q(uud) = \tfrac23+\tfrac23-\tfrac13 = +1 =
-Q(e)$, and hydrogen is exactly neutral.  Physics intuition: the most
precisely verified coincidence in nature --- atomic neutrality at one
part in $10^{21}$ --- is here a trace identity: quarks and leptons
share a multiplet, so their charges must balance within it, the way the
entries of any traceless matrix must sum against each other.

\medskip\noindent\textbf{G.4.2. Verify all four anomaly-cancellation
sums for one family, using only the computed hypercharges.}

\noindent\emph{Solution.}  Step 1: $[\SU(3)]^2\,\U(1)_Y$ sums $Y$ over
color representations: $2\cdot\tfrac16 + (-\tfrac23) + \tfrac13 =
\tfrac13 - \tfrac23 + \tfrac13 = 0$.  Step 2: $[\SU(2)]^2\,\U(1)_Y$
sums $Y$ over doublets: $3\cdot\tfrac16 + (-\tfrac12) = \tfrac12 -
\tfrac12 = 0$ --- the quark doublet's three colors against the one
lepton doublet.  Step 3: the cubic and gravitational sums over all
fifteen Weyl fields:
$[\U(1)_Y]^3 = 6(\tfrac16)^3 + 3(-\tfrac23)^3 + 3(\tfrac13)^3 +
2(-\tfrac12)^3 + 1^3 = \tfrac{1}{36} - \tfrac{32}{36} + \tfrac{4}{36} -
\tfrac{9}{36} + \tfrac{36}{36} = 0$, and
$\mathrm{grav}^2\,\U(1)_Y = 6\cdot\tfrac16 - 3\cdot\tfrac23 +
3\cdot\tfrac13 - 2\cdot\tfrac12 + 1 = 1 - 2 + 1 - 1 + 1 = 0$; as a
bonus, the family contains four $\SU(2)$ doublets (three colors plus
one lepton), an even number, so the global Witten anomaly vanishes too.
Physics intuition: in the bottom-up construction these four zeros are
miracles that constrain model-builders' dreams; in the export they are
the shadow of one consistent bulk multiplet --- the same sums, computed
upstairs once, descend as identities rather than as checks that might
have failed.

\medskip\noindent\textbf{G.5.1. Compute $\operatorname{Tr} Y^2$,
$\operatorname{Tr} T_3^2$, and $\operatorname{Tr} Q^2$ over one family,
and extract both the $5/3$ normalization and $\sin^2\theta_W = 3/8$.}

\noindent\emph{Solution.}  Step 1:
$\operatorname{Tr} Y^2 = \tfrac{10}{3}$ by the term-by-term sum of
Eq.~\eqref{eq:trace-ratio-course}, and $\operatorname{Tr} T_3^2 =
8\cdot\tfrac14 = 2$ from the eight doublet states.  Step 2: the ratio
gives the unified normalization $\operatorname{Tr} Y^2 /
\operatorname{Tr} T_3^2 = \tfrac53$, i.e.\ $g_1^2 = \tfrac53 g'^2$
\eqref{eq:su5norm-app}.  Step 3: with $Q = T_3 + Y$ and the cross-term
$\operatorname{Tr}(T_3 Y) = \tfrac16 \cdot 3 \cdot (\tfrac12-\tfrac12)
+ \dots = 0$ vanishing doublet by doublet,
$\operatorname{Tr} Q^2 = \operatorname{Tr} T_3^2 + \operatorname{Tr}
Y^2 = 2 + \tfrac{10}{3} = \tfrac{16}{3}$, so $\sin^2\theta_W =
\operatorname{Tr} T_3^2 / \operatorname{Tr} Q^2 = \tfrac{2}{16/3} =
\tfrac38$ at the seam \eqref{eq:sin2-gut-app} --- the same $3/8$ the
gauge chapter's problem D.1.2 met from the coupling side, now derived
from pure representation traces.  Physics intuition: $3/8$ is what the
weak angle looks like when the three forces are one undressed force ---
a ratio of counting weights, no dynamics anywhere --- and the descent
from $3/8$ at the seam to the rational $25/108$ at $M_Z$ is the desert,
crossed twice by this course from opposite ends.

\medskip\noindent\textbf{G.8.1. Prove that no gauge-invariant
polynomial in the Wilson-line phase exists, and state what this kills.}

\noindent\emph{Solution.}  Step 1: under a large gauge transformation
--- one winding of the compact circle --- the holonomy phase shifts by
a full period, $\theta_H \to \theta_H + 2\pi$
\eqref{eq:largegauge-course}, while all physics is unchanged; any
admissible effective operator $f(\theta_H)$ must therefore satisfy
$f(\theta_H + 2\pi) = f(\theta_H)$.  Step 2: a non-constant polynomial
cannot be periodic --- $(\theta_H + 2\pi)^n \ne \theta_H^n$ --- so
every polynomial term, and in particular the mass operator
$\mu^2\,\theta_H^2$, is forbidden as an operator; the admissible
potential is a Fourier series, $V_{\mathrm{eff}} \propto \cos\theta_H +
\dots$ \eqref{eq:Veff-cos-app}, whose one-loop coefficients are finite
and set by the compactification scale.  Step 3: consequence --- the
quadratic divergence of the textbook hierarchy problem has no invariant
operator to feed: there is no primitive scalar mass to drag to the
cutoff, so the fine-tuning question dissolves at the level of the
operator list, before any cancellation mechanism is needed.  Physics intuition:
 this is the entire hierarchy debate in three lines --- you
can renormalize a length, but you cannot quadratically renormalize an
angle; periodicity is the symmetry that scalar-mass counterterms cannot
break, and the Hosotani mechanism is that protection made dynamical.

\medskip\noindent\textbf{G.11.1. From memory: name the zeroth and the six silent
axioms of the textbook Lagrangian, and state which stage of the export
chain replaces each.}

\noindent\emph{Solution.}  Step 1: the axioms --- (S0) that a
four-dimensional Lagrangian, with its field content and derivative
order, is the starting point at all; (S1) the gauge group;
(S2) the fifteen representations and their hypercharges; (S3) three
families; (S4) one scalar doublet with a chosen potential and a tuned
$\mu^2 < 0$; (S5) the operator list itself, ``all renormalizable
gauge-invariant terms''; (S6) the twenty-eight measured coefficients.
Step 2: the replacements --- S0 by
the earned bulk action of S-1a (form derived at lowest derivative
order; content on the Minimal-Matter Lemma); S1 by the splitting of S-3
\eqref{eq:splitting-course}; S2 by the zero-mode computation of S-2
(one traceless matrix and its pair sums); S3 by the \PaperPTwo{} pincer
(conditional); S4 by the Wilson line and its forced Fourier potential
(S-6); S5 by the export principle with the absences derived (S-8); S6
by the codebook of the derivation Parts, graded $6/14/8$.  Step 3: the
one-sentence audit --- every structural choice is replaced by a theorem
or a graded conditional, the absences included, and the register of
each replacement is stated next to it.  Physics intuition: if the
reader can reproduce this mapping without looking, the interlude has
done its work: the first page of the textbook has become the last page
of a derivation, and the reader knows, item by item, exactly how strong
each link in that inversion is.

\GGtakeaway{\emph{Standard result:} the Standard-Model Lagrangian is
postulated whole --- group, representations, families, scalar sector,
operator list, and twenty-eight coefficients all chosen --- and fifty
years of data confirm the choices without explaining any of them.
\emph{What it assumed:} everything on that list; the zeroth and the six
silent axioms.
\emph{The geometric reading:} the Lagrangian is the terminal compression
of a six-dimensional descent --- its bulk action earned brace by brace
at stated registers (S-1a), structures exported, coefficients
registered at the seam, absences (the $\theta$-term, the proton-decay
operators, the scalar-mass dial) derived as theorems --- assembled here
term by term with the register of every claim stated, and as strong as
the descent behind it, no stronger.}

\section{Fermion masses and the Yukawa hierarchy}
\label{app:ch-fermion}

\GGteachbox{The twelve fermion masses --- the most
arbitrary-looking block of the Standard Model, spanning twelve
orders of magnitude from the top quark to the lightest neutrino:
the textbook Yukawa mechanism; the staggering freedom and
unexplained hierarchy it leaves behind; and the geometric
cascade in which the hierarchy becomes depth counting on the
bulk profile, with the per-fermion depths carried at the
cascade-depth lemma's stated grade --- structure forced, last
digits riding on the named lemma.}
\GGreadalong{\citet{Griffiths2008} for the phenomenological
first pass; \citet{Langacker2017} for the flavor sector at
depth.}

This is the third sector appendix of the course. Its subject is the origin of the
twelve fermion masses --- the most arbitrary-looking block of the Standard Model,
spanning twelve orders of magnitude from the top quark to the lightest neutrino.
We follow the three-step plan: the textbook Yukawa mechanism; the staggering
freedom and unexplained hierarchy it leaves behind; and the geometric reading in
which the masses become the boundary spectrum of one family codebook, ordered by
exponential cascade tolls. This is also the appendix where the course is most
candid about what is \emph{not} yet a theorem: only the top Yukawa is fully
derived, and the per-fermion cascade depths remain a named open lemma. A standard treatment of the fermion families and the Yukawa sector is given in \citet{Griffiths2008}.

\subsection{Step 1 --- the Yukawa sector as a graduate student meets it}
\label{app:fermion-step1}

\paragraph{How a fermion gets its mass.}
A Dirac mass term \(m\bar\psi_L\psi_R\) joins a left-handed and a right-handed
field. But in the Standard Model the left-handed fermions are \(\SU(2)_L\)
doublets and the right-handed ones are singlets, so a bare mass term is forbidden
by gauge invariance. Mass can only come through the Higgs, in a gauge-invariant
\emph{Yukawa} coupling,
\begin{equation}
\mathcal L_{\mathrm{Yuk}}
=-\,y_f\,\bar{L}\,\Phi\,f_R+\mathrm{h.c.},
\label{eq:yukawa-app}
\end{equation}
which, after the Higgs acquires its vacuum value, becomes a mass
\begin{equation}
m_f=\frac{y_f\,v}{\sqrt2}.
\label{eq:mf-yukawa-app}
\end{equation}
\emph{(Physics intuition: a fermion has no mass of its own; it borrows one from
the Higgs condensate. The Yukawa coupling \(y_f\) is how tightly that fermion
grips the condensate as it moves through it. A heavy particle grips hard; a light
one barely touches.)}

\paragraph{The hierarchy.}
There is one Yukawa per charged fermion --- nine numbers --- plus the neutrino
sector. With \(v/\sqrt2\simeq 174\ \mathrm{GeV}\), the measured couplings range
from \(y_t\simeq 1\) for the top quark down to \(y_e\simeq 3\times10^{-6}\) for
the electron, and far smaller still for the neutrinos. The fermion masses
therefore span
\begin{equation}
\underbrace{m_t\simeq 173\ \mathrm{GeV}}_{\text{top}}
\quad\longrightarrow\quad
\underbrace{m_\nu\lesssim 0.1\ \mathrm{eV}}_{\text{neutrino}},
\label{eq:hierarchy-span-app}
\end{equation}
a spread of more than twelve orders of magnitude. In the Standard Model these are
simply tabulated.

\paragraph{Neutrinos: Dirac, Majorana, and the seesaw.}
Neutrinos are special: being electrically neutral, they may carry a Majorana mass
\(\tfrac12 M\,\nu_L^{\mathsf T}C\nu_L\) in addition to (or instead of) a Dirac
mass. If a heavy Majorana scale \(M\) sits above the electroweak scale, the
seesaw mechanism gives the light eigenvalue
\begin{equation}
m_\nu\sim\frac{(y_\nu v)^2}{M},
\label{eq:seesaw-app}
\end{equation}
naturally tiny because \(M\) is large.
\emph{(Physics intuition: the seesaw is a lever. A very heavy right-handed partner
on the long arm forces the observed neutrino onto the short arm, pushing its mass
far down. The heavier the partner, the lighter the neutrino.)}

\paragraph{Worked example (STANDARD TEXTBOOK PEDAGOGY).}
Invert \eqref{eq:mf-yukawa-app} for the top quark with \(m_t\simeq 173\
\mathrm{GeV}\):
\begin{equation}
y_t=\frac{\sqrt2\,m_t}{v}
=\frac{1.414\times 173}{246}\simeq 0.99 .
\label{eq:yt-extract-app}
\end{equation}
The top Yukawa is, to within a percent, exactly \emph{one} --- a striking fact the
Standard Model records as a coincidence. For the electron the same inversion gives
\(y_e\simeq 2.9\times10^{-6}\). The textbook framework reproduces every mass ---
\emph{given} the nine Yukawa couplings as inputs. Why they range over twelve
orders of magnitude, and why the top sits at unity, the Standard Model does not say.

\subsection{Step 2 --- what the Yukawa sector assumes but cannot explain}
\label{app:fermion-step2}

This is the least constrained sector of the Standard Model, and the honest list of
open questions is long. The nine charged-fermion Yukawas are pure inputs, and with
the neutrino masses and mixings they make up the single largest block of the
twenty-eight parameters. Three failures stand out.

First, \emph{the hierarchy itself}. Twelve orders of magnitude separate the top
from the lightest neutrino, and the Standard Model offers no organizing principle:
the couplings are not related by any symmetry, not ordered by any rule, not bounded
by any argument. That \(y_t\simeq 1\) while \(y_e\simeq 10^{-6}\) is, in the
bottom-up theory, twelve unrelated accidents.

Second, \emph{the family number}. The three families are exact copies
distinguished only by mass, and nothing selects three rather than one or
seventeen. The replication is imposed by observation, not derived.

Third, \emph{the lightness of neutrinos}. The seesaw \eqref{eq:seesaw-app}
explains small neutrino masses only by positing a heavy scale \(M\) by hand, and it
leaves the Dirac-versus-Majorana question, the absolute mass scale, and the
ordering undetermined.
\emph{(Physics intuition: the Yukawa sector is where the Standard Model most openly
admits defeat. Every other sector has at least a structural story --- gauge
invariance, the Higgs mechanism --- but flavor is a spreadsheet of measured
numbers. A theory that could turn that spreadsheet into a short rule would be saying
something genuinely new.)}

\subsection{Step 3 --- the geometric derivation: masses as a cascade on one family surface}
\label{app:fermion-step3}

In \GGTheory{} the Yukawa entries are not primitive. Each is an overlap coefficient --- the reading of effective Yukawa couplings as wavefunction overlaps in the bulk, established in \ATwo{} and \AFive{} (\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}) and worked out in full for the charged-lepton ladder in \PaperPThree{} \citep{ArisakaP3Electron}
--- the integral of two fermion wavefunctions against the Higgs profile --- on a
single geometric family stratum, the complex projective surface
\(\mathbb{C}P^2\) doubled by \(\SU(2)_L\). A mass table is then not twelve unrelated
insertions but the boundary spectrum of \emph{one} codebook, and the twelve-order
hierarchy is the signature of an exponential cascade across that surface. We give
the geometric reading first, then the explicit codebook arithmetic.

\subsubsection{The cascade tolls: why the hierarchy is exponential}
\label{app:fermion-cascade}

The family stratum is explored by the \GRIP{} cascade, and moving down it from one
family basin to the next is not free: each step across a \GSSB{} gate pays a
Bogomolnyi toll \(e^{-S}\) with action \(S=\pi k\), \(k\) the integer winding of the
gate. The quark Yukawas therefore descend as integer powers of the elementary toll
\(e^{-\pi}\),
\begin{equation}
y_t=1,\quad
y_b\simeq\tfrac38 e^{-\pi},\quad
y_c\simeq 3\,e^{-2\pi},\quad
y_s\simeq 6\,e^{-3\pi},\quad
y_d\simeq 6\,e^{-4\pi},\quad
y_u\simeq 3\,e^{-4\pi},
\label{eq:quark-tolls-app}
\end{equation}
in which the \emph{exponent} \(n=0,1,2,3,4\) sets the order of magnitude and the
small-integer prefactor (\(1,\tfrac38,3,6,6,3\)) is an overlap multiplicity.
\emph{(Physics intuition: picture the three families as three terraces down a
hillside, separated by walls. To stand on a lower terrace a fermion must tunnel
through a wall, and each wall costs the universal toll \(e^{-\pi}\approx 0.043\) ---
a factor of about twenty-three in mass. The twelve-order spread is not twelve random
numbers; it is four tolls of the same wall, \(e^{-\pi},e^{-2\pi},e^{-3\pi},
e^{-4\pi}\), with small integer crowds of states riding through each gate.)}

\subsubsection{The top Yukawa: the zero-toll anchor}
\label{app:fermion-top}

The top quark sits at the head of the cascade, at zero winding, and therefore pays
\emph{no} toll. Its Yukawa is the bare geometric unit,
\begin{equation}
\boxed{\,y_t=1\,,}
\label{eq:yt-one-app}
\end{equation}
which predicts \(m_t=v/\sqrt2\simeq 174\ \mathrm{GeV}\) at first order, matching the
measured \(\simeq 173\ \mathrm{GeV}\) after the quantum-defect transport factor
\(127/128\) is applied. This is a genuine parameter-free \emph{prediction}, and in
the honesty triage \(y_t=1\) is one of the six \textbf{Derived} entries.
\emph{(Physics intuition: the top is the only fermion that grips the Higgs with the
full, undiscounted strength of the geometry --- it stands on the top terrace and
tunnels through nothing. That the strongest possible Yukawa is exactly unity, and
that nature places one quark exactly there, is the cleanest single hint that the
Yukawa couplings are geometric overlaps rather than free dials.)}

\subsubsection{The charged-lepton ladder: a dyadic code}
\label{app:fermion-leptons}

The charged leptons live on the same surface but, carrying no color winding, are
read on a dyadic--rational lattice rather than through \(e^{-\pi}\) tolls,
\begin{equation}
y_\tau=\frac{21}{2048},\qquad
y_\mu=\frac{5}{8192},\qquad
y_e=\frac{3}{1024000},
\label{eq:lepton-codes-app}
\end{equation}
with denominators that are powers of two (\(2048=2^{11}\), \(8192=2^{13}\)) dressed
by small rational overlap factors. The lepton hierarchy is thus a binary-lattice
reading of the same family geometry that gives the quarks their tolls.

\subsubsection{The neutrinos: a geometric seesaw}
\label{app:fermion-neutrinos}

The neutrino sector realizes the seesaw geometrically: the high scale is not a free
\(M\) but the grand-unification seam \(\EGUT\), and the common neutrino scale is
\begin{equation}
m_\star=\pi\,\frac{v^2}{\EGUT}\simeq 8.36\ \mathrm{meV},
\label{eq:seesaw-geom-app}
\end{equation}
with first-order eigenvalues in the integer ratio
\((m_1,m_2,m_3)=(0,1,6)\,m_\star\). The neutrinos are light for the same reason the
electroweak scale is small: a ratio of fixed geometric scales, here \(v^2/\EGUT\),
rather than a tuned input.
\emph{(Physics intuition: the seesaw lever's long arm is the GUT seam itself. Once
the boundary law fixes \(\EGUT=\EPl e^{-2\pi}\), the neutrino scale
\(\pi v^2/\EGUT\) is fixed too --- the meV scale of neutrino mass is the geometric
shadow of the \(10^{16}\ \mathrm{GeV}\) seam, not a new parameter.)}

\subsubsection{Honesty: the cascade-depth lemma is the dominant open work}
\label{app:fermion-honesty}

It must be said plainly where this stands. The \emph{pattern} of
\eqref{eq:quark-tolls-app} --- integer tolls \(e^{-n\pi}\) with small-integer
prefactors --- is striking and reduces a twelve-order spread to a handful of
integers. But the \emph{assignment} of a definite winding \(n\) and prefactor to
each fermion is read from the codebook, not yet proved: it requires a per-fermion
equivariant-cohomology calculation on \(\mathbb{C}P^2\) doubled by \(\SU(2)_L\) ---
the \emph{angular-cascade-depth lemma}. Until that lemma is proved, the eight
charged-fermion Yukawas other than the top are recorded in the codebook
(\S\ref{sec:codebook-provenance}) as \textbf{Asserted}, and the cascade-depth lemma
is identified as the dominant genuinely-open task of the program, the natural
subject of a successor paper. The top alone, at zero toll, is \textbf{Derived}.
This is the most incomplete sector of the derivation, and the course states it
without softening: the geometry strongly suggests the cascade, the numbers fit the
cascade, but the cascade is not yet a theorem.

The explicit arithmetic of all twelve masses --- the first-order codes, the
quantum-defect transport factors, and the resulting masses --- is recorded next,
so that every number in the main text can be reconstructed by hand.

\subsection{Detailed arithmetic of the fermion mass codebook}
\label{app:fermion-arithmetic}

This appendix records the elementary arithmetic behind the 12-mass block
so that no numerical step is hidden.

\subsection{Common electroweak normalization}

For every Dirac fermion at first order,
\[
m_f^{(1)}=\frac{v^{(1)}}{\sqrt2}\,y_f^{(1)}.
\]
Using
\[
v^{(1)}=246.331354~\mathrm{GeV},
\]
one obtains the common normalization
\begin{equation}
\frac{v^{(1)}}{\sqrt2}=174.182570832~\mathrm{GeV}.
\label{eq:vroot2-appendix-main}
\end{equation}
This is the single number against which all first-order charged-fermion
codes are evaluated.

\subsection{Charged-lepton arithmetic}

The charged-lepton Yukawa codes are
\[
y_\tau^{(1)}=\frac{21}{2048}=0.01025390625,
\qquad
y_\mu^{(1)}=\frac{5}{8192}=0.0006103515625,
\]
\[
y_e^{(1)}=\frac{3}{1024000}=2.9296875\times 10^{-6}.
\]
Multiplying by \eqref{eq:vroot2-appendix-main} gives
\[
m_\tau^{(1)} = 174.182570832\times 0.01025390625
=1.7860517517~\mathrm{GeV},
\]
\[
m_\mu^{(1)} = 174.182570832\times 0.0006103515625
=0.1063126043~\mathrm{GeV},
\]
\[
m_e^{(1)} = 174.182570832\times 2.9296875\times10^{-6}
=5.10300500\times10^{-4}~\mathrm{GeV}.
\]

The QD transport factors are
\[
\mathcal R_\tau=\frac{637}{640}=0.9953125,
\qquad
\mathcal R_\mu=\frac{159}{160}=0.99375,
\qquad
\mathcal R_e=\frac{641}{640}=1.0015625.
\]
Hence
\[
m_\tau^{\mathrm{QD}}=\frac{637}{640}\,m_\tau^{(1)}=1.7776796341~\mathrm{GeV},
\]
\[
m_\mu^{\mathrm{QD}}=\frac{159}{160}\,m_\mu^{(1)}=0.1056481505~\mathrm{GeV},
\]
\[
m_e^{\mathrm{QD}}=\frac{641}{640}\,m_e^{(1)}=5.11097845\times10^{-4}~\mathrm{GeV}.
\]

\subsection{Quark arithmetic}

The first-order quark Yukawa codes are
\[
y_t^{(1)}=1,
\qquad
y_b^{(1)}=\frac{3}{8}\e^{-\pi}=0.01620587656,
\]
\[
y_c^{(1)}=3\e^{-2\pi}=0.00560238352,
\qquad
y_s^{(1)}=6\e^{-3\pi}=4.84264774\times10^{-4},
\]
\[
y_d^{(1)}=6\e^{-4\pi}=2.09238845\times10^{-5},
\qquad
y_u^{(1)}=3\e^{-4\pi}=1.04619422\times10^{-5}.
\]
Multiplying by \eqref{eq:vroot2-appendix-main} yields the first-order
masses quoted in the main text:
\[
m_t^{(1)}=174.182570832~\mathrm{GeV},
\]
\[
m_b^{(1)}=2.8226667671~\mathrm{GeV},
\qquad
m_c^{(1)}=0.9758279277~\mathrm{GeV},
\]
\[
m_s^{(1)}=8.4338696612\times10^{-2}~\mathrm{GeV},
\qquad
m_d^{(1)}=3.64460554\times10^{-3}~\mathrm{GeV},
\]
\[
m_u^{(1)}=1.82230277\times10^{-3}~\mathrm{GeV}.
\]

The QD transport factors are
\[
\mathcal R_t=\frac{127}{128},
\qquad
\mathcal R_b=\frac32,
\qquad
\mathcal R_c=\frac43,
\qquad
\mathcal R_s=\frac98,
\qquad
\mathcal R_u=\mathcal R_d=\frac54.
\]
Therefore
\[
m_t^{\mathrm{QD}}=\frac{127}{128}m_t^{(1)}=172.821770~\mathrm{GeV},
\]
\[
m_b^{\mathrm{QD}}=\frac32 m_b^{(1)}=4.23400015~\mathrm{GeV},
\qquad
m_c^{\mathrm{QD}}=\frac43 m_c^{(1)}=1.30110390~\mathrm{GeV},
\]
\[
m_s^{\mathrm{QD}}=\frac98 m_s^{(1)}=94.8810337~\mathrm{MeV},
\qquad
m_d^{\mathrm{QD}}=\frac54 m_d^{(1)}=4.55575693~\mathrm{MeV},
\]
\[
m_u^{\mathrm{QD}}=\frac54 m_u^{(1)}=2.27787846~\mathrm{MeV}.
\]

\subsection{Neutrino arithmetic}

The common first-order neutrino scale is
\[
m_\star^{(1)}=\pi\frac{[v^{(1)}]^2}{\EGUT}
=8.36113789~\mathrm{meV}.
\]
The first-order eigenvalues are
\[
(m_1,m_2,m_3)^{(1)}=(0,1,6)m_\star^{(1)}
=(0,8.36113789,50.1668273)~\mathrm{meV}.
\]
Their squared differences are
\[
\Delta m_{21}^{2\,(1)}=(8.36113789\times10^{-3})^2
=6.990864\times10^{-5}~\mathrm{eV}^2,
\]
\[
\Delta m_{31}^{2\,(1)}=(50.1668273\times10^{-3})^2
=2.516711\times10^{-3}~\mathrm{eV}^2.
\]

At QD order,
\[
m_\star=\pi\frac{v_{\mathrm{QD}}^2}{\EGUT}=8.35358030~\mathrm{meV},
\]
and
\[
m_1^{\mathrm{QD}}=\frac{1}{128}m_\star=0.0652623~\mathrm{meV},
\]
\[
m_2^{\mathrm{QD}}=\frac{36}{35}m_\star=8.59225345~\mathrm{meV},
\]
\[
m_3^{\mathrm{QD}}=6\frac{129}{128}m_\star=50.5130555~\mathrm{meV}.
\]
Therefore
\[
\Delta m_{21}^{2\,\mathrm{QD}}
=
(8.59225345^2-0.0652623^2)\times10^{-6}
=
7.382256\times10^{-5}~\mathrm{eV}^2,
\]
\[
\Delta m_{31}^{2\,\mathrm{QD}}
=
(50.5130555^2-0.0652623^2)\times10^{-6}
=
2.551565\times10^{-3}~\mathrm{eV}^2.
\]

\subsection{Theory-error arithmetic}

For every first-order comparison entry, the theory error is the omitted
QD correction:
\[
\sigma_{\theory}^{(1)} = |m^{\mathrm{QD}}-m^{(1)}|.
\]
For example,
\[
\sigma_{\theory}^{(1)}(m_t)
=
|172.821770-174.182571|
=
1.360801~\mathrm{GeV}.
\]
For every QD entry,
\[
\sigma_{\theory}^{(2)}=\frac18|m^{\mathrm{QD}}-m^{(1)}|.
\]
Thus
\[
\sigma_{\theory}^{(2)}(m_t)=0.170100~\mathrm{GeV},
\qquad
\sigma_{\theory}^{(2)}(m_\tau)=0.0010465~\mathrm{GeV},
\]
and similarly for the other particles.

The purpose of this appendix is simple: every number quoted in the main
text can be reconstructed by hand from a compact chain of formulas.


\subsection{Detailed arithmetic of the CKM codebook}
\label{app:ckm-arithmetic}

This appendix reconstructs the CKM block numerically from the quark code
of Section~\ref{sec:quark-masses}. Its purpose is the same as that of
Appendix~\ref{app:fermion-arithmetic}: every number that appears in the main
text should be reproducible by hand from a compact chain of formulas.

\subsection{Sectoral overlap angles from the quark codebook}

From the first-order quark Yukawas,
\[
y_t^{(1)}=1,\qquad
y_b^{(1)}=\frac{3}{8}\e^{-\pi},\qquad
y_c^{(1)}=3\e^{-2\pi},\qquad
y_s^{(1)}=6\e^{-3\pi},\qquad
y_u^{(1)}=3\e^{-4\pi},\qquad
y_d^{(1)}=6\e^{-4\pi},
\]
the four sectoral nearest-neighbor overlap ratios needed for CKM are
\begin{equation}
\sqrt{\frac{y_u^{(1)}}{y_c^{(1)}}}=\e^{-\pi}=0.0432139183,
\qquad
\sqrt{\frac{y_d^{(1)}}{y_s^{(1)}}}=\e^{-\pi/2}=0.207879576,
\label{eq:appendix-u12d12-main}
\end{equation}
\begin{equation}
\sqrt{\frac{y_c^{(1)}}{y_t^{(1)}}}=\sqrt3\,\e^{-\pi}=0.0748496212,
\qquad
\sqrt{\frac{y_s^{(1)}}{y_b^{(1)}}}=4\,\e^{-\pi}=0.172855673.
\label{eq:appendix-u23d23-main}
\end{equation}
These four numbers are the basic geometric ingredients of the CKM block.

\subsection{Cabibbo arithmetic and the golden-ratio coefficient}

The branch-complement phase is
\[
\varphi_{12}=\frac{3\pi}{5},
\qquad
\cos\varphi_{12}=\cos\frac{3\pi}{5}=-\frac{\sqrt5-1}{4}.
\]
Hence
\begin{align}
\sin^2\theta_{12}^{(1)}
&=
\left|\e^{-\pi/2}-e^{\,i3\pi/5}\e^{-\pi}\right|^2 \notag\\
&=
\e^{-\pi}+\e^{-2\pi}-2\e^{-3\pi/2}\cos\frac{3\pi}{5} \notag\\
&=
\e^{-\pi}+\e^{-2\pi}+\frac{\sqrt5-1}{2}\e^{-3\pi/2}.
\label{eq:appendix-cabibbo-exact-main}
\end{align}
Numerically,
\[
\e^{-\pi}=0.0432139183,\qquad
\e^{-2\pi}=0.00186744273,\qquad
\frac{\sqrt5-1}{2}\e^{-3\pi/2}=0.00555197873,
\]
so
\[
\sin^2\theta_{12}^{(1)}=0.0506333397,
\qquad
\sin\theta_{12}^{(1)}=0.225018533.
\]
The QD-protected value follows by multiplication with the combined
dyadic/throat factor \(1-1/(K2^J)=1-1/4480\):
\[
\sin\theta_{12}^{\mathrm{QD}}
=
\left(1-\frac{1}{4480}\right)0.225018533
=
0.224968306.
\]

\subsection{\texorpdfstring{\(2\!\leftrightarrow\!3\)}{2<->3} and \texorpdfstring{\(1\!\leftrightarrow\!3\)}{1<->3} arithmetic}

The first-order \(2\!\leftrightarrow\!3\) angle is
\begin{align}
\sin\theta_{23}^{(1)}
&=
\pi
\left(\sqrt3\,\e^{-\pi}\right)
\left(4\,\e^{-\pi}\right)\notag\\
&=
4\pi\sqrt3\,\e^{-2\pi}
=
0.0406459973.
\label{eq:appendix-s23-main}
\end{align}
The first-order \(1\!\leftrightarrow\!3\) angle is
\begin{equation}
\sin\theta_{13}^{(1)}
=
\sqrt3\,\e^{-2\pi}
=
0.00323450569.
\label{eq:appendix-s13-main}
\end{equation}

For the QD values we use the already-derived quark masses:
\[
m_t^{\mathrm{QD}}=172.821770~\mathrm{GeV},\quad
m_b^{\mathrm{QD}}=4.23400015~\mathrm{GeV},\quad
m_c^{\mathrm{QD}}=1.30110390~\mathrm{GeV},
\]
\[
m_s^{\mathrm{QD}}=94.8810337~\mathrm{MeV},\quad
m_u^{\mathrm{QD}}=2.27787846~\mathrm{MeV}.
\]
Then
\[
\sin\theta_{23}^{\mathrm{QD}}
=
\pi\sqrt{
\frac{(94.8810337\times10^{-3})(1.30110390)}
 {(4.23400015)(172.821770)}
}
=
0.0408057070,
\]
and
\[
\sin\theta_{13}^{\mathrm{QD}}
=
\sqrt{
\frac{2.27787846\times10^{-3}}{172.821770}
}
=
0.00363049673.
\]

\subsection{Phase arithmetic, Jarlskog invariant, and matrix reconstruction}

The first-order physical phase is
\[
\delta_{\rm CKM}^{(1)}
=
\frac{2\pi}{5}
=
1.25663706144,
\]
while the QD phase is
\[
\delta_{\rm CKM}^{\mathrm{QD}}
=
\frac{13\pi}{35}
=
1.16687727133.
\]
Using the QD angles,
\[
s_{12}=0.224968306,\qquad
s_{23}=0.0408057070,\qquad
s_{13}=0.00363049673,
\]
one obtains
\[
c_{12}=0.974359685,\qquad
c_{23}=0.999167112,\qquad
c_{13}=0.999993410.
\]
The Jarlskog invariant is therefore
\begin{align}
J^{\mathrm{QD}}
&=
c_{12}c_{23}c_{13}^2
s_{12}s_{23}s_{13}\sin\delta_{\rm CKM}^{\mathrm{QD}}\notag\\
&=
(0.974359685)(0.999167112)(0.999986820)\notag\\
&\qquad\times
(0.224968306)(0.0408057070)(0.00363049673)\sin(1.16687727133)\notag\\
&=
2.98351\times10^{-5}.
\label{eq:appendix-j-main}
\end{align}

Substituting the same angles and phase into the standard CKM
parameterization yields
\[
|V_{\rm CKM}^{\mathrm{QD}}|
=
\begin{pmatrix}
0.974360 & 0.224967 & 0.003630\\
0.224838 & 0.973541 & 0.040805\\
0.008442 & 0.040087 & 0.999161
\end{pmatrix}.
\]
From this one reads the unitarity-triangle apex
\[
\bar\rho+i\bar\eta
=
-\frac{V_{ud}V_{ub}^\ast}{V_{cd}V_{cb}^\ast}
=
0.151746 + 0.354449\,i,
\]
and therefore
\[
(\bar\rho,\bar\eta)^{\mathrm{QD}}=(0.151746,\ 0.354449).
\]

\subsection{Residual theory uncertainties used in the main tables}

For the primary CKM observables, the first-order theory uncertainties are
\[
\sigma_{\theory}^{(1)}(\sin\theta_{12})=5.02274\times10^{-5},\qquad
\sigma_{\theory}^{(1)}(\sin\theta_{23})=1.59710\times10^{-4},
\]
\[
\sigma_{\theory}^{(1)}(\sin\theta_{13})=3.95991\times10^{-4},\qquad
\sigma_{\theory}^{(1)}(\delta_{\rm CKM})=8.97598\times10^{-2},
\]
obtained by taking the difference between the QD and first-order values.
The residual QD uncertainties are simply one eighth of these values:
\[
\sigma_{\theory}^{(2)}(\sin\theta_{12})=6.27842\times10^{-6},\qquad
\sigma_{\theory}^{(2)}(\sin\theta_{23})=1.99637\times10^{-5},
\]
\[
\sigma_{\theory}^{(2)}(\sin\theta_{13})=4.94989\times10^{-5},\qquad
\sigma_{\theory}^{(2)}(\delta_{\rm CKM})=1.121997\times10^{-2}.
\]
The propagated uncertainties for \(|V_{ij}|\), \(A\), \(\bar\rho\),
\(\bar\eta\), and \(J\) in Table~\ref{tab:ckm-derived-main} are obtained
from these four residual QD uncertainties by standard quadrature of the
linearized sensitivities.

The practical lesson of this appendix is that the CKM arithmetic is no
less explicit than the mass arithmetic. Every number in the quark-mixing
block follows from a finite chain of formulas whose inputs are already
fixed by the preceding sections of the paper.

\subsection{Detailed arithmetic of the PMNS/Majorana codebook}
\label{app:pmns-arithmetic}

This appendix records the arithmetic behind the PMNS/Majorana block so that
every number quoted in Section~\ref{sec:pmns} can be checked line by line.
The purpose is the same as in the earlier fermion and CKM appendices: the
final numerical tables should be visibly traceable to short exact formulas.

\subsection{Solar and reactor arithmetic}

The solar angle begins from
\[
\sin^2\theta_{12}^{(1)}=\frac{L^2}{3N^3}=\frac{25}{81}=0.3086419753.
\]
The QD correction uses the factor
\[
1-\frac{1}{K2^{2N+1}}=1-\frac{1}{35\cdot2^7}=1-\frac{1}{4480}=0.999776786,
\]
so that
\[
\sin^2\theta_{12}^{\rm QD}=\frac{25}{81}\left(1-\frac{1}{4480}\right)=0.3085730820.
\]

The reactor angle begins from
\[
\sin^2\theta_{13}^{(1)}=\frac{N}{4K}=\frac{3}{140}=0.02142857143.
\]
Its QD multiplier is
\[
1+\frac{1}{2K}+\frac{1}{2^{2N+1}}=1+\frac{1}{70}+\frac{1}{128}=1.022098214,
\]
which gives
\[
\sin^2\theta_{13}^{\rm QD}=\frac{3}{140}\left(1+\frac{1}{70}+\frac{1}{128}\right)=0.0219021046.
\]

\subsection{Atmospheric and phase arithmetic}

The first-order atmospheric angle is
\[
\sin^2\theta_{23}^{(1)}=\frac12\left(1+\frac17\right)=\frac47=0.5714285714.
\]
The QD correction is
\[
\sin^2\theta_{23}^{\rm QD}=\frac12\left(1+\frac17-\frac{1}{35}\right)=\frac{39}{70}=0.5571428571.
\]

For the Dirac phase we first define
\[
\psi_\nu=\arcsin\!\left(\frac17\right)=0.1433475689\ \mathrm{rad}.
\]
Hence
\[
\delta_{\rm PMNS}^{(1)}=\pi+2\psi_\nu=3.4282877914\ \mathrm{rad}=196.426421^\circ.
\]
The QD phase slip is
\[
\frac{\pi}{2K(2N+1)}=\frac{\pi}{490}=0.00641141358\ \mathrm{rad}=0.367346939^\circ,
\]
so that
\[
\delta_{\rm PMNS}^{\rm QD}=3.4282877914-0.00641141358=3.4218763778\ \mathrm{rad}=196.059074^\circ.
\]

\subsection{Majorana phases, PMNS matrices, Jarlskog invariant, and neutrinoless-double-beta scale}

The Majorana phases are protected at
\[
(\alpha_{21},\alpha_{31})=(0,\pi).
\]
Using the first-order angles and phase yields
\[
|U_{\rm PMNS}^{(1)}|=
\begin{pmatrix}
0.822522 & 0.549571 & 0.146385\\
0.276669 & 0.603548 & 0.747786\\
0.496901 & 0.577669 & 0.647602
\end{pmatrix},
\qquad
J_{\rm PMNS}^{(1)}=-9.26009\times10^{-3}.
\]
Using the QD angles and phase yields
\[
|U_{\rm PMNS}^{\rm QD}|=
\begin{pmatrix}
0.822364 & 0.549377 & 0.147994\\
0.282542 & 0.612560 & 0.738201\\
0.493849 & 0.568292 & 0.658147
\end{pmatrix},
\qquad
J_{\rm PMNS}^{\rm QD}=-9.18733\times10^{-3}.
\]
For the effective neutrinoless-double-beta mass one obtains
\[
m_{\beta\beta}^{(1)}=1.72386\ \mathrm{meV},
\qquad
m_{\beta\beta}^{\rm QD}=1.79925\ \mathrm{meV}.
\]
Thus the PMNS/Majorana arithmetic is fully explicit, from the exact rational
codes to the final oscillation observables and the derived Majorana test
quantity.

\subsection{Residual theory uncertainties used in the PMNS tables}

The first-order omitted-QD shifts are
\[
\sigma_{\theory}^{(1)}(\sin^2\theta_{12})=6.88933\times10^{-5},
\qquad
\sigma_{\theory}^{(1)}(\sin^2\theta_{13})=4.73533\times10^{-4},
\]
\[
\sigma_{\theory}^{(1)}(\sin^2\theta_{23})=1.42857\times10^{-2},
\qquad
\sigma_{\theory}^{(1)}(\delta_{\rm PMNS})=0.367347^\circ.
\]
The residual QD uncertainties are one eighth of these values:
\[
\sigma_{\theory}^{(2)}(\sin^2\theta_{12})=8.61166\times10^{-6},
\qquad
\sigma_{\theory}^{(2)}(\sin^2\theta_{13})=5.91916\times10^{-5},
\]
\[
\sigma_{\theory}^{(2)}(\sin^2\theta_{23})=1.78571\times10^{-3},
\qquad
\sigma_{\theory}^{(2)}(\delta_{\rm PMNS})=0.0459184^\circ.
\]
For the protected Majorana phases,
\[
\sigma_{\theory}^{(1)}(\alpha_{21})=\sigma_{\theory}^{(1)}(\alpha_{31})=0.0401786^\circ,
\qquad
\sigma_{\theory}^{(2)}(\alpha_{21})=\sigma_{\theory}^{(2)}(\alpha_{31})=0.00502232^\circ.
\]
These are exactly the numbers used in the comparison tables of the main
text.


\subsection{The full 28-parameter census (consolidated in the Summary and audit Part)}
\label{app:full28}

The complete 28-parameter census --- every parameter's first-order
prediction, QD-completed value, observed benchmark, and fractional
deviation collected in one place --- now lives in a single canonical home:
the \emph{Summary and audit} Part (Part~\ref{part:XIII}). The first-order
codebook is Table~\ref{tab:first28-summary-v08}
(\S\ref{sec:first-order-summary-canonical}); its QD-completed refinement is
Table~\ref{tab:second28-summary-v08}
(\S\ref{sec:second-order-summary-canonical}); and the
\MSMF{}/\OGN{} per-parameter provenance audit there grades each entry and
licenses the $6/14/8$ honesty triage. The twenty-eight parameters are tabulated once and
graded once, in those two tables and nowhere else, so there is no second copy
that could drift away from them. The compression claim those tables make good on
is the same one stated throughout: one is not being asked to accept 28
unrelated empirical inputs, but to test whether 28 low-energy quantities can
be read as one coherent seam export of the \GG{}/\GDOS{} geometry.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{H.1.1. Extract the top Yukawa and
recognize the anchor.}

\noindent\emph{Solution.}  Step 1: from
\eqref{eq:mf-yukawa-app}, $y_f = \sqrt2\,m_f/v$.  Step 2: for
the top, $y_t = \sqrt2\times172.7/246.2 = 0.992$
\eqref{eq:yt-extract-app} --- unity to better than one
percent, the only order-one Yukawa in the theory.  Step 3: in
the Standard Model this is an unexplained accident; in the
cascade of Step 3 it is the anchor --- the top sits at zero
winding and pays no toll, $y_t = 1$ exactly at first order
\eqref{eq:yt-one-app}, with the percent-level residual priced
in the QD section.  Physics intuition: when a dial reads
$1.000$ to three digits, suspect that it is not a dial ---
the top's Yukawa is the course's loudest hint that the
flavor block is geometry, and every other fermion's smallness
is measured \emph{downhill} from this anchor.

\medskip\noindent\textbf{H.1.2. Span the ladder: from the top
to the electron to the neutrinos.}

\noindent\emph{Solution.}  Step 1: the electron:
$y_e = \sqrt2\times0.000511/246.2 = 2.94\times10^{-6}$.
Step 2: the ratio $y_t/y_e = 3.4\times10^{5}$; including the
neutrinos ($m_\nu \lesssim 0.1$~eV,
\eqref{eq:hierarchy-span-app}) the full span exceeds
$10^{12}$.  Step 3: count the decades: twelve orders of
magnitude across one interaction type with one vev --- the
Standard Model's least explained structure, and the chapter's
Step 2 lists it as pure input: twelve dials.  Physics intuition:
 no other sector of physics tolerates couplings of
one mechanism spread across twelve decades without a
generating principle; the cascade of the next problems is the
claim that the principle was always there --- the decades are
\emph{windings}, not choices.

\medskip\noindent\textbf{H.7.1. Read the quark ladder: extract
the winding number of each rung.}

\noindent\emph{Solution.}  Step 1: the cascade prices each
\GSSB{} gate at the Bogomolnyi toll $e^{-\pi k}$
\eqref{eq:quark-tolls-app}: $y_t = 1$ ($k=0$);
$y_b \simeq \tfrac38 e^{-\pi}$ ($k=1$); $y_c \simeq 3e^{-2\pi}$
($k=2$); $y_s \simeq 6e^{-3\pi}$ ($k=3$); $y_d \simeq
6e^{-4\pi}$, $y_u \simeq 3e^{-4\pi}$ ($k=4$).  Step 2: verify
the collapse: take logarithms --- $\ln(1/y)$ advances by
$\approx \pi$ per rung, with the small rational prefactors
($\tfrac38, 3, 6$) as the ``crowd factors'' of states sharing
each gate.  Step 3: twelve orders of magnitude in the quark
sector reduce to one number ($\pi$), five windings
($0$--$4$), and small integers --- the freedom count drops
from six continuous dials to a handful of integers.  Physics intuition:
 the ladder converts \emph{why is the up quark so
light?} into \emph{why does it sit four gates downhill?} ---
a question about topology, answerable, and answered upstream;
the residuals against measured masses are priced honestly in
the chapter's theory-error subsection, not hidden in the
prefactors.

\medskip\noindent\textbf{H.6.1. Evaluate the dyadic lepton
codes against the measured masses.}

\noindent\emph{Solution.}  Step 1: the colorless leptons read
on the dyadic--rational lattice \eqref{eq:lepton-codes-app}:
$y_\tau = 21/2048$, $y_\mu = 5/8192$, $y_e = 3/1024000$.
Step 2: convert with $m = yv/\sqrt2$:
$m_\tau = 1.785$~GeV (measured $1.777$),
$m_\mu = 106.3$~MeV (measured $105.7$),
$m_e = 0.5100$~MeV (measured $0.5110$).  Step 3: three masses
across four decades, each within $0.2$--$0.6\%$, from three
rational codes and no continuous parameter --- with the
residuals carried by the QD dressing at its stated grade.
Physics intuition: the leptons, carrying no color winding,
cannot pay $e^{-\pi}$ tolls --- so the same geometry that
gives quarks a transcendental ladder gives leptons a
\emph{rational} one: two number species, one mechanism
distinction (winding vs.\ none), exactly as F.10.1's census
demands.

\medskip\noindent\textbf{H.8.1. Evaluate the geometric seesaw
scale.}

\noindent\emph{Solution.}  Step 1: the standard seesaw
\eqref{eq:seesaw-app} leaves the heavy scale $M$ free; the
geometric version pins it to the unification seam:
$m_\star = \pi v^2/\EGUT$ \eqref{eq:seesaw-geom-app}.
Step 2: with $\EGUT = \MPl e^{-2\pi} = 2.28\times10^{16}$~GeV:
$m_\star = \pi\times246.2^2/2.28\times10^{16}\,\mathrm{GeV}
= 8.35$~meV.  Step 3: the common neutrino scale lands in
the meV decade --- the right neighborhood for the measured
splittings ($\sqrt{\Delta m^2_{21}} \approx 8.7$~meV) --- with
no free $M$ anywhere; the ordering prediction that follows is
the laboratory kill row.  Physics intuition: the cosmology
companion's generic seesaw (its problem I.1.1) ran the logic
backwards --- from $m_\nu$ up to a scale; here the scale is
supplied and the mass comes \emph{down} as a prediction: the
same lever, held at the other end, and the two chapters'
numbers agree because the seam is one.

\medskip\noindent\textbf{H.12.1. Evaluate the exact Cabibbo
expression --- golden ratio included.}

\noindent\emph{Solution.}  Step 1: the pentagonal holonomy
gives the $1$--$2$ overlap
\eqref{eq:appendix-cabibbo-exact-main}:
$\sin^2\theta_{12} = e^{-\pi} + e^{-2\pi} +
\tfrac{\sqrt5 - 1}{2}\,e^{-3\pi/2}$, the golden-ratio
coefficient arriving from $\cos(3\pi/5)$.  Step 2: evaluate:
$0.0432136 + 0.0018674 + 0.61803\times0.0089833 =
0.0506335$, so $\sin\theta_{12} = 0.225019$.  Step 3: the
measured Wolfenstein parameter is $\lambda = 0.22500(67)$:
agreement at the $10^{-4}$ level from an expression containing
$\pi$, $e$, and $\varphi$ and no dial.  Physics intuition:
the Cabibbo angle --- flavor physics' oldest measured number
--- is here two tolls and a pentagon; the golden ratio is not
numerology but the cosine of the pentagonal phase doing its
one job, and the QD dressing (the $1/4480$ of the next
subsection) is priced separately and openly.

\medskip\noindent\textbf{H.13.1. Evaluate the small mixings
and state their register.}

\noindent\emph{Solution.}  Step 1: the first-order $2$--$3$
overlap is $\sin\theta_{23}^{(1)} = 4\pi\sqrt3\,e^{-2\pi} =
0.040646$ \eqref{eq:appendix-s23-main}, against the measured
$0.0405 \pm 0.0006$: inside $1\sigma$ at first order.
Step 2: the $1$--$3$ overlap begins at
$\sin\theta_{13}^{(1)} = \sqrt3\,e^{-2\pi} = 0.0032345$
\eqref{eq:appendix-s13-main}, against the measured
$0.00365$: the first-order value carries an $11\%$ deficit.
Step 3: the register, exactly as the chapter states it: the
QD completion lifts $s_{13}$ to $0.0036305$ (the value used
in H.14.1) --- structure at first order, last digits at QD,
uncertainties in the residual-theory subsection.  Physics intuition:
 the smallest mixing is the most
dressing-sensitive --- it is a double toll, so every
correction matters relatively more; that its QD value lands
on the data is a nontrivial second-order success, and saying
so \emph{as} a second-order success is what the honesty
register is for.

\medskip\noindent\textbf{H.14.1. Assemble the Jarlskog
invariant and compare with the global fit.}

\noindent\emph{Solution.}  Step 1: with the QD-completed
values $s_{12} = 0.224968$, $s_{23} = 0.0408057$,
$s_{13} = 0.0036305$, $\delta = 1.16688$, form
$J = c_{12}c_{23}c_{13}^2\,s_{12}s_{23}s_{13}\sin\delta$
\eqref{eq:appendix-j-main}.  Step 2: evaluate:
$J = 2.9835\times10^{-5}$.  Step 3: the global-fit value is
$J = (3.08^{+0.15}_{-0.13})\times10^{-5}$: the assembled
invariant sits within $\sim\!0.7\sigma$ --- and, as the
P5 companion's problem E.2.1 proves, \emph{every}
CP-violating observable in the quark sector is proportional
to this one number: one pentagon, all of CP.  Physics intuition:
 $J$ is the area of the unitarity triangle --- the
cascade fixes its three sides from tolls and a pentagonal
phase, and the area comes out right; this is the flavor
sector's single sharpest test of the whole construction,
which is why the falsification catalog keys on it.

\GGtakeaway{\emph{Standard result:} twelve masses from twelve
Yukawa couplings --- a perfect fit with zero explanation.
\emph{What it assumed:} the couplings themselves, spanning
twelve orders of magnitude, plus the absence of any principle
relating them.  \emph{The geometric reading:} the hierarchy is
depth counting on the bulk profile --- exponentials of integers
--- with the per-fermion depths graded honestly on the
cascade-depth lemma, and the whole block tabulated once in the
canonical census.}

\section{Quark mixing: the CKM matrix and CP violation}
\label{app:ckm}

\GGteachbox{The CKM matrix from scratch --- three angles, one
CP-violating phase, the Wolfenstein hierarchy, and the Jarlskog
invariant --- the four unexplained parameters the textbook
leaves free; and the geometric derivation of the pentagonal
phase and the multiplicative suppression, with the forced
structure separated honestly from the last-digit values that
ride on the per-fermion cascade depths.}
\GGreadalong{\citet{Thomson2013} for the experiment-aware
introduction; \citet{KobayashiMaskawa1973} and
\citet{Cabibbo1963} for the originals.}

This is the fourth sector appendix of the course. Its subject is the
Cabibbo--Kobayashi--Maskawa matrix --- the three angles and one CP-violating phase
that relate the quark mass eigenstates to the weak eigenstates. We follow the
three-step plan: the textbook CKM matrix; the four unexplained parameters it leaves
free; and the geometric derivation, in which mixing becomes the misalignment of two
quark family projectors and the Cabibbo angle emerges from a pentagonal holonomy with
the golden ratio in plain view. The textbook treatment of quark mixing and the CKM matrix is given in \citet{Thomson2013}.

\subsection{Step 1 --- the CKM matrix as a graduate student meets it}
\label{app:ckm-step1}

\paragraph{Where mixing comes from.}
Quark masses and quark weak couplings are diagonal in different bases. Diagonalizing
the up-type and down-type mass matrices requires separate unitary rotations
\(U_{uL},U_{dL}\); the charged current then sees only their mismatch,
\begin{equation}
V_{\mathrm{CKM}}=U_{uL}^{\dagger}U_{dL},
\label{eq:vckm-app}
\end{equation}
a \(3\times3\) unitary matrix. Of its nine parameters, five are unphysical quark
phases, leaving \emph{four} physical ones: three mixing angles
\(\theta_{12},\theta_{23},\theta_{13}\) and one CP-violating phase \(\delta\).
\emph{(Physics intuition: if the up and down quarks agreed on which family was
``first,'' there would be no mixing. They do not quite agree, and the CKM matrix is
the small rotation between their two opinions.)}

\paragraph{The Wolfenstein hierarchy.}
Experimentally the matrix is strongly hierarchical, well captured by the Wolfenstein
expansion in \(\lambda=\sin\theta_{12}\simeq 0.225\),
\begin{equation}
V_{\mathrm{CKM}}\approx
\begin{pmatrix}
1-\lambda^2/2 & \lambda & A\lambda^3(\rho-i\eta)\\
-\lambda & 1-\lambda^2/2 & A\lambda^2\\
A\lambda^3(1-\rho-i\eta) & -A\lambda^2 & 1
\end{pmatrix},
\label{eq:wolfenstein-app}
\end{equation}
so the angles fall off as powers of \(\lambda\):
\(\sin\theta_{12}\!:\!\sin\theta_{23}\!:\!\sin\theta_{13}\approx\lambda:\lambda^2:\lambda^3\).
The single phase \(\eta\) is the origin of all CP violation in the quark sector --- the insight, due to Kobayashi and Maskawa \citep{KobayashiMaskawa1973}, that three generations admit exactly one irreducible CP-violating phase ---
quantified rephasing-invariantly by the Jarlskog determinant
\(J\simeq 3\times10^{-5}\).
\emph{(Physics intuition: \(\lambda\approx0.22\) is the master small number of the
quark world; nearest-neighbor mixing is order \(\lambda\), next-neighbor order
\(\lambda^2\), and the corner element order \(\lambda^3\). One number organizes the
whole matrix.)}

\paragraph{Worked example (STANDARD TEXTBOOK PEDAGOGY).}
The Cabibbo angle is read directly from strangeness-changing decays:
\(|V_{us}|=\sin\theta_{12}\simeq 0.2243\), whence \(\lambda\simeq 0.225\). The
next element gives \(A=|V_{cb}|/\lambda^2\simeq 0.041/0.0506\simeq 0.81\). With these
two numbers the entire matrix \eqref{eq:wolfenstein-app} is reconstructed --- given
\(\lambda\), \(A\), \(\rho\), \(\eta\) from data. Why \(\lambda\simeq 0.225\), and why
there is a CP phase at all, the Standard Model does not say.

\subsection{Step 2 --- what the CKM matrix assumes but cannot explain}
\label{app:ckm-step2}

The four physical CKM parameters are four more free inputs, and they raise three
questions the Standard Model cannot answer. First, \emph{the hierarchy}: why is
mixing small and ordered in powers of \(\lambda\), rather than either absent
(\(V=\mathbf 1\)) or maximal? Nothing forbids large quark mixing. Second, \emph{the
origin of CP violation}: the single phase \(\delta\) is inserted by hand; the
Standard Model guarantees only that three families \emph{permit} one irreducible
phase, not that it is nonzero or of any particular size. Third, \emph{why CP
violation is small but not tiny}: \(J\sim10^{-5}\) is a specific number with no
structural account.
\emph{(Physics intuition: the CKM matrix is the quark sector's second spreadsheet,
after the masses. It is more constrained than the masses --- it is nearly the identity
--- but the smallness of its off-diagonal entries, and the lone complex phase that
makes matter and antimatter behave differently, are simply measured, not understood.)}

\subsection{Step 3 --- the geometric derivation: mixing as projector misalignment}
\label{app:ckm-step3}

In \GGTheory{} the CKM matrix is not an independent input. By
\eqref{eq:vckm-app} it is the misalignment between the up-type and down-type quark
family projectors --- the off-diagonal transport that survives after the common family
rotations are quotiented out --- on the \emph{same} stabilized quark packet whose
diagonal masses were derived in the previous appendix. We give the geometric core and
then point to the explicit arithmetic.

\subsubsection{The Cabibbo angle from pentagonal holonomy}
\label{app:ckm-cabibbo}

The \(1\!\leftrightarrow\!2\) (Cabibbo) channel, named for the rotation Cabibbo introduced in 1963 \citep{Cabibbo1963}, is the least suppressed. The full projector-misalignment construction is given in the main text (\S\ref{sec:CKM}); here we follow the Cabibbo channel in detail. The up-type
and down-type nearest-neighbor overlap angles are read straight off the quark mass
codebook \eqref{eq:quark-tolls-app},
\begin{equation}
\theta_{12}^{(u)}=\sqrt{\frac{y_u}{y_c}}=e^{-\pi},
\qquad
\theta_{12}^{(d)}=\sqrt{\frac{y_d}{y_s}}=e^{-\pi/2}.
\label{eq:ud-angles-app}
\end{equation}
If these two amplitudes carried no relative phase, their difference would miss the
measured Cabibbo angle. The geometry supplies the phase. Because the only compact
holonomy direction in the flavor sector is the single circle \(S^1_\theta\), and the
Higgs doublet keeps the fraction \(2/L\) of the \(\SU(5)\) block, the complementary
branch phase is forced to the pentagonal value
\begin{equation}
\varphi_{12}=\pi\left(1-\frac{2}{L}\right)=\frac{3\pi}{5}.
\label{eq:phi12-app}
\end{equation}
The first-order Cabibbo angle is then the modulus of the difference of the two
nearest-neighbor amplitudes,
\begin{equation}
\boxed{\;
\sin\theta_{12}^{(1)}
=\left|\,e^{-\pi/2}-e^{\,i3\pi/5}\,e^{-\pi}\right|
=0.225019 ,\;}
\label{eq:s12-app}
\end{equation}
and using \(\cos(3\pi/5)=-(\sqrt5-1)/4\) it can be written in closed form as
\begin{equation}
\sin^2\theta_{12}^{(1)}
=e^{-\pi}+e^{-2\pi}+\frac{\sqrt5-1}{2}\,e^{-3\pi/2}.
\label{eq:s12sq-app}
\end{equation}
This is within \(0.01\%\) of the measured \(0.2250\) before any sharpening, and it is
the thirteenth benchmarked closed form of the codebook.
\emph{(Physics intuition: the up and down nearest-neighbor links lie on complementary
arcs of one pentagon --- the \(L=5\) circle. The Higgs keeps two-fifths of the block,
so the flavor-complement phase is the remaining three-fifths of a half-circle,
\(3\pi/5\). The golden ratio in \eqref{eq:s12sq-app} is not numerology; it is the
trigonometry of a pentagon, the literal signature of \(L=5\) showing up in the quark
mixing angle. The Cabibbo angle is the interference of two known tunneling
amplitudes across a pentagonal gate.)}

\subsubsection{The smaller angles and the CP phase}
\label{app:ckm-smaller}

The \(2\!\leftrightarrow\!3\) angle is much smaller because the left-handed doublet is
shared between the up and down sectors: a common \(2\!\leftrightarrow\!3\) rotation is
unobservable and cancels in \(U_{uL}^{\dagger}U_{dL}\), so what survives is only the
\emph{bilinear} mismatch of the two sectoral overlaps --- multiplicative, not
additive, hence the extra power of \(\lambda\). The deep \(1\!\leftrightarrow\!3\)
channel is suppressed further still.

The CP phase is the sharpest structural point. There is \emph{no second flavor torus}
on which an independent quark CP angle could float: the same single circle
\(S^1_\theta\) that supplies the pentagonal phase \(3\pi/5\) supplies the rephasing
step \(\pi/5\), and the physical CP phase is the topological-projection remainder left
after the unphysical quark phases are removed. CP violation is therefore not an
inserted parameter but a forced consequence of the pentagonal holonomy --- it is
nonzero precisely because \(3\pi/5\) is not a multiple of \(\pi\).
\emph{(Physics intuition: matter and antimatter differ in the quark sector for a
geometric reason --- the flavor pentagon is chiral. A phase of \(3\pi/5\) cannot be
rotated away to \(0\) or \(\pi\), so a complex residue always remains, and that residue
is CP violation. The same pentagon that sets the size of the Cabibbo angle sets the
existence of CP violation.)}

\subsubsection{Explicit arithmetic and honesty}
\label{app:ckm-honesty}

The full sectoral-overlap reconstruction of the \(2\!\leftrightarrow\!3\) and
\(1\!\leftrightarrow\!3\) angles, the Dirac phase, and the precision comparison
against the measured matrix are recorded in the CKM codebook arithmetic of
Appendix~\ref{app:ckm-arithmetic}, and the underlying minimal family-link algebra
\(V_{\mathrm{CKM}}=U_{uL}^{\dagger}U_{dL}\) is set out in full in the main text
(\S\ref{sec:CKM}). In the honesty triage the Cabibbo angle stands with the
best-derived quantities --- a parameter-free closed form matching data to \(0.01\%\) ---
while the smaller angles and the Dirac phase inherit their precision from the same
quark cascade depths \eqref{eq:quark-tolls-app}, and so share the status of the
cascade-depth lemma discussed in the previous appendix: the \emph{structure} (pentagonal
phase, multiplicative suppression, topological CP remainder) is forced, while the
last-digit values ride on the per-fermion depths that the angular-cascade-depth lemma
would close.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{I.1.1. Count the physical parameters of the
CKM matrix, and locate the one that violates CP.}

\noindent\emph{Solution.}  Step 1: a $3\times3$ unitary matrix carries
nine real parameters; three are the angles of an orthogonal rotation,
so six are phases.  Step 2: the six quark fields can each absorb a
phase, but one overall common phase does nothing to
\eqref{eq:vckm-app}, so $2N-1 = 5$ phases are removable.  Step 3: what
survives is $6 - 5 = 1$ physical phase beside the three angles --- four
parameters, exactly as the chapter states; the general-$N$ counting
(ledger-chapter problem M.11.1) shows this single phase is a
three-family novelty.  Physics intuition: CP violation lives in the
one corner of the matrix that no amount of quark rephasing can reach
--- a residue of complexity that exists only because the family space
is three-dimensional; two families would have none, and the 1973
Kobayashi--Maskawa argument ran this very counting backwards to
predict a third generation nobody had seen.

\medskip\noindent\textbf{I.1.2. Reconstruct the Wolfenstein
parameters, reading $A$ from the \emph{derived} angle.}

\noindent\emph{Solution.}  Step 1: the expansion
\eqref{eq:wolfenstein-app} organizes the matrix in powers of
$\lambda = \sin\theta_{12} \simeq 0.225$, with
$\sin\theta_{23} = A\lambda^2$.  Step 2: instead of the measured
$|V_{cb}|$, insert the codebook's own derived
$\sin\theta_{23} = 4\pi\sqrt3\,e^{-2\pi} = 0.040646$
\eqref{eq:appendix-s23-main}: $A = 0.040646/0.225^2 = 0.803$, against
the global-fit $A \simeq 0.81$ --- the derived second angle lands the
Wolfenstein slope within one percent.  Step 3: the corner elements
follow as $|V_{ub}| \sim A\lambda^3\sqrt{\rho^2+\eta^2} \approx
0.0036$ and $|V_{td}| \sim A\lambda^3\sqrt{(1-\rho)^2+\eta^2} \approx
0.0084$ with the global-fit apex --- the whole hierarchical matrix
hangs on numbers the codebook supplies.  Physics intuition: in the
textbook, $\lambda$ and $A$ are two more measured dials; here
$\lambda$ is a pentagonal interference (I.3.1) and $A$ rides the
double-toll $e^{-2\pi}$ --- the Wolfenstein hierarchy is the cascade's
toll schedule written in matrix form.

\medskip\noindent\textbf{I.3.1. Work the Cabibbo interference to six
digits, both routes.}

\noindent\emph{Solution.}  Step 1: the two nearest-neighbor
amplitudes \eqref{eq:ud-angles-app} are $e^{-\pi/2} = 0.2078796$
(down-type) and $e^{-\pi} = 0.0432139$ (up-type), and the pentagonal
phase \eqref{eq:phi12-app} carries $\cos(3\pi/5) = -0.3090170$.
Step 2: the modulus \eqref{eq:s12-app}:
$\sin^2\theta_{12} = e^{-\pi} + e^{-2\pi}
- 2e^{-3\pi/2}\cos(3\pi/5) = 0.0432139 + 0.0018674 + 0.0055520 =
0.0506333$, so $\sin\theta_{12} = 0.225018$.  Step 3: the closed form
\eqref{eq:s12sq-app} reproduces it exactly, because
$-2\cos(3\pi/5) = (\sqrt5-1)/2 = 0.618034$ --- the golden ratio
conjugate --- so the cross term is
$\tfrac{\sqrt5-1}{2}e^{-3\pi/2} = 0.618034 \times 0.0089833 =
0.0055520$; against the measured $0.22500(7)$ the closed form sits at
$+0.01\%$.  Physics intuition: the oldest measured number of flavor
physics is here two tunneling amplitudes interfering across a
pentagonal gate --- and the golden ratio is not decoration but the
literal trigonometry of $L = 5$: work problem I.3.2 to see what
happens to the agreement if the pentagon is traded for a square or a
hexagon.

\medskip\noindent\textbf{I.3.2. Falsify the wrong polygons: recompute
the Cabibbo angle with $L = 4$ and $L = 6$.}

\noindent\emph{Solution.}  Step 1: the branch phase
\eqref{eq:phi12-app} is $\varphi_{12} = \pi(1 - 2/L)$; for $L = 4$
(square) it is $\pi/2$, killing the cross term:
$\sin\theta_{12} = \sqrt{e^{-\pi} + e^{-2\pi}} = 0.21232$, off the
measurement by $-5.6\%$.  Step 2: for $L = 6$ (hexagon)
$\varphi_{12} = 2\pi/3$, $\cos = -\tfrac12$:
$\sin\theta_{12} = \sqrt{0.0432139 + 0.0018674 + 0.0089833} =
0.23252$, off by $+3.3\%$.  Step 3: only the pentagon, $L = 5$, lands
at $+0.01\%$ --- the agreement is not generic to the construction but
specific to the ledger's unified block, and a measured Cabibbo angle a
percent away would have falsified $L = 5$ outright.  Physics intuition:
 this is what separates a derivation from a fit --- there is
no dial to turn; the polygon is fixed upstream by the same integer
that sets the $\mathrm{SU}(5)$ spine and the $2/5$ doublet fraction,
and the Cabibbo angle is its trigonometric fingerprint at the fourth
decimal place.

\medskip\noindent\textbf{I.3.3. Explain why $\theta_{23}$ is
quadratically small and why CP violation is compulsory.}

\noindent\emph{Solution.}  Step 1: the left-handed doublet is shared
between the up and down sectors, so a \emph{common}
$2\!\leftrightarrow\!3$ family rotation cancels in
$V = U_{uL}^\dagger U_{dL}$ \eqref{eq:vckm-app}; only the bilinear
mismatch of the two sectoral overlaps survives, hence
$\sin\theta_{23} \propto e^{-2\pi}$ --- one toll from each sector,
multiplicative, not additive --- and the deep $1\!\leftrightarrow\!3$
channel is suppressed further still.  Step 2: the CP phase cannot be
dialed away: the flavor sector owns exactly one compact circle, its
rephasing step is $\pi/5$, and the branch phase $3\pi/5$ is not a
multiple of $\pi$, so a complex remainder survives every allowed
rephasing.  Step 3: consequence --- $J \ne 0$ is \emph{forced}, and its
size is assembled from the same tolls in the fermion chapter's problem
H.14.1; the companion paper's problem E.2.1 (\PaperPFive{}) then shows
every CP-violating quark observable is proportional to this one $J$.
Physics intuition: matter differs from antimatter, on this reading,
because a pentagon is chiral --- $3\pi/5$ can face neither $0$ nor
$\pi$ --- so the universe's CP violation is not an optional decoration
but a topological remainder; and the hierarchy $\lambda : \lambda^2 :
\lambda^3$ is simply one, then two, then three tolls paid.




\medskip\noindent\textbf{I.2.1. Show that the CKM phase convention is
unphysical, and find what is not.}

\noindent\emph{Solution.}  Step 1: rephasing the six quark fields,
$q \to e^{i\phi_q}q$, changes individual $V_{ij}$ by phases while
leaving all observables fixed.  Step 2: therefore no single matrix
element's phase is physical; what is physical is any rephasing-invariant
combination, the simplest being the modulus $|V_{ij}|$ and the quartet
$V_{ud}V_{cs}V^*_{us}V^*_{cd}$, whose imaginary part is the Jarlskog
invariant.  Step 3: hence a statement like ``$V_{ub}$ has phase
$-\gamma$'' is convention-dependent, while ``$J = 3\times10^{-5}$'' is
not.  Physics intuition: whenever a quantity in a gauge theory can be
changed by a field redefinition, it is bookkeeping.  The discipline of
asking ``is this rephasing-invariant?'' before quoting a phase is the
same discipline that makes the Majorana counting of problem K.1.4
tractable, and it is why the Jarlskog invariant, not $\delta$, is the
right thing to derive.

\medskip\noindent\textbf{I.2.2. Estimate how much of the CKM matrix is
actually independent data.}

\noindent\emph{Solution.}  Step 1: the matrix has nine complex entries,
eighteen real numbers, but unitarity imposes nine real conditions and
rephasing removes five phases: four physical parameters remain.  Step 2:
yet the Particle Data Group quotes ten independent $|V_{ij}|$
measurements plus several angle and side determinations of the
unitarity triangle --- far more measurements than parameters.  Step 3:
the excess is the content of the unitarity tests, and their agreement
at the percent level is the strongest evidence that there is no fourth
family and no significant new flavor physics below the multi-TeV
scale.  Physics intuition:
 an over-determined system is where a theory earns its
keep.  Four numbers predicting fifteen measurements is a much stronger
statement than four numbers fitting four, and this is the right frame
for judging the codebook of this paper too --- count the outputs against
the inputs, not the successes against the failures.

\medskip\noindent\textbf{I.2.3. Ask why the quark and lepton sectors
must be compared, not treated separately.}

\noindent\emph{Solution.}  Step 1: both mixing matrices arise the same
way --- as the misalignment $U_L^\dagger U'_L$ of two family bases
\eqref{eq:vckm-app}.  Step 2: yet one is nearly the identity and the
other nearly democratic, and no Standard-Model input distinguishes the
two cases: the gauge structure, the number of families and the
mechanism are identical.  Step 3: therefore the contrast is a
\emph{single} datum about family structure, not two independent ones,
and any framework that explains one without the other has explained
half a fact.  Physics intuition: this is the most important thing the
quark chapter has to say to the lepton chapters, and it is the reason
they sit consecutively in this course.  ``CKM small, PMNS large'' should
be read as one sentence with a contrast in it, and a theory should be
judged on whether it produces the contrast from one construction ---
which is exactly the claim App.~\ref{app:pmns-angles} makes and the
claim a reader should test hardest.


\GGtakeaway{\emph{Standard result:} the CKM matrix --- three
angles, one phase --- fits every flavor observable.  \emph{What
it assumed:} all four numbers, and the origin of CP violation
itself.  \emph{The geometric reading:} the pentagonal phase and
the multiplicative suppression are forced structure; the
last-digit values await the same cascade-depth lemma as the
masses --- one lemma, two sectors, stated in one register.}

\section{Neutrino masses and PMNS mixing}
\label{app:pmns}

\GGteachbox{The lepton sector that inverts the quark story ---
tiny masses, large mixing angles. Step~1 develops the oscillation
formalism from scratch (including the matter effect, the
three-flavor \CP{}-odd term, and the three independent absolute-mass
probes) and derives the seesaw in all three of its types. Step~2 lists
ten quantities the Standard Model must take from experiment. Step~3
gives the geometric reading, in which one reused scale sets the meV
floor, a dyadic-against-democratic projector misalignment forces the
large angles as ratios of small integers, and the ordering becomes a
near-term kill row --- with the one place the ledger visibly strains,
the solar splitting, identified and quantified rather than averaged
into a ratio.}
\GGreadalong{\citet{GiuntiKim2007} Ch.~7--8 and \citet{Zuber2020} for
the formalism and the experiments; \citet{Langacker2017} Ch.~8 for the
seesaw; \citet{Fukuda1998} and \citet{Ahmad2002} for the discovery
papers; \citet{Thomson2013} Ch.~13 for the cleanest short derivation
of the oscillation probability.}

This is the fifth sector appendix of the course, and the first of three
consecutive chapters on the lepton sector --- this one on masses and
mixing, the next (App.~\ref{app:majorana}) on the Majorana question and
the \CP{} phases, the third (App.~\ref{app:leptogenesis}) on what the
same structure does to the early universe. Its subject inverts the quark
story of the previous two appendices in every particular: the masses are
not merely hierarchical but \emph{absolutely} tiny, six orders of
magnitude below the electron; and the mixing angles are not small but
large, two of them within a factor of two of maximal. We follow the
three-step plan.

\subsection{Step 1 --- neutrino masses and lepton mixing as a graduate student meets it}
\label{app:pmns-step1}

\subsubsection{Why the Standard Model predicts a massless neutrino}
\label{app:pmns-massless}

It is worth being precise about this, because ``the Standard Model does
not accommodate neutrino mass'' is often said and is not quite what is
true. The statement is structural, and it has three steps.

\emph{Step (a): the field content.} The minimal Standard Model contains
the left-handed doublet \(L=(\nu_L,e_L)^{\mathsf T}\) and the
right-handed singlet \(e_R\), and \emph{no} \(\nu_R\). A Dirac mass term
\(m\,\bar\nu_L\nu_R\) therefore has nothing to pair with.

\emph{Step (b): the gauge quantum numbers.} Could one write a Majorana
mass \(\tfrac12 m\,\overline{\nu_L^{c}}\,\nu_L\) instead? The bilinear
\(\nu_L\nu_L\) sits in \(L L\), which carries weak isospin
\(\mathbf2\otimes\mathbf2 = \mathbf3\oplus\mathbf1\) and hypercharge
\(2Y_L=-1\). It is not a gauge singlet, so no dimension-three mass term
is gauge invariant.

\emph{Step (c): what \emph{is} allowed, and at what dimension.} Contract
\(LL\) with two Higgs doublets. The unique gauge-invariant, Lorentz-scalar
operator built from Standard-Model fields alone that gives the neutrino a
mass is
\begin{equation}
\mathcal O_5=\frac{c_5}{\Lambda}\,(L^{\mathsf T}\!\epsilon H)(L^{\mathsf T}\!\epsilon H)+\mathrm{h.c.},
\label{eq:weinberg-app}
\end{equation}
of \emph{dimension five} --- the celebrated Weinberg operator
\citep{Weinberg1979}. After electroweak breaking it delivers
\(m_\nu = c_5 v^2/\Lambda\).

Three consequences follow immediately and are worth stating separately,
because each is a fact about the Standard Model rather than about any
model beyond it. First, \(\mathcal O_5\) is the \emph{only} dimension-five
operator in the whole Standard Model: neutrino mass is the leading
signature of \emph{anything} above the electroweak scale, whatever that
thing is. Second, \(\mathcal O_5\) violates lepton number by two units,
so a nonzero neutrino mass of this origin makes the neutrino its own
antiparticle. Third, the suppression is \emph{automatic}: a mass
generated at dimension five is smaller than an ordinary Yukawa mass by
\(v/\Lambda\), which is why the neutrino is light without anybody tuning
anything.
\emph{(Physics intuition: the Standard Model does not forbid neutrino
mass --- it \emph{postpones} it. Every other fermion mass is allowed at
dimension four and therefore arrives at full strength; the neutrino's is
allowed only at dimension five and therefore arrives suppressed by one
power of a high scale. The neutrino is light not because its coupling is
small but because its mass operator is the first one that has to reach
outside the theory to exist at all. Its smallness is a measurement of
how far outside.)}

\subsubsection{Oscillations, derived}
\label{app:pmns-osc}

\paragraph{The two-flavor calculation, with no steps skipped.}
Suppose two flavor states are related to two mass states by a rotation,
\begin{equation}
\begin{pmatrix}\nu_e\\ \nu_\mu\end{pmatrix}
=\begin{pmatrix}\cos\theta & \sin\theta\\ -\sin\theta & \cos\theta\end{pmatrix}
\begin{pmatrix}\nu_1\\ \nu_2\end{pmatrix}.
\label{eq:twoflav-app}
\end{equation}
A source produces \(|\nu_e\rangle = \cos\theta\,|\nu_1\rangle +
\sin\theta\,|\nu_2\rangle\). Each mass eigenstate propagates with its own
phase \(e^{-i(E_it-p_iL)}\). For ultrarelativistic neutrinos of common
momentum \(p\simeq E\),
\begin{equation}
E_i=\sqrt{p^2+m_i^2}\simeq p+\frac{m_i^2}{2p},
\label{eq:disp-app}
\end{equation}
so over a distance \(L\simeq t\) the two states accumulate a
\emph{relative} phase
\begin{equation}
\Delta\phi=(E_2-E_1)t-(p_2-p_1)L\;\simeq\;\frac{\Delta m^2_{21}L}{2E},
\qquad \Delta m^2_{21}\equiv m_2^2-m_1^2 .
\label{eq:relphase-app}
\end{equation}
Projecting the evolved state back onto \(\langle\nu_\mu|\) and squaring,
the cross term gives
\begin{equation}
\boxed{\;
P(\nu_e\to\nu_\mu)=\sin^2 2\theta\,\sin^2\!\left(\frac{\Delta m^2_{21}L}{4E}\right).\;}
\label{eq:posc-app}
\end{equation}
Two features are the whole subject in miniature. The \emph{amplitude} of
the oscillation is set by the mixing angle and the \emph{frequency} by
the mass-squared difference: an oscillation experiment measures
\(\theta\) and \(\Delta m^2\), and \emph{cannot} measure \(m_1\) or
\(m_2\) separately, because the absolute phase common to both states
cancels in \eqref{eq:relphase-app}.

\paragraph{The conversion constant, derived.}
Restoring \(\hbar\) and \(c\): with \(1\ \mathrm{km}=5.0677\times10^{18}\
\mathrm{GeV}^{-1}\) and \(1\ \mathrm{eV}^2=10^{-18}\ \mathrm{GeV}^2\),
\begin{equation}
\frac{\Delta m^2L}{4E}
=\frac{5.0677}{4}\,
\frac{\Delta m^2[\mathrm{eV}^2]\;L[\mathrm{km}]}{E[\mathrm{GeV}]}
=1.2669\,\frac{\Delta m^2 L}{E},
\label{eq:conv-app}
\end{equation}
so the first oscillation maximum sits at
\(L_{\max}= (\pi/2)/1.2669 \times E/\Delta m^2 = 1.240\,E/\Delta m^2\)
in these units, and the full oscillation length is
\(L_{\rm osc}=2.480\,E/\Delta m^2\).
\emph{(Physics intuition: memorize \(1.27\). Every neutrino baseline ever
built --- a reactor experiment at a kilometer, a solar experiment at an
astronomical unit, an accelerator at several hundred kilometers --- is
this one number solved for \(L\) with a different \(\Delta m^2\) in the
denominator. The experiments look nothing alike and are all the same
experiment.)}

\paragraph{The caveat the plane-wave derivation hides.}
Equation~\eqref{eq:posc-app} was obtained by giving the two mass states a
common momentum, which is not consistent: they have different masses and
are produced by a process with definite energy--momentum bookkeeping. The
honest derivation uses wave packets \citep{GiuntiKim2007}, and it returns
\eqref{eq:posc-app} multiplied by a coherence factor that stays near one
provided the packet separation \(\Delta v\,L\) remains small compared to
the packet width. For every terrestrial experiment this is satisfied by
many orders of magnitude; for supernova and solar neutrinos it is not,
and the oscillating term averages to \(\tfrac12\). A student should know
that the standard formula is a limit, and know which limit.

\paragraph{Three flavors.}
With three families the same computation gives, for
\(\Delta_{ij}\equiv\Delta m^2_{ij}L/4E\),
\begin{equation}
P(\nu_\alpha\!\to\!\nu_\beta)=\delta_{\alpha\beta}
-4\sum_{i>j}\mathrm{Re}\!\left(U^{*}_{\alpha i}U_{\beta i}U_{\alpha j}U^{*}_{\beta j}\right)\sin^2\Delta_{ij}
+2\sum_{i>j}\mathrm{Im}\!\left(U^{*}_{\alpha i}U_{\beta i}U_{\alpha j}U^{*}_{\beta j}\right)\sin 2\Delta_{ij}.
\label{eq:three-app}
\end{equation}
The last term is odd under \(\alpha\leftrightarrow\beta\) and changes sign
between neutrinos and antineutrinos: it is the \CP{}-violating term, and it
collapses to a single rephasing-invariant,
\begin{equation}
P(\nu_\alpha\!\to\!\nu_\beta)-P(\bar\nu_\alpha\!\to\!\bar\nu_\beta)
=16\,\JPMNS{}\,\sin\Delta_{21}\sin\Delta_{31}\sin\Delta_{32},
\label{eq:cpasym-app}
\end{equation}
with the leptonic Jarlskog invariant
\begin{equation}
\JPMNS{}=c_{12}s_{12}\,c_{23}s_{23}\,c_{13}^2 s_{13}\,\sin\dPMNS{}.
\label{eq:jarlskog-app}
\end{equation}
\emph{(Physics intuition: \CP{} violation in oscillations needs
\emph{all three} splittings to be active at once --- the product of three
sines in \eqref{eq:cpasym-app} vanishes if any two mass states are
degenerate or if any angle is zero. That is why the discovery of a
nonzero \(\theta_{13}\) in 2012 was the event that made leptonic \CP{}
violation an experimental program rather than a hope: it lifted the last
factor off the floor.)}

\subsubsection{Matter: the MSW effect, and why it matters twice}
\label{app:pmns-msw}

Neutrinos crossing matter forward-scatter off electrons. All three
flavors scatter through \(Z\) exchange, which contributes a common phase
and drops out; only \(\nu_e\) additionally scatters through \(W\)
exchange off the electrons themselves. That difference is a potential
\citep{Wolfenstein1978,MikheyevSmirnov1985}
\begin{equation}
V_{\rm CC}=\sqrt2\,G_F n_e ,
\label{eq:vcc-app}
\end{equation}
which enters the effective Hamiltonian exactly like a mass-squared term,
\begin{equation}
H_{\rm eff}=\frac{1}{4E}\begin{pmatrix}
-\Delta m^2\cos2\theta+2EV_{\rm CC} & \Delta m^2\sin2\theta\\
\Delta m^2\sin2\theta & \Delta m^2\cos2\theta-2EV_{\rm CC}\end{pmatrix}.
\label{eq:heff-app}
\end{equation}
The in-medium mixing angle is maximal --- \emph{whatever the vacuum angle}
--- when the diagonal vanishes:
\begin{equation}
2E\,V_{\rm CC}=\Delta m^2\cos 2\theta
\quad\Longleftrightarrow\quad
n_e^{\rm res}=\frac{\Delta m^2\cos2\theta}{2\sqrt2\,G_F E}.
\label{eq:msw-res-app}
\end{equation}
Numerically, for the solar parameters and a \(10\)~MeV \(^8\)B neutrino,
\(n_e^{\rm res}=1.14\times10^{25}\ \mathrm{cm^{-3}}\simeq 19\,N_A\
\mathrm{cm^{-3}}\), comfortably below the solar core density
\(n_e\simeq 5.8\times10^{25}\ \mathrm{cm^{-3}}\). The high-energy solar
neutrinos are therefore \emph{born above the resonance}, and if the
density profile varies slowly enough for the evolution to be adiabatic
they emerge as the pure heavier state \(\nu_2\), giving a survival
probability \(P_{ee}\to\sin^2\theta_{12}\simeq 0.31\) --- against the
vacuum-averaged \(1-\tfrac12\sin^2 2\theta_{12}=0.57\) that the low-energy
\(pp\) neutrinos show. Both are observed, and the transition between them
across the \(^8\)B spectrum is the signature of the effect.

The MSW effect matters twice. It is how the solar solution was
\emph{identified}, and --- because \(V_{\rm CC}\) flips sign for
antineutrinos and its interference with \(\Delta m^2_{31}\) depends on the
\emph{sign} of that splitting --- it is also the handle by which a
long-baseline accelerator experiment determines the mass ordering.
\emph{(Physics intuition: matter is a flavor-dependent index of
refraction for neutrinos, and it is flavor-dependent for the mundane
reason that ordinary matter contains electrons and not muons. That
accident of chemistry is what lets an experiment on Earth read the sign of
a mass-squared difference --- and it is also the reason a long-baseline
\CP{} measurement must disentangle a genuine \CP{} asymmetry from the
matter-induced fake one, which is the single hardest systematic in the
field.)}

\subsubsection{What each experiment actually measures}
\label{app:pmns-experiments}

\begin{table}[htbp]
\centering
\small
\caption[The measurement landscape.]{The measurement landscape. Each row is an oscillation channel
with its characteristic \(L/E\); the phase column is
\eqref{eq:conv-app} evaluated at the quoted baseline and energy, and
shows that every one of these experiments was built to sit near the
first oscillation maximum of the splitting it targets --- $\pi/2 =
1.571$. This is the single most useful table in the subject: it explains
why the baselines are what they are.}
\label{tab:osc-landscape}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.21\textwidth}rr>{\raggedright\arraybackslash}p{0.15\textwidth}r>{\raggedright\arraybackslash}p{0.20\textwidth}@{}}
\toprule
\textbf{Channel} & \(L\) & \(E\) & \textbf{Splitting} & \textbf{Phase} & \textbf{Fixes}\\
\midrule
Reactor, short (Daya Bay, RENO) & 1.65~km & 4~MeV & \(\Delta m^2_{31}\) & 1.31 & \(\theta_{13}\)\\
Reactor, medium (JUNO) & 52.5~km & 4~MeV & \(\Delta m^2_{21}\) & 1.25 & \(\theta_{12}\), \(\Delta m^2_{21}\), ordering\\
Reactor, long (KamLAND) & 180~km & 4~MeV & \(\Delta m^2_{21}\) & 4.29 & \(\Delta m^2_{21}\) (past first max)\\
Accelerator (T2K) & 295~km & 0.6~GeV & \(\Delta m^2_{31}\) & 1.56 & \(\theta_{23}\), \(\dPMNS{}\)\\
Accelerator (NOvA) & 810~km & 2~GeV & \(\Delta m^2_{31}\) & 1.29 & \(\theta_{23}\), ordering\\
Accelerator (DUNE) & 1300~km & 2.5~GeV & \(\Delta m^2_{31}\) & 1.65 & \(\dPMNS{}\), ordering\\
Atmospheric (Super-K, IceCube) & $10$--$10^4$~km & 1~GeV & \(\Delta m^2_{31}\) & spectrum & \(\theta_{23}\), octant\\
Solar ($^8$B, $pp$) & 1~AU & 0.1--15~MeV & \(\Delta m^2_{21}\) (MSW) & averaged & \(\theta_{12}\), sign of \(\Delta m^2_{21}\)\\
\bottomrule
\end{tabular}
\end{table}

The current determinations, which this course scores its predictions
against throughout, are the global fit \citep{NuFIT61} --- cross-checked
against the independent global analysis of \citet{Capozzi2025}, which
uses a different treatment of the atmospheric and solar data and agrees
on every parameter except the atmospheric octant --- together with the
two measurements that most sharpen it: JUNO's direct determination of the
solar parameters \citep{JUNO2026} and the T2K--NOvA joint fit
\citep{T2KNOvA2025}. The reader should note two live disagreements rather
than a settled table. The atmospheric angle is reported in the
\emph{lower} octant by the global fit and the \emph{upper} octant by the
accelerator joint fit, a split treated at length in
\S\ref{sec:theta23-split}; and the ordering is preferred normal at only
\(2.2\)--\(2.3\sigma\) after JUNO \citep{EstebanJUNO2026}. A student
scoring any lepton-sector prediction should score it against both octants
and say which.

\subsubsection{The absolute scale: three independent probes}
\label{app:pmns-absolute}

Oscillations are blind to the absolute scale. Three quite different
experiments are not, and they measure three \emph{different} combinations
--- which is why all three are needed and why a discrepancy between them
would be informative rather than merely awkward.
\begin{enumerate}\itemsep3pt
\item \textbf{Kinematic (single \(\beta\) decay).} The endpoint of the
tritium spectrum is displaced by the incoherent sum
\(m_\beta^2=\sum_i |U_{ei}|^2 m_i^2\). This is model-independent --- it
uses only energy conservation --- and therefore the most robust. KATRIN
reports \(m_\beta<0.45\)~eV \citep{KATRIN2025}.
\item \textbf{Cosmological.} Massive neutrinos free-stream and suppress
structure below their free-streaming scale, so a galaxy survey plus the
CMB constrains \(\Summnu{}=\sum_i m_i\). This is the most \emph{sensitive}
probe by two orders of magnitude and the most model-dependent: DESI~DR2
gives \(\Summnu{}<64.2\)~meV in \(\Lambda\)CDM but \(<163\)~meV once the
dark-energy equation of state is allowed to vary \citep{DESI2025}. The
tightening is recent and worth registering: the DR1 analysis
\citep{DESI2024} gave \(<72\)~meV on the same assumptions, so this bound
has moved by more than ten per cent in one data release --- which is the
right frame for judging how much headroom any prediction pressed against
it actually has.
\item \textbf{Neutrinoless double beta decay.} The rate is controlled by
the coherent, phase-sensitive combination
\(\mbb{}=\bigl|\sum_i U_{ei}^2 m_i\bigr|\), and is nonzero only if the
neutrino is Majorana. Treated in full in App.~\ref{app:majorana}.
\end{enumerate}
\emph{(Physics intuition: the three probes weight the same three masses
with a square, a first power, and a first power with phases. A theory
that predicts the spectrum predicts all three, and they are not
independent tests of it --- but they \emph{are} independent tests of each
other's systematics, which for a quantity this small is the more useful
service. If cosmology reported a sum incompatible with a measured
\(\mbb{}\), the news would be about cosmology.)}

\subsubsection{The seesaw, in its three types}
\label{app:pmns-seesaw}

Where does \eqref{eq:weinberg-app} come from? Integrating out a heavy
state produces it, and at tree level there are exactly three ways
\citep{MohapatraPal2004}: the heavy state can be a fermion singlet, a
scalar triplet, or a fermion triplet.

\paragraph{Type I: a heavy fermion singlet.}
Add \(N_R\) with a Dirac Yukawa and a Majorana mass
\citep{Minkowski1977,Yanagida1979,GellMannRamondSlansky1979,MohapatraSenjanovic1980}.
In the \((\nu_L,N_R^{c})\) basis the mass matrix is
\begin{equation}
\mathcal M=\begin{pmatrix}0 & m_D\\ m_D^{\mathsf T} & M_N\end{pmatrix},
\qquad m_D=\Ynu{}\,v .
\label{eq:seesawmatrix-app}
\end{equation}

\paragraph{A convention worth pinning, because it is a factor of two.}
Textbooks divide here on whether the Dirac mass is \(\Ynu{} v\) or
\(\Ynu{} v/\sqrt2\), and the two differ in \(m_\nu\) by a factor of two.
\emph{This paper uses \(m_D = \Ynu{} v\) with \(v = 246.33\)~GeV
throughout}, which is the convention under which the effective washout
mass is \(\widetilde m = \Ynu{}^2 v^2/M_N\) and under which the lock-scale
matching of \S\ref{sec:lg-seesaw} is an identity rather than an identity
up to two. Note that the Davidson--Ibarra bound below is conventionally
written with \(\bar v = 174\)~GeV: that is a property of \emph{that}
formula, not a second convention for \(m_D\), and the two do not
interfere. A student comparing numbers across papers should check this
first --- more disagreements in the seesaw literature come from it than
from physics.
Block-diagonalizing for \(M_N\gg m_D\) gives the two eigenvalues
\begin{equation}
m_\nu \simeq -m_D M_N^{-1} m_D^{\mathsf T},
\qquad m_{\rm heavy}\simeq M_N ,
\label{eq:seesaw-typeI-app}
\end{equation}
the light state suppressed by exactly the ratio the heavy state is
enhanced by --- which is the image the name records.

\paragraph{Type II: a heavy scalar triplet.} A triplet \(\Delta\) with
hypercharge~1 couples as \(f\,L^{\mathsf T}\!\Delta L\) and acquires an
induced vev \(v_\Delta\simeq \mu v^2/M_\Delta^2\), giving
\(m_\nu = 2f v_\Delta\). \paragraph{Type III: a heavy fermion triplet.}
Identical in structure to type~I with the singlet replaced by an
\(\SU(2)\) triplet \(\Sigma\); the light mass is again
\(m_\nu\sim v^2/M_\Sigma\).

\paragraph{The parametrization that shows what is free.}
For type~I with three heavy states, \citet{CasasIbarra2001} showed that
the most general Dirac Yukawa reproducing a given light spectrum and
mixing matrix is
\begin{equation}
\Ynu{}=\frac{1}{v}\,\sqrt{\widehat M_N}\;R\;\sqrt{\widehat m_\nu}\;U^\dagger_{\rm PMNS},
\label{eq:casasibarra-app}
\end{equation}
with \(R\) an arbitrary \emph{complex} orthogonal matrix. This equation is
the honest statement of the sector's freedom, and every student should
internalize it: the low-energy data fix \(\widehat m_\nu\) and \(U\) and
say \emph{nothing whatever} about \(\widehat M_N\) or \(R\). The three
heavy masses and the three complex angles of \(R\) --- nine real
parameters --- are entirely unconstrained by oscillation experiments. It
is this freedom that App.~\ref{app:leptogenesis} will find is exactly what
a leptogenesis prediction needs and cannot get.

\paragraph{Worked example (STANDARD TEXTBOOK PEDAGOGY).}
Take the atmospheric splitting and assume \(m_1\approx 0\), so that
\(m_3\approx\sqrt{|\Delta m^2_{31}|}=\sqrt{2.511\times10^{-3}}
= 50.1\)~meV. Insert this in \eqref{eq:seesaw-typeI-app} with a Dirac Yukawa of
order the top's, \(\Ynu{}\sim1\), so \(m_D = \Ynu{} v = 246\)~GeV in the
convention pinned above:
\begin{equation}
M_N\sim\frac{m_D^2}{m_\nu}=\frac{(246.33~\mathrm{GeV})^2}{5.01\times10^{-11}~\mathrm{GeV}}
=1.2\times10^{15}~\mathrm{GeV},
\label{eq:seesawscale-app}
\end{equation}
within an order of magnitude of the grand-unification scale --- a
coincidence the Standard Model records and cannot explain. Note carefully
what was assumed to get it: \(\Ynu{}\sim1\), for no reason except that the
top quark's is. Choose \(\Ynu{}\sim10^{-6}\) instead, and the same
neutrino mass comes from \(M_N\sim1\)~TeV --- which is, within a factor
of a few, exactly where App.~\ref{app:leptogenesis} will find the
geometry putting it, with the Yukawa derived rather than chosen. \emph{The seesaw fixes only the
combination \(m_D^2/M_N\); it does not fix the scale, and any textbook
statement that ``the seesaw points at grand unification'' is a statement
about the assumed Yukawa, not about the data.} This will matter a great
deal in App.~\ref{app:leptogenesis}.

\subsection{Step 2 --- what the lepton sector assumes but cannot explain}
\label{app:pmns-step2}

The lepton sector is the least constrained part of the Standard Model.
Where the quark sector left four numbers free, this one leaves ten, and
adds two structural puzzles that no choice of parameters resolves.

\begin{enumerate}\itemsep3pt
\item \textbf{The absolute mass scale.} Free. Equivalently, \(\Lambda\) in
\eqref{eq:weinberg-app} is free.
\item \textbf{The operator coefficient \(c_5\).} Free, and degenerate with
\(\Lambda\) --- only the ratio is observable.
\item \textbf{The two mass-squared splittings.} Two independent measured
inputs. Nothing explains why their ratio is \(\sim 33\) rather than
\(\sim 1\) or \(\sim 10^6\).
\item \textbf{The mass ordering.} Not predicted; the sign of
\(\Delta m^2_{31}\) is an experimental question.
\item \textbf{Dirac or Majorana.} Not predicted. The Standard Model
extended by \(\nu_R\) gives Dirac; extended by \eqref{eq:weinberg-app}
gives Majorana; nothing selects.
\item \textbf{The three mixing angles.} Three measured inputs.
\item \textbf{The Dirac phase.} One measured input, currently determined
only to a broad interval.
\item \textbf{The two Majorana phases.} Two inputs, unmeasured and
unconstrained by any experiment yet performed.
\item \textbf{The heavy sector.} \(\widehat M_N\) and \(R\) in
\eqref{eq:casasibarra-app}: nine further real parameters, invisible to
low-energy data.
\item \textbf{The number of light species.} Three, from the \(Z\) width and
cosmology --- observed, not derived.
\end{enumerate}

\noindent And the two structural puzzles, which are the more important
half of the list because no amount of data will resolve them:

\paragraph{Puzzle A: why are the lepton angles large when the quark angles
are small?} This is the \emph{opposite} of the quark puzzle, in the same
theory, with the same gauge structure and the same number of families.
Both facts are recorded and neither is explained.

\paragraph{Puzzle B: why is the neutrino mass scale six orders of
magnitude below the electron's?} The seesaw explains how a small mass can
arise without a small coupling, but --- as
\eqref{eq:seesawscale-app} showed --- it converts one unexplained number
into two, and fixes only their ratio.

\emph{(Physics intuition: the quark mixing matrix is nearly the identity;
the lepton one is nearly democratic. The Standard Model has no reason for
either, and certainly no reason for the striking contrast between them. A
theory that produced ``CKM small, PMNS large'' from \emph{one}
construction would be explaining a headline fact of particle physics ---
and it would be doing so in the one sector where the data are still
arriving, which is the only interesting place for a theory to make a
claim.)}

\subsection{Step 3 --- the geometric derivation: a Majorana floor and a democratic misalignment}
\label{app:pmns-step3}

In \GGTheory{} both the smallness of the neutrino masses and the largeness
of the lepton angles come from one construction, with no new scale and no
right-handed neutrino sector. Every item in Step~2's list of ten is
addressed; two of them are addressed \emph{less well than the others}, and
this section identifies which and by how much rather than quoting an
average.

\subsubsection{The neutrino floor from the minimal Majorana channel}
\label{app:pmns-floor}

\BIsix{} forbids any new intermediate mass scale and any second compact
loop, so the neutrino mass must come from the one constitutionally
admitted operator --- the Weinberg dimension-five \((LH)(LH)\) of
\eqref{eq:weinberg-app} --- reusing the already-derived grand-unification
seam \(\EGUT\) as its scale. This is the whole of the answer to Step~2's
items~1 and~2: \(\Lambda\) is not free because it is not new. The common
neutrino scale is therefore fixed,
\begin{equation}
m_\star=\pi\,\frac{v^2}{\EGUT}
=\frac{\pi\times(246.33~\mathrm{GeV})^2}{2.280\times10^{16}~\mathrm{GeV}}
= 8.36114\ \mathrm{meV},
\label{eq:nu-floor-app}
\end{equation}
the same expression recorded in the codebook arithmetic
(Eq.~\ref{eq:seesaw-geom-app}), with first-order eigenvalues in the
integer ratio \((m_1,m_2,m_3)=(0,1,6)\,m_\star\).

Two structural predictions follow with no further input. First, the
Majorana ledger is \emph{rank two}, so there are exactly two independent
mass-squared splittings and the absolute \(m_1\) enters only through the
operator coefficient --- precisely the observed pattern, and the answer to
item~3's first half. Second, the cascade-depth assignment places the
ground state (the \GGA{} of the Majorana stratum) at \(\nu_1\), forcing the
\emph{normal} ordering \(m_1<m_2<m_3\) --- item~4 --- and an established
inverted ordering would falsify the assignment outright.

\paragraph{Where the ledger strains, stated with the number.} The integer
ratio \(36 = (2N)^2\) is the framework's answer to item~3's second half,
and it should be scored honestly. Against \citet{NuFIT61} the measured
ratio is \(\Delta m^2_{31}/\Delta m^2_{21}=33.35\), so the first-order
\(36\) is \(+7.96\%\) high; the QD-completed spectrum gives \(34.56\), or
\(+3.65\%\). But the ratio is the wrong quantity to look at, because it
averages two very different agreements:
\begin{equation}
\Delta m^{2\,(1)}_{31}=2.5167\times10^{-3}~\mathrm{eV}^2\ \ (+0.23\%),
\qquad
\Delta m^{2\,(1)}_{21}=6.9909\times10^{-5}~\mathrm{eV}^2\ \ (-7.16\%).
\label{eq:splittings-scored-app}
\end{equation}
\emph{The atmospheric splitting is right to two parts in a thousand and
essentially all of the discrepancy sits in the solar one.} The QD layer
moves the solar splitting to \(-1.96\%\) and the atmospheric to
\(+1.62\%\), which is an improvement in the ratio bought partly by
degrading the quantity that was already right. A student should read this
as the framework's clearest live tension in the lepton sector: the
integer \(6\) in \((0,1,6)\) is doing very well and the integer \(1\) is
doing less well, and with JUNO now measuring \(\Delta m^2_{21}\) to
\(1.6\%\) \citep{JUNO2026} this is a discrepancy that is no longer
hiding inside the error bar.
\emph{(Physics intuition: the neutrinos are light for the same reason the
electroweak scale is small --- a ratio of fixed geometric scales, here
\(v^2/\EGUT\), not a tuned seesaw mass. The meV scale of neutrino mass is
the geometric shadow of the \(10^{16}\ \mathrm{GeV}\) seam, and the two
most extreme mass scales ever measured become one ratio. And because the
Majorana ledger has rank two, nature can hand the oscillation experiments
only two beat frequencies --- the absence of a third is a prediction, not
a gap.)}

\subsubsection{The forced triplet: why $(D,S,A)$ and not some other basis}
\label{app:pmns-dsa}

Before the angles, the basis. The neutrino block is a rank-two Majorana
packet on a three-dimensional family space, and three facts fix its
natural basis with no freedom left.

\emph{Rank two singles out a vector plus a plane.} A rank-two symmetric
form on a three-dimensional space has a one-dimensional kernel; the
decomposition is therefore \(\mathbf 3 = \mathbf 1 \oplus \mathbf 2\), a
distinguished direction and its orthogonal plane. \emph{Permutation
symmetry picks the vector.} The Majorana channel is generated by the
single operator \eqref{eq:weinberg-app} acting identically on all three
families, so the distinguished direction must be invariant under family
permutations: it is the democratic vector. \emph{The charged-lepton
hierarchy splits the plane.} The remaining two-dimensional space is
resolved by the only structure that distinguishes families, the dyadic
charged-lepton ladder, into the \(\mu\!-\!\tau\)-even and
\(\mu\!-\!\tau\)-odd combinations. The result is forced:
\begin{equation}
|D\rangle=\tfrac{1}{\sqrt3}(1,1,1)^{\mathsf T},\quad
|S\rangle=\tfrac{1}{\sqrt6}(2,-1,-1)^{\mathsf T},\quad
|A\rangle=\tfrac{1}{\sqrt2}(0,1,-1)^{\mathsf T}.
\label{eq:DSA-app}
\end{equation}
\emph{(Physics intuition: this is the same argument that gives the
democratic, symmetric and antisymmetric combinations in any three-body
problem with an interaction that does not know which body is which ---
and the point is that the neutrino sector's operator genuinely does not
know, because it is one operator, while the charged-lepton sector
emphatically does, because it is three very different Yukawas. The PMNS
matrix is the angle between a sector that cannot tell the families apart
and a sector that can barely do anything else.)}

\subsubsection{Large angles from a dyadic-versus-democratic misalignment}
\label{app:pmns-angles}

Reading the misalignment of the dyadic charged-lepton basis against
\eqref{eq:DSA-app} gives the three angles as closed-form rationals,
\begin{equation}
\boxed{\;
\sin^2\theta_{12}=\frac{L^2}{3N^3}=\frac{25}{81},
\qquad
\sin^2\theta_{23}=\frac12\Bigl(1+\frac1J\Bigr)=\frac47,
\qquad
\sin^2\theta_{13}=\frac{N}{4K}=\frac{3}{140},\;}
\label{eq:pmns-angles-app}
\end{equation}
each built from the ledger integers \((M,N,L;J,K)=(1,3,5;7,35)\) and no
others. Each deserves a sentence of its own geometry, because the three
are structurally different and a student who reads them as three fitted
fractions has missed the chapter.

\paragraph{The solar angle: a democratic overlap, depressed.} A pure
democratic overlap gives exactly \(1/3\). The ledger gives
\begin{equation}
\frac{25}{81}=\frac13\Bigl(1-\frac{2}{27}\Bigr)=0.308642,
\label{eq:solar-depress-app}
\end{equation}
a \(7.4\%\) depression of the democratic value, arising because the
charged-lepton plane is not exactly orthogonal to \(|D\rangle\). This
number now decides between hypotheses: JUNO measures
\(0.3092\pm0.0087\) \citep{JUNO2026}, so the pure democratic \(1/3\) sits
\(2.8\sigma\) high and the depressed value sits \(0.07\sigma\) away. The
measurement has become precise enough to \emph{see} the \(2/27\), and it
sees it. Tri-bimaximal models, which predict the undepressed \(1/3\),
predict the value the data have moved off.

\paragraph{The reactor angle: the one residue of sharpness.} The factor of
four in \(N/4K\) is two factors of \(1/\sqrt2\) --- the two independent
\(\mu\!-\!\tau\) projections through which the dyadic comb reaches the
democratic sector --- and \(K=35\) is the full closure depth. Small
because the reach is second order; nonzero because the comb is not
orthogonal to the packet.

\paragraph{The atmospheric angle: a tilt about maximal, not about zero.}
This is the structurally most interesting of the three and the one most
often misread. The ledger does not predict a small angle plus a
correction; it predicts a \emph{traceless tilt about maximal mixing},
\begin{equation}
\sin^2\theta_{23}=\frac12\Bigl(1+\frac1J\Bigr),\qquad J=7,
\label{eq:atm-tilt-app}
\end{equation}
so the deviation from \(1/2\) is \(1/(2J)=1/14\) and its \emph{sign} is
forced by which direction the charged-lepton hierarchy distinguishes ---
upward, into the upper octant. Maximal mixing is the symmetric point and
\(J\) is what breaks it. Consequently the octant is not a detail of this
prediction, it is the prediction: \(4/7=0.5714\) is inside the accelerator
joint fit's \(0.56^{+0.03}_{-0.05}\) and more than four sigma from the
global fit's \(0.470^{+0.017}_{-0.014}\). \S\ref{sec:theta23-split} treats
the split, and F-SM-3e registers its resolution as a kill criterion.

\emph{(Physics intuition: ``CKM small, PMNS large'' is not two
constitutions but one. On the quark side both sectors are hierarchical, so
their family bases nearly agree and the mixing is small. On the lepton
side a sharp dyadic comb is laid over a near-uniform democratic
background, and two combs of such different teeth can only mesh at large
angles. The small reactor angle is the one residue of the charged-lepton
sharpness that survives into the democratic sector --- and this single
picture answers Puzzle~A of Step~2, which is the reason the chapter
exists.)}

\subsubsection{Phases, and the honest status of the lepton sector}
\label{app:pmns-honesty}

The Dirac phase is fixed by the same compact-angle holonomy logic as the
quark sector rather than by a new continuous parameter, and the two
Majorana phases are parity data of the rank-two ledger rather than dials.
Both are derived in the next chapter (App.~\ref{app:majorana}), where they
are the subject rather than a postscript; the full projector
construction and the precision comparison against the global fit are in
the main text (\S\ref{sec:neutrino-masses}, \S\ref{sec:PMNS}), and the
explicit mass arithmetic is in the codebook arithmetic of the
fermion-mass appendix (\S\ref{app:fermion-neutrinos}).

Scoring Step~2's ten items honestly: items 1, 2, 4, 5, 6, 7, 8 and 10 are
structural outputs of the construction, and both puzzles are answered.
Item~9 --- the nine parameters of the heavy sector --- is answered by the
lock-scale construction of App.~\ref{app:leptogenesis}, where it becomes
the chapter's central claim and also its most conditional one. Item~3 is
answered in its first half (\emph{why two}) and answered with a visible
strain in its second (\emph{why 33}), quantified in
\eqref{eq:splittings-scored-app}. And the angle values inherit whatever
residual freedom the charged-lepton dyadic codes carry --- the same
cascade-depth lemma that governs the charged fermions, item~13 of the
Part's open list.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{J.1.1. Derive the oscillation probability, and
the constant $1.27$.}

\noindent\emph{Solution.}  Step 1: expand the dispersion relation
\eqref{eq:disp-app} for $m \ll p$ and form the relative phase
\eqref{eq:relphase-app}: only the \emph{difference} of
$m_i^2$ survives, because the common $p$ cancels between the two mass
states.  Step 2: project and square --- the interference term carries
$\sin^2 2\theta$ from the two rotations and
$\sin^2(\Delta m^2 L/4E)$ from the phase, giving
\eqref{eq:posc-app}.  Step 3: restore units.  With $1\,\mathrm{km} =
5.0677\times10^{18}~\mathrm{GeV}^{-1}$ and
$1~\mathrm{eV}^2 = 10^{-18}~\mathrm{GeV}^2$ the phase is
$(5.0677/4)\,\Delta m^2 L/E = 1.2669\,\Delta m^2[\mathrm{eV}^2]
L[\mathrm{km}]/E[\mathrm{GeV}]$.  Physics intuition: the entire
subject is one interferometer formula and one conversion constant;
every baseline in Table~\ref{tab:osc-landscape} is $1.27$ solved for
$L$.  Memorize the constant and you can reconstruct the design of any
neutrino experiment ever built from its target splitting alone.

\medskip\noindent\textbf{J.1.2. Design the experiments: check that every
baseline sits at its first maximum.}

\noindent\emph{Solution.}  Step 1: the first maximum is at phase
$\pi/2 = 1.571$.  Step 2: evaluate \eqref{eq:conv-app} for T2K
($L = 295$~km, $E = 0.6$~GeV, $\Delta m^2_{31} = 2.511\times10^{-3}$):
$1.2669 \times 2.511\times10^{-3} \times 295/0.6 = 1.564$ --- within
half a percent of $\pi/2$.  For Daya Bay ($1.65$~km, $4$~MeV, same
splitting): $1.31$.  For JUNO ($52.5$~km, $4$~MeV,
$\Delta m^2_{21} = 7.53\times10^{-5}$): $1.25$.  For DUNE
($1300$~km, $2.5$~GeV): $1.65$.  Step 3: every entry in
Table~\ref{tab:osc-landscape} lands between $1.25$ and $1.65$ except
KamLAND at $4.29$ --- and KamLAND is the exception that proves the
rule, since it was a reactor experiment sited at an existing cavern
rather than built to a baseline, and reads $\Delta m^2_{21}$ from the
\emph{spectral} distortion of several oscillations rather than from a
single dip.  Physics intuition: a neutrino experiment is a tuned
instrument, and the tuning is visible in one arithmetic line.  When a
proposal quotes a baseline, this is the calculation that tells you what
it is really for --- and, for DUNE at $1.65$, that it is deliberately
placed just \emph{past} the maximum, where the matter effect separates
the orderings most cleanly.

\medskip\noindent\textbf{J.1.3. Invert the seesaw --- and then show the
inversion proves less than it appears to.}

\noindent\emph{Solution.}  Step 1: the Weinberg operator gives
$m_\nu \sim v^2/\Lambda$; invert, $\Lambda \sim v^2/m_\nu$.  Step 2:
with $m_\nu = 0.05$~eV and $v = 246$~GeV,
$\Lambda \sim (246)^2/(5\times10^{-11})~\mathrm{GeV} =
1.2\times10^{15}$~GeV --- the grand-unification decade.  Step 3: now
repeat the argument with the type-I mass \eqref{eq:seesaw-typeI-app},
$M_N = m_D^2/m_\nu$, and notice that $m_D = \Ynu{} v$ contains a
Yukawa nobody has measured.  With $\Ynu{} \sim 1$ one gets
$1.2\times10^{15}$~GeV; with $\Ynu{} \sim 10^{-6}$ one gets $1$~TeV; both
give the same neutrino mass.  Physics intuition: this is the most
commonly overstated inference in the field.  The data fix
$m_D^2/M_N$ and nothing else, so ``neutrino mass points at grand
unification'' is a statement about an \emph{assumed} Yukawa of order one
--- assumed because the top quark's is.  A framework that derives
$\Ynu{}$ instead of assuming it can put the heavy scale anywhere the
geometry says, and App.~\ref{app:leptogenesis} puts it at $3.85$~TeV.

\medskip\noindent\textbf{J.1.4. Count the physical phases of the lepton
sector, Dirac versus Majorana, at general $n$.}

\noindent\emph{Solution.}  Step 1: an $n\times n$ unitary matrix has
$n^2$ real parameters, of which $n(n-1)/2$ are angles and $n(n+1)/2$
are phases.  Step 2: if the neutrinos are Dirac, both sectors absorb
phases and $2n-1$ are removable, leaving
$n(n+1)/2 - (2n-1) = (n-1)(n-2)/2$ physical phases: zero for $n = 2$,
one for $n = 3$.  If the neutrinos are Majorana, $\nu \to e^{i\phi}\nu$
would change the Majorana mass term and is forbidden, so only the $n$
charged-lepton phases are available and $n(n+1)/2 - n = n(n-1)/2$
survive: one for $n = 2$, \emph{three} for $n = 3$ --- one Dirac plus
two Majorana.  Step 3: the two extra phases are invisible in
oscillations, which depend only on the rephasing-invariant products
$U^{\vphantom{*}}_{\alpha i}U^*_{\beta i}$, and surface only in
lepton-number-violating processes.  Physics intuition: a Majorana
particle cannot hide a phase in its own redefinition because it
\emph{is} its own antiparticle; the phases that quarks shrug off become
physical bookkeeping.  Counting them is the cleanest way to see that
``Dirac or Majorana?'' is not metaphysics but two more measurable
numbers --- and the next chapter measures them.

\medskip\noindent\textbf{J.1.5. Derive the solar MSW resonance density,
and show the Sun is dense enough.}

\noindent\emph{Solution.}  Step 1: the charged-current potential
\eqref{eq:vcc-app} is $V_{\rm CC} = \sqrt2 G_F n_e$, and the
in-medium mixing is maximal when it cancels the vacuum diagonal:
$n_e^{\rm res} = \Delta m^2 \cos 2\theta/(2\sqrt2 G_F E)$.  Step 2:
with $\Delta m^2_{21} = 7.53\times10^{-5}$~eV$^2$,
$\cos2\theta_{12} = 1 - 2(0.308) = 0.384$, $E = 10$~MeV and
$G_F = 1.166\times10^{-5}~\mathrm{GeV}^{-2}$:
$n_e^{\rm res} = 8.6\times10^{-17}~\mathrm{GeV}^3 =
1.14\times10^{25}~\mathrm{cm^{-3}} = 19\,N_A~\mathrm{cm^{-3}}$.
Step 3: the solar core has $\rho \simeq 150~\mathrm{g\,cm^{-3}}$ and
$Y_e \simeq 0.64$, so $n_e \simeq 5.8\times10^{25}~\mathrm{cm^{-3}}$
--- a factor five \emph{above} resonance.  A $10$~MeV $^8$B neutrino is
therefore born on the far side of the resonance and, if the profile is
adiabatic, exits as pure $\nu_2$ with $P_{ee} \to \sin^2\theta_{12} =
0.31$; the $pp$ neutrinos at $0.3$~MeV never reach resonance and
average to $1 - \tfrac12\sin^2 2\theta_{12} = 0.57$.  Physics intuition:
both numbers are measured, and the factor-of-two step between
them across the solar spectrum is the fingerprint.  The Sun works as a
neutrino spectrometer because its density happens to bracket the
resonance for the energies it emits --- had the core been ten times
less dense, the solar neutrino problem would have been a flat deficit
with no structure and far harder to diagnose.

\medskip\noindent\textbf{J.1.6. Show that CP violation in oscillations
requires all three splittings and all three angles.}

\noindent\emph{Solution.}  Step 1: the \CP{}-odd difference
\eqref{eq:cpasym-app} is $16 \JPMNS{} \sin\Delta_{21}\sin\Delta_{31}
\sin\Delta_{32}$, and $\JPMNS{}$ itself \eqref{eq:jarlskog-app} is
$c_{12}s_{12}c_{23}s_{23}c_{13}^2 s_{13}\sin\delta$.  Step 2: the
product vanishes if any one of the three angles is $0$ or $\pi/2$, if
$\delta$ is $0$ or $\pi$, or if any two masses are degenerate (which
kills one of the three sines).  Step 3: historically this is why
$\theta_{13}$ mattered so much: before 2012 it was consistent with
zero, and a zero would have closed the leptonic \CP{} program
entirely.  Its measured value $\sin^2\theta_{13} \simeq 0.0225$ is
small but decisively nonzero, and it is the smallest factor in the
product --- the leptonic \CP{} asymmetry is small \emph{because of the
reactor angle}, not because of the phase.  Physics intuition: \CP{}
violation is an area, and an area needs all three sides.  A degenerate
pair collapses the triangle onto a line; a vanishing angle collapses
it onto a point.  Nature gave us a triangle with one short side, which
is why DUNE needs a megawatt beam and a decade.

\medskip\noindent\textbf{J.1.7. Read the three absolute-mass probes as
three different weightings.}

\noindent\emph{Solution.}  Step 1: write the three combinations:
$m_\beta^2 = \sum_i |U_{ei}|^2 m_i^2$ (incoherent, squares),
$\Summnu{} = \sum_i m_i$ (unweighted, first power), and
$\mbb{} = |\sum_i U_{ei}^2 m_i|$ (coherent, first power, with phases).
Step 2: with $|U_{e1}|^2 = c_{12}^2c_{13}^2$,
$|U_{e2}|^2 = s_{12}^2c_{13}^2$, $|U_{e3}|^2 = s_{13}^2$ and the
derived first-order spectrum $(0,\,8.36114,\,50.1668)$~meV:
$m_\beta = \sqrt{0.3020\times69.909 + 0.02143\times2516.7}
= 8.66$~meV, while $\Summnu{} = 58.53$~meV and (next chapter)
$\mbb{} = 1.72$~meV.  Step 3: the same spectrum gives three numbers
spanning a factor of thirty-four, and only the middle one is within
reach of a current experiment --- $8.66$~meV is fifty times below
KATRIN's $450$~meV limit and below even its ultimate design
sensitivity.  Physics intuition: the ordering of the three numbers is
itself a prediction.  The coherent combination is the smallest because
the phases interfere destructively; the kinematic one is small because
squaring a spectrum dominated by its top eigenvalue and then weighting
that eigenvalue by $s_{13}^2 = 0.021$ throws most of it away.
A framework that gets the spectrum right must get all three, and the
one that will decide is the sum.

\medskip\noindent\textbf{J.1.8. Locate the Casas--Ibarra freedom, and
count it.}

\noindent\emph{Solution.}  Step 1: \eqref{eq:casasibarra-app} writes
the Dirac Yukawa as $\sqrt{\widehat M_N} R \sqrt{\widehat m_\nu}
U^\dagger$ with $R$ complex orthogonal.  Step 2: count.  A complex
orthogonal $3\times3$ matrix has three complex angles, so six real
parameters; the three heavy masses add three more; nine in total,
\emph{none} of which appears in any oscillation observable, because
$m_\nu = -m_D M_N^{-1} m_D^{\mathsf T}$ is invariant under
$R \to R'$ by construction.  Step 3: consequence --- any statement of
the form ``the measured PMNS phase implies the sign of the baryon
asymmetry'' is false in the general type-I seesaw, because $R$ can
supply an arbitrary extra phase.  Physics intuition: Casas--Ibarra is
the field's honesty device.  It makes visible, in one equation,
exactly which parameters low-energy experiments can never touch --- and
therefore exactly what a theory must supply from elsewhere if it wants
to predict leptogenesis rather than accommodate it.  Hold this count
of nine; App.~\ref{app:leptogenesis} returns every one of them.

\medskip\noindent\textbf{J.2.1. State what a global fit is fitting, and
why the octant disagreement is not a mistake.}

\noindent\emph{Solution.}  Step 1: a global fit minimizes a
$\chi^2$ over six oscillation parameters against several hundred data
points from experiments with different $L/E$, different channels, and
different systematics.  Step 2: the atmospheric octant is determined
mainly by two competing handles --- the $\nu_\mu$ disappearance
amplitude, which depends on $\sin^2 2\theta_{23}$ and so is
\emph{octant-blind}, and the $\nu_e$ appearance rate, which depends on
$\sin^2\theta_{23}$ and is not, but is entangled with $\dPMNS{}$ and
the ordering.  Step 3: the global fit \citep{NuFIT61} weights the
atmospheric data heavily and lands at $0.470$; the accelerator joint
fit \citep{T2KNOvA2025} weights appearance and lands at $0.56$.  Both
are correct analyses of different data.  Physics intuition: this is
what a genuine experimental tension looks like from inside, and a
student should resist the urge to average.  The disagreement is
\emph{information}: it says the octant is being determined by a
sub-leading effect that two datasets see differently, and it will be
settled by a measurement (Hyper-K, DUNE) rather than by a weighted
mean.  A prediction that lands in one octant should say so and wait.

\medskip\noindent\textbf{J.2.2. Show that the seesaw explains the
smallness but not the scale.}

\noindent\emph{Solution.}  Step 1: the seesaw formula
\eqref{eq:seesaw-typeI-app} contains two unknowns, $m_D$ and $M_N$, and the
data determine one combination.  Step 2: therefore, for any $M_N$ in
the range $[10^{2}, 10^{16}]$~GeV there is a Yukawa
$\Ynu{} = \sqrt{2 m_\nu M_N}/v$ that reproduces the measured mass:
fourteen orders of magnitude of heavy scale are equally consistent
with all oscillation data.  Step 3: the constraint that does bite is
not from the neutrino sector at all --- it is that thermal
leptogenesis in the standard picture requires $M_N \gtrsim 10^9$~GeV
(next chapter), and even that is a bound on a \emph{mechanism}, not on
the seesaw.  Physics intuition: the seesaw is a mechanism, not a
prediction.  It converts ``why is the neutrino light?'' into ``what is
the heavy scale, and what is the Yukawa?'' --- two questions where
there was one.  That is real progress, because the new questions are
about structure rather than about a small number; but a course should
not let a student leave believing the seesaw predicts $10^{15}$~GeV.

\medskip\noindent\textbf{J.2.3. Estimate what would have to be true for
the mass ordering to be inverted.}

\noindent\emph{Solution.}  Step 1: inverted ordering means
$m_3 < m_1 < m_2$, so the two nearly degenerate states are the heavy
pair and $\Summnu{} \gtrsim 2\sqrt{|\Delta m^2_{31}|} = 100$~meV.
Step 2: DESI DR2 in $\Lambda$CDM gives $\Summnu{} < 64.2$~meV
\citep{DESI2025}, which is already \emph{below} the inverted-ordering
minimum --- cosmology disfavors inverted ordering independently of
any oscillation data.  Step 3: the caveat that keeps this honest: the
same data allow $\Summnu{} < 163$~meV once $w_0$ and $w_a$ float, and
the inverted minimum survives comfortably there.  So the correct
statement is that inverted ordering is disfavored \emph{conditional on
$\Lambda$CDM}.  Physics intuition: this is the cleanest example in
contemporary physics of a particle-physics question being settled by
cosmology --- and of why the settlement is provisional.  A student
should be able to say, in one sentence, which assumption is carrying
the conclusion; here it is the dark-energy equation of state, which
has nothing to do with neutrinos and everything to do with the answer.

\medskip\noindent\textbf{J.3.1. Evaluate the neutrino floor from the
reused seam.}

\noindent\emph{Solution.}  Step 1: the one admissible operator reuses
$\EGUT = \EPl\,e^{-2\pi} = 2.280\times10^{16}$~GeV --- no new scale ---
so the floor \eqref{eq:nu-floor-app} is $m_\star = \pi v^2/\EGUT$.
Step 2: with the first-order $v = 246.33$~GeV
\eqref{eq:v-arith-app}: $m_\star = \pi \times 60678/2.280\times10^{16}
~\mathrm{GeV} = 8.36114$~meV.  Step 3: compare the fermion chapter's
problem H.8.1 and the cosmology companion's problem I.1.1 (\ATen{}):
the same lever read from three directions --- here the scale is
\emph{supplied} and the mass descends as a prediction, with no free
right-handed mass anywhere in the construction, which is the concrete
content of the answer to problem J.1.3.  Physics intuition: the meV
decade of neutrino physics is the geometric shadow of the
$10^{16}$-GeV seam --- the two most extreme mass scales ever measured
are one ratio, and the $\pi$ out front is the same compact circle that
runs the whole flavor sector.

\medskip\noindent\textbf{J.3.2. Score the rank-two ledger honestly: the
two splittings separately, not their ratio.}

\noindent\emph{Solution.}  Step 1: the first-order eigenvalues
$(m_1, m_2, m_3) = (0, 1, 6)\,m_\star$ give
$\Delta m^2_{21} = m_\star^2 = 6.9909\times10^{-5}$~eV$^2$ and
$\Delta m^2_{31} = 36\,m_\star^2 = 2.5167\times10^{-3}$~eV$^2$.
Step 2: score each against \citet{NuFIT61}
($7.53\times10^{-5}$ and $2.511\times10^{-3}$): the atmospheric
splitting is $+0.23\%$, the solar one $-7.16\%$.  The ratio $36$
against the measured $33.35$ is $+7.96\%$, and the QD-completed
spectrum gives $34.56$, or $+3.65\%$.  Step 3: the honest reading is
that quoting the \emph{ratio} hides the structure --- one integer of
$(0,1,6)$ is performing to two parts in a thousand and the other is
off by seven percent, and it is the solar one.  With JUNO now
measuring $\Delta m^2_{21}$ to $1.6\%$ \citep{JUNO2026} the gap is
$4.2\sigma$ against JUNO alone and no longer inside anybody's error bar.  Physics intuition:
 a framework that reports an average when its two
components disagree is hiding from its own data.  The right response
is to name the strained integer, which is the $1$ in $(0,1,6)$, and to
say what would close it --- the QD layer already moves it two-thirds of
the way, so the remaining gap is a question about the QD completion of
the Majorana channel and not about the channel itself.

\medskip\noindent\textbf{J.3.3. Test the ordering prediction against the
one number that could kill it.}

\noindent\emph{Solution.}  Step 1: the cascade-depth assignment puts the
\GGA{} at $\nu_1$, forcing $m_1 < m_2 < m_3$.  Step 2: the current
evidence is $2.2$--$2.3\sigma$ for normal after JUNO
\citep{EstebanJUNO2026}, and JUNO alone reaches $3\sigma$ around 2032
by a method independent of matter effects.  Step 3: this is a genuine
one-sided kill row --- confirmation adds little, since normal is
already preferred, but an established inverted ordering falsifies the
depth assignment outright and with it the whole cascade reading of the
neutrino block.  Physics intuition: the most valuable predictions are
the ones that can only lose.  This one has no free parameter to
absorb a surprise: the ground state of a cascade is its lightest
member, and if nature puts the lightest state at the top of the
spectrum then the object being described is not a cascade.

\medskip\noindent\textbf{J.3.4. Derive the $(D,S,A)$ triplet rather than
adopting it.}

\noindent\emph{Solution.}  Step 1: a rank-two symmetric form on a
three-dimensional family space has a one-dimensional kernel, so the
decomposition is $\mathbf3 = \mathbf1 \oplus \mathbf2$ --- one
distinguished direction and a plane.  Step 2: the Majorana channel is
generated by the \emph{single} operator \eqref{eq:weinberg-app} acting
identically on all three families, so the distinguished direction is
permutation-invariant: $|D\rangle = (1,1,1)^{\mathsf T}/\sqrt3$.
Step 3: the residual plane is resolved by the only family-distinguishing
structure available, the dyadic charged-lepton ladder, into
$\mu\!-\!\tau$-even $|S\rangle$ and $\mu\!-\!\tau$-odd $|A\rangle$;
nothing is chosen at any step.  Physics intuition: the basis is not
an ansatz, it is a consequence of the rank and of the fact that one
operator cannot tell three identical families apart.  This is the
whole reason the lepton angles come out large: the neutrino sector's
natural basis is maximally spread over the families while the charged
sector's is maximally concentrated, and the PMNS matrix is the angle
between ``spread'' and ``concentrated''.

\medskip\noindent\textbf{J.3.5. Score the closed-form angles against the
global fit --- and against the octant split.}

\noindent\emph{Solution.}  Step 1: the misalignment
\eqref{eq:pmns-angles-app} gives $\sin^2\theta_{12} = 25/81 = 0.30864$
against JUNO's $0.3092 \pm 0.0087$: $0.07\sigma$.  Step 2:
$\sin^2\theta_{13} = 3/140 = 0.021429$ against the reactor
$0.02248^{+0.00055}_{-0.00059}$: $1.8\sigma$ low.  And
$\sin^2\theta_{23} = 4/7 = 0.57143$ scored \emph{twice} --- inside the
accelerator joint fit's $0.56^{+0.03}_{-0.05}$, and more than four
sigma above the global fit's $0.470^{+0.017}_{-0.014}$.  Step 3: all
three are ratios of the ledger integers $(M,N,L;J,K) = (1,3,5;7,35)$
with no others available, and the famous contrast ``CKM small, PMNS
large'' becomes one statement: hierarchical-against-hierarchical
meshes finely (chapter~\ref{app:ckm}),
hierarchical-against-democratic can only mesh coarsely.  Physics intuition:
 when three angles that dominated two decades of
experimental effort come out as fractions a student can write on one
line, the right response is not belief but the falsification channels
--- and the atmospheric row is currently the sharpest of them, because
the two datasets disagree and the prediction sits squarely in one of
the two camps.

\medskip\noindent\textbf{J.3.6. Show that the solar angle now
discriminates between hypotheses.}

\noindent\emph{Solution.}  Step 1: the pure democratic overlap gives
$\sin^2\theta_{12} = 1/3 = 0.33333$, which is what tri-bimaximal
mixing predicts.  Step 2: JUNO's direct measurement is
$0.3092 \pm 0.0087$, so the democratic value is $2.8\sigma$ high.  The
ledger's $25/81 = 0.30864 = \tfrac13(1 - 2/27)$ is $0.07\sigma$ away.
Step 3: the discriminating quantity is the $7.4\%$ depression $2/27$,
and it is now resolved by experiment --- the measurement crossed the
threshold of being able to see it somewhere around the 2020s and JUNO
has made it decisive.  Physics intuition: for twenty years
$\sin^2\theta_{12} \simeq 1/3$ was the most-quoted approximate
relation in flavor physics, and a whole model-building literature was
built on making it exact.  The data have moved off it.  That is the
best kind of experimental result --- one that kills a beautiful
approximation --- and any framework claiming this sector must now
predict the \emph{deviation}, not the value.

\medskip\noindent\textbf{J.3.7. Show why the atmospheric angle is a
tilt about maximal, and why the sign is not free.}

\noindent\emph{Solution.}  Step 1: write the prediction in the form
\eqref{eq:atm-tilt-app}, $\sin^2\theta_{23} = \tfrac12(1 + 1/J)$: the
central value is \emph{maximal} mixing and the ledger supplies a
traceless tilt of size $1/(2J) = 1/14 = 0.0714$.  Step 2: the sign is
fixed by which direction the charged-lepton hierarchy distinguishes,
so the prediction is $0.5 + 0.0714 = 0.5714$ and \emph{not}
$0.5 - 0.0714 = 0.4286$ --- the two would be experimentally
indistinguishable in $\sin^2 2\theta_{23}$ and are very distinguishable
in appearance.  Step 3: therefore the octant is the prediction, and
F-SM-3e registers it: a global determination settling in the lower
octant at $5\sigma$ kills the assignment, and the same measurement in
the upper octant confirms it.  Physics intuition: most flavor
predictions are about a magnitude; this one is about a \emph{sign}, and
signs are much harder to accommodate after the fact.  Note also what
the form says structurally --- maximal mixing is the symmetric point,
$\mu$--$\tau$ exchange is the symmetry, and $J = 7$ is what breaks it.
That is a rather different statement from ``$\theta_{23}$ happens to be
near $45^\circ$''.

\medskip\noindent\textbf{J.3.8. Compute $\Summnu{}$ and locate it under
the cosmological bound.}

\noindent\emph{Solution.}  Step 1: the first-order spectrum sums to
$\Summnu{}^{(1)} = 0 + 8.36114 + 50.1668 = 58.528$~meV; the
QD-completed spectrum gives $59.171$~meV.  Step 2: against DESI DR2's
$\Lambda$CDM bound $\Summnu{} < 64.2$~meV \citep{DESI2025} the
prediction sits at $92\%$ of the bound, with $5.0$~meV of headroom.
Step 3: this is F-SM-3c, and it is the channel most likely to fire
first: the next cosmological data release either pushes the bound
below $59$~meV and kills the spectrum, or measures a nonzero
$\Summnu{}$ and tests it directly.  Physics intuition: the prediction
is not merely consistent with the bound, it is \emph{pressed against}
it, which is the most dangerous place a prediction can sit and the
most informative.  There is no version of this framework in which
$\Summnu{}$ is comfortably small --- the spectrum is nearly hierarchical
and its sum is nearly the atmospheric mass, so the number is what it
is, and the experiment either finds it in the next few years or
removes it.

\medskip\noindent\textbf{J.3.9. Compute $m_\beta$ and explain why
$\beta$ decay will not see it.}

\noindent\emph{Solution.}  Step 1: $m_\beta^2 = \sum_i |U_{ei}|^2m_i^2$
with $|U_{e2}|^2 = s_{12}^2c_{13}^2 = 0.30203$ and
$|U_{e3}|^2 = s_{13}^2 = 0.021429$.  Step 2: with the derived spectrum,
$m_\beta^2 = 0.30203 \times 69.909 + 0.021429 \times 2516.7 =
21.11 + 53.93 = 75.04~\mathrm{meV}^2$, so $m_\beta = 8.66$~meV.
Step 3: KATRIN's current limit is $450$~meV \citep{KATRIN2025} and its
design sensitivity is about $200$~meV --- a factor of twenty-three
above the prediction.  No planned kinematic experiment reaches it.
Physics intuition: it is worth being explicit that a prediction can be
unfalsifiable in one channel and sharply falsifiable in another.  The
same spectrum that hides from KATRIN by a factor of twenty-three sits
at $92\%$ of the DESI bound.  A student choosing a thesis should read
this pair of numbers as a map of where the decisive measurements are
--- and they are in cosmology and in double beta decay, not at the
tritium endpoint.

\medskip\noindent\textbf{J.3.10. Audit the chapter: which of Step~2's
ten items are actually closed?}

\noindent\emph{Solution.}  Step 1: list the closures.  Items 1 and 2
(the scale and the operator coefficient) close together, because
$\Lambda = \EGUT$ is reused rather than introduced.  Item~3 closes in
its first half --- rank two forces exactly two splittings --- and
strains in its second, at $-7.16\%$ on the solar splitting
\eqref{eq:splittings-scored-app}.  Items 4 (ordering), 5
(Majorana), 6 (three angles), 7 (Dirac phase), 8 (Majorana phases) and
10 (three families, imported from \PaperPTwo{}) are structural
outputs.  Step 2: item~9, the nine Casas--Ibarra parameters of problem
J.1.8, is answered in App.~\ref{app:leptogenesis} --- and answered
\emph{conditionally}, on a hypothesis whose crux is open.  Step 3: so
the honest tally is eight closed, one closed with a stated strain, one
closed conditionally; and both structural puzzles answered.  Physics intuition:
 the point of running this audit at the end of the
chapter is that it is very easy, in a framework with many derived
quantities, to let the successes carry the failures.  Writing the tally
down forces the strained solar splitting and the conditional heavy
sector to appear next to the eight clean closures, at the same size,
which is where a reader deciding what to believe needs them.


\GGtakeaway{\emph{Standard result:} oscillations, two mass splittings,
three large angles, and a sector the Standard Model must leave massless
at dimension four. \emph{What it assumed:} ten quantities, plus the
nine invisible parameters of the heavy sector, plus both structural
puzzles. \emph{The geometric reading:} one reused scale sets the meV
floor; a rank-two Majorana ledger forces exactly two splittings and the
normal ordering; a dyadic-against-democratic misalignment forces the
large angles as ratios of the ledger integers; the atmospheric angle is
a tilt about maximal whose \emph{sign} is the prediction. \emph{Where it
strains:} the solar splitting, at $-7.2\%$, named and quantified rather
than averaged into a ratio.}



\section{Majorana neutrinos, leptonic \texorpdfstring{\CP{}}{CP} violation, and neutrinoless double beta decay}
\label{app:majorana}

\GGteachbox{The three questions the previous chapter deferred. Step~1
builds the spinor technology from the Lorentz algebra --- Weyl, Dirac
and Majorana fields, and why only a \emph{neutral} fermion may be its
own antiparticle --- then derives the seesaw exactly, counts the
physical phases at general generation number, sets out \CP{} violation
in oscillations with its degeneracies, and obtains the
\(0\nu\beta\beta\) amplitude with the propagator step that produces
\(\mbb{}\). Step~2 lists eight quantities left open. Step~3 derives all
three phases as \emph{parity data of a rank-two ledger} rather than as
dials, works \(\mbb{}\) term by term, and reports the uncomfortable
consequence: the derived parity puts the prediction near the
\emph{destructive} end of its own accessible envelope.}
\GGreadalong{\citet{Bilenky2018} for the phase counting;
\citet{MohapatraPal2004} Ch.~5--7 for the seesaw variants and
\(0\nu\beta\beta\); \citet{Srednicki2007} Ch.~33--36 for two-component
spinor technology; \citet{GiuntiKim2007} Ch.~6 for the
charge-conjugation bookkeeping done carefully.}

This is the sixth sector appendix and the second of the three lepton
chapters. It takes up the three questions the previous chapter posed and
deferred: \emph{is the neutrino its own antiparticle}, \emph{what are the
three leptonic \CP{} phases}, and \emph{what would it take to measure
them}. The three are one question, because the only laboratory access to
the two Majorana phases runs through a process that exists only if the
neutrino is Majorana.

\subsection{Step 1 --- Majorana fermions and \texorpdfstring{\CP{}}{CP} violation as a graduate student meets them}
\label{app:majorana-step1}

\subsubsection{Weyl, Dirac, Majorana: three ways to be a spin-half field}
\label{app:majorana-spinors}

The Lorentz algebra \(\mathfrak{so}(3,1)\simeq
\mathfrak{su}(2)\oplus\mathfrak{su}(2)\) has two inequivalent
two-dimensional representations, \((\tfrac12,0)\) and \((0,\tfrac12)\) ---
the left- and right-handed \emph{Weyl} spinors \(\psi_L,\psi_R\). A Dirac
field is the pair \(\Psi=(\psi_L,\psi_R)^{\mathsf T}\), four complex
components, and its mass term \(m\bar\Psi\Psi=m(\bar\psi_L\psi_R+\mathrm{h.c.})\)
couples the two.

The Majorana observation is that one does not need two Weyl fields to
write a mass. Charge conjugation
\begin{equation}
\Psi^{c}\equiv C\bar\Psi^{\mathsf T},\qquad C=i\gamma^2\gamma^0,
\label{eq:cconj-app}
\end{equation}
maps a left-handed field to a \emph{right-handed} one:
\((\psi_L)^{c}\) transforms in \((0,\tfrac12)\). So \(\psi_L\) and
\((\psi_L)^{c}\) are exactly the two ingredients a mass term needs, and
one may write
\begin{equation}
\mathcal L_M=-\tfrac12\,m\,\overline{(\psi_L)^{c}}\,\psi_L+\mathrm{h.c.}
=-\tfrac12\,m\,\psi_L^{\mathsf T}C\psi_L+\mathrm{h.c.}
\label{eq:majmass-app}
\end{equation}
A field satisfying \(\Psi=\Psi^{c}\) is a \emph{Majorana} field. It has
two, not four, independent complex components --- the particle and the
antiparticle are the same state, distinguished only by helicity.
\emph{(Physics intuition: a Dirac fermion is a fermion with a
conserved arrow --- something that distinguishes it from its
antiparticle, and that something is always a charge. Remove the charge
and the arrow has nothing to point along; the particle and antiparticle
become two helicity states of one object. The factor of \(\tfrac12\) in
\eqref{eq:majmass-app} is the same \(\tfrac12\) that appears in front of
any real field's mass term, and for the same reason: you are
double-counting if you do not put it there.)}

\paragraph{The one-sentence reason this is a neutrino question.}
\eqref{eq:majmass-app} takes \emph{two} \(\psi_L\)'s, so it carries
\emph{twice} whatever charge \(\psi_L\) carries. It is therefore
forbidden for every charged fermion by electromagnetic gauge invariance
alone, and permitted for exactly one Standard-Model field: the neutrino.
The Dirac-or-Majorana question exists for the neutrino and for nothing
else in the observed particle content --- which is why an answer would be
a structural fact and not a detail.

\subsubsection{The seesaw, diagonalized exactly}
\label{app:majorana-seesaw}

Take one generation for clarity. With \(\nu_L\) and a singlet \(N_R\), the
Lagrangian permits a Dirac mass \(m_D\) and, since \(N_R\) is a total
gauge singlet, a Majorana mass \(M\) for \(N_R\) --- but \emph{not} for
\(\nu_L\), by the argument above. In the basis
\((\nu_L,\,N_R^{c})\) this is
\begin{equation}
\mathcal M=\begin{pmatrix}0&m_D\\ m_D&M\end{pmatrix}.
\label{eq:seesaw2x2-app}
\end{equation}
The eigenvalues are exactly
\begin{equation}
\lambda_\pm=\frac{M}{2}\left(1\pm\sqrt{1+\frac{4m_D^2}{M^2}}\right),
\label{eq:seesaweig-app}
\end{equation}
and for \(m_D\ll M\) these expand to
\begin{equation}
\lambda_-\simeq-\frac{m_D^2}{M},\qquad \lambda_+\simeq M+\frac{m_D^2}{M},
\qquad
\tan2\vartheta=\frac{2m_D}{M}\ \Rightarrow\ \vartheta\simeq\frac{m_D}{M}.
\label{eq:seesawexp-app}
\end{equation}
\emph{Both} eigenstates are Majorana --- the mass matrix
\eqref{eq:seesaw2x2-app} is symmetric, and a symmetric mass matrix for
neutral fields has Majorana eigenstates. That is the essential structural
point and it is easy to lose: the light neutrino is Majorana not because
somebody added a Majorana mass for it, but because the heavy singlet's
Majorana mass propagates down. The minus sign in \(\lambda_-\) is
absorbed into the Majorana phase, which is the first hint that these
phases are bookkeeping about \emph{signs} rather than free angles.
\emph{(Physics intuition: the determinant of \eqref{eq:seesaw2x2-app} is
\(-m_D^2\), fixed; so the product of the two eigenvalues is fixed. Push
one up and the other must come down by exactly the same factor. That is
the entire mechanism, and it is why the name is right: a seesaw has a
fixed fulcrum, and the fulcrum here is \(m_D\), not \(M\).)}

\subsubsection{Counting the physical phases}
\label{app:majorana-counting}

An \(n\times n\) unitary matrix has \(n^2\) real parameters:
\(n(n-1)/2\) angles and \(n(n+1)/2\) phases. How many phases survive
rephasing depends on whether the neutrino mass term permits
\(\nu\to e^{i\phi}\nu\):
\begin{equation}
N_{\rm phase}^{\rm Dirac}=\frac{(n-1)(n-2)}{2},
\qquad
N_{\rm phase}^{\rm Majorana}=\frac{n(n-1)}{2}.
\label{eq:phasecount-app}
\end{equation}
For \(n=3\): one phase if Dirac, \emph{three} if Majorana --- one Dirac
phase \(\dPMNS{}\) and two Majorana phases \(\alpha_{21},\alpha_{31}\),
entering as
\begin{equation}
U=\tilde U\,\mathrm{diag}\!\left(1,\,e^{i\alpha_{21}/2},\,e^{i\alpha_{31}/2}\right).
\label{eq:majphases-app}
\end{equation}
The half-angles are not a convention chosen for elegance: the mass term
\eqref{eq:majmass-app} is quadratic in the field, so a phase \(\phi\) on
the field appears as \(2\phi\) in the mass. Writing \(\alpha/2\) in the
matrix makes \(\alpha\) the physical, mass-term phase --- and it is
\(\alpha\), not \(\alpha/2\), that Step~3 derives.
\emph{(Physics intuition: the two extra phases are exactly the two
relative signs --- more generally, the two relative phases --- among the
three Majorana mass eigenvalues, which a Dirac theory would have rotated
away into the fields. They are invisible in oscillations because an
oscillation probability depends on \(U_{\alpha i}U^*_{\beta i}\), in which
the diagonal phase matrix cancels identically. Any process that is to see
them must be quadratic in \(U\) without the conjugate --- and that means
lepton-number violating.)}

\subsubsection{\texorpdfstring{\CP{}}{CP} violation in oscillations, and the three degeneracies}
\label{app:majorana-cpviol}

The Dirac phase is measurable in oscillations through the asymmetry
\eqref{eq:cpasym-app}. In practice the measurement is hard for three
reasons, and a student should know all three because they are what the
next decade of experiments is designed around.

\emph{The matter fake.} The charged-current potential
\eqref{eq:vcc-app} has opposite sign for antineutrinos, so matter
generates an asymmetry that has nothing to do with \CP{}. At DUNE's
\(1300\)~km the matter effect is comparable to the \CP{} effect, and the
two are separated by their different energy dependence, not by a single
measurement.

\emph{The octant degeneracy.} The appearance rate depends on
\(\sin^2\theta_{23}\sin^22\theta_{13}\), so an upper-octant
\(\theta_{23}\) with one \(\dPMNS{}\) can mimic a lower-octant
\(\theta_{23}\) with another. This is the same octant question that
\S\ref{sec:theta23-split} finds the datasets disagreeing about.

\emph{The ordering degeneracy.} Flipping the sign of \(\Delta m^2_{31}\)
changes the matter term, and can be partially compensated by a shift in
\(\dPMNS{}\).

The current status reflects all three: the T2K--NOvA joint fit
\citep{T2KNOvA2025} gives \(\dPMNS{}=203.4^{+63.0}_{-37.8}\)~degrees for
normal ordering and excludes \(\dPMNS{}=+\pi/2\) at \(3\sigma\) --- but
notably \emph{not} \(3\pi/2\), the value that maximizes \CP{} violation
with the opposite sign.

\subsubsection{Neutrinoless double beta decay}
\label{app:majorana-0nubb}

\paragraph{The process and the amplitude.} Ordinary double beta decay,
\((A,Z)\to(A,Z+2)+2e^-+2\bar\nu_e\), is a second-order weak process
observed in about a dozen isotopes. Its neutrinoless partner
\begin{equation}
(A,Z)\to(A,Z+2)+2e^-
\label{eq:0nubb-app}
\end{equation}
violates lepton number by two units and is forbidden unless the neutrino
is Majorana. The amplitude joins the two vertices with a neutrino
propagator; because the exchanged neutrino is virtual with momentum of
order the nuclear Fermi scale \(|p|\sim 100\)~MeV \(\gg m_i\), the
propagator's \emph{mass} term is what survives the chirality projection,
\begin{equation}
\frac{\slashed p+m_i}{p^2-m_i^2}\;\xrightarrow{\ \text{chirality}\ }\;
\frac{m_i}{p^2},
\label{eq:prop-app}
\end{equation}
and each mass eigenstate contributes with weight \(U_{ei}^2\) --- the
square, \emph{not} \(|U_{ei}|^2\), because no conjugate appears. Summing,
the whole particle-physics content is one number:
\begin{equation}
\boxed{\;
\mbb{}=\left|\sum_i U_{ei}^2\,m_i\right|
=\left|\,m_1c_{12}^2c_{13}^2
+m_2s_{12}^2c_{13}^2e^{i\alpha_{21}}
+m_3s_{13}^2e^{i(\alpha_{31}-2\dPMNS{})}\right| .\;}
\label{eq:mbb-def-app}
\end{equation}
The rate is
\begin{equation}
\bigl[T^{0\nu}_{1/2}\bigr]^{-1}=G^{0\nu}\,\bigl|M^{0\nu}\bigr|^2\,
\frac{\mbb{}^2}{m_e^2},
\label{eq:0nubb-rate-app}
\end{equation}
with \(G^{0\nu}\) an exactly calculable phase-space factor and
\(M^{0\nu}\) a nuclear matrix element whose value differs between
many-body methods by a factor of two to three --- the dominant theory
uncertainty in the entire field.
\emph{(Physics intuition: the reason \(\mbb{}\) is a coherent sum with
phases, while the \(\beta\)-decay observable \(m_\beta\) is an incoherent
sum of squares, is that here the three mass eigenstates are exchanged as
\emph{virtual} particles inside one amplitude and there interfere, while
in \(\beta\) decay they are produced as \emph{real} particles in
distinguishable final states and there do not. Interference is exactly
what makes the Majorana phases observable --- and exactly what allows the
answer to be small through cancellation.)}

\paragraph{The two bands, and the funnel.} Plotting \eqref{eq:mbb-def-app}
against the lightest mass with the phases scanned gives two allowed bands.
For \emph{inverted} ordering the two dominant states are nearly
degenerate and their contributions cannot fully cancel, so there is a
floor: \(\mbb{}\gtrsim 19\)~meV, computed in problem~K.1.7. For
\emph{normal} ordering there is no floor --- the band passes through zero
in a narrow ``funnel'' near a lightest mass of a few meV, where the three
terms can cancel exactly. \emph{This asymmetry is the field's strategic
fact:} a tonne-scale experiment covers the inverted band completely and
therefore either finds a signal or excludes inverted ordering, whereas
the normal band is only partially covered and a null result there proves
nothing.

\paragraph{Current limits.} The strongest are
\(T^{0\nu}_{1/2}>3.8\times10^{26}\)~yr in \(^{136}\)Xe
\citep{KamLANDZen2024}, \(>1.9\times10^{26}\)~yr in \(^{76}\)Ge
\citep{LEGEND2025} and \(>3.5\times10^{25}\)~yr in \(^{130}\)Te
\citep{CUORE2025}, corresponding to \(\mbb{}\) limits in the few tens of
meV with the nuclear-matrix-element spread dominating. The next
generation, LEGEND-1000 and CUPID, targets \(9\)--\(21\)~meV.

\paragraph{The black-box theorem, and its honest size.}
\citet{SchechterValle1982} proved that \emph{any} mechanism producing
\eqref{eq:0nubb-app} generates a Majorana neutrino mass, by closing the
process's own diagram into a self-energy. So an observation of
\(0\nu\beta\beta\) proves the neutrino is Majorana whatever the
underlying mechanism. The theorem is qualitative and must be quoted as
such: the mass so generated is of order \(10^{-24}\)~eV, thirteen orders
of magnitude below the observed scale, so the theorem establishes
\emph{that} the neutrino is Majorana and says nothing about \emph{how
much} of the observed rate comes from light-neutrino exchange.

\paragraph{Worked example (STANDARD TEXTBOOK PEDAGOGY).}
The rate \eqref{eq:0nubb-rate-app} scales as \(\mbb{}^2\), so
\emph{half-life sensitivity} scales as \(\mbb{}^{-2}\). In the
background-limited regime a half-life reach grows only as
\(\sqrt{Mt/(b\,\Delta E)}\) --- the square root of exposure over
background. Chaining the two: improving the \(\mbb{}\) reach by a factor
\(f\) costs \(f^2\) in half-life and \(f^4\) in exposure. From the
next generation's \(9\)~meV to a hypothetical \(1.8\)~meV is \(f=5\),
hence \(25\) in half-life and \(\mathbf{625}\) in exposure. That single
arithmetic line is the reason a meV-scale \(\mbb{}\) is not an
incremental goal, and Step~3 will show it is exactly where this framework
puts its prediction.

\subsection{Step 2 --- what the Majorana sector assumes but cannot explain}
\label{app:majorana-step2}

\begin{enumerate}\itemsep3pt
\item \textbf{Whether the neutrino is Majorana at all.} Not predicted;
both completions are consistent.
\item \textbf{The Dirac phase \(\dPMNS{}\).} A free parameter, currently
constrained only to a broad interval.
\item \textbf{The two Majorana phases.} Free, and \emph{unconstrained by
any measurement ever made}.
\item \textbf{The value of \(\mbb{}\).} With the phases free, even a
perfectly known spectrum leaves \(\mbb{}\) uncertain by the full width of
its envelope --- typically a factor of two to three.
\item \textbf{Whether light-neutrino exchange dominates the rate.} Other
lepton-number-violating mechanisms contribute; the black-box theorem does
not separate them.
\item \textbf{The nuclear matrix elements.} A factor two to three of
theory uncertainty, unrelated to particle physics and currently
irreducible.
\item \textbf{Why leptonic \CP{} violation should be large when the
quark one is small.} Recorded, not explained.
\item \textbf{Whether any of this connects to the baryon asymmetry.} As
problem~J.1.8 showed, the Casas--Ibarra matrix \(R\) severs the
connection in the general seesaw.
\end{enumerate}

\emph{(Physics intuition: items 2, 3 and 7 are the same item seen three
ways. The Standard Model's answer to ``why is there a \CP{} phase?'' is
``because three families permit one'' --- a statement about what is
\emph{allowed}. Nothing anywhere in the framework says what the phase
\emph{is}, and for the two Majorana phases nothing even says they are
not both zero. A theory that derived all three would be making its
strongest possible claim in its least constrained sector.)}

\subsection{Step 3 --- the geometric derivation: phases as parity data}
\label{app:majorana-step3}

\subsubsection{Why the parities are data and not dials}
\label{app:majorana-parity}

The framework's answer to Step~2's items 1, 2 and 3 rests on a single
structural fact established in the previous chapter: the Majorana ledger
is \emph{rank two}. A rank-two symmetric form on the three-family space
has a one-dimensional kernel, and the relative phases among its
eigenvalues are therefore not continuous parameters at all --- they are
the \emph{signs} that a rank-two form is allowed to carry, which is to
say a discrete parity assignment. The framework derives
\begin{equation}
\boxed{\;(\alpha_{21},\,\alpha_{31})=(0,\,\pi).\;}
\label{eq:parities-app}
\end{equation}
Two things follow at once. The neutrino is Majorana --- item~1 --- because
the operator \eqref{eq:weinberg-app} is the only admitted mass channel and
it is lepton-number violating by construction. And the phases are not
tunable: \eqref{eq:parities-app} is a choice between a finite number of
sign assignments, not a point in a two-torus, so the framework cannot
adjust \(\mbb{}\) to taste.

\paragraph{The half-angle, and why a parity of \(\pi\) is not the same as a
phase of \(\pi\).} Because the Majorana phases enter \eqref{eq:majphases-app}
halved, the parity \(\alpha_{31}=\pi\) appears in the mixing matrix as
\begin{equation}
e^{i\alpha_{31}/2}=e^{i\pi/2}=i .
\label{eq:halfangle-app}
\end{equation}
A parity of \(\pi\) in the mass term is a factor of \(i\) --- a
\emph{quarter} turn --- in the amplitude. In \(\mbb{}\), which is
quadratic in \(U\), the \(i\) squares back to \(-1\) and flips the sign of
the third term; in any amplitude \emph{linear} in \(U\) it would instead
rotate by ninety degrees. The leptogenesis chapter uses precisely this,
and \S\ref{sec:lg-halfangle} is where the sign it sets is cashed.
\emph{(Physics intuition: this is the single most useful thing to
understand about Majorana phases. They are not angles you dial; they are
square roots of signs. A relative minus sign between two Majorana masses
--- the only thing a rank-two ledger is free to have --- appears in the
mixing matrix as a factor of \(i\), and whether you see a sign flip or a
quarter turn depends entirely on whether your observable is quadratic or
linear in \(U\). That is why the same parity makes \(\mbb{}\) smaller and
makes the \CP{} asymmetry of leptogenesis nonzero.)}

\subsubsection{The Dirac phase, derived}
\label{app:majorana-dirac}

The Dirac phase is fixed by the same compact-angle holonomy logic that
fixes the quark phase, read on the lepton sector's reference channel. The
result is the closed form
\begin{equation}
\dPMNS{}=\pi+2\arcsin\frac1J=\pi+2\arcsin\frac17=196.426421^{\circ},
\label{eq:dirac-closed-app}
\end{equation}
with the QD-completed value \(196.059074^{\circ}\). Three features
deserve comment. The base value is \(\pi\), \emph{not} zero, because the
conventional parameterization refers the phase to a channel entering with
a negative real coefficient; converting to a positive real axis is a
rotation by half a turn. It is worth being explicit that this is
\emph{not} the Majorana parity \(\alpha_{31} = \pi\) reappearing: the two
are both \(\pi\), and they may well share a geometric origin in the
parity-flipped reference channel, but they are distinct observables and
K.1.5 shows that only the first is visible to oscillation. The
residual is \(2\arcsin(1/J)\) and not \(\arcsin(1/J)\), because the phase
is read on a channel traversed twice, which is the same doubling that put
the half-angle in \eqref{eq:halfangle-app}. And \(J=7\) is the ledger's
own integer, the same one that tilts the atmospheric angle in
\eqref{eq:atm-tilt-app} --- so the atmospheric angle and the Dirac phase
are two readings of one number, and cannot be adjusted independently.

Against data: the T2K--NOvA joint fit gives
\(203.4^{+63.0}_{-37.8}\)~degrees \citep{T2KNOvA2025}, so
\(196.4^{\circ}\) sits well inside the measured interval --- but the
interval is \(100\) degrees wide, and this is a prediction awaiting an
experiment rather than one that has passed a test. F-SM-3d registers it:
a \(5\sigma\) determination of \(\dPMNS{}\) outside
\([186^{\circ},207^{\circ}]\) kills it, and DUNE and Hyper-K are the
facilities.

\subsubsection{$\mbb{}$, worked}
\label{app:majorana-mbb}

Rather than quote the result, work it; the way the three terms combine is
the content. At first order \(m_1=0\), so the first term of
\eqref{eq:mbb-def-app} vanishes identically and the answer is a two-term
interference. With \(s_{12}^2=25/81\), \(s_{13}^2=3/140\) and hence
\(c_{13}^2=137/140\), and with \(\alpha_{21}=0\) so that the second term
is real and positive:
\begin{equation}
m_2 s_{12}^{2}c_{13}^{2}
=8.36114\times\tfrac{25}{81}\times\tfrac{137}{140}=+2.52530~\mathrm{meV}.
\label{eq:mbb-t2-app}
\end{equation}
The third term carries the phase
\(\alpha_{31}-2\dPMNS{}=\pi-2(196.426421^{\circ})\) and the modulus
\(m_3s_{13}^2=50.1668\times 3/140=1.07500\)~meV, giving
\begin{equation}
m_3 s_{13}^{2}e^{i(\alpha_{31}-2\dPMNS{})}=-0.90307+0.58317\,i~\mathrm{meV}.
\label{eq:mbb-t3-app}
\end{equation}
\emph{The real parts oppose}: \(+2.52530\) against \(-0.90307\). Adding
and taking the modulus,
\begin{equation}
\mbb{}^{(1)}=\bigl|\,2.52530-0.90307+0.58317\,i\,\bigr|=1.72386~\mathrm{meV},
\label{eq:mbb-first-app}
\end{equation}
and at QD order the first term is no longer zero
(\(m_1^{\rm QD}=0.0653\)~meV contributes \(+0.04416\)~meV), giving
\(\mbb{}^{\rm QD}=1.79925\)~meV.

\subsubsection{The finding: the derived parity sits near the worst case}
\label{app:majorana-envelope}

The cancellation in \eqref{eq:mbb-first-app} is not incidental --- it is
most of the answer, and it is caused by the derived parity. Two
comparisons on the \emph{same} derived spectrum and angles make its size
concrete.
\begin{itemize}\itemsep2pt
\item With the other admissible parity, \(\alpha_{31}=0\), the third
term's real part changes sign and adds: \(\mbb{}\) would be
\(3.62\)~meV, twice as large.
\item Scanning both Majorana phases freely over the full circle --- which
is what an experiment analyzing a generic normal-ordering spectrum must
allow --- gives the envelope \(\mbb{}\in[1.44,\,3.74]\)~meV.
\end{itemize}
The derived value \(1.80\)~meV therefore lands in the \emph{lowest
quarter} of the accessible range, near the maximally destructive end.
\textbf{This framework's Majorana prediction is harder to confirm than a
generic one}, and the difficulty is a direct consequence of the parity
the framework derives rather than of any choice made to avoid a
constraint. The cost is paid for the parity being \emph{derived}: a
framework free to choose \(\alpha_{31}\) would place its prediction at the
top of the envelope and claim a better chance of confirmation, about a
number it had fitted.

Two consequences. The prediction is one-sided in the sense of F-SM-3b --- a
confirmed \(0\nu\beta\beta\) line anywhere in the tonne-scale window of
\(9\)--\(21\)~meV kills it, while a null result there proves nothing ---
and by the exposure arithmetic of the Step~1 worked example, reaching
\(1.8\)~meV needs \(625\) times the exposure of an experiment sensitive to
\(9\)~meV. This is not a prediction any planned experiment will confirm.
It is a prediction that a positive result in the approved program would
\emph{refute}, and it is offered on that basis.

\subsubsection{The two unitarity triangles, at true relative area}
\label{app:majorana-triangles}

The Jarlskog invariant is twice the area of the unitarity triangle, so the
two sectors can be compared directly. This paper derives both:
\begin{equation}
\frac{\bigl|\JPMNS{}\bigr|}{\bigl|J_{\rm CKM}\bigr|}
=\frac{9.187\times10^{-3}}{2.984\times10^{-5}}=308
\qquad\text{(QD order; \(338\) at first order).}
\label{eq:triangle-ratio-app}
\end{equation}
\emph{The leptonic unitarity triangle has about three hundred times the
area of the quark one}, and because both numbers are outputs of the same
ledger the ratio is a prediction rather than an observation. The quark
side compares two hierarchical Dirac sectors and gets a thin triangle;
the lepton side compares a dyadic charged-lepton sector against a nearly
democratic Majorana one and gets a fat one. \emph{Same constitution,
different projector geometry} --- and three hundred is how much
different.

The caveat matters and is stated wherever this ratio is used: three
hundred is \emph{not} ten orders of magnitude, so this is not why the
lepton sector can pay for baryogenesis when the quark sector cannot. That
is the \(\Delta L=2\) door and the out-of-equilibrium lock crossing, and
the next chapter finds that the \emph{unflavored} invariant --- the one
\(\JPMNS{}\) belongs to --- drops out of the matter asymmetry entirely.

\subsubsection{Honest status}
\label{app:majorana-honesty}

Items 1, 2, 3, 4 and 7 of Step~2 are answered: the neutrino is Majorana
by construction, all three phases are derived, and \(\mbb{}\) follows
with no envelope left. Item~8 is answered in the next chapter, and
answered conditionally. Items 5 and 6 are \emph{not} answered and are not
the framework's to answer --- whether light-neutrino exchange dominates,
and what the nuclear matrix elements are, are questions for the
lepton-number-violation and nuclear-structure communities respectively,
and the second of them is currently the larger uncertainty on any
\(\mbb{}\) limit quoted anywhere.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{K.1.1. Show that charge conjugation turns a
left-handed Weyl field into a right-handed one, and count the degrees of
freedom.}

\noindent\emph{Solution.}  Step 1: under the Lorentz algebra
$\psi_L \in (\tfrac12,0)$; the operation \eqref{eq:cconj-app} involves
complex conjugation, which exchanges the two $\mathfrak{su}(2)$
factors, so $(\psi_L)^c \in (0,\tfrac12)$.  Step 2: therefore
$\overline{(\psi_L)^c}\psi_L$ is a Lorentz scalar built from
\emph{one} Weyl field, and \eqref{eq:majmass-app} is a legitimate mass
term.  Step 3: count.  A Dirac field is two Weyl fields, four complex
components, describing particle and antiparticle each with two
helicities.  A Majorana field is one Weyl field, two complex
components, and the ``antiparticle'' is the opposite helicity state of
the same object.  Physics intuition: mass, for a fermion, is whatever
couples left to right.  Nature offers two ways to find a right-handed
partner for $\psi_L$ --- import one ($\psi_R$, Dirac) or manufacture one
by conjugation ($(\psi_L)^c$, Majorana) --- and the second is free.
It is available only when there is no charge to conjugate, which is
why this is a neutrino question and only a neutrino question.

\medskip\noindent\textbf{K.1.2. Prove that no charged fermion can have a
Majorana mass.}

\noindent\emph{Solution.}  Step 1: the term \eqref{eq:majmass-app} is
$\psi_L^{\mathsf T}C\psi_L$, bilinear in $\psi_L$ with \emph{no}
conjugate.  Step 2: under a $\U(1)_{\rm EM}$ rotation
$\psi_L \to e^{iq\theta}\psi_L$, the term picks up $e^{2iq\theta}$, so
it is invariant only for $q = 0$.  Step 3: hence the electron, muon,
tau and every quark are excluded by electromagnetic gauge invariance
alone, and among Standard-Model fermions only the neutrino survives ---
and even it survives only at dimension five \eqref{eq:weinberg-app},
because $\nu_L$ sits in a doublet carrying weak isospin and
hypercharge.  Physics intuition: ``is it its own antiparticle?'' is
really ``does it carry any conserved charge at all?''  A photon is its
own antiparticle for the same reason.  The neutrino is the only
\emph{fermion} that could be, and whether it actually is depends on
whether $B - L$ --- the last global symmetry that could still forbid it
--- is a symmetry of nature or an accident of the Standard Model's
field content.

\medskip\noindent\textbf{K.1.3. Diagonalize the seesaw exactly, then
check the expansion.}

\noindent\emph{Solution.}  Step 1: the eigenvalues of
\eqref{eq:seesaw2x2-app} solve
$\lambda^2 - M\lambda - m_D^2 = 0$, giving \eqref{eq:seesaweig-app}.
Step 2: expand for $m_D \ll M$ using
$\sqrt{1+x} \simeq 1 + x/2 - x^2/8$ with $x = 4m_D^2/M^2$:
$\lambda_- \simeq -m_D^2/M$ and $\lambda_+ \simeq M + m_D^2/M$, and the
mixing angle is $\vartheta \simeq m_D/M$.  Step 3: check the invariant
--- $\det\mathcal M = -m_D^2$ exactly, so
$\lambda_+\lambda_- = -m_D^2$ at every order, which is the seesaw
identity itself.  Physics intuition: the determinant is the thing that
cannot move.  Whatever you do to the heavy mass, the product of the two
eigenvalues is pinned to $m_D^2$, so the light state descends by exactly
the factor the heavy state ascends.  Notice also that the light
eigenvalue is \emph{negative}: that minus sign is not physical by
itself, it is absorbed into a Majorana phase --- your first sighting of a
Majorana phase as a bookkeeping sign rather than a free angle.

\medskip\noindent\textbf{K.1.4. Derive the phase counting at general
$n$, and find where the two cases first differ.}

\noindent\emph{Solution.}  Step 1: an $n \times n$ unitary matrix has
$n^2$ real parameters, $n(n-1)/2$ of them angles, leaving
$n(n+1)/2$ phases.  Step 2: Dirac --- $2n$ fields absorb phases but one
overall phase is inert, so $2n-1$ are removable and
$n(n+1)/2 - (2n-1) = (n-1)(n-2)/2$ survive.  Majorana --- only the $n$
charged leptons may be rephased, leaving $n(n-1)/2$.  Step 3: at
$n = 2$ the counts are $0$ and $1$: two generations of Majorana
neutrinos already admit one physical phase, while two generations of
Dirac fermions admit none.  Physics intuition: this is a sharper
statement than the familiar ``three families are needed for \CP{}
violation'', which is a \emph{Dirac} statement.  Majorana neutrinos
violate \CP{} with two families --- there is simply no oscillation
experiment that can see it, because the phase lives in the sector
oscillations cancel.  The Kobayashi--Maskawa counting and the Majorana
counting are different theorems and a student should never conflate
them.

\medskip\noindent\textbf{K.1.5. Show that the Majorana phases cancel in
every oscillation probability.}

\noindent\emph{Solution.}  Step 1: write
$U = \tilde U P$ with $P = \mathrm{diag}(1, e^{i\alpha_{21}/2},
e^{i\alpha_{31}/2})$ as in \eqref{eq:majphases-app}.  Step 2: every
term of \eqref{eq:three-app} is a product
$U^*_{\alpha i}U_{\beta i}U_{\alpha j}U^*_{\beta j}$, in which index $i$
appears once conjugated and once not, and likewise $j$: each phase
$e^{i\alpha_i/2}$ meets its own conjugate and cancels.  Step 3: hence
\emph{no} oscillation experiment, at any baseline, in any channel, can
ever measure $\alpha_{21}$ or $\alpha_{31}$ --- this is a theorem, not a
sensitivity statement.  Physics intuition: oscillations measure the
neutrino's \emph{propagation}, and propagation cannot tell a particle
from an antiparticle if the two are the same object.  The phases that
record ``the same object'' are therefore invisible to it by
construction.  Only a process that converts a neutrino into an
antineutrino --- that is, one violating lepton number --- can see them,
which is why $0\nu\beta\beta$ is not merely the best probe but
essentially the only one.

\medskip\noindent\textbf{K.1.6. Derive $\mbb{}$ from the propagator,
and explain the square.}

\noindent\emph{Solution.}  Step 1: the two vertices of
\eqref{eq:0nubb-app} each emit a $\nu$ (not a $\bar\nu$), so the
internal line must convert one into the other --- possible only for a
Majorana propagator, and only through its mass term.  Step 2: with
$|p| \sim 100$~MeV $\gg m_i$ the chirality projectors kill the
$\slashed p$ piece and leave $m_i/p^2$ \eqref{eq:prop-app}; each
vertex contributes $U_{ei}$ with \emph{no} conjugation, since neither
is an absorption.  Step 3: summing over $i$ gives
\eqref{eq:mbb-def-app}, with $U_{ei}^2$ and not $|U_{ei}|^2$ --- which
is exactly why the Majorana phases survive here and nowhere else.
Physics intuition: the whole of $0\nu\beta\beta$ phenomenology comes
from one chirality argument.  The exchanged neutrino must flip
helicity to be absorbed at the second vertex, the only helicity-flipping
object in the propagator is the mass, and so the amplitude is
proportional to mass --- which is why a process that would be
spectacular if neutrinos were heavy is agonisingly rare because they
are light.

\medskip\noindent\textbf{K.1.7. Compute the inverted-ordering floor, and
explain why normal ordering has none.}

\noindent\emph{Solution.}  Step 1: for inverted ordering with
$m_3 \approx 0$, the two heavy states are nearly degenerate at
$m \approx \sqrt{|\Delta m^2_{31}|} = 50.1$~meV, and
\eqref{eq:mbb-def-app} reduces to
$\mbb{} \approx m\,c_{13}^2|c_{12}^2 + s_{12}^2e^{i\alpha_{21}}|$.
Step 2: the worst case is $\alpha_{21} = \pi$, giving
$\mbb{}^{\min} \approx m\,c_{13}^2|c_{12}^2 - s_{12}^2| =
m\,c_{13}^2\cos2\theta_{12} = 50.1 \times 0.978 \times 0.384 =
19$~meV.  Step 3: for \emph{normal} ordering the dominant term is
$m_2 s_{12}^2 c_{13}^2$ and the two others can, at a lightest mass of
a few meV, be arranged to cancel it exactly --- so the normal band
passes through zero and there is no floor.  Physics intuition: the
floor exists because in inverted ordering the two big contributions
are weighted by $c_{12}^2$ and $s_{12}^2$, which are $0.69$ and $0.31$
--- unequal, so they cannot cancel.  Had $\theta_{12}$ been maximal, the
inverted floor would have vanished too and the entire strategic case
for tonne-scale experiments with it.  The solar angle's being far from
$45^\circ$ is what makes this a decidable question.

\medskip\noindent\textbf{K.1.8. Price the exposure: what a factor of
five in $\mbb{}$ costs.}

\noindent\emph{Solution.}  Step 1: the rate \eqref{eq:0nubb-rate-app}
goes as $\mbb{}^2$, so half-life reach goes as $\mbb{}^{-2}$: a factor
$f$ in mass reach is $f^2$ in half-life.  Step 2: in the
background-limited regime the half-life reach grows as
$\sqrt{Mt/(b\,\Delta E)}$, so $f^2$ in half-life is $f^4$ in exposure.
Step 3: from $9$~meV to $1.8$~meV is $f = 5$: $25$ in half-life and
$625$ in exposure.  Physics intuition: this quartic is the hardest fact
in the subject and it is worth carrying around.  It means a meV-scale
$\mbb{}$ is not reachable by scaling the current program, and that the
only routes are a zero-background technology --- where reach grows
linearly in exposure rather than as its square root, changing $f^4$ to
$f^2$ --- or a different isotope with a much larger matrix element.  A
prediction sitting at $1.8$~meV is therefore a prediction about what the
next generation will \emph{not} see, and honest reporting says so
rather than quoting the number as though it were around the corner.

\medskip\noindent\textbf{K.2.1. Show that the Casas--Ibarra freedom
breaks the link between $\dPMNS{}$ and the baryon asymmetry.}

\noindent\emph{Solution.}  Step 1: the \CP{} asymmetry of leptogenesis
is built from $\Ynu{}^\dagger\Ynu{}$, which by
\eqref{eq:casasibarra-app} equals
$U\sqrt{\widehat m}\,R^\dagger\widehat M_N R\sqrt{\widehat m}\,U^\dagger$
in the relevant combination.  Step 2: the complex angles of $R$ enter
this expression and the low-energy phases in $U$ do not enter it alone,
so a shift in $\dPMNS{}$ can be exactly compensated by a shift in $R$.
Step 3: hence in the general type-I seesaw no measurement of
$\dPMNS{}$ constrains the sign or magnitude of the baryon asymmetry.
Physics intuition: this is the reason the frequently heard slogan
``measuring leptonic \CP{} violation would confirm leptogenesis'' is
false as stated.  It would confirm that \emph{a} leptonic \CP{} phase
exists, which is necessary but very far from sufficient.  Only a
framework that supplies $R$ and $\widehat M_N$ from elsewhere can make
the link, and the next chapter's central claim is precisely that the
geometry supplies both.

\medskip\noindent\textbf{K.2.2. Explain why the nuclear matrix element
is the largest uncertainty, and what that does to a limit.}

\noindent\emph{Solution.}  Step 1: from \eqref{eq:0nubb-rate-app},
$\mbb{} \propto 1/|M^{0\nu}|$ at fixed half-life.  Step 2: many-body
methods --- shell model, QRPA, energy-density functional --- disagree by a
factor of two to three on $M^{0\nu}$ for the same isotope, so a single
measured half-life limit maps onto an $\mbb{}$ limit spanning that same
factor: this is why KamLAND-Zen's $3.8\times10^{26}$~yr is quoted as a
\emph{range} of tens of meV rather than a number.  Step 3: consequence
for falsification --- a criterion written against $\mbb{}$ inherits the
spread, so a well-posed kill row must be stated either against the
half-life directly or against the full quoted range.  Physics intuition:
 the bottleneck in the most sensitive lepton-number
experiment ever built is not the detector, it is a nuclear-structure
calculation.  A student who wants to make the largest possible
difference to this field should consider that the highest-leverage
work in it may be many-body nuclear theory rather than instrumentation.

\medskip\noindent\textbf{K.3.1. Derive the parity pair from rank two,
and count what is \emph{not} free.}

\noindent\emph{Solution.}  Step 1: a rank-two symmetric form on the
three-family space has a one-dimensional kernel, so its eigenvalue
structure admits relative \emph{signs} between eigenvalues but no
continuous relative phases --- a continuum would require rank three.
Step 2: hence $(\alpha_{21},\alpha_{31})$ takes values in a finite set
of parity assignments, and the ledger selects
\eqref{eq:parities-app}, $(0,\pi)$.  Step 3: count what has been
removed: a generic Majorana model has a two-torus of $(\alpha_{21},
\alpha_{31})$ and therefore a factor-of-2.6 envelope in $\mbb{}$
(problem K.3.4); here there is no envelope at all, only a discrete
choice already made.  Physics intuition: the difference between a
parameter and a parity is the difference between a theory that can be
adjusted after the measurement and one that cannot.  A discrete
prediction can be wrong in a way a continuous one cannot --- and it is
worth noticing that this discreteness is what makes the m$_{\beta\beta}$
result of problem K.3.4 uncomfortable rather than convenient.

\medskip\noindent\textbf{K.3.2. Cash the half-angle: show that a parity
of $\pi$ is a factor of $i$.}

\noindent\emph{Solution.}  Step 1: the Majorana phases enter
\eqref{eq:majphases-app} halved, because the mass term is quadratic in
the field while the mixing matrix is linear.  Step 2: therefore
$\alpha_{31} = \pi$ appears in $U$ as $e^{i\pi/2} = i$
\eqref{eq:halfangle-app} --- a quarter turn, not a sign flip.  Step 3:
in $\mbb{}$, which is quadratic in $U$, the $i$ squares to $-1$ and
the third term's contribution reverses (that is the
$-0.90307$ of \eqref{eq:mbb-t3-app}); in the \CP{} asymmetry of
leptogenesis, which is built from an \emph{interference} linear in the
relative phase, the same parity supplies a nonzero imaginary part
where a parity of $0$ would have supplied none.  Physics intuition:
one discrete assignment, two completely different-looking consequences,
because the two observables are different powers of the same matrix.
This is the cleanest illustration in the paper of why the phases had to
be derived once rather than fitted twice --- a framework that fitted
$\alpha_{31}$ to $\mbb{}$ would have had nothing left to say about the
baryon asymmetry.

\medskip\noindent\textbf{K.3.3. Score the derived Dirac phase, and say
what would kill it.}

\noindent\emph{Solution.}  Step 1: the closed form
\eqref{eq:dirac-closed-app} is
$\dPMNS{} = \pi + 2\arcsin(1/7)$; evaluate:
$\arcsin(1/7) = 8.2132^\circ$, so
$\dPMNS{} = 180 + 16.4266 = 196.4264^\circ$, with the QD value
$196.0591^\circ$.  Step 2: against the T2K--NOvA joint fit
$203.4^{+63.0}_{-37.8}$ \citep{T2KNOvA2025} the prediction sits
$0.11\sigma$ from the central value --- but the interval is a hundred
degrees wide, so this is agreement with a very weak measurement and
should be reported as such.  Step 3: F-SM-3d states the kill row: a
$5\sigma$ determination outside $[186^\circ, 207^\circ]$ falsifies it,
and DUNE and Hyper-K are the facilities.  Physics intuition: notice
what the closed form does \emph{not} have --- a free parameter.  The
same $J = 7$ that tilts the atmospheric angle in
\eqref{eq:atm-tilt-app} sets the residual here, so the two cannot be
adjusted independently; a measurement of either constrains the other.
That coupling is the prediction's real content, and it is more
falsifiable than the number itself.

\medskip\noindent\textbf{K.3.4. Work $\mbb{}$ term by term, then scan
the phases and locate the derived value in its own envelope.}

\noindent\emph{Solution.}  Step 1: at first order $m_1 = 0$, so
\eqref{eq:mbb-def-app} has two terms.  With
$m_2 = 8.36114$~meV, $s_{12}^2 = 25/81$, $c_{13}^2 = 137/140$ and
$\alpha_{21} = 0$: the second term is $+2.52530$~meV
\eqref{eq:mbb-t2-app}.  With $m_3 s_{13}^2 = 1.07500$~meV and phase
$\pi - 2(196.426421^\circ)$: the third is
$-0.90307 + 0.58317i$~meV \eqref{eq:mbb-t3-app}, so
$\mbb{} = |1.62223 + 0.58317i| = 1.72386$~meV.  Step 2: now scan.  On
the QD spectrum the three moduli are $0.04416$, $2.59327$ and
$1.10634$~meV, so free phases give
$\mbb{} \in [1.4428, 3.7438]$~meV, and the other admissible parity
$\alpha_{31} = 0$ gives $3.6225$~meV.  Step 3: the derived
$1.79925$~meV sits at $(1.799-1.443)/(3.744-1.443) = 15\%$ of the way
up its own envelope --- the lowest quarter, near maximal destructive
interference.  Physics intuition: state the uncomfortable version
plainly.  A framework free to choose $\alpha_{31}$ would have put its
prediction at $3.6$~meV, twice as large and twice as findable, and
would have been quoting a number it had fitted.  This one is stuck at
the bottom of the range because the parity is derived, and the honest
report is that the derivation has made the prediction \emph{harder} to
confirm.  That is what it costs to have no dials.

\medskip\noindent\textbf{K.3.5. Compare the two unitarity triangles, and
say exactly what the comparison does not prove.}

\noindent\emph{Solution.}  Step 1: this paper derives both Jarlskog
invariants, so the two triangle areas can be compared at true relative
scale: $|\JPMNS{}|/|J_{\rm CKM}| = 9.187\times10^{-3}/
2.984\times10^{-5} = 308$ at QD order, $338$ at first order
\eqref{eq:triangle-ratio-app}.  Step 2: the structural reading ---
hierarchical-against-hierarchical gives a thin triangle,
dyadic-against-democratic a fat one; same constitution, different
projector geometry.  Step 3: the caveat.  It is often said that quark
\CP{} violation falls short of the observed baryon asymmetry by ten
orders of magnitude and that the lepton sector must therefore supply
it.  Three hundred is not ten orders of magnitude, so
\eqref{eq:triangle-ratio-app} does \emph{not} close that gap --- what
closes it is the $\Delta L = 2$ door and the out-of-equilibrium lock
crossing, and the next chapter finds the unflavored invariant
$\JPMNS{}$ belongs to drops out of the asymmetry entirely.  Physics intuition:
 a factor of three hundred is a striking prediction
about flavor geometry and a completely inadequate explanation of
cosmology, and it is worth being able to hold both statements at once.
The temptation to let the first stand in for the second is exactly the
kind of overclaim a course should teach a student to catch --- including
in the framework the course is teaching.

\medskip\noindent\textbf{K.3.6. Check that the three phases are not
independently adjustable.}

\noindent\emph{Solution.}  Step 1: list what each phase is built from.
$\alpha_{21}$ and $\alpha_{31}$ are the parity data of the rank-two
ledger \eqref{eq:parities-app} --- a discrete assignment with no
continuous freedom.  $\dPMNS{}$ is $\pi + 2\arcsin(1/J)$ with $J = 7$
\eqref{eq:dirac-closed-app}.  Step 2: the $\pi$ in $\dPMNS{}$ is the
\emph{same} parity that sets $\alpha_{31}$, and the $J$ in its residual
is the \emph{same} integer that tilts $\sin^2\theta_{23} =
\tfrac12(1+1/J)$.  Step 3: hence a measurement of the atmospheric
octant constrains $\dPMNS{}$, and the parity that fixes $\mbb{}$ fixes
the base of $\dPMNS{}$ --- three phases and one angle from two ledger
inputs.  Physics intuition: this interlocking is the difference between
a derivation and a parametrization, and it is where the framework is
most exposed.  Four measurements now have to agree with each other in a
specific pattern, and any one of them landing outside its window takes
the other three with it.  A student assessing the framework should
count the constraints, not the successes.

\medskip\noindent\textbf{K.3.7. Audit Step~2: which items close and
which are not ours to close?}

\noindent\emph{Solution.}  Step 1: items 1--4 and 7 close --- the
neutrino is Majorana by construction, all three phases are derived,
$\mbb{}$ follows with no envelope, and the largeness of leptonic
\CP{} violation relative to the quark sector is the triangle ratio.
Step 2: item 8 is answered in the next chapter and answered
\emph{conditionally}.  Step 3: items 5 and 6 --- whether light-neutrino
exchange dominates the rate, and the nuclear matrix elements --- are not
answered and are not this framework's to answer; they belong to the
lepton-number-violation and nuclear-structure literatures, and the
second is currently the dominant uncertainty on every published
$\mbb{}$ limit.  Physics intuition: a framework should be as clear
about what is outside its scope as about what is inside it.  Claiming
$\mbb{} = 1.80$~meV while staying silent about a factor-of-three
nuclear uncertainty would make the prediction look sharper than the
experiment can ever be; saying both keeps the comparison honest.

\medskip\noindent\textbf{K.3.8. Reconcile the three lepton observables
on one spectrum.}

\noindent\emph{Solution.}  Step 1: the same derived spectrum gives
$\Summnu{} = 59.17$~meV (QD), $m_\beta = 8.66$~meV and
$\mbb{} = 1.80$~meV --- a factor of thirty-three from top to bottom.
Step 2: check the ordering makes sense.  $\Summnu{}$ is an unweighted
sum, so it is the largest; $m_\beta$ weights by $|U_{ei}|^2$ and takes
a square root, throwing away most of $m_3$; $\mbb{}$ weights by
$U_{ei}^2$ \emph{with phases} and suffers the destructive interference
of problem K.3.4.  Step 3: therefore the experimental prospects are
inverted relative to intuition --- the largest number sits at $92\%$ of
the DESI bound and could fire within a few years, the middle one is
twenty-three times below KATRIN's design reach, and the smallest is
$625$ exposure-units beyond the next generation.  Physics intuition:
one spectrum, three observables, three completely different
experimental verdicts, and the reason is entirely in how each observable
weights the same three numbers.  If a student takes one thing from this
chapter, it should be that ``the neutrino mass'' is not a quantity ---
there are at least three of them, they are not equal, and which one an
experiment measures determines everything about whether it can see this
framework at all.


\GGtakeaway{\emph{Standard result:} a neutral fermion may be its own
antiparticle; if it is, two extra phases become physical and
\(0\nu\beta\beta\) becomes possible with rate \(\propto\mbb{}^2\).
\emph{What it assumed:} the Majorana nature itself, all three phases,
and hence \(\mbb{}\) to within a factor of nearly three. \emph{The
geometric reading:} the rank-two ledger makes the Majorana phases
\emph{parities} rather than dials, fixing \((0,\pi)\); the half-angle
turns that parity into a factor of \(i\); the Dirac phase closes as
\(\pi+2\arcsin(1/7)\) on the same integer that tilts the atmospheric
angle. \emph{What it costs:} the derived parity puts \(\mbb{}\) at
\(1.80\)~meV --- the lowest quarter of its own envelope, and \(625\)
exposure-units beyond the approved program. Derived, and harder to
confirm for being derived.}

\section{Leptogenesis and the origin of the baryon asymmetry}
\label{app:leptogenesis}

\GGteachbox{The chapter in which the lepton sector meets cosmology.
Step~1 measures \etaB{} two independent ways, states Sakharov's three
conditions and scores the Standard Model against each, derives the
\(B+L\) anomaly and the sphaleron conversion factor \(28/79\) from
chemical equilibrium, and then builds thermal leptogenesis from the
one-loop \CP{} asymmetry through the Davidson--Ibarra bound to the
Boltzmann system. Step~2 lists eight quantities the standard picture
must choose. Step~3 gives the lock-scale construction, in which every
one of the eight descends --- and states, at every site and in the
takeaway, that the whole chain rests on one open transport condition.
\emph{This is the most conditional chapter in the course and it is
written to be read that way.}}
\GGreadalong{\citet{DavidsonNardiNir2008} for the whole subject at
review depth; \citet{BuchmullerDiBariPlumacher2005} for the efficiency
factors; \citet{KolbTurner1990} Ch.~3--5 for the thermodynamics;
\citet{Weinberg2008} Ch.~3 for what \etaB{} is as a measurement.}

This is the seventh sector appendix and the last of the three lepton
chapters. Its subject is a single number --- the ratio of baryons to
photons in the universe --- and the claim that the same geometry which
fixed the neutrino block in App.~\ref{app:pmns} and the Majorana parities
in App.~\ref{app:majorana} also writes it.

A warning before the chapter starts, because it governs how everything
in it should be read. The two previous chapters derived quantities and
scored them; this one derives a quantity and scores it \emph{subject to a
hypothesis that is not proved}. The hypothesis is named
(\Hphi{}), its six conditions are tabulated in the main text
(Table~\ref{tab:lg-c1c6}), and the crux --- condition C4 --- is
\textbf{open at the derivation level}. Everything below is stated
conditionally, and a reader who finds an unconditional claim anywhere in
this chapter has found an error.

\subsection{Step 1 --- baryogenesis and leptogenesis as a graduate student meets them}
\label{app:lepto-step1}

\subsubsection{The number, measured twice}
\label{app:lepto-number}

The baryon asymmetry is quoted as
\begin{equation}
\etaB{}\equiv\frac{n_B-n_{\bar B}}{n_\gamma}=(6.12\pm0.04)\times10^{-10},
\label{eq:etaB-obs-app}
\end{equation}
and its credibility rests on being measured two entirely different ways.

\emph{From primordial nucleosynthesis.} The light-element abundances
produced in the first three minutes depend on the baryon density through
the nuclear reaction rates; deuterium is the sensitive one, its abundance
falling roughly as \(\etaB{}^{-1.6}\). A measured D/H in a metal-poor
absorption system therefore reads \(\etaB{}\) at a redshift of
\(10^{9}\).

\emph{From the microwave background.} The acoustic peak heights depend on
the baryon-to-photon ratio through the photon--baryon fluid's inertia;
Planck measures \(\Omega_bh^2=0.02237\), and
\begin{equation}
\eta_{10}\equiv 10^{10}\etaB{}=273.9\,\Omega_bh^2=6.13 ,
\label{eq:eta10-app}
\end{equation}
reading \(\etaB{}\) at a redshift of \(10^{3}\).

The two agree. That agreement is one of the strongest quantitative
results in cosmology, and it is what makes \(\etaB{}\) a target worth
predicting rather than a fitted parameter.
\emph{(Physics intuition: what does \(6\times10^{-10}\) mean physically?
Before the quark--antiquark annihilation there was roughly one quark per
photon and one antiquark per photon. Nearly all of them annihilated. For
every \(1.63\times10^{9}\) antiquarks there were
\(1.63\times10^{9}+1\) quarks, and every atom in every star and every
person is built from that one survivor per billion and a half. The
question of baryogenesis is the question of where the ``\(+1\)'' came
from --- and note that it is a question about a \emph{tiny} number, which
is why it is a question about a small effect operating on an enormous
population rather than about a large effect.)}

\subsubsection{Sakharov's three conditions, and the Standard Model's score}
\label{app:lepto-sakharov}

\citet{Sakharov1967} observed that a universe starting baryon-symmetric
can end asymmetric only if three conditions hold.

\paragraph{(1) Baryon-number violation.} \emph{Satisfied} in the Standard
Model, and this surprises most students: \(B\) is violated
non-perturbatively (\S\ref{app:lepto-sphaleron}).

\paragraph{(2) \(C\) and \CP{} violation.} \emph{Satisfied but far too
small.} The Standard Model's only source is the CKM phase, and every
\CP{}-violating quark observable is proportional to the Jarlskog
invariant times the mass-splitting Jacobian, which at electroweak
temperatures suppresses the effect to roughly \(10^{-20}\) --- ten orders
of magnitude short of \eqref{eq:etaB-obs-app}.

\paragraph{(3) Departure from thermal equilibrium.} \emph{Not
satisfied.} In equilibrium \CPT{} guarantees equal number densities for
particle and antiparticle, so any asymmetry generated is washed out. The
Standard Model's candidate is the electroweak phase transition --- but
for the measured Higgs mass of \(125\)~GeV the transition is a smooth
\emph{crossover}, not first order, and provides no departure at all.

\medskip

\noindent Two of the three legs fail, and each failure is quantitative
and well established. \emph{Baryogenesis therefore requires physics
beyond the Standard Model --- this is not a preference, it is a
theorem-strength conclusion from a measured Higgs mass and a measured
Jarlskog invariant.}

\subsubsection{The anomaly and the sphaleron}
\label{app:lepto-sphaleron}

Classically \(B\) and \(L\) are separately conserved accidental
symmetries. Quantum mechanically the corresponding currents are anomalous
\citep{Adler1969,BellJackiw1969,tHooft1976}:
\begin{equation}
\partial_\mu j^\mu_{B}=\partial_\mu j^\mu_{L}
=\frac{N_f}{32\pi^2}\Bigl(g^2W^a_{\mu\nu}\widetilde W^{a\,\mu\nu}
-g'^2B_{\mu\nu}\widetilde B^{\mu\nu}\Bigr).
\label{eq:anomaly-app}
\end{equation}
The two anomalies are \emph{identical}, so
\begin{equation}
\partial_\mu j^\mu_{B-L}=0,\qquad
\partial_\mu j^\mu_{B+L}\neq0 :
\label{eq:bml-app}
\end{equation}
\(B-L\) is exactly conserved and \(B+L\) is not. A topological transition
changing the Chern--Simons number by one unit changes \(B\) and \(L\) by
\(N_f=3\) each.

At zero temperature the transition proceeds by tunneling and is
suppressed by \(e^{-8\pi^2/g^2}\sim10^{-170}\) --- utterly negligible. At
high temperature it proceeds by thermal activation over the barrier,
whose top is the \emph{sphaleron} \citep{Manton1979,KlinkhamerManton1984},
a static saddle-point configuration of energy
\(E_{\rm sph}\simeq(4\pi v/g)B(\lambda/g^2)\simeq 9\)~TeV at \(T=0\).
Above the electroweak crossover the barrier vanishes and the rate is
unsuppressed \citep{KuzminRubakovShaposhnikov1985}; sphalerons freeze out
at \(T_{\rm sph}\simeq132\)~GeV.
\emph{(Physics intuition: the vacuum of a non-Abelian gauge theory is not
a point but a periodic landscape, and each step between neighboring
minima drags three units of baryon number and three of lepton number
along with it. Baryon number is not conserved in the Standard Model; it
is merely \emph{very difficult} to change at zero temperature and
completely free to change at a few hundred GeV. The entire strategy of
leptogenesis is to exploit that freedom: make an asymmetry in \(L\),
which is easy, and let the sphalerons convert part of it into \(B\),
which is free.)}

\paragraph{The conversion factor, derived.} If \(B-L\) is conserved and
sphalerons are in equilibrium, chemical-equilibrium bookkeeping over all
Standard-Model species fixes the fraction of \(B-L\) that ends up in
\(B\) \citep{HarveyTurner1990}:
\begin{equation}
\boxed{\;
B=\frac{8N_f+4N_H}{22N_f+13N_H}\,(B-L)
=\frac{28}{79}\,(B-L)=0.354430\,(B-L)\;}
\label{eq:csph-app}
\end{equation}
for \(N_f=3\) families and \(N_H=1\) Higgs doublet. Problem~L.1.4 works
the counting. This factor is exact, model-independent given the field
content, and one of the six assembly factors of \S\ref{app:lepto-assembly}.

\subsubsection{Thermal leptogenesis}
\label{app:lepto-thermal}

\citet{FukugitaYanagida1986} observed that the seesaw's heavy Majorana
neutrinos supply all three Sakharov conditions at once. Add
\begin{equation}
\mathcal L\supset-\Ynu{}_{\alpha i}\,\bar L_\alpha \tilde H N_i
-\tfrac12 M_i\,\overline{N_i^{c}}N_i+\mathrm{h.c.}
\label{eq:leptolag-app}
\end{equation}
The Majorana mass violates \(L\) by two units --- Sakharov~(1), through
\(B-L\); the complex Yukawa matrix violates \CP{} --- Sakharov~(2); and
the heavy states decay as the universe cools below their mass, so their
abundance falls out of equilibrium --- Sakharov~(3).

\paragraph{The \CP{} asymmetry.} The decay asymmetry
\begin{equation}
\epsilon_i\equiv
\frac{\Gamma(N_i\to LH)-\Gamma(N_i\to\bar L\bar H)}
{\Gamma(N_i\to LH)+\Gamma(N_i\to\bar L\bar H)}
\label{eq:eps-def-app}
\end{equation}
vanishes at tree level and arises at one loop from the interference of
the tree amplitude with the \emph{vertex} and \emph{self-energy}
corrections. For hierarchical heavy states,
\begin{equation}
\epsilon_1=\frac{1}{8\pi}\frac{1}{(\Ynu{}^\dagger\Ynu{})_{11}}
\sum_{j\neq1}\mathrm{Im}\Bigl[\bigl((\Ynu{}^\dagger\Ynu{})_{1j}\bigr)^2\Bigr]\,
f\!\left(\frac{M_j^2}{M_1^2}\right),
\label{eq:eps1-loop-app}
\end{equation}
with \(f\) the sum of the two loop functions. Two features matter. The
asymmetry needs the \emph{square} of an off-diagonal element, so it
vanishes if \(\Ynu{}^\dagger\Ynu{}\) is real --- a phase is essential, as
Sakharov requires. And it requires at least two heavy states: with one
\(N\) there is no \(j\neq1\) and no interference.

\paragraph{The Davidson--Ibarra wall.} \citet{DavidsonIbarra2002} proved
that in the hierarchical regime \eqref{eq:eps1-loop-app} is bounded:
\begin{equation}
|\epsilon_1|\;\le\;\frac{3}{16\pi}\,\frac{M_1\,(m_3-m_1)}{v^2},
\qquad v=174~\mathrm{GeV}.
\label{eq:di-app}
\end{equation}
This is one of the most consequential inequalities in the subject,
because it converts an observed asymmetry into a \emph{lower bound on the
heavy scale}. Problem~L.1.6 runs it: reproducing
\eqref{eq:etaB-obs-app} at the most favorable efficiency requires
\(M_1\gtrsim6\times10^{8}\)~GeV, quoted in the literature as
\(\sim10^{9}\)~GeV. \emph{The consequence is that standard thermal
leptogenesis is untestable}: it demands a reheating temperature above
\(10^{9}\)~GeV, a heavy sector nine orders of magnitude beyond any
collider, and it predicts nothing measurable at low energy.

\paragraph{Escaping the wall: resonance.} Equation~\eqref{eq:di-app}
assumes hierarchical heavy states. If two are nearly degenerate the
self-energy contribution is resonantly enhanced
\citep{Pilaftsis1997,PilaftsisUnderwood2004}: the loop function develops
a Breit--Wigner factor
\begin{equation}
\epsilon^{\rm res}\;\propto\;\frac{\Delta M\,\Gamma}{\Delta M^2+\Gamma^2/4},
\label{eq:bw-app}
\end{equation}
maximized at \(\Delta M=\Gamma/2\), where the Yukawa suppression cancels
and \(\epsilon\) can approach order unity. Resonant leptogenesis can
therefore operate at the TeV scale --- \emph{at the price of a degeneracy
that must be arranged}, and the arrangement is the standard picture's
least attractive feature.

\subsubsection{The Boltzmann system, washout, and efficiency}
\label{app:lepto-boltzmann}

Producing an asymmetry is not enough; it must survive. The competing
processes are inverse decay \(LH\to N\) and \(\Delta L=2\) scattering,
both of which erase what the decays make. Their strength relative to the
expansion is the \emph{washout parameter}
\begin{equation}
K_{\rm wash}=\frac{\Gamma_N}{H(T=M_N)}=\frac{\widetilde m}{m_\star},
\qquad
\widetilde m\equiv\frac{(\Ynu{}^\dagger\Ynu{})_{11}v^2}{M_1},
\quad m_\star\simeq1.07\times10^{-3}~\mathrm{eV},
\label{eq:washout-app}
\end{equation}
where \(m_\star\) is the ``equilibrium neutrino mass'' --- the value at
which decay and expansion rates coincide. The surviving fraction is the
\emph{efficiency} \(\kappa\), obtained by integrating the coupled
Boltzmann equations for \(Y_N\) and \(Y_{B-L}\) in the variable
\(z=M_1/T\).

The two regimes are worth knowing by heart. \emph{Weak washout}
(\(K_{\rm wash}\ll1\)): \(\kappa\to1\), but the result depends on the
unknown initial \(N\) abundance. \emph{Strong washout}
(\(K_{\rm wash}\gg1\)): \(\kappa\sim1/K_{\rm wash}\) up to logarithms,
and the initial condition is \emph{erased} --- the result is independent
of how the universe started. Strong washout is the theoretically
preferred regime for exactly that reason: it trades efficiency for
predictivity.
\emph{(Physics intuition: washout is not a nuisance, it is the mechanism
that makes leptogenesis a prediction rather than an initial condition. A
strongly washed-out scenario forgets everything about its own beginning
and remembers only its couplings, which is precisely the property one
wants of an explanation.)}

\subsubsection{Flavor}
\label{app:lepto-flavour}

Below about \(10^{12}\)~GeV the tau Yukawa interaction is fast enough to
bring the tau flavor into equilibrium, and below \(10^{9}\)~GeV the muon
too. When that happens the lepton doublet produced in an \(N\) decay is
no longer a coherent superposition but a \emph{measured} flavor state,
and the Boltzmann system must be written flavor by flavor: each flavor
carries its own asymmetry \(\epsilon_\alpha\) and suffers its own washout
\(K_\alpha\). The consequence is important and non-obvious: the
\emph{sum} \(\sum_\alpha\epsilon_\alpha\) may vanish while the individual
\(\epsilon_\alpha\) do not, and a flavor with weak washout can then
carry an asymmetry that the unflavored treatment would have reported as
zero. Step~3 uses exactly this.

\paragraph{Worked example (STANDARD TEXTBOOK PEDAGOGY).}
Assemble the standard estimate. The heavy states are relativistic and in
equilibrium before they decay, so their abundance per unit entropy is
\(Y_N^{\rm eq}\simeq0.42/\gstar{}\) per Majorana species. Each decay
deposits \(\epsilon_1\) of asymmetry in \(B-L\), a fraction \(\kappa\)
survives washout, a fraction \(28/79\) is converted to \(B\), and
converting from entropy to photons multiplies by \(s/n_\gamma=7.04\):
\begin{equation}
\etaB{}\simeq
\frac{s}{n_\gamma}\;\csph{}\;Y_N^{\rm eq}\;\epsilon_1\;\kappa .
\label{eq:etaB-assembly-app}
\end{equation}
With \(\gstar{}=106.75\) and one species, the prefactor is
\(7.04\times0.3544\times0.42/106.75=9.8\times10^{-3}\), so reproducing
\(6.12\times10^{-10}\) needs \(\epsilon_1\kappa\simeq6\times10^{-8}\).
That is the whole of standard leptogenesis in one line, and every model in
the literature is an account of how its \(\epsilon_1\) and \(\kappa\)
multiply to \(6\times10^{-8}\).

\subsection{Step 2 --- what leptogenesis assumes but cannot explain}
\label{app:lepto-step2}

\begin{enumerate}\itemsep3pt
\item \textbf{The heavy mass scale \(M_N\).} Free above the
Davidson--Ibarra wall, and therefore free over seven orders of magnitude.
\item \textbf{The mass splitting \(\Delta M\).} Free. In resonant models
it must be tuned to \(\Gamma/2\), a degeneracy of order \(10^{-10}\)
arranged by hand.
\item \textbf{The Dirac Yukawa \(\Ynu{}\).} Free --- the six real
parameters of the Casas--Ibarra matrix \(R\), invisible to all
low-energy data (problem~J.1.8).
\item \textbf{The \CP{} phases entering \(\epsilon_i\).} Free, and by
problem~K.2.1 not fixed by \(\dPMNS{}\).
\item \textbf{The reheating temperature.} Free, and constrained from
above by the gravitino problem in supersymmetric completions.
\item \textbf{The initial heavy-neutrino abundance.} Free, unless the
washout is strong enough to erase it.
\item \textbf{Which flavor regime applies.} Determined by \(M_N\), which
is free.
\item \textbf{The sign.} No standard model of leptogenesis predicts
whether the universe ends up made of matter or antimatter; the sign
follows the sign of an unmeasured phase.
\end{enumerate}

\noindent The honest summary is that thermal leptogenesis is an
\emph{existence proof} --- it demonstrates that the seesaw can produce an
asymmetry of the right order --- rather than a prediction. Item~8 is the
sharpest illustration: a mechanism that cannot say whether we are made of
matter has explained less than it appears to.

\subsection{Step 3 --- the geometric derivation: the door at the lock}
\label{app:lepto-step3}

\subsubsection{The heavy scale, not chosen}
\label{app:lepto-lock}

The framework has no freedom in item~1. The one scale available between
the electroweak seam and the grand-unification seam is the \OGNfive{}
lock, and it is fixed by the ledger integer \(K=35\):
\begin{equation}
\Mlock{}=\tfrac12\,\EPl{}\,e^{-K}=\tfrac12\,\EPl{}\,e^{-35}=3.85~\mathrm{TeV},
\qquad
\Delta M=\frac{\Mlock{}}{K}=110~\mathrm{GeV}.
\label{eq:lock-app}
\end{equation}
Both numbers are outputs of a ledger fixed upstream by the gauge sector,
and neither was adjusted for this chapter. The splitting is not tuned ---
it is \(\Mlock{}/K\), the same \(K\) that sets the scale.

\subsubsection{The selection rule that permits this at all}
\label{app:lepto-selection}

One structural point separates this framework from every other
leptogenesis scenario and belongs before the mechanism rather than after
it. In the Standard Model, sphalerons are simply available. Here they
are the \emph{only} thing available, and narrowly so.

The bulk \(\UoneB{}\) is gauged and its two non-vanishing mixed anomaly
coefficients are absorbed by the \BIsix{} Green--Schwarz inflow; the
gauge boson takes a Stueckelberg mass and an exact \emph{global} baryon
number survives on the boundary. Every local operator with
\(\Delta B \notin 3\mathbb{Z}\) therefore has identically vanishing
coefficient, to all orders: \(\Delta B = 1\) (proton decay) and
\(\Delta B = 2\) (neutron--antineutron oscillation) are forbidden
outright, which is the \(\tau_p = \infty\) theorem of
\S\ref{subsec:fsm4}. What survives is not an operator at all but a
topological transition --- the \(\SU(2)_L\) vertex at
\(\Delta B = N_{\rm gen}n = 3n\), weighted \(e^{-4\pi/\alpha_W}\approx
10^{-162}\) at zero temperature and unsuppressed above the electroweak
crossover.

Two consequences follow, and they are the same consequence read at two
temperatures.

\emph{No \(\Delta B = \pm1\) mechanism exists here.} Any baryogenesis
scenario that mints baryon number directly --- GUT baryogenesis,
post-sphaleron \(\Delta B = 2\) baryogenesis --- is not suppressed in
this framework; it is unwritable. The asymmetry must be minted in a
currency the selection rule conserves, which means in the lepton sector,
as \(B-L\), and converted at the door.

\emph{So leptogenesis is not chosen; it is forced.} That is a stronger
statement than any efficiency argument, and it is worth holding beside
the chain's conditionality: \emph{the class} of mechanism is forced by
the selection rule, while \emph{this instance} of it rests on C4. If the
lock-scale chain fails, the class survives and some other leptogenesis
must be found; if the class fails --- if baryogenesis were ever shown to
require \(\Delta B = \pm1\) minting --- the framework's proton theorem
and its account of matter become mutually inconsistent, and it dies of
the contradiction.

\emph{(Physics intuition: the same equation read at two temperatures. Cold,
the door is welded and the proton is immortal; hot, the door swings and
the baryons get made. A framework in which protons decay would have had
an easier time making them, and a framework in which baryon number were
exactly conserved with no exception could not have made them at all. This
one threads the gap with a rule that admits multiples of three and
nothing else --- and that rule was written for the proton, long before
anyone asked it about cosmology.)}

\subsubsection{The seam Yukawa, and the width}
\label{app:lepto-yukawa}

Item~3 closes the same way. The Dirac Yukawa is not free because the
Majorana channel is the seam operator of App.~\ref{app:pmns}, so
\begin{equation}
\Ynu{}^2=\pi\,\frac{\Mlock{}}{\EGUT}=5.303\times10^{-13},
\qquad \Ynu{}=7.28\times10^{-7},
\label{eq:seamyuk-app}
\end{equation}
whence the decay width
\begin{equation}
\Gamma_N=\frac{\Ynu{}^2\Mlock{}}{8\pi}=8.12\times10^{-11}~\mathrm{GeV}.
\label{eq:width-app}
\end{equation}
\emph{And here is the difficulty, stated before the resolution.}
Comparing with the derived splitting,
\begin{equation}
\frac{\Delta M}{\Gamma_N}=\frac{110~\mathrm{GeV}}{8.12\times10^{-11}~\mathrm{GeV}}
=1.4\times10^{12},
\label{eq:ratio-bare-app}
\end{equation}
whereas the resonance \eqref{eq:bw-app} needs this ratio near
\emph{unity}. The framework's derived numbers miss the resonant condition
by twelve orders of magnitude, and \emph{the chapter says so before it
says anything else}. Everything that follows is an argument that the
relevant width is not \(\Gamma_N\).

\subsubsection{The regulator question, and the open condition}
\label{app:lepto-regulator}

The claim --- and it is a claim, not a derivation --- is that the width
appearing in the Breit--Wigner denominator is not the vacuum decay width
but an \emph{effective} width set by the thermal bath's flavor-resolving
interactions, dominated by the tau Yukawa. That gives the closed form
\begin{equation}
\frac{\Delta M}{\Geff{}}=\frac{1}{K\,c_\tau\,y_\tau^2}
=(2.7\text{--}5.3)\times10^{4},
\qquad y_\tau(\Mlock{})=0.0104,
\label{eq:regulator-app}
\end{equation}
with \(c_\tau\in[5,10]\times10^{-3}\) the one order-unity coefficient the
argument does not fix. Table~\ref{tab:lg-regulator} of the main text compares five candidate
regulators spanning sixteen orders of magnitude, drawn in
Fig.~\ref{fig:lg-regulator-landscape}; exactly one lands in the required
decade.

\paragraph{The six conditions, and which one is open.} The identification
is packaged as the named hypothesis \Hphi{} with conditions C1--C6
(Table~\ref{tab:lg-c1c6}). Five are argued. \textbf{C4 --- that the
thermal bath resolves the two heavy states as a flavor measurement on
the timescale the resonance requires --- is not derived}, and closing it
demands a full Kadanoff--Baym resummation that nobody has performed. Its
home is \ATen{} Appendix~I; \PaperPFive{} carries the phase routing.

\emph{This is the single most valuable calculation anyone could do to
this chain, in either direction}, and the framework's own falsification
channel F-SM-7 is written around it: unlike every other channel in the
paper, F-SM-7 waits on a calculation rather than on a facility, and could
be settled today.

\subsubsection{The \texorpdfstring{\CP{}}{CP} asymmetry: flavored, with the unflavored part vanishing}
\label{app:lepto-cp}

Items 4 and 8 close through the parities of App.~\ref{app:majorana}. Two
results are structural.

\emph{The flavor sum vanishes exactly.} On the derived (real
Casas--Ibarra) texture the trace of the asymmetry matrix is zero,
\(\mathrm{Tr}\,S=0\), so the \emph{unflavored} \CP{} asymmetry --- the
one built from \(\JPMNS{}\) --- is identically zero. The matter asymmetry
is carried entirely by the differences between flavors. This is why
App.~\ref{app:majorana}'s factor of three hundred between the two
unitarity triangles, striking as it is, is \emph{not} the reason the
lepton sector can pay for baryogenesis: the invariant it measures drops
out.

\emph{The sign comes from the half-angle.} With
\(\alpha_{31}=\pi\), the mixing matrix carries
\(e^{i\alpha_{31}/2}=i\) \eqref{eq:halfangle-app}, and it is that factor
of \(i\) --- linear in \(U\), so a quarter turn rather than a sign flip
--- that supplies the imaginary part \eqref{eq:eps1-loop-app} needs. A parity of
zero would have given none. \emph{The derived Majorana parity therefore
determines the sign of the matter asymmetry}, which is item~8, and no
standard model of leptogenesis can say the same.

\subsubsection{Washout, flavor by flavor: the electron door}
\label{app:lepto-washout}

Computed from the derived masses and the derived PMNS matrix, the
flavored washouts are
\begin{equation}
K_\alpha=(3.5,\;29.3,\;22.5)\quad\text{for }(e,\mu,\tau),
\qquad K_{\rm wash}=47 ,
\label{eq:kalpha-app}
\end{equation}
against \(m_\star^{\rm wash}=1.07\times10^{-3}\)~eV. The electron flavor
is washed out an order of magnitude less than the other two: it is the
\emph{door} through which the asymmetry escapes, and its weakness is a
consequence of the derived reactor angle being small, not an assumption.
The overall efficiency from the Boltzmann integration is
\(\kappa=1.0\times10^{-2}\).

\subsubsection{The six factors, and the assembly}
\label{app:lepto-assembly}

Every factor in \eqref{eq:etaB-assembly-app} is now either derived or
standard:
\begin{equation}
\etaB{}=\underbrace{7.04}_{s/n_\gamma}\times
\underbrace{\frac{28}{79}}_{\csph{}}\times
\underbrace{\frac{1.25}{\gstar{}}}_{Y_N^{\rm eq},\ \text{triplet}}\times
\underbrace{e^{-K/2}}_{\epsres{}\kappa}
=7.34\times10^{-10},
\qquad \gstar{}=106.75 ,
\label{eq:etaB-pred-app}
\end{equation}
against \(6.12\times10^{-10}\) observed --- high by \(20\%\).

\subsubsection{The throughput identity, and the exponent that is postulated}
\label{app:lepto-throughput}

The fourth factor is the claim of the chapter. The Boltzmann integration
and the resonance together produce
\begin{equation}
\epsres{}\,\kappa\;\simeq\;e^{-K/2}=e^{-17.5}=2.511\times10^{-8},
\label{eq:throughput-app}
\end{equation}
so the entire dynamical content of the chain compresses to \emph{half the
ledger integer in the exponent}. That compression is elegant and it is
\textbf{postulated, not derived}: the exponent \(K/2\) is read off the
result rather than obtained from the transport equations, and it is
exactly the statement that condition C4 would have to establish. A reader
who wants one sentence for what this chapter owes, it is this one.

\subsubsection{The error bar, at three levels}
\label{app:lepto-errorbar}

A factor-three band is honest only if it is derived. The main text
(\S\ref{sec:lg-errorbar}) states it at three levels and this chapter
quotes them without softening:
\begin{itemize}\itemsep3pt
\item \textbf{Level A --- the ledger identity.}
\(\etaB{}^{\rm A}=7.34\times10^{-10}\times3^{\pm1}\), band
\((2.4\text{--}22)\times10^{-10}\). Span of nine; the exponent is
postulated.
\item \textbf{Level B --- the Boltzmann computation.}
\(\etaB{}^{\rm B}=0.86^{+0.52}_{-0.32}\times10^{-10}\), band width
\(\times2.6\). Sits a factor \(E=7.1\) \([4.4,11.4]\) \emph{below}
observation.
\item \textbf{Level C --- the completed inventory.} Folding in four
standard transport effects omitted from Level~B, all pointing up:
\(\etaB{}^{\rm C}=2.8^{+2.0}_{-1.2}\times10^{-10}\). The observation
sits \(1.3\sigma\) above the median on a logarithmic scale: consistent,
with a central deficit of a factor \(2.2\).
\end{itemize}
The three numbers answer three different questions, and the gap between
Level~A and Level~C --- about a factor of two and a half --- is the
measure of how much of the chain is still owed.
\emph{All three levels are conditional on C4.}

\subsubsection{What this chapter owes}
\label{app:lepto-owes}

Stated plainly, and repeated because it is the point. This chapter does
not own the finite-temperature transport argument or the conditions
C1--C6 --- their home is \ATen{} Appendix~I. It does not own the
Boltzmann integration or the phase routing. Condition C4 is open there,
and no statement in this chapter is stronger than that condition.

What the chapter \emph{does} establish, subject to C4, is that all eight
of Step~2's free quantities descend from a ledger fixed by the gauge
sector: the scale, the splitting, the Yukawa, the phases, the flavor
regime, the efficiency, and the sign. Item~5 (reheating) and item~6
(initial abundance) are removed by the strong-washout regime the derived
couplings put the chain in. That is a great deal to get from one integer
--- which is exactly why the open condition matters as much as it does.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{L.1.1. Convert the CMB baryon density into
$\etaB{}$, and check it against BBN.}

\noindent\emph{Solution.}  Step 1: the standard conversion is
$\eta_{10} = 273.9\,\Omega_bh^2$ \eqref{eq:eta10-app}, which comes from
$n_B = \Omega_b\rho_c/m_N$ divided by
$n_\gamma = (2\zeta(3)/\pi^2)T_0^3$ at $T_0 = 2.725$~K.  Step 2: with
Planck's $\Omega_bh^2 = 0.02237$,
$\eta_{10} = 273.9 \times 0.02237 = 6.13$.  Step 3: the deuterium
abundance in metal-poor systems independently gives
$\eta_{10} \simeq 6.1$, from physics at a redshift of $10^9$ rather
than $10^3$.  Physics intuition: two measurements separated by six
decades of cosmic expansion, using completely different physics ---
nuclear reaction rates in one case, acoustic oscillations of a
photon--baryon fluid in the other --- agree to better than a percent.
That agreement is why $\etaB{}$ is a number worth trying to predict.
It is also the reason a prediction cannot hide in the error bar: the
observational uncertainty is $0.7\%$.

\medskip\noindent\textbf{L.1.2. Count the survivors.}

\noindent\emph{Solution.}  Step 1: at $T \gg 1$~GeV quarks and
antiquarks are both relativistic and roughly as abundant as photons, so
$n_q \simeq n_{\bar q} \simeq n_\gamma$.  Step 2: the asymmetry is
$(n_q - n_{\bar q})/n_\gamma = \etaB{} = 6.12\times10^{-10}$, so the
fractional excess is one part in $1/6.12\times10^{-10} =
1.63\times10^{9}$.  Step 3: after annihilation only the excess
survives, so every baryon in the present universe is one of those
survivors.  Physics intuition: baryogenesis is not the problem of
making matter --- there was plenty --- it is the problem of making
matter and antimatter \emph{slightly} unequal.  A mechanism that
produced an asymmetry of order one would be as badly wrong as one that
produced none, and this is worth remembering when assessing claims: the
target is a small number, and small numbers are easy to produce by
accident and hard to produce for a reason.

\medskip\noindent\textbf{L.1.3. Score the Standard Model against
Sakharov's three conditions.}

\noindent\emph{Solution.}  Step 1: baryon-number violation --- satisfied,
by the anomaly \eqref{eq:anomaly-app} and unsuppressed sphaleron
transitions above the electroweak crossover.  Step 2: \CP{} violation
--- present, but the CKM contribution is suppressed by the Jarlskog
invariant times a Jacobian of quark mass-squared differences over
$T^{12}$, giving of order $10^{-20}$ against the required
$10^{-10}$.  Step 3: departure from equilibrium --- absent, because at
$m_H = 125$~GeV the electroweak transition is a crossover rather than
first order.  Physics intuition: two failures, both quantitative and
both settled by measurements (the Jarlskog invariant, the Higgs mass).
This is one of the very few places where ``the Standard Model is
incomplete'' is not a philosophical preference but a conclusion from
data --- and note it became a conclusion only in 2012, when the Higgs
mass was measured and the transition was thereby shown to be a
crossover.

\medskip\noindent\textbf{L.1.4. Derive the sphaleron conversion factor
$28/79$.}

\noindent\emph{Solution.}  Step 1: above the electroweak crossover,
assign a chemical potential to each species and impose equilibrium for
every fast reaction --- gauge interactions, Yukawa interactions, and the
sphaleron process itself, which enforces
$\sum_i(3\mu_{q_i} + \mu_{\ell_i}) = 0$.  Step 2: impose also
hypercharge neutrality of the plasma, and solve the linear system for
$B$ in terms of the conserved $B - L$.  Step 3: the general result is
$B = (8N_f + 4N_H)/(22N_f + 13N_H)(B-L)$; at $N_f = 3$, $N_H = 1$ this
is $28/79 = 0.354430$ \eqref{eq:csph-app}.  Physics intuition: the
factor is not $1$ because sphalerons redistribute rather than convert
--- they push the plasma toward the equilibrium partition of a conserved
$B - L$ between baryons and leptons, and the partition is fixed by how
many species of each there are to share it among.  The appearance of
$N_H$ is worth noting: a two-Higgs-doublet model gives $8/23$ instead,
so this ``standard'' factor is a statement about field content.

\medskip\noindent\textbf{L.1.5. Show that $B - L$ is anomaly-free and
explain why leptogenesis needs it to be.}

\noindent\emph{Solution.}  Step 1: the anomalies of $j^\mu_B$ and
$j^\mu_L$ in \eqref{eq:anomaly-app} are \emph{identical}, being
proportional to the same $N_f$ and the same field strengths.  Step 2:
therefore $\partial_\mu j^\mu_{B-L} = 0$ exactly, while
$\partial_\mu j^\mu_{B+L} \ne 0$.  Step 3: consequence --- an asymmetry
stored in $B + L$ would be erased by the very sphalerons that are
supposed to convert it, whereas one stored in $B - L$ is immune and
survives to be partitioned by \eqref{eq:csph-app}.  Physics intuition:
this is the hinge of the whole subject.  Leptogenesis works
because $L$ violation feeds the \emph{conserved} combination, so the
sphalerons act as a redistribution channel rather than a drain.  Any
mechanism that made an asymmetry in $B + L$ alone before the
electroweak crossover would have nothing left to show for it, and this
is exactly why several early baryogenesis proposals failed.

\medskip\noindent\textbf{L.1.6. Run the Davidson--Ibarra bound and get
the $10^9$~GeV wall.}

\noindent\emph{Solution.}  Step 1: from the assembly
\eqref{eq:etaB-assembly-app} the required asymmetry is
$\epsilon_1 \simeq \etaB{}/(9.8\times10^{-3}\kappa)$.  Take the theoretical maximum
efficiency $\kappa \to 1$: then
$\epsilon_1 \simeq 6.12\times10^{-10}/9.8\times10^{-3} =
6.2\times10^{-8}$.  Step 2: invert
\eqref{eq:di-app} for $M_1$ with $m_3 = 50$~meV and $v = 174$~GeV:
$M_1 \ge 16\pi\epsilon_1 v^2/(3m_3) =
50.27 \times 6.2\times10^{-8} \times 30276/(1.5\times10^{-10}) =
6\times10^{8}$~GeV.  Step 3: quoted in the literature as
$M_1 \gtrsim 10^9$~GeV, and it forces a reheating temperature of the
same order.  Physics intuition: this single inequality is why standard
thermal leptogenesis is beautiful and untestable.  It puts the
mechanism nine orders of magnitude beyond any conceivable experiment
and leaves no low-energy signature, so the theory can be neither
confirmed nor refuted --- which is the condition a framework should be
most suspicious of in itself, and the reason the resonant escape route
\eqref{eq:bw-app} attracts so much attention.

\medskip\noindent\textbf{L.1.7. Show why the resonance needs
$\Delta M \sim \Gamma/2$, and why that is uncomfortable.}

\noindent\emph{Solution.}  Step 1: the self-energy contribution carries
the Breit--Wigner factor \eqref{eq:bw-app},
$\Delta M\Gamma/(\Delta M^2 + \Gamma^2/4)$.  Step 2: maximize --- set the
derivative to zero and find $\Delta M = \Gamma/2$, where the factor
equals $1$ and the Yukawa suppression of $\epsilon$ cancels entirely.
Step 3: for a TeV-scale $N$ with a seesaw-sized Yukawa,
$\Gamma/M \sim 10^{-13}$, so the required degeneracy is one part in
$10^{13}$ --- arranged by hand in every standard resonant model.
Physics intuition: resonant leptogenesis trades one implausibility for
another.  It escapes the $10^9$~GeV wall and becomes testable, but it
does so by demanding a degeneracy nobody can motivate.  A framework
that \emph{derived} the splitting would be making a genuinely different
claim --- which is what Step~3 attempts, and why the twelve-order
mismatch of \eqref{eq:ratio-bare-app} has to be confronted rather than
buried.

\medskip\noindent\textbf{L.1.8. Contrast weak and strong washout, and
say which is preferable and why.}

\noindent\emph{Solution.}  Step 1: $K_{\rm wash} = \widetilde m/m_\star$
\eqref{eq:washout-app} compares the decay rate to the expansion rate at
$T = M_N$.  Step 2: for $K_{\rm wash} \ll 1$ the efficiency
$\kappa \to 1$ but the result depends on the unknown initial abundance
of $N$; for $K_{\rm wash} \gg 1$, $\kappa \sim 1/K_{\rm wash}$ and any
pre-existing asymmetry \emph{and} the initial abundance are both erased.
Step 3: strong washout is preferable despite being less efficient,
because it makes the answer a function of couplings alone.  Physics intuition:
 this is a general lesson about early-universe
model-building and worth carrying beyond this chapter.  A process that
erases its own initial conditions converts a cosmological accident into
a particle-physics prediction; one that does not requires you to
specify what happened before, which is usually a way of hiding a free
parameter in a place nobody will check.

\medskip\noindent\textbf{L.1.9. Explain how a vanishing flavor sum can
still give a nonzero asymmetry.}

\noindent\emph{Solution.}  Step 1: below $10^{12}$~GeV the tau Yukawa
equilibrates and the lepton doublet from an $N$ decay is measured into
flavor states; each flavor then has its own $\epsilon_\alpha$ and its
own $K_\alpha$.  Step 2: suppose $\sum_\alpha\epsilon_\alpha = 0$.  The
surviving asymmetry is $\sum_\alpha \epsilon_\alpha\kappa(K_\alpha)$,
and since $\kappa$ differs between flavors, this is \emph{not} zero.
Step 3: the physical picture --- the decays deposit equal and opposite
asymmetries in different flavors, and washout destroys them at
different rates, leaving a residue.  Physics intuition: an unflavored
calculation of such a scenario would report exactly zero and conclude
the mechanism fails.  This is one of the clearest cases in cosmology of
a ``more careful treatment'' changing an answer from nothing to
something, and it is the reason the flavored Boltzmann system became
standard after 2006.  Step~3 of this chapter is built on it.

\medskip\noindent\textbf{L.1.10. Assemble the standard estimate and find
what $\epsilon\kappa$ must be.}

\noindent\emph{Solution.}  Step 1: the chain is
$\etaB{} = (s/n_\gamma)\csph{}Y_N^{\rm eq}\epsilon_1\kappa$
\eqref{eq:etaB-assembly-app}, with $s/n_\gamma = 7.04$,
$\csph{} = 28/79$ and $Y_N^{\rm eq} = 0.42/\gstar{}$ per species.
Step 2: with $\gstar{} = 106.75$ the prefactor is
$7.04 \times 0.3544 \times 0.42/106.75 = 9.8\times10^{-3}$.  Step 3:
matching $6.12\times10^{-10}$ needs
$\epsilon_1\kappa = 6.2\times10^{-8}$.  Physics intuition: every
leptogenesis paper ever written is, at bottom, an account of how its
$\epsilon$ and its $\kappa$ multiply to $6\times10^{-8}$.  Knowing this
one line lets a reader evaluate any such paper in a minute: find its
$\epsilon$, find its $\kappa$, multiply, and see whether the claimed
agreement is a calculation or a choice.  Apply the test to Step~3 of
this chapter as well.

\medskip\noindent\textbf{L.2.1. Show that low-energy \CP{} measurements
do not test leptogenesis in the general seesaw.}

\noindent\emph{Solution.}  Step 1: $\epsilon_1$ \eqref{eq:eps1-loop-app} is
built from $\mathrm{Im}[((\Ynu{}^\dagger\Ynu{})_{1j})^2]$, and by
Casas--Ibarra \eqref{eq:casasibarra-app} the combination
$\Ynu{}^\dagger\Ynu{}$ depends on $R$ as well as on $U$.  Step 2: the
six real parameters of $R$ can supply an arbitrary phase, so
$\epsilon_1$ can be given any value, including zero and either sign,
at fixed $\dPMNS{}$.  Step 3: hence measuring $\dPMNS{}$ constrains
leptogenesis only in models that fix $R$ by some other principle.
Physics intuition: it is worth being precise about a claim one hears
often.  Measuring leptonic \CP{} violation would establish that the
lepton sector \emph{has} a \CP{} phase, which is a necessary
ingredient; it would not establish that this phase is the one that made
the baryons.  A framework wanting to claim the link must derive $R$,
and the derivation is then the thing to scrutinize.

\medskip\noindent\textbf{L.2.2. Ask what a mechanism that cannot predict
the sign has explained.}

\noindent\emph{Solution.}  Step 1: in the standard picture the sign of
$\etaB{}$ follows the sign of an unmeasured phase in $R$, so both signs
are equally available.  Step 2: therefore the mechanism predicts
$|\etaB{}|$ of the right order and nothing about whether we are made of
matter or antimatter.  Step 3: since ``the universe is made of matter''
is one of the two facts to be explained --- the other being the
magnitude --- an existence proof that reproduces the magnitude for
either sign has answered half the question.  Physics intuition: this is
a useful discipline to apply generally.  Ask of any mechanism not only
``does it give the right size?'' but ``how many of the observed
features does it \emph{not} get to choose?''  The magnitude is one
number and the sign is one bit, and a framework that fixes both from
structure is making a claim twice as strong as one that fixes only the
first.

\medskip\noindent\textbf{L.3.1. Derive the lock scale and its splitting
from the ledger.}

\noindent\emph{Solution.}  Step 1: the \OGNfive{} closure integer is
$K = LJ = 5 \times 7 = 35$, fixed by the gauge sector upstream and not
by anything in this chapter.  Step 2: the lock sits at
$\Mlock{} = \tfrac12\EPl{}e^{-K}$; with
$\EPl{} = 1.2209\times10^{19}$~GeV and $e^{-35} = 6.305\times10^{-16}$
this is $3.85$~TeV \eqref{eq:lock-app}.  Step 3: the splitting is
$\Delta M = \Mlock{}/K = 3849/35 = 110$~GeV --- the same integer
appearing twice, so the degeneracy is not an independent choice.
Physics intuition: contrast this with problem L.1.7, where a resonant
model had to arrange $\Delta M/M \sim 10^{-13}$ by hand.  Here
$\Delta M/M = 1/K = 1/35$, which is not small at all.  That is either
the chapter's most interesting feature or its central difficulty, and
problem L.3.2 shows it is both.

\medskip\noindent\textbf{L.3.2. Confront the twelve-order mismatch
before accepting the resolution.}

\noindent\emph{Solution.}  Step 1: the derived Yukawa
\eqref{eq:seamyuk-app} gives $\Ynu{}^2 = \pi\Mlock{}/\EGUT =
5.303\times10^{-13}$, hence the vacuum width
$\Gamma_N = \Ynu{}^2\Mlock{}/8\pi = 8.12\times10^{-11}$~GeV
\eqref{eq:width-app}.  Step 2: compare to the derived splitting:
$\Delta M/\Gamma_N = 110/8.12\times10^{-11} = 1.4\times10^{12}$, while
the resonance \eqref{eq:bw-app} wants order unity.  \emph{The
framework's own derived numbers miss the resonant condition by twelve
orders of magnitude.}  Step 3: the proposed resolution is that the
relevant width is not the vacuum width but a thermal effective width
\eqref{eq:regulator-app}, $\Delta M/\Geff{} = 1/(Kc_\tau y_\tau^2) =
(2.7\text{--}5.3)\times10^4$ --- eight orders closer, and in the decade
the chain needs.  Physics intuition: a student should sit with the
mismatch before accepting the fix.  A twelve-order-of-magnitude gap
closed by identifying a different width is exactly the kind of move
that deserves the most scrutiny in the paper, and the framework's own
assessment agrees: condition C4, which is what would justify the
identification, is marked open, and F-SM-7 is written so that a single
Kadanoff--Baym calculation settles it either way.

\medskip\noindent\textbf{L.3.3. Locate the one regulator that works
among five candidates, and price the coincidence.}

\noindent\emph{Solution.}  Step 1: the required window is
$\Delta M/\Gamma \sim 10^4$--$10^5$, from the throughput the assembly
needs.  Step 2: Table~\ref{tab:lg-regulator} of the main text lists five
candidate widths --- the $N$ widths on the $S$-matrix prescription, the
flavor-measurement rate $c_\tau y_\tau^{2}T$, the oscillation rate
$\Delta M$ itself, gauge damping $g^{2}T$, and the Hubble rate --- spanning
sixteen orders of magnitude in $\Delta M/\Geff{}$
(Fig.~\ref{fig:lg-regulator-landscape}).  Exactly one, the
flavor-measurement width, lands in the decade.  Step 3: this is presented
in the paper as a \emph{selection}, not a derivation, and the distinction
matters --- five candidates spread over sixteen orders sit about three
decades apart, so one of them falling inside the required decade is a
coincidence of roughly the odds of a fair die, not a derivation.  Physics intuition: the
honest reading is that the regulator identification is a hypothesis
with a testable consequence, not a calculation with a result.  What
makes it more than numerology is that the same identification fixes
the flavor structure and the sign independently, and those are
separate checks; what would make it a derivation is C4.

\medskip\noindent\textbf{L.3.4. Show that the unflavored asymmetry
vanishes, and why that is a feature.}

\noindent\emph{Solution.}  Step 1: on the derived texture the
Casas--Ibarra matrix is real, so the asymmetry matrix $S$ has
$\mathrm{Tr}\,S = 0$ exactly.  Step 2: hence the unflavored \CP{}
asymmetry --- the one proportional to the invariant $\JPMNS{}$ --- is
identically zero, and the entire matter asymmetry is carried by
\emph{differences} between the flavored $\epsilon_\alpha$ acting
through different washouts (problem L.1.9).  Step 3: the consequence for
the previous chapter is sharp --- the factor of three hundred between
the two unitarity triangles \eqref{eq:triangle-ratio-app}, striking as
it is, is not why the lepton sector can pay for baryogenesis, because
the invariant it measures drops out of the answer.  Physics intuition:
this is a good example of a framework's own result cutting
against a rhetorically attractive claim.  It would have been very
convenient to point at $J_{\rm PMNS}/J_{\rm CKM} = 308$ and say ``there
is the missing \CP{} violation''; the calculation says that quantity
contributes exactly nothing, and the paper says so in both chapters.

\medskip\noindent\textbf{L.3.5. Cash the half-angle for the sign of the
universe.}

\noindent\emph{Solution.}  Step 1: the derived parity is
$\alpha_{31} = \pi$, which enters the mixing matrix halved as
$e^{i\pi/2} = i$ \eqref{eq:halfangle-app}.  Step 2: $\epsilon_1$
\eqref{eq:eps1-loop-app} needs the imaginary part of a product of Yukawa
combinations; the factor of $i$ --- linear in $U$ here, unlike in
$\mbb{}$ where it squared --- supplies exactly that imaginary part,
and its sign fixes whether the surviving asymmetry is in leptons or
antileptons.  Step 3: hence the derived parity determines the sign of
$\etaB{}$, answering item~8 of Step~2, which no standard leptogenesis
model does.  Physics intuition: this is the single most striking
structural claim in the chapter.  One discrete parity, forced by the
rank of the Majorana ledger, fixes $\mbb{} = 1.80$~meV in the previous
chapter and the sign of the matter content of the universe in this one
--- the same bit read through two different powers of the same matrix.
It is also, of course, conditional on C4 like everything else here, and
the right attitude is that this is what makes C4 worth someone's year.

\medskip\noindent\textbf{L.3.6. Find the electron door.}

\noindent\emph{Solution.}  Step 1: the flavored washouts computed from
the derived masses and derived PMNS matrix are
$K_\alpha = (3.5,\,29.3,\,22.5)$ for $(e,\mu,\tau)$
\eqref{eq:kalpha-app}, against the unflavored $K_{\rm wash} = 47$.
Step 2: the electron flavor is washed out roughly an order of
magnitude less than the other two, so its asymmetry survives with an
order of magnitude better efficiency: the electron is the door.
Step 3: trace the cause --- $K_e$ is small because the electron row of
the PMNS matrix is dominated by the small reactor angle, so the derived
$\sin^2\theta_{13} = 3/140$ of App.~\ref{app:pmns} is what opens the
door.  Physics intuition: a reactor experiment measuring a mixing angle
at a kilometer baseline, and the survival of the matter asymmetry at
$10^{-11}$ seconds, are here the same number.  Whether or not the chain
holds up, that is the kind of connection a unification claim is
supposed to make, and it is worth noting it is not an adjustable one:
$\theta_{13}$ was measured before this was computed.

\medskip\noindent\textbf{L.3.7. Assemble $\etaB{}$ from the six factors
and check the arithmetic.}

\noindent\emph{Solution.}  Step 1: list the six --- $s/n_\gamma = 7.04$;
$\csph{} = 28/79 = 0.354430$; $Y_N^{\rm eq} = 1.25/\gstar{}$ for the
triplet; $\gstar{} = 106.75$; and $\epsres{}\kappa = e^{-K/2}$.
Step 2: multiply.  $7.04 \times 0.354430 = 2.4952$; $\times 1.25 =
3.1190$; $/106.75 = 2.9218\times10^{-2}$; $\times e^{-17.5} =
2.9218\times10^{-2} \times 2.5110\times10^{-8} = 7.34\times10^{-10}$
\eqref{eq:etaB-pred-app}.  Step 3: against
$6.12\times10^{-10}$ this is high by $19.9\%$, or $+20\%$.  Physics intuition:
 four of the six factors are textbook thermodynamics
that any leptogenesis calculation uses; the fifth is the field content;
the entire content of the framework's claim is the sixth,
$\epsres{}\kappa = e^{-K/2}$.  When assessing this result, that is the
only factor to argue about --- and the next problem is about why it is
the one that is not derived.

\medskip\noindent\textbf{L.3.8. Identify precisely what is postulated.}

\noindent\emph{Solution.}  Step 1: the throughput identity
\eqref{eq:throughput-app} states $\epsres{}\kappa \simeq e^{-K/2} =
e^{-17.5} = 2.511\times10^{-8}$, compressing all the dynamics into half
the ledger integer.  Step 2: the exponent $K/2$ is \emph{read off the
result} rather than obtained from the transport equations --- it is not
a consequence of anything derived in the chapter.  Step 3: what would
derive it is condition C4 of \Hphi{}, namely that the thermal bath
resolves the two heavy states as a flavor measurement on the required
timescale; establishing it needs a full Kadanoff--Baym resummation that
has not been done.  Physics intuition: a student wanting one sentence
for what this chain owes should take this one.  Everything else --- the
scale, the splitting, the Yukawa, the washouts, the sign --- is
computed; the compression into $e^{-K/2}$ is asserted.  A framework
that quietly let this look like the rest of the chapter would be
misrepresenting itself, which is why C4 is named in the table, in the
takeaway, in the falsification channel, and here.

\medskip\noindent\textbf{L.3.9. Read the three error-bar levels without
averaging them.}

\noindent\emph{Solution.}  Step 1: Level~A takes the ledger identity at
face value: $7.34\times10^{-10}\times3^{\pm1}$, band
$(2.4\text{--}22)\times10^{-10}$, a span of nine.  Step 2: Level~B
integrates the Boltzmann system and scans the four remaining freedoms:
$0.86^{+0.52}_{-0.32}\times10^{-10}$, band width $\times2.6$ --- a
factor $E = 7.1$ $[4.4, 11.4]$ \emph{below} observation.  Level~C folds
in four standard transport effects Level~B omits, all pointing up:
$2.8^{+2.0}_{-1.2}\times10^{-10}$, with the observation $1.3\sigma$
above the median on a logarithmic scale.  Step 3: they answer three
different questions --- what the identity gives, what the declared
dynamics produces, and what the completed inventory would produce ---
and the factor of two and a half between A and C measures what is
still owed.  Physics intuition: resist averaging them.  The instinct to
collapse three numbers into one is exactly wrong here, because their
\emph{spread} is the informative quantity: it says the chain's
arithmetic and the chain's dynamics currently disagree by a factor of
a few, which is a much more useful statement than any single number
would be.

\medskip\noindent\textbf{L.3.10. Audit Step~2: eight items, and the one
condition they all hang on.}

\noindent\emph{Solution.}  Step 1: items 1 (scale), 2 (splitting), 3
(Yukawa), 4 (phases), 7 (flavor regime) and 8 (sign) descend from the
ledger; items 5 (reheating) and 6 (initial abundance) are removed
because the derived couplings put the chain in strong washout, which
erases both.  Step 2: so all eight close --- \emph{subject to} the
regulator identification, whose condition C4 is open.  Step 3: the
correct summary statement is therefore conditional and should always be
written that way: \emph{given} that the thermal bath supplies the
flavor-resolving regulator, the baryon asymmetry descends from the
same integer $K = 35$ that fixes the lock scale and the neutrino
floor.  Physics intuition: eight free parameters replaced by one
integer is a very large claim, and one open analytic condition is a
very small-looking price --- which is exactly why it must be stated
every time.  The useful question for a student is not whether the claim
is impressive but what would settle it, and here the answer is unusually
concrete: one finite-temperature calculation, no new data, and F-SM-7 is
written so that it decides the matter in either direction.


\GGtakeaway{\emph{Standard result:} sphalerons convert a lepton
asymmetry into a baryon one at the exact rate \(28/79\); the seesaw
supplies all three Sakharov conditions; and the Davidson--Ibarra bound
then puts the mechanism above \(10^{9}\)~GeV, where it is untestable.
\emph{What it assumed:} eight quantities, including the sign of the
universe. \emph{The geometric reading:} the lock scale
\(\Mlock{}=\tfrac12\EPl{}e^{-K}=3.85\)~TeV, its splitting
\(\Mlock{}/K\), the seam Yukawa, the flavored washouts and the sign
all descend from \(K=35\); the unflavored asymmetry vanishes
identically; the electron flavor is the door, opened by the derived
reactor angle. \emph{What it owes:} the compression
\(\epsres{}\kappa\simeq e^{-K/2}\) is \textbf{postulated}, and
condition C4 --- the finite-temperature transport step --- is
\textbf{open}. Every number in this chapter is conditional on it, and
F-SM-7 is written so that one calculation settles it.}

\section{Constitutional ledgers and Standard-Model architecture}
\label{app:ch-ledger}

\GGteachbox{The audit chapter: the constitutional ledger of
exactly what the visible derivation consumed --- which preprints,
which restricted inputs, which kill criteria --- and the
architecture cards that compare the constitutional reading with
its stabilized boundary projection.  The course's provenance
record: to be checked rather than worked.}

\subsection{Constitutional ledger of the present gauge/Higgs, fermion-mass, CKM, PMNS/Majorana, and topological-CP derivation}
\label{app:constitutional-ledger}

Because the visible derivation relies only on the written Arisaka-GG preprints and on the restricted input set, it is useful to
close the appendix sequence with a compact ledger of exactly what the
present paper does. Table~\ref{tab:constitutional-ledger} lists each
major stage of the argument, what it is allowed to consume, and what it is
forbidden to borrow from observation.

\begin{center}
\begin{longtable}{p{0.18\textwidth}p{0.23\textwidth}p{0.25\textwidth}p{0.21\textwidth}}
\caption[Constitutional ledger for the present gauge/Higgs, fermion-mass, CKM, PMNS/Majorana, and topological-CP derivation.]{Constitutional ledger for the present gauge/Higgs,
fermion-mass, CKM, PMNS/Majorana, and topological-CP derivation. Each stage of the derivation is one row, and each row names what may be used, what comes out, and what would be a shortcut at that particular stage. The last column changes from row to row, which is the point: an input that is legitimate at a later stage can be circular at an earlier one, and a ledger naming only a single global list of forbidden moves would miss exactly that.}
\label{tab:constitutional-ledger}\\
\toprule
\textbf{Stage} & \textbf{Allowed inputs} & \textbf{Outputs produced} & \textbf{Forbidden shortcut at this stage}\\
\midrule
\endfirsthead
\toprule
\textbf{Stage} & \textbf{Allowed inputs} & \textbf{Outputs produced} & \textbf{Forbidden shortcut at this stage}\\
\midrule
\endhead
Constitutional setup & \AOne{} axioms/postulates and \AFive{} observation rules & Meaning of derivation, admissible inputs, role of GDOS/GFF & Smuggling in measured SM parameters as if they were theoretical primitives\\
GG-6 stage selection & \ATwo{} geometry, BI-6 discipline, OGN-5 ledger & Minimal bulk \(M_4\times I_r\times S^1_\theta\), composite connection, throat depth & Adding extra dimensions, supersymmetry, or string states not present in the written preprints\\
Scale chain & \(c,h,G_N\), \(\pi\), \(\e\), and OGN-5 integers & \(\EPl\), \(\EGUT\), \(\Mlock\) & Using \(G_F\), \(m_Z\), or any electroweak datum to set the weak scale\\
GDOS/GFF layer & \AFive{} open-system dynamics and GFF functional & Meaning of stationary seams, vacuum selection, measurement map & Treating observables as external empirical labels without projection theory\\
Radial RG layer & Warped metric and logarithmic radial coordinate & Transport equation \(\partial_r g_i=-\beta_i\) and seam evaluation logic & Importing textbook beta functions as free phenomenological inputs divorced from GG-6 geometry\\
Wilson--Hosotani layer & Compact holonomy and orbifold parity structure & Geometric Higgs origin, finite holonomy potential, naturalness reformulation & Introducing an elementary 4D scalar with an arbitrary bare mass parameter\\
Gauge/Higgs first order & Same restricted constants plus OGN/BI-6 rational structure & Leading formulas for \(\alphahatinv(M_Z)\), \(\sin^2\theta_W\), \(\alpha_s(M_Z)\), \(v\), and \(m_h\) & Fitting first-order coefficients directly to benchmark central values\\
Gauge/Higgs QD & Finite residual seam corrections within the same framework & Second-order sharpened gauge/Higgs predictions and theory truncation errors & Replacing the second-order layer by ad hoc numerical regression\\
Charged-fermion overlap block & Yukawa overlap theorem from \ATwo{}/3, fixed \(v\), OGN dyadic/winding codebook & First-order charged-lepton and quark Yukawas and masses & Inserting empirical Yukawa matrices or family-specific fit constants\\
Neutrino Majorana block & Allowed \((LH)(LH)\) operator, fixed \(\EGUT\), fixed \(v\), OGN family projector & First-order and QD neutrino masses and oscillation scales & Choosing an arbitrary seesaw scale or measured neutrino mass as an input\\
Fermion QD transport & Same dyadic and rational closure logic as above & QD-completed fermion masses and truncation errors & Independently fitting each fermion mass with a separate floating correction factor\\
CKM block & Fixed quark masses, family-link algebra, compact-angle phase logic, and the same OGN/BI-6 ledger & First-order and QD CKM angles, phase, matrix elements, and Jarlskog invariant & Importing measured CKM angles or a generic fitted unitary flavor matrix\\
PMNS/Majorana block & Fixed charged-lepton codebook, fixed Majorana neutrino block, family projector algebra, compact-angle phase logic, and the same OGN/BI-6 ledger & First-order and QD PMNS angles, leptonic Dirac phase, Majorana phases, full matrix, and \(m_{\beta\beta}\) & Importing measured PMNS angles/phases or a generic fitted lepton mixing matrix\\
Comparison stage & Only now: benchmark observations in the matching scheme & Dev and Pull tables, assessment of agreement, identification of residual work & Feeding the comparison numbers back into the derivation as hidden inputs\\
\bottomrule
\end{longtable}
\end{center}

The value of this ledger is methodological. It clarifies why the present
manuscript is a genuine first-principles construction in the sense
adopted here. Every stage is allowed to use only the written
constitutional language, the minimal conversion constants, the geometric
transcendentals, and the \OGNfive{} integers. No stage is allowed to consume a
measured Standard-Model number in order to make the subsequent algebra
work.

This point also explains why the comparison tables are placed late in the
paper. In an empirical fit paper, the comparison table often acts as an
input filter: one varies unknown constants until the table looks good. In
the present paper the comparison table is a post-derivational auditor.

\subsection{Explicit \texorpdfstring{\(\SU(5)\)}{SU(5)} decomposition, hypercharge embedding, and charge assignments}
\label{app:su5-decomposition-detail}

This appendix supports the new phenomenological prologue by collecting the basic
\(\SU(5)\) decomposition formulae used implicitly in the main text. None of the
formulae are novel by themselves; the point is to display them in the notation of
\ANine{} so that the reader can see exactly how the historical Georgi--Glashow
structure is being lifted into \GG{}.

\subsection{Adjoint decomposition}

The adjoint of \(\SU(5)\) has dimension 24. Under the Standard-Model subgroup it
decomposes as
\begin{equation}
\label{eq:adjoint-24-detail}
\mathbf{24}
\rightarrow
(\mathbf 8,\mathbf 1)_0
\oplus
(\mathbf 1,\mathbf 3)_0
\oplus
(\mathbf 1,\mathbf 1)_0
\oplus
(\mathbf 3,\mathbf 2)_{-5/6}
\oplus
(\bar{\mathbf 3},\mathbf 2)_{5/6}.
\end{equation}
The interpretation is immediate. The octet gives the gluons, the triplet gives
the weak bosons, the singlet gives the hypercharge boson, and the mixed-color
weak doublets correspond to the heavy \(X\) and \(Y\) gauge bosons.

\subsection{Matter-family decomposition}

A single visible family is organized into
\begin{equation}
\label{eq:matter-reps-detail}
\bar{\mathbf 5}\oplus \mathbf{10},
\end{equation}
with the decompositions
\begin{equation}
\label{eq:5bar-detail}
\bar{\mathbf 5}
\rightarrow
(\bar{\mathbf 3},\mathbf 1)_{1/3}
\oplus
(\mathbf 1,\mathbf 2)_{-1/2},
\end{equation}
\begin{equation}
\label{eq:10-detail}
\mathbf{10}
\rightarrow
(\mathbf 3,\mathbf 2)_{1/6}
\oplus
(\bar{\mathbf 3},\mathbf 1)_{-2/3}
\oplus
(\mathbf 1,\mathbf 1)_1.
\end{equation}
These are precisely the familiar quark and lepton assignments of one family.

\subsection{Hypercharge generator and electric charge}

Inside \(\SU(5)\), the conventional hypercharge generator is represented by
\begin{equation}
\label{eq:y-generator-detail}
Y=\diag\!\left(-\frac13,-\frac13,-\frac13,\frac12,\frac12\right),
\end{equation}
with the trace condition \(\Tr Y=0\). Electric charge is then
\begin{equation}
\label{eq:q-electric-detail}
Q=T_3+Y.
\end{equation}
One immediately obtains the standard charge pattern shown in
Table~\ref{tab:su5-charge-table-detail}.

\begin{table}[H]
\centering
\caption[Electric charges from the \(\SU(5)\) embedding.]{Electric charges from the \(\SU(5)\) embedding. Each row gives a field, its representation with the weak-isospin component where that matters, the electric charge that follows, and the sector the field belongs to. The charges are outputs here and not assignments: they are read off the embedding, and the fractional values in the quark rows follow from the hypercharge normalization rather than from a separate postulate.}
\label{tab:su5-charge-table-detail}
\small
\begin{tabular}{p{0.18\textwidth}p{0.24\textwidth}p{0.18\textwidth}p{0.24\textwidth}}
\toprule
\textbf{State} & \textbf{SM representation} & \textbf{\(Q\)} & \textbf{Interpretation}\\
\midrule
Left quark doublet upper component & \((\mathbf 3,\mathbf 2)_{1/6}\) with \(T_3=+1/2\) & \(+2/3\) & Up-type quark\\
Left quark doublet lower component & \((\mathbf 3,\mathbf 2)_{1/6}\) with \(T_3=-1/2\) & \(-1/3\) & Down-type quark\\
Right up antiquark field & \((\bar{\mathbf 3},\mathbf 1)_{-2/3}\) & \(-2/3\) on conjugate field & Up-type singlet sector\\
Right charged antilepton field & \((\mathbf 1,\mathbf 1)_1\) & \(+1\) on conjugate field & Charged-lepton singlet sector\\
Lepton doublet upper component & \((\mathbf 1,\mathbf 2)_{-1/2}\) with \(T_3=+1/2\) & 0 & Neutrino\\
Lepton doublet lower component & \((\mathbf 1,\mathbf 2)_{-1/2}\) with \(T_3=-1/2\) & \(-1\) & Charged lepton\\
\bottomrule
\end{tabular}
\end{table}

The table is elementary, but it is valuable because it shows why charge
quantization is already explained once the unified embedding is fixed.

\subsection{Baryon number in the \GG{} extension}

The historical \(\SU(5)\) model does not by itself make local baryon number a
fundamental gauge symmetry. \GG{} extends the visible gauge sector by local
\(\U(1)_B\). At the seam this gives the familiar assignments
\begin{equation}
\label{eq:baryon-number-detail}
B(q)=\frac13,
\qquad
B(\ell)=0.
\end{equation}
The importance of writing Eq.~\eqref{eq:baryon-number-detail} explicitly is that
it clarifies why baryon bookkeeping is not treated in \ANine{} as an accidental
approximate global symmetry. It belongs to the constitutional visible gauge
structure.

\subsection{Spin, family counting, and why three generations are minimal under \MSMF{}}
\label{app:family-counting-detail}

The new main-text sections argued that three generations are the minimal
nontrivial family number compatible with hierarchical mixing and CP violation.
This appendix records the underlying logic more explicitly.

\subsection{Spin counting from a 6D spinor to a 4D massive fermion}

A six-dimensional Dirac spinor carries more components than a four-dimensional
Dirac spinor, but the boundary reduction and chiral projection remove unwanted
light-sector components. Schematically,
\begin{equation}
\label{eq:spinor-decomp-detail}
\Psi_{6D}
\longrightarrow
\psi_{L,4D}\oplus \psi_{R,4D}\oplus \text{heavy mirror/KK components}.
\end{equation}
After electroweak mass generation, a massive 4D fermion supports two physical
spin projections,
\begin{equation}
\label{eq:spin-states-detail}
s_z=\pm\frac12.
\end{equation}
This is the precise technical meaning of the phrase ``half spin, spin up/down.

\subsection{Flavor counting versus state counting}

For one generation the visible matter content contains four flavor types:
up-type quark, down-type quark, charged lepton, and neutrino. With three
generations the flavor content therefore becomes twelve. State counting,
however, is richer because quarks carry color and all massive fermions carry two
spin projections.

Table~\ref{tab:flavor-vs-state-counting-detail} keeps these notions separate.

\begin{table}[H]
\centering
\caption[Why ``six quarks and six leptons'' refers to flavor counting rather than full state counting.]{Why ``six quarks and six leptons'' refers to flavor counting rather than
full state counting. The distinction matters because the two counts are used interchangeably in the literature and differ by the color and spin multiplicities. The table separates them line by line, so that the twelve of ``six quarks and six leptons'' can be traced to flavor counting and the larger numbers that appear in thermodynamic and anomaly arguments can be traced to state counting. Nothing here is derived; it is bookkeeping made explicit so that later counts cannot be misread.}
\label{tab:flavor-vs-state-counting-detail}
\small
\begin{tabular}{p{0.24\textwidth}p{0.17\textwidth}p{0.17\textwidth}p{0.28\textwidth}}
\toprule
\textbf{Object counted} & \textbf{Per family} & \textbf{For three families} & \textbf{Comment}\\
\midrule
Quark flavors & 2 & 6 & Up-type and down-type across three generations\\
Lepton flavors & 2 & 6 & Charged lepton and neutrino across three generations\\
Quark color states & \(2\times 3\) & \(6\times 3\) & Color multiplies state count but not flavor count\\
Spin states for massive fermions & factor of 2 & factor of 2 & Spin doubles physical state multiplicity for each massive species\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Why one generation is too poor}

With one generation there is no nontrivial flavor mixing. The family space is
one-dimensional, so any would-be mixing matrix is trivial. This immediately
fails to account for the observed pattern of quark and lepton misalignment.

\subsection{Why two generations are still insufficient}

With two generations one can have one real mixing angle, but no irreducible
Kobayashi--Maskawa phase. Put differently, all apparent complex phases can be
removed by rephasing, so there is no physical Jarlskog invariant. In that sense
two generations are the first nontrivial mixing case but not the first full flavor
case.

\subsection{Why three generations are the first fully nontrivial case}

With three generations the family space is large enough to support
\begin{enumerate}[label=(\roman*),leftmargin=2.5em]
\item nontrivial hierarchical masses,
\item nontrivial quark and lepton misalignment,
\item a physical CP-odd invariant,
\item and, in the neutrino case, genuinely distinct Majorana-phase structure.
\end{enumerate}
This is the real meaning of ``minimal but nontrivial'' in the present paper.
\MSMF{} does not merely say ``choose the smallest number because it is small.'' It
says ``choose the smallest number that can actually carry the observed structure.''
For flavor, that number is three.

\subsection{Relation to the later codebook construction}

The later sections of \ANine{} make this argument constructive by writing down
minimal family projectors and overlap codebooks. The present appendix does not
repeat all of that arithmetic. Its role is to record the logical reason why the
family number should already be expected to be three before the detailed codebook
is displayed.

\subsection{Packet cards for the unified and Standard-Model seam descriptions}
\label{app:sip-cards-sm}

The main text introduced both a unified packet and a Standard-Model seam packet.
This appendix presents them in compact ``packet card'' form for quick reference.
The purpose is practical: it lets the reader see in one place how the abstract \SIP{}
formalism is instantiated in the particle-physics context.

\subsection{Unified visible packet card}

\begin{table}[htbp]
\centering
\small
\caption[The unified visible packet before branch selection, in the compact card form used through this appendix.]{The unified visible packet before branch selection, in the compact card form used through this appendix. Each field of the abstract formalism is instantiated for the particle-physics case, so that the correspondence can be read in one place rather than reconstructed from the general definitions. The kill criteria are the operative row: they say what would falsify the instantiation, and they are stated before the branch is chosen rather than after it.}
\label{tab:packet-unified}
\begin{tabular}{p{0.26\textwidth}p{0.62\textwidth}}
\toprule
\textbf{Field} & \textbf{Unified visible packet before branch selection}\\
\midrule
State manifold & Unified gauge connection, compact-angle holonomy, radial transport variables, matter multiplets, anomaly-compatible boundary data\\
Ports & Boundary conditions, bulk-to-boundary projection channels, UV/IR matching interfaces, admissible preparation/readout channels\\
Generator & Warped-bulk transport plus projected open-system evolution\\
\GFF{} ledger & Unified gauge + gravity + matter + corrective scalar ledger\\
Observables & Holonomy class, anomaly checks, surviving zero modes, projected response functions\\
Kill criteria & Anomaly failure, wrong surviving algebra, failure of positivity/projection, proton-instability operators\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Standard-Model seam packet card}

\begin{table}[htbp]
\centering
\small
\caption[The Standard-Model seam packet after branch selection, in the same card form and with the same fields as the unified packet above.]{The Standard-Model seam packet after branch selection, in the same card form and with the same fields as the unified packet above. The two cards are meant to be read together and compared field by field. What changes across the branch selection is as informative as what survives it, and the fields that survive unchanged are the ones on which the selection has no purchase at all.}
\label{tab:packet-sm-seam}
\begin{tabular}{p{0.26\textwidth}p{0.62\textwidth}}
\toprule
\textbf{Field} & \textbf{Standard-Model seam packet after branch selection}\\
\midrule
State manifold & Running gauge couplings, Wilson-line/Higgs order parameter, family projectors, renormalized masses and mixings\\
Ports & Scattering channels, decay channels, oscillation baselines, detector readout and reconstruction pipelines\\
Generator & Effective seam RG transport plus low-energy interaction dynamics\\
\GFF{} ledger & Scalar budget governing branch stability and observable extraction\\
Observables & \(\alphahatinv\), \(\sin^2\theta_W\), \(\alpha_s\), \(v\), \(m_h\), fermion masses, CKM, PMNS, Majorana phases\\
Kill criteria & Wrong gauge pattern, wrong stability class, catastrophic failure of predicted observables beyond theory error, inconsistency with proton stability\\
\bottomrule
\end{tabular}
\end{table}

The comparison between the two cards makes the logic of the manuscript explicit.
The second is not independent of the first. It is its stabilized boundary
projection.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{M.3.1. Decompose the adjoint $\mathbf{24}$
and read off the leptoquark charges.}

\noindent\emph{Solution.}  Step 1: count the decomposition
\eqref{eq:adjoint-24-detail}: $(8,1)_0 \oplus (1,3)_0 \oplus (1,1)_0
\oplus (3,2)_{-5/6} \oplus (\bar 3,2)_{+5/6}$ gives
$8+3+1+6+6 = 24$ states --- the gluons, the weak triplet, the
hypercharge boson, and twelve leptoquark directions $X, Y$.  Step 2:
the leptoquark hypercharge is a pair difference of the generator
\eqref{eq:y-generator-detail}: a color row against a weak row gives
$Y = -\tfrac13 - \tfrac12 = -\tfrac56$; then $Q = T_3 + Y$
\eqref{eq:q-electric-detail} yields $Q(X) = -\tfrac43$ and
$Q(Y) = -\tfrac13$.  Step 3: these twelve are exactly the directions
the boundary conditions remove from the four-dimensional gauge sector
--- bulk-only in \GG{} --- which is why proton decay by $X, Y$
exchange never reaches the boundary (chapter L).  Physics intuition:
the adjoint is the unified force's full keyboard; the boundary gets
to play only the $8+3+1$ white keys, and the twelve black keys ---
the ones whose chords turn quarks into leptons --- stay locked in the
bulk.

\medskip\noindent\textbf{M.6.1. Audit the anomalies of gauged baryon
number, and find the one that bites.}

\noindent\emph{Solution.}  Step 1: with the assignments
\eqref{eq:baryon-number-detail} ($B = \tfrac13$ on quark fields, $0$
on leptons), three sums vanish identically per family:
$[\SU(3)]^2\,\UoneB{}$: $2(\tfrac13) - \tfrac13 - \tfrac13 = 0$;
$\mathrm{grav}^2\,\UoneB{}$: $6(\tfrac13) - 3(\tfrac13) - 3(\tfrac13)
= 0$; $[\UoneB{}]^3$: $\tfrac{6}{27} - \tfrac{3}{27} - \tfrac{3}{27}
= 0$.  Step 2: two do not: $[\SU(2)]^2\,\UoneB{}$ sums $B$ over
doublets --- three colored quark doublets at $\tfrac13$ against a
lepton doublet at $0$ --- giving $1 \ne 0$; and the mixed
$\UoneB{}\,[\U(1)_Y]^2$ sum gives $-\tfrac12 \ne 0$.  Step 3: this is
why baryon number in the Standard Model is only an \emph{accidental}
symmetry, and why gauging it --- as \GG{} does --- is impossible in
four dimensions alone: the surviving anomalies must be canceled by
the \BIsix{} Green--Schwarz inflow, which is precisely the
load-bearing hypothesis (H3) of the proton-stability theorem
(Theorem~\ref{thm:proton-stability-a9}).  Physics intuition: the
proton's guardian symmetry is anomalous on the boundary and can only
be rescued from the bulk --- so proton stability, in this framework,
is not a low-energy accident but a six-dimensional bookkeeping
identity; the assembly interlude's problem G.3.2 is the same audit
run for hypercharge, where all four sums close on their own.

\medskip\noindent\textbf{M.11.1. Prove that two generations admit no
CP violation: the general-$N$ counting.}

\noindent\emph{Solution.}  Step 1: an $N\times N$ unitary mixing
matrix carries $N^2$ parameters, of which $N(N-1)/2$ are angles and
the rest phases; $2N-1$ phases are removable by quark rephasing.
Step 2: physical phases number
$\tfrac{N^2+N}{2}\,{-}\,(2N-1) = \tfrac{(N-1)(N-2)}{2}$: for $N = 1$,
no mixing at all; for $N = 2$, one angle (Cabibbo's 1963 rotation)
and \emph{zero} phases --- every complex entry can be rephased away,
so no Jarlskog invariant exists; for $N = 3$, three angles and one
phase --- the first $N$ with room for CP.  Step 3: this is leg one of
the \MSMF{} minimality argument of this chapter: three generations
are not decoration but the smallest family space that can carry
hierarchical masses, nontrivial misalignment, \emph{and} a CP-odd
invariant --- and the closed two-sided theorem lives in \PaperPTwo{}
(\S\ref{sec:N3-theorem}).  Physics intuition: nature needed CP
violation to make more matter than antimatter, and CP violation
needed three families to exist at all --- the counting on this page,
run forwards, is the anthropic-sounding fact made mechanical: a
two-family universe would have been perfectly symmetric, and perfectly
empty.

\medskip\noindent\textbf{M.2.1. Show that the ledger integers are not
independently adjustable.}

\noindent\emph{Solution.}  Step 1: the ledger is $(M,N,L;J,K) =
(1,3,5;7,35)$ with the closure relation $K = LJ$.  Step 2: so $K$ is not
a fifth input --- fixing $L$ and $J$ fixes it, and $K = 35$ then appears
in the reactor angle $N/4K$, in the lock scale
$\tfrac12\EPl{}e^{-K}$, and in the throughput exponent $e^{-K/2}$.
Step 3: consequently a change to $L$ or $J$ propagates into the lepton
mixing, the collider scale and the baryon asymmetry simultaneously; they
cannot be tuned one at a time.  Physics intuition: this is where the
framework is most exposed and most testable.  Five integers, only four
of them independent, feeding twenty-eight parameters --- so the
falsification channels are not independent either, and a single decisive
failure in one sector cannot be repaired locally.  A reader assessing
the program should look for this interlocking rather than for
individual agreements.

\medskip\noindent\textbf{M.2.2. Count the inputs against the outputs.}

\noindent\emph{Solution.}  Step 1: the ledger supplies four independent
integers plus the constitutional structure.  Step 2: the codebook
delivers twenty-eight Standard-Model parameters, and Stage-5 material
adds the baryon asymmetry, the two Majorana phases and $\mbb{}$ ---
conditionally in the first case.  Step 3: the ratio of outputs to
inputs is therefore of order seven to one, and it is that ratio, not
the precision of any single entry, that is the claim.  Physics intuition:
 a framework with as many parameters as observables explains
nothing however well it fits, and one with far fewer is falsifiable
even if some entries are imprecise.  When judging any unification
proposal, count first and admire second --- and apply the count
honestly, which here means noting that the conditional entries should be
weighted less than the derived ones.

\medskip\noindent\textbf{M.2.3. Locate where the architecture would
break if $N$ were four.}

\noindent\emph{Solution.}  Step 1: $N = 3$ is imported from
\PaperPTwo{} rather than derived here, on three independent legs.
Step 2: trace the consequences of $N = 4$: the cascade-closure bound
$N(N-1)/2 \le N$ fails at $N = 4$; the reactor angle $N/4K$ would move
by a third; the family projector geometry on $\mathbb{C}P^{N-1}$ would
change dimension; and the Majorana ledger's rank-two structure, which
gives exactly two splittings, would admit a third.  Step 3: so a fourth
family is not an extension of this framework but a refutation of it,
and the observation of one falsifies the whole chain.  Physics intuition:
 the useful question about any structural input is not
``is it true?'' but ``what does the theory look like if it is false?''
Here the answer is that there is no theory left, which is a much
stronger statement than an adjustable parameter would allow --- and it
is why the $N = 3$ derivation is carried by a dedicated paper rather
than assumed in passing.

\medskip\noindent\textbf{M.2.4. Distinguish a ledger from a fit by a
test a reader can apply.}

\noindent\emph{Solution.}  Step 1: state the test --- for each derived
quantity, ask whether any ledger input was chosen \emph{after} that
quantity was known, and whether the same input is used elsewhere with
no freedom to differ.  Step 2: apply it to the reactor angle
$\sin^2\theta_{13} = N/4K$: $N$ comes from the family count and $K$
from the closure $LJ$, both fixed by the gauge sector, and both reused
in the lock scale and the throughput --- so the angle has no private
dial.  Step 3: apply it to a place where the answer is less
comfortable: the Weinberg-operator coefficient of
App.~\ref{app:pmns}, recorded in the provenance table as
\textbf{Derived (cond.)} precisely because the projector ratio $2N$ was
selected against the measured splitting ratio.  Physics intuition: the
test is worth applying to every row of the codebook, and the honest
outcome is a mixture rather than a clean sweep.  A framework's
credibility rests less on how many rows pass than on whether its own
provenance table reports the failures --- which is what the
\textbf{Derived (cond.)} marker exists to do.



\section{Force laws, anomaly cancellation, and proton stability}
\label{app:ch-forcelaws}

\GGteachbox{Three structural facts the Standard Model presents
as facts and the geometry reads as consequences: the
inverse-square form of the classical force laws (from boundary
Green's functions), the family-by-family cancellation of gauge
anomalies (inherited from the bulk identity rather than imposed),
and the stability of the proton (a gauged baryon symmetry rather
than an accident) --- with the Super-Kamiokande bound as the
worked example.}
\GGreadalong{\citet{HalzenMartin1984} for the cross-section and
decay estimates; \citet{Adler1969} and \citet{BellJackiw1969}
for the anomaly originals.}

This appendix collects three structural features that the Standard Model presents as
facts and \GGTheory{} reads as consequences of the boundary geometry: the
inverse-square form of the classical force laws, the family-by-family cancellation of
gauge anomalies, and the stability of the proton. We follow the three-step plan, with
the explicit Green's-function and \(\U(1)_B\) derivations carried in the detailed
subsections that follow this framing. The classical force laws and proton phenomenology are developed in \citet{HalzenMartin1984}.

\subsection{Step 1 --- force laws, anomalies, and proton decay as a graduate student meets them}
\label{app:forcelaws-step1}

\paragraph{The inverse-square laws.}
Both Newtonian gravity and the Coulomb force fall off as \(1/r^2\). This is not an
independent postulate but a consequence of a massless mediator in three spatial
dimensions: the static potential is the Green's function of the Laplacian,
\(\nabla^2\phi=-\rho\), whose solution \(\phi\propto 1/r\) gives a force
\(\propto 1/r^2\). The explicit derivation from the three-dimensional Gauss law is
given in the subsection below.
\emph{(Physics intuition: field lines from a point source spread over a sphere whose
area grows as \(r^2\), so their density --- the field strength --- falls as \(1/r^2\).
The inverse-square law is the geometry of three-dimensional space, not a separate force
rule.)}

\paragraph{Anomaly cancellation.}
A chiral gauge theory is consistent only if its gauge anomalies cancel. In the Standard
Model the triangle anomalies, identified by Adler and by Bell and Jackiw \citep{Adler1969,BellJackiw1969} and shown to cancel family-by-family in the Standard Model by Bouchiat, Iliopoulos, and Meyer \citep{BouchiatIliopoulosMeyer1972} --- \([\SU(3)]^2\U(1)\), \([\SU(2)]^2\U(1)\),
\([\U(1)]^3\), and the mixed gravitational \([\mathrm{grav}]^2\U(1)\) --- all cancel
\emph{within each generation}, a delicate conspiracy between the quark and lepton
hypercharges that also forces electric-charge quantization (why \(|q_e|=|q_p|\) to
\(10^{-21}\)).
\emph{(Physics intuition: anomaly cancellation is the quantum consistency tax. Each
fermion contributes to a potential breakdown of gauge invariance at the loop level, and
only because the quark and lepton charges add up to exactly zero in every channel does
the theory survive. That the charges of utterly different particles sum to zero is one
of the deepest hints that a family is a single object.)}

\paragraph{Proton decay in grand unification.}
Grand unified theories place quarks and leptons in common multiplets, so they generally
contain heavy \(X,Y\) gauge bosons that mediate baryon-number-violating decays such as
\(p\to e^+\pi^0\). Minimal \(\SU(5)\) \citep{GeorgiGlashow1974} predicts a lifetime
\(\tau_p\sim 10^{30}\)--\(10^{31}\ \mathrm{yr}\), already excluded by the
Super-Kamiokande bound \(\tau_p>1.6\times10^{34}\ \mathrm{yr}\).
\emph{(Physics intuition: unify quarks with leptons and you open a door between them ---
a quark can turn into a lepton --- and the proton can fall apart. The non-observation of
proton decay is therefore a sharp constraint on how, and whether, the forces unify.)}

\paragraph{Worked example (STANDARD TEXTBOOK PEDAGOGY).}
Dimensionally, \(\tau_p\sim M_X^4/(\alpha_{\mathrm{GUT}}^2 m_p^5)\). With
\(M_X\sim 10^{15}\ \mathrm{GeV}\) and \(\alpha_{\mathrm{GUT}}\sim 1/40\) this gives
\(\tau_p\sim 10^{31}\ \mathrm{yr}\) --- below the experimental bound, which is exactly
why minimal \(\SU(5)\) is ruled out and why the proton's stability needs an
explanation.

\subsection{Step 2 --- what these structural facts assume but cannot explain}
\label{app:forcelaws-step2}

The Standard Model uses all three facts without grounding them. The inverse-square law
is inherited from working in three spatial dimensions, which is itself unexplained.
Anomaly cancellation \emph{works}, but the theory does not say why the fermion charges
take the precise values that make it work --- it is imposed as a consistency condition,
not derived. And the proton's stability rests on a coincidence: baryon number \(B\) and
lepton number \(L\) are only \emph{accidental} global symmetries of the renormalizable
Standard-Model Lagrangian, not gauged and not fundamental, so nothing protects the
proton once the theory is embedded in anything larger. Generic unification breaks \(B\)
and predicts a decay that is not seen.
\emph{(Physics intuition: the proton is stable in the Standard Model only by an
accident of which operators happen to be renormalizable. Lean on the theory from above
--- unify it --- and the accident evaporates. A framework that made baryon number exact
rather than accidental would protect the proton on purpose.)}

\subsection{Step 3 --- the geometric derivation: gauged baryon number and boundary force laws}
\label{app:forcelaws-step3}

In \GGTheory{} all three facts descend from the boundary geometry. The inverse-square
laws are the boundary Green's functions of the seam Laplacian, derived explicitly from
the three-dimensional Gauss law in the next subsection. Anomaly freedom is not an
imposed condition but is inherited from upstream: \BIsix{} (\AThree{}) proves the
parent bundle anomaly-free before any low-energy descent, so the family-by-family
cancellation is a theorem of the legal parent theory rather than a numerical accident.
Most decisively, baryon number is \emph{gauged}: the \GG{} extension carries a local
\(\U(1)_B\) that makes \(B\) an exact rather than accidental symmetry, excluding the
\(X,Y\)-type operators that would let the proton decay. The proton is therefore not
merely long-lived but \emph{absolutely stable} in the minimal construction --- a sharp,
falsifiable difference from ordinary four-dimensional grand unification. The detailed
mechanism, and its contrast with minimal 4D unification, is set out in the
proton-stability subsection below.
\emph{(Physics intuition: where ordinary unification opens a door between quarks and
leptons and then must explain why the proton does not walk through it, the geometric
construction never builds the door. Local baryon number is a gauged wall, not an
accidental fence, so the proton is stable for the same kind of reason the photon is
massless --- a symmetry protects it.)}

\subsection{Green's functions, Gauss laws, and the inverse-square form of Newton and Coulomb}
\label{app:newton-coulomb-detail}

This appendix gives the compact mathematical reason why both Newton's and
Coulomb's laws take inverse-square form in three spatial dimensions.

\subsection{Green's function of the 3D Laplacian}

Let \(G(\mathbf x)\) solve
\begin{equation}
\label{eq:laplacian-green-detail}
\nabla^2 G(\mathbf x)=-\delta^{(3)}(\mathbf x).
\end{equation}
The rotationally symmetric solution is
\begin{equation}
\label{eq:green-3d-detail}
G(r)=\frac{1}{4\pi r}.
\end{equation}
Therefore any static massless field whose source equation reduces to a Poisson
problem in three dimensions produces a \(1/r\) potential and a \(1/r^2\) force.

\subsection{Application to Newtonian gravity}

For a point mass \(M\),
\begin{equation}
\label{eq:phi-grav-green-detail}
\Phi_N(\mathbf x)
=
-4\pi G_N M\,G(r)
=
-\frac{G_N M}{r}.
\end{equation}
Differentiating gives
\begin{equation}
\label{eq:force-grav-green-detail}
\mathbf F_N=-m\nabla\Phi_N=-G_N\frac{mM}{r^2}\,\hat{\mathbf r}.
\end{equation}

\subsection{Application to electrostatics}

For a point charge \(q\),
\begin{equation}
\label{eq:phi-em-green-detail}
\phi(\mathbf x)
=
\frac{q}{\varepsilon_0}G(r)
=
\frac{q}{4\pi\varepsilon_0 r},
\end{equation}
so the electric field is
\begin{equation}
\label{eq:e-field-green-detail}
\mathbf E=-\nabla\phi=\frac{q}{4\pi\varepsilon_0 r^2}\,\hat{\mathbf r},
\end{equation}
and the force on a second charge \(q'\) is
\begin{equation}
\label{eq:force-em-green-detail}
\mathbf F_C=q'\mathbf E=\frac{qq'}{4\pi\varepsilon_0 r^2}\,\hat{\mathbf r}.
\end{equation}

\subsection{Why the \GG{} viewpoint adds more than this elementary derivation}

The Green's-function derivation explains the inverse-square form, but not why the
relevant massless fields should exist or why one should have both a geometric and
a gauge sector in the first place. The \GG{} framework adds exactly that missing
layer. It says that the two long-range sectors are boundary exports of one
composite geometry--gauge constitution. The familiar inverse-square laws are
therefore not merely historical laws; they are the infrared trace of a deeper
unified structure.


\subsection{Phenomenological comparison between historical unification language and the \GG{} seam language}
\label{app:historical-vs-gg6}

This appendix supports the professional phenomenological tone of the revised
manuscript by making explicit how the present paper relates to familiar historical
languages of unification. The intent is not polemical. The intent is to help a
reader trained in ordinary gauge-theory phenomenology locate exactly what is
standard, what is reinterpreted, and what is genuinely new in \GG{}/\GDOS{}.

\subsection{Why this comparison is useful}

A phenomenological paper is easier to evaluate when the reader can quickly answer
three questions:
\begin{enumerate}[label=(\roman*),leftmargin=2.5em]
\item Which features are simply inherited from the successful Standard-Model and
historical GUT traditions?
\item Which features are retained but geometrically reinterpreted?
\item Which features are genuinely additional structures introduced by the new
framework?
\end{enumerate}
The present appendix answers those questions in a compact but explicit way.

\subsection{The historical baseline}

The baseline traditional story is familiar. One starts from a 4D gauge theory,
asks whether the gauge couplings suggest unification at high energy, and then
studies simple groups such as \(\SU(5)\) or larger groups as candidate UV
symmetries. The Higgs sector is usually introduced by scalar multiplets, proton
stability is a strong empirical constraint, and flavor is often left largely as a
separate problem.

This story contains genuine achievements. It correctly emphasizes gauge
symmetry, representation theory, anomaly cancellation, RG running, and symmetry
breaking. The present paper does not discard any of those achievements.
Instead it asks whether they can be embedded in a more constrained constitutional
and open-system setting.

\subsection{What remains standard in \GG{}}

Several core ingredients remain completely standard at the phenomenological
level.

\begin{enumerate}[label=(\arabic*),leftmargin=2.5em]
\item The low-energy gauge group visible in laboratory physics is still the
Standard-Model group before electroweak breaking and its familiar remnants after
breaking.
\item Representation theory still matters. Quarks and leptons are still read in
a language inherited from unified group embeddings.
\item Renormalization-group running still matters. The present paper does not
reject RG thinking; it geometrizes it.
\item CKM and PMNS matrices remain the correct low-energy observable language for
flavor mixing.
\item Benchmark comparison to precision data still uses the ordinary observable
basis collected by the PDG and neutrino global analyses.
\end{enumerate}

These points are worth stating because they show that \GG{} is not attempting to
replace successful phenomenology by unfamiliar jargon. It is attempting to give
a deeper origin to structures phenomenology already knows are real.

\subsection{What is reinterpreted rather than discarded}

The clearest examples of reinterpretation are the following.

\paragraph{Unification scale.}
In ordinary language, the unification scale is where couplings approximately meet.
In \GG{}, the relevant ultraviolet seam is tied to the six-dimensional
stage and to the scale chain \(\EPl\to\EGUT\to\Mlock\). The observable meeting
of couplings is thus a seam consequence rather than the sole definition of the UV
scale.

\paragraph{Higgs mechanism.}
In ordinary language, the Higgs is a scalar doublet with a symmetry-breaking
potential. In \GG{}, this language survives only as a low-energy shadow. The
fundamental variable is the Wilson-line holonomy and the selected vacuum branch.

\paragraph{Yukawa sector.}
In ordinary language, Yukawa matrices are phenomenological inputs. In \GG{} they
are effective seam images of overlaps and family projectors.

\paragraph{Mass.}
In ordinary language, mass may be introduced sector by sector. In \GG{}, mass is
recast as a geometric spectral gap and an \GFF{}-curvature quantity.

\paragraph{Observable.}
In ordinary phenomenology, an observable is often treated as a renormalized
number quoted in a review table. In \GDOS{} it is a registered packet export under
a lawful boundary map.

\subsection{What is genuinely additional in \GG{}/\GDOS{}}

The following structures are not merely reinterpretations of standard jargon;
they are genuine additions.

\begin{enumerate}[label=(\arabic*),leftmargin=2.5em]
\item The constitutional layer of four axioms and seven postulates.
\item The six-dimensional bulk as the minimal admissible stage.
\item The \BIsix{} anomaly-cancellation structure with OGN-fixed coefficients.
\item Local \(\U(1)_B\) and local \(\U(1)_{Qi}\) as part of the visible
unified group.
\item The \OGGEP{}/\GDOS{} observational layer that turns the UV theory into an
explicit open-system boundary theory.
\item The \SIP{}/\GFF{}/\DGI{}/\GSSB{}/GA/\QiFour{} runtime vocabulary used to organize branch
selection and measurement.
\end{enumerate}

These are the distinctive conceptual features that the present paper asks the
reader to evaluate.

\subsection{A compact comparison table}

\begin{center}
\begin{longtable}{p{0.23\textwidth}p{0.29\textwidth}p{0.31\textwidth}}
\caption[Phenomenological comparison between historical unification language and the GG-6/GDOS reinterpretation.]{Phenomenological comparison between historical unification language and
the GG-6/GDOS reinterpretation. The left column is stated in the language its own authors used, so that the comparison is between positions rather than between caricatures. The table is a reading aid for the Parts that follow and not an argument against the historical language: on several rows the two columns describe the same physics, and the difference is in what each treats as primitive.}
\label{tab:historical-vs-gg6}\\
\toprule
\textbf{Phenomenological issue} & \textbf{Historical unification language} & \textbf{GG-6/GDOS language}\\
\midrule
\endfirsthead
\toprule
\textbf{Phenomenological issue} & \textbf{Historical unification language} & \textbf{GG-6/GDOS language}\\
\midrule
\endhead
Unified gauge structure & Simple 4D group such as \(\SU(5)\) & Composite 6D geometry--gauge structure with \(\SU(5)\times\U(1)_B\times\U(1)_{Qi}\) on a bulk\\
Scale running & 4D RG flow of effective field theory & Radial transport in \(r=\ln(\EGUT/\mu)\) plus seam evaluation\\
Electroweak breaking & Scalar potential and VEV & Wilson--Hosotani holonomy selection and lock-seam export\\
Flavor masses & Yukawa inputs & Overlap/projector codebook with QD completion\\
CKM/PMNS & Observable unitary matrices & Observable matrices derived from family-projector misalignment\\
Proton stability & Strong empirical constraint, often problematic in minimal GUTs & Local \(\U(1)_B\) and operator-selection discipline\\
Measurement theory & Usually implicit or external to the field theory & Explicit OGGEP/GDOS boundary map\\
Open-system language & Rare in particle phenomenology & Built in from AX-3 onward\\
Hierarchy/naturalness & Bare-scalar sensitivity problem & Finite holonomy potential and multiplicative throat hierarchy\\
Status of parameters & Often free inputs of the effective theory & Seam observables of a constrained packet\\
\bottomrule
\end{longtable}
\end{center}

\subsection{Why the comparison supports the present paper's ambitions}

The comparison table helps clarify the ambition level of \ANine{}. The paper is
not content to say that a few Standard-Model numbers happen to be reproduced by a
creative formula. Its stronger claim is that once the phenomenological target is
re-embedded in a more constrained geometric and operational framework, a large
part of the Standard-Model parameter problem becomes derivable rather than merely
fittable.

Whether one ultimately agrees with every detailed formula in the later sections,
this is the proper standard by which the paper should be judged. It is a paper
about the origin of the parameter structure, not just about numerical coincidence.

\subsection{Final interpretive remark}

A useful way to summarize the appendix is the following. Historical unification
language already knew that gauge symmetry and representation theory are powerful.
\GG{}/\GDOS{} asks whether gauge symmetry, geometry, anomaly freedom, scale,
measurement, and open-system dynamics can all be treated as one non-separable
chain. The present revised manuscript is an attempt to show what happens when that
question is answered in the affirmative and then carried all the way through the
phenomenology of couplings, masses, and mixings.


\subsection{Proton stability, local \texorpdfstring{\(\U(1)_B\)}{U(1)B}, and why the \GG{} Standard-Model foundation differs from minimal 4D unification}
\label{app:proton-stability-foundation}

A recurring phenomenological objection to simple grand unification is proton
instability. The objection is entirely reasonable: any credible UV completion of
the Standard Model must respect the extraordinary empirical longevity of the
proton. Because the foundation sections of the present paper emphasize historical \(\SU(5)\) as the correct conceptual starting point, it is useful to
state explicitly why the \GG{} realization differs in its treatment of baryon
number and proton decay.

\subsection{Why proton stability is a foundational rather than secondary issue}

Proton stability is not merely another observable among many. It is one of the
sharpest filters on theories that unify quarks and leptons. If a candidate
unification scheme predicts rapid baryon-number violation, then it fails before
one even reaches the level of detailed flavor fitting. This is why the present
paper treats local \(\U(1)_B\) as part of the visible unified structure rather than
as an optional afterthought.

In other words, proton stability belongs to the Standard-Model foundation section
for the same reason anomaly freedom does: it is part of the admissibility filter
that any later parameter derivation must inherit.

\subsection{The usual tension in minimal 4D unification}

In minimal 4D \(\SU(5)\)-type unification, the same unified structure that puts
quarks and leptons into common multiplets also tends to introduce heavy bosons or
effective operators that mediate baryon-number violation. The tension is famous:
the unification idea is elegant, but proton decay becomes a powerful empirical
threat.

The present paper does not need to rehearse every historical model-building route
around this tension. It is enough to emphasize the conceptual point: a unified
group alone does not automatically guarantee sufficiently strong protection of the
proton. Additional structure is required.

\subsection{Why \GG{} treats baryon number differently}

\GG{} provides that additional structure by elevating baryon number to a local
\(\U(1)_B\) factor in the visible unified sector. At the conceptual level this
changes the problem from ``how can we suppress dangerous baryon-violating
operators well enough?'' to ``which operators are constitutionally admissible once
local baryon number is part of the gauge structure?''

That is a major shift. It means proton stability is not merely an accidental
numerical consequence of a high mass scale. It is tied to the admissible operator
content of the theory.

\subsection{Why this matters for a parameter paper}

One might wonder why an appendix on proton stability belongs in a paper devoted to
Standard-Model parameters. The answer is that the parameter paper is not being
written inside an arbitrary effective theory. It is being written inside a very
specific ultraviolet framework. The credibility of the entire parameter
construction depends on whether that framework respects the strongest known
structural constraints on the visible world. Proton stability is one of those
constraints.

This appendix therefore supports the main text in a simple way: it reminds the
reader that the same framework used to derive couplings, masses, and mixings also
addresses one of the most severe historical problems of unified model building.

\subsection{Compact comparison}

\begin{table}[htbp]
\centering
\small
\caption[Proton stability in minimal historical four-dimensional unification and in the geometric foundation this paper uses, question by question.]{Proton stability in minimal historical four-dimensional unification and in the geometric foundation this paper uses, question by question. The comparison is included because proton decay is the constraint that ended the first generation of unified models, and a framework that did not address it would be incomplete however well it did elsewhere. The table states the contrast; the quantitative treatment is delegated to the companion paper that owns it.}
\label{tab:proton-comparison}
\begin{tabular}{p{0.28\textwidth}p{0.26\textwidth}p{0.30\textwidth}}
\toprule
\textbf{Question} & \textbf{Minimal historical 4D unification} & \textbf{\GG{} foundation used in this paper}\\
\midrule
Status of baryon number & Often accidental or approximate at low energy & Local \(\U(1)_B\) in the visible unified structure\\
Proton-decay issue & Potentially severe phenomenological tension & Reframed as an operator-admissibility question under local baryon number\\
Role in parameter derivation & Often treated as a separate model-building constraint & Built into the foundation inherited by the parameter derivation\\
\bottomrule
\end{tabular}
\end{table}

The practical moral is straightforward. In the present paper, proton stability is
not disconnected from the rest of the phenomenology. It is one of the reasons the
\GG{} route to the Standard Model is being taken seriously in the first place.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{N.4.1. Derive both inverse-square laws from
one Green's function.}

\noindent\emph{Solution.}  Step 1: the static potential of a point
source obeys $\nabla^2 G = \delta^3(\mathbf r)$, whose solution
\eqref{eq:green-3d-detail} is $G = -1/4\pi r$ --- fixed entirely by
the dimensionality of space, via Gauss: the flux of $\nabla G$
through a sphere is constant, and the sphere's area grows as
$4\pi r^2$.  Step 2: feed the two static sectors the same function:
gravity's potential $\phi = -G_N M/r$ gives
$F = G_N M m/r^2$ \eqref{eq:force-grav-green-detail};
electrostatics' $\phi = q/4\pi\epsilon_0 r$ gives Coulomb's
$F = q_1 q_2/4\pi\epsilon_0 r^2$ \eqref{eq:force-em-green-detail}.
Step 3: in $D$ spatial dimensions the same argument gives
$F \propto r^{-(D-1)}$ --- so the observed $r^{-2}$, tested from
micrometers to solar-system scales, is a live measurement of
$D = 3$: the boundary force laws are a standing check that the
\emph{visible} world has exactly the dimensionality the seam assigns
it, with the extra directions compactified out of the flux count.
Physics intuition: an inverse-square law is not a law about forces
but a law about real estate --- flux shared over the surface of a
growing sphere; two interactions as different as gravity and
electricity obey the same power because they rent area in the same
three-dimensional space.

\medskip\noindent\textbf{N.4.2. Compute the $10^{39}$ and say what
kind of explanation could reach it.}

\noindent\emph{Solution.}  Step 1: form the dimensionless ratio of
the electric to gravitational force between a proton and an electron
(any separation --- both fall as $r^{-2}$):
$e^2/4\pi\epsilon_0 \div G_N m_p m_e$.  Step 2: numerically,
$\alpha\hbar c = 2.31\times10^{-28}$~N\,m$^2$ against
$G_N m_p m_e = 1.02\times10^{-67}$~N\,m$^2$: ratio
$2.3\times10^{39}$.  Step 3: no perturbative mechanism generates
$10^{39}$ from order-one inputs, but exponentials of modest integers
do it effortlessly --- $e^{91} \simeq 3\times10^{39}$ --- and the
paper's worked instance is the hierarchy identity
$v/\EPl = \tfrac12 e^{-35}(2/5)^3$ (assembly-interlude register:
integers of the ledger, not tuned decimals); gravity is feeble
because it alone couples through the full exponential depth of the
radial dimension.  Physics intuition: the most extreme number in
everyday physics --- the reason a toy magnet beats the whole planet's
gravity --- is, on this reading, a \emph{length} in the fifth
dimension measured in $e$-folds; hierarchy problems dissolve when
ratios stop being dials and become distances.

\medskip\noindent\textbf{N.6.1. Kill classic $\mathrm{SU}(5)$ by
arithmetic, and state why the export is immune.}

\noindent\emph{Solution.}  Step 1: four-dimensional $X, Y$ exchange
generates the $\Delta B = 1$ operator $qqql/M_X^2$, giving the
standard estimate $\Gamma_p \sim \alpha_G^2\,m_p^5/M_X^4$.  Step 2:
with $\alpha_G \simeq 1/25$ and $M_X \simeq 10^{15}$~GeV:
$\Gamma_p \simeq (1/625)(0.938)^5/10^{60}~\mathrm{GeV} =
1.2\times10^{-63}$~GeV, i.e.\ $\tau_p = \hbar/\Gamma_p \simeq
1.8\times10^{31}$~yr --- three orders of magnitude below the
Super-Kamiokande bound $\tau(p \to e^+\pi^0) > 2.4\times10^{34}$~yr:
the 1974 theory is excluded by a wide margin, not a technicality.
Step 3: the export escapes at the operator level, not by raising
$M_X$: the $X, Y$ are bulk-only (M.3.1), so no four-dimensional
exchange exists, and every $\Delta B \ne 0$ Wilson coefficient
vanishes under the exact global baryon number of
Theorem~\ref{thm:proton-stability-a9} --- conditional on its stated
hypotheses, with (H3) the anomaly-inflow audit of M.6.1.  Physics intuition:
 classic unification put the quark and the lepton in one
multiplet and paid with the proton's life; the six-dimensional
version keeps the marriage but moves the matchmakers out of reach
--- the difference between suppressing an operator and never writing
it is the difference between $10^{31}$ years and forever.

\medskip\noindent\textbf{N.6.2. Show that even the anomaly cannot
kill a proton: $\Delta B \in 3\mathbb{Z}$.}

\noindent\emph{Solution.}  Step 1: the one exact non-conservation of
baryon number is the electroweak anomaly,
$\partial_\mu J_B^\mu = N_{\rm gen}(g^2/32\pi^2)\,
\mathrm{tr}\,W\widetilde W$ --- each family contributes once, so any
topological transition changes $B$ by $N_{\rm gen}\,n = 3n$.
Step 2: a proton decay is $\Delta B = -1$; but $-1 \notin
3\mathbb{Z}$, so no anomalous process, however violent, can remove a
single nucleon --- the minimal event costs three baryons at once
(and three leptons with it, preserving $B - L$).  Step 3: the rate
is in any case tunneling-suppressed by the instanton action
$S = 8\pi^2/g^2 \approx 187$ --- utterly negligible today --- but the
$3\mathbb{Z}$ selection rule is the structural point: it ties proton
safety to the same $N = 3$ that the family pincer fixed.  Physics intuition:
 even the one legal loophole in baryon conservation is
quantized in family units --- the anomaly can reshuffle the cosmic
books only three baryons at a time, so the generation count
literally stands guard over every atom; $N = 3$ is not flavor
trivia, it is nuclear security.

\medskip\noindent\textbf{N.6.3. Defend the Stueckelberg mechanism
with the massive-photon analogy.}

\noindent\emph{Solution.}  Step 1: the objection --- ``your
$\UoneB{}$ boson is massive, so baryon number is broken and the
proton should decay.''  Step 2: separate the two things a gauge
symmetry does: the \emph{local} redundancy (broken --- the $Z'_B$ is
massive, its longitudinal mode the eaten Stueckelberg scalar, whose
couplings are derivative and $B$-symmetric) and the \emph{global}
charge (exact --- every operator still carries its $B$, and
$\Delta B \ne 0$ coefficients vanish identically).  Step 3: the
controlled analogy: give the photon a mass and electric charge is
still exactly conserved --- a boson mass gaps the force, not the
bookkeeping; hence the prediction is ``exact global $B$, residual to
a Stueckelberg-massive $\UoneB{}$,'' with the massive $Z'_B$ itself
the falsifiable trace (the collider and fifth-force windows of the
falsification catalog).  Physics intuition: symmetries have a body
and a soul --- the gauge boson is the body, the conservation law the
soul; Stueckelberg kills the body's long range and leaves the soul
untouched, and the proton answers to the soul.


\GGtakeaway{\emph{Standard result:} inverse-square laws,
anomaly-free generations, and a proton stable to
$>10^{34}$~yr.  \emph{What it assumed:} anomaly cancellation
imposed as a consistency check, and baryon number protected
only by accident.  \emph{The geometric reading:} Green's
functions on the boundary give the force laws; anomaly freedom
is inherited from the bulk identity; and a gauged $U(1)_B$
makes proton stability absolute --- a prediction, not a
tolerance.}



\section{The strong-CP problem, confinement, and the mass-energy dictionary}
\label{app:ch-strongcp}

\GGteachbox{The protected corners of the Standard Model ---
$\bar\theta$ and the neutron-EDM bound; color confinement and
the flux tube; and the mass-energy dictionary that keeps one
ontology for mass across sectors: the textbook statements, the
two famous non-explanations (why $\bar\theta$ is small, why
quarks never appear alone), and the geometric readings --- the
protected topological zero, and mass as entries of one seam
ledger.}
\GGreadalong{\citet{PeskinSchroeder1995} for the instanton and
$\theta$-vacuum background; \citet{PecceiQuinn1977} for the
axion alternative this framework does not need.}

This appendix takes up two of the protected corners of the Standard Model --- the
strong-CP angle and color confinement --- together with the unified language for
mass that the framework uses throughout. Each is treated in the three-step course
form: the textbook statement, what it leaves unexplained, and the geometric reading.
We begin with the twenty-eighth and last free parameter, the strong-CP angle. The instanton and topological background to this section is developed in \citet{PeskinSchroeder1995}.

\subsection{Step 1 --- the strong-CP problem as a graduate student meets it}
\label{app:strongcp-step1}

\paragraph{The \texorpdfstring{$\bar\theta$}{theta-bar} term.}
Quantum chromodynamics admits a gauge-invariant, renormalizable term that violates
CP,
\begin{equation}
\mathcal L_\theta=\theta\,\frac{g_s^2}{32\pi^2}\,G^a_{\mu\nu}\widetilde G^{a\,\mu\nu},
\label{eq:thetaterm-app}
\end{equation}
a total derivative that nonetheless contributes through topologically nontrivial
gauge configurations (instantons). Chiral rotations of the quark fields shift
\(\theta\) by the phase of the quark mass determinant, so the physical, basis-independent
quantity is
\begin{equation}
\bar\theta=\theta_{\mathrm{QCD}}+\arg\det(M_uM_d).
\label{eq:thetabar-app}
\end{equation}
\emph{(Physics intuition: \(\bar\theta\) measures how much the strong vacuum
distinguishes left from right. There is no symmetry reason for it to vanish, and on
general grounds a dimensionless angle should be of order one.)}

\paragraph{The experimental bound.}
A nonzero \(\bar\theta\) would give the neutron an electric dipole moment
\(d_n\sim\bar\theta\times 6\times10^{-17}\ e\,\mathrm{cm}\). The measured bound
\(d_n<1.8\times10^{-26}\ e\,\mathrm{cm}\) forces
\begin{equation}
\bar\theta\lesssim 10^{-10}.
\label{eq:thetabar-bound-app}
\end{equation}
The strong-CP problem is the question of why this angle is ten orders of magnitude
below its natural value. The leading proposed answer is the Peccei--Quinn axion \citep{PecceiQuinn1977,Weinberg1978,Wilczek1978}, taken up within the cohort as the cold-dark-matter Kaxion of \PaperCOne{} \citep{ArisakaC1Kaxion}: a
new symmetry that promotes \(\bar\theta\) to a dynamical field relaxing to zero,
predicting a light pseudoscalar that has gone undiscovered through more than forty
years of searches.
\emph{(Physics intuition: the neutron is a sensitive CP antenna. If the strong vacuum
were even slightly chiral, the neutron would carry a measurable dipole; that it does
not, to one part in \(10^{10}\), is the puzzle.)}

\subsection{Step 2 --- what the strong-CP sector assumes but cannot explain}
\label{app:strongcp-step2}

In the Standard Model \(\bar\theta\) is simply the twenty-eighth free parameter, tuned
to zero by hand to one part in \(10^{10}\) with no protecting symmetry. The axion
mechanism would explain the smallness, but at the cost of a new global symmetry of
uncertain quality and a new particle that remains unobserved. The Standard Model, on
its own, offers no reason for \(\bar\theta\) to be small.

\subsection{Step 3 --- the geometric derivation: a protected topological readout}
\label{app:strongcp-step3}

In \GGTheory{} \(\bar\theta\) is not tuned and needs no axion. Both terms of
\eqref{eq:thetabar-app} vanish at first order for structural reasons, and the infrared region suppresses any residue to far below detection. The first term vanishes because
the visible four-dimensional Lagrangian is \emph{exported} from the six-dimensional
bulk, not assembled by listing all gauge invariants: the topological density
\(G\widetilde G\) can appear only if the bulk descent generates a coefficient for it,
and \ATwo{}/\AThree{} introduce no such coefficient, so \(\theta_{\mathrm{top}}^{(1)}=0\).
The second term vanishes because, in the first-order quark codebook, the singular
values are positive and the physical phases live only in the misalignment matrices
\(L_u^\dagger L_d\), giving \(\arg\det U_u+\arg\det U_d=0\). Hence
\(\bar\theta^{(1)}=0\). The smallest constitutionally natural residue is the triple
product
\begin{equation}
\varepsilon_{\mathrm{topo}}^{\mathrm{QD}}
=\underbrace{\sin(2\pi/L)}_{\text{minimal CP-odd compact phase}}
\;\underbrace{2^{-J}}_{\text{minimal dyadic leakage}}
\;\underbrace{e^{-K}}_{\text{deep topological suppression}}
\simeq 4.7\times10^{-18},
\label{eq:thetabar-residue-app}
\end{equation}
some eight orders of magnitude below the experimental bound
\eqref{eq:thetabar-bound-app}; the explicit reduction is recorded in
Appendix~\ref{app:theta-detail}.
\emph{(Physics intuition: strong-CP is solved here not by adding a particle but by the
\emph{absence of a knob}. The bulk simply does not hand the boundary a \(\theta\)-term,
and the first-order quark phases cancel, so the angle starts at zero; the only thing
that can reopen it is a deep topological leakage through the throat, and that is
suppressed by the same \(e^{-K}\) scale and \(2^{-J}\) dyadic factors that appear
everywhere else --- down to \(10^{-18}\). The neutron antenna stays silent because there
is nothing to drive it.)} In the honesty triage \(\bar\theta\) is recorded as
\textbf{Derived (cond.)}: the determinant-phase cancellation is not yet proved
end-to-end (the named lemma L9), and at first order the codebook ``happens to be real.''

\subsection{Explicit topological-phase reduction for the strong-\texorpdfstring{CP}{CP} sector}
\label{app:theta-detail}

This appendix records the logic of the strong-CP reduction in a more
algebraic form. The main text emphasized the physical combination
\(\bar\theta=\theta_{\rm top}+\arg\det(M_uM_d)\). Two claims were used:
first, there is no independent first-order seam \(\theta_{\rm top}\) term;
second, the determinant phases of the first-order quark blocks cancel.

The first claim is a statement about admissible descent. The visible seam
Lagrangian is not written by listing all 4D gauge-invariant terms. It is
exported from the 6D bulk. Therefore the 4D topological density
\(G\widetilde G\) can appear only if the descent of the admissible bulk
action generates a free coefficient multiplying it. The written \ATwo{}
and \AThree{} papers do not introduce such a coefficient. Hence
\(\theta_{\rm top}^{(1)}=0\).

The second claim may be expressed in terms of the polar decomposition. Write
\begin{equation}
M_f = H_f U_f,
\qquad
H_f=(M_fM_f^\dagger)^{1/2},
\qquad
U_f\in U(3).
\label{eq:polar-mf-app}
\end{equation}
Then
\begin{equation}
\arg\det M_f = \arg\det U_f.
\label{eq:argdet-u-app}
\end{equation}
In the present first-order codebook the singular values are all positive and
the nontrivial phases enter only in relative misalignment matrices such as
\(L_u^\dagger L_d\), not in an independent determinant phase of \(U_f\).
Therefore \(\arg\det U_u + \arg\det U_d=0\) at first order.

The QD estimate may be derived by the same three-factor logic used in the
main text:
\begin{equation}
\varepsilon_{\rm topo}^{\rm QD}
=
\underbrace{\sin(2\pi/L)}_{\text{minimal CP-odd compact phase}}
\underbrace{2^{-J}}_{\text{minimal dyadic leakage}}
\underbrace{e^{-K}}_{\text{deep topological suppression}},
\label{eq:theta-product-app}
\end{equation}
which for \((L,J,K)=(5,7,35)\) gives \(4.68\times10^{-18}\). The value is
not claimed to be the final output of a completed nonperturbative
topological computation. It is the smallest constitutionally natural scale
available to the finished parameter paper.

\subsection{Arithmetic of the derived electroweak constants}
\label{app:constants-detail}

The main text derived the familiar electroweak constants from the primary
ledger. Here the arithmetic is recorded explicitly.

From the primary gauge/Higgs block one has
\[
\alpha_{\rm em}^{-1}=128,
\qquad
\sin^2\theta_W=\frac{25}{108}.
\]
Therefore
\begin{equation}
\alpha_2^{-1}
=
\alpha_{\rm em}^{-1}\sin^2\theta_W
=
128\times\frac{25}{108}
=
\frac{800}{27}
=
29.6296296,
\label{eq:alpha2-arith-app}
\end{equation}
and
\begin{equation}
\alpha_Y^{-1}
=
\alpha_{\rm em}^{-1}-\alpha_2^{-1}
=
128-\frac{800}{27}
=
\frac{2656}{27}
=
98.3703704.
\label{eq:alphay-arith-app}
\end{equation}
Hence
\begin{equation}
g_2=\sqrt{4\pi\alpha_2}=0.651241,
\qquad
g_Y=\sqrt{4\pi\alpha_Y}=0.357415,
\qquad
e=\sqrt{4\pi\alpha_{\rm em}}=0.313329.
\label{eq:g-arith-app}
\end{equation}

For the weak scale,
\begin{equation}
v
=
\frac{\EPl}{2e^{35}}\left(\frac25\right)^3
=
246.331354~\mathrm{GeV}
\quad\text{(first order)},
\label{eq:v-arith-app}
\end{equation}
which gives
\begin{equation}
G_F^{(1)}
=
\frac{1}{\sqrt2\,v^2}
=
1.16532111\times10^{-5}\ \mathrm{GeV}^{-2}.
\label{eq:gf-arith-app}
\end{equation}
Likewise
\begin{equation}
m_W^{(1)}=\frac12 g_2v=80.2106~\mathrm{GeV},
\qquad
m_Z^{(1)}=\frac12 v\sqrt{g_2^2+g_Y^2}=91.4965~\mathrm{GeV}.
\label{eq:mw-mz-arith-app}
\end{equation}

The same steps can be repeated with the QD values
\[
\alpha_{\rm em}^{-1}=127.9296875,
\qquad
v^{\rm QD}=246.220000~\mathrm{GeV},
\]
leading to
\[
g_2^{\rm QD}=0.651420,
\qquad
g_Y^{\rm QD}=0.357513,
\qquad
G_F^{\rm QD}=1.16637539\times10^{-5}\ \mathrm{GeV}^{-2},
\]
\[
m_W^{\rm QD}=80.3700~\mathrm{GeV},
\qquad
m_Z^{\rm QD}=91.1900~\mathrm{GeV}.
\]
These are the numbers quoted in the main text.

\subsection{Step 1 --- confinement and the mass-energy of matter as a graduate student meets it}
\label{app:confine-step1}

\paragraph{Asymptotic freedom and dimensional transmutation.}
The strong coupling weakens at high energy and grows at low energy
(Appendix~\ref{app:gauge-strong}). As one runs downward, \(\alpha_s\) becomes large
near a scale at which perturbation theory breaks down, and quarks and gluons are
confined into color-singlet hadrons. That scale is generated from the dimensionless
coupling by \emph{dimensional transmutation},
\begin{equation}
\Lambda_{\mathrm{QCD}}=\mu\,\exp\!\left(-\frac{8\pi^2}{b_0\,g_3^2(\mu)}\right)
\simeq 200\ \mathrm{MeV},
\label{eq:lambdaqcd-step1-app}
\end{equation}
a mass scale conjured from a pure number.
\emph{(Physics intuition: QCD has no mass in its Lagrangian for the light quarks, yet
it produces the proton. The trick is that the running coupling hits strong coupling at
a definite energy, and that energy --- where the force can no longer be ignored --- is
itself the scale of hadron masses.)}

\paragraph{Most mass is binding energy.}
The proton weighs \(\simeq 938\ \mathrm{MeV}\), but its three valence quarks carry only
a few MeV of current mass. More than \(98\%\) of the proton's mass --- and hence of all
visible matter --- is the energy of the confined gluon field, \(E=mc^2\) made concrete.
\emph{(Physics intuition: you are made almost entirely of binding energy. The quark
masses contribute a percent; the rest is the stored energy of the strong field holding
the quarks together. Mass, here, is mostly trapped motion and field.)}

\paragraph{Worked example (STANDARD TEXTBOOK PEDAGOGY).}
Running \(\alpha_s(M_Z)=0.118\) down with one-loop \(b_0\) for QCD gives
\(\Lambda_{\mathrm{QCD}}\) of order a few hundred MeV --- the observed hadronic scale ---
confirming that the proton mass is fixed by where the coupling runs strong, not by the
Higgs Yukawa of the light quarks.

\subsection{Step 2 --- what confinement assumes but cannot explain}
\label{app:confine-step2}

Two things are taken for granted. First, confinement itself is not analytically
proven: that Yang--Mills theory has a mass gap and confines is one of the Clay
Millennium Problems, established only numerically on the lattice, via Wilson's lattice formulation of gauge theory \citep{Wilson1974}. Second, the value of
\(\Lambda_{\mathrm{QCD}}\) relative to the electroweak or Planck scale is, in the
bottom-up theory, just where the measured coupling happens to run strong --- an emergent
input with no deeper account.
\emph{(Physics intuition: the Standard Model can compute how the strong coupling runs,
but it cannot say why it runs strong at \(200\ \mathrm{MeV}\) rather than at some other
scale, nor prove from first principles that quarks must be permanently jailed.)}

\subsection{Step 3 --- the geometric derivation: confinement as a scale window}
\label{app:confine-step3}

In \GGTheory{} the running of the couplings is radial transport in the sixth dimension, and the confinement scale is set by the geometry rather than left as an
emergent accident. The remaining subsections record the explicit formulas: the radial
renormalization-group law, the dimensional-transmutation scale read as a \emph{scale window}, the role of local \(\U(1)_B\) in protecting the baryon sector, and the unified
mass-energy dictionary in which every sector's mass --- weak-boson, Higgs, fermion,
baryon, gravitational --- is one and the same seam spectral gap in the \GFF{} ledger.
\emph{(Physics intuition: the strong coupling runs because moving in energy is moving
radially through the infrared region, and the place where it runs strong --- the
confinement scale --- is a fixed window of that throat. The proton's mass is then a
geometric quantity: the spectral gap of the strong seam, the same ledger entry that, in
other sectors, reads out as the \(W\) mass or the Higgs mass.)}

\subsection{Running couplings, dimensional transmutation, and the baryonic confinement window}
\label{app:rg-confinement-detail}

The main text tied radial transport to renormalization-group flow. This
appendix records a few useful formulas explicitly.

For a generic gauge factor,
\begin{equation}
\mu\frac{dg_i}{d\mu}
=
\beta_i(g_i)
=
-\frac{b_i}{16\pi^2}g_i^3+\cdots,
\label{eq:generic-beta-app}
\end{equation}
which is equivalent to
\begin{equation}
\frac{d}{dr}\frac{1}{g_i^2}
=
\frac{b_i}{8\pi^2}+\cdots,
\qquad
r=\ln\!\left(\frac{\EGUT}{\mu}\right).
\label{eq:generic-r-app}
\end{equation}
Thus the sign of \(b_i\) determines whether the seam coupling strengthens
or weakens as one moves to larger \(r\).

For QED the beta function is positive in \(\alpha\), so the electric charge
is screened in the infrared and unscreened in the ultraviolet. For QCD the
sign flips because of non-Abelian self-interactions, and one obtains the
RG-invariant scale
\begin{equation}
\Lambda_{\rm QCD}
=
\mu\exp\!\left(
-\frac{8\pi^2}{b_0g_3^2(\mu)}
\right)
\quad\text{(one loop)}.
\label{eq:lambdaqcd-app}
\end{equation}
This is dimensional transmutation: the strong interaction generates a mass
scale from a dimensionless coupling.

The additional \GG{} point is the role of local \(U(1)_B\). Confinement by
itself requires asymptotic color singlets. Local baryon number then
organizes which singlets are baryonic and excludes operators that would
violate the baryonic label. One may therefore speak of a \emph{baryonic
confinement window}: the regime in which QCD has already forced color
neutralization while local \(U(1)_B\) still protects the baryon sector from
unified-theory decay operators.

\subsection{The unified mass-energy dictionary}
\label{app:mass-dictionary-detail}

The mass-origin manuscripts repeatedly emphasize that ``mass'' is a
sector-dependent word. This appendix summarizes the dictionary used in the
present paper.

\begin{center}
\begin{longtable}{@{}p{0.15\textwidth}p{0.24\textwidth}p{0.29\textwidth}p{0.20\textwidth}@{}}
\caption[Dictionary from sector-specific mass language to the unified GG-6/GFF language.]{Dictionary from sector-specific mass language to the unified
GG-6/GFF language. Each row takes a mass statement as its sector normally makes it, restates it in the unified language, and gives the formula carrying the restatement. The fourth column is what makes the restatements more than verbal: the same geometric object --- a holonomy commutator, a curvature, an overlap depth --- appears in the formula for a boson mass and for a fermion mass, and the dictionary is what makes that visible.}
\label{tab:mass-dictionary-app}\\
\toprule
\textbf{Sector language} & \textbf{Standard statement} & \textbf{GG-6/GFF reinterpretation} & \textbf{Representative formula}\\
\midrule
\endfirsthead
\toprule
\textbf{Sector language} & \textbf{Standard statement} & \textbf{GG-6/GFF reinterpretation} & \textbf{Representative formula}\\
\midrule
\endhead
Weak boson mass & Higgs VEV gives \(W\) and \(Z\) masses & Holonomy commutator in \(F_{\mu\theta}\) opens a vector spectral gap & \(m_V^2\propto [T_H,\cdot]^2\)\\
Higgs mass & Curvature of scalar potential & Curvature of finite holonomy potential & \(m_h^2=f_H^{-2}V''(\Theta_0)\)\\
Fermion mass & Yukawa coupling times VEV & Overlap integral of left, Higgs, and right internal profiles & cf.~\S\ref{sec:deep-mass}\\
Baryon mass & QCD energy plus anomaly & GFF split \(E_{\rm GEP}+E_{\rm CEP}\) & cf.~\S\ref{sec:deep-mass}\\
Gravitational mass-energy & Source of curvature & Same seam action varied with respect to \(g_{\mu\nu}\) & \(T_{\mu\nu}=-2\delta S_{\rm eff}/\delta g^{\mu\nu}\)\\
\bottomrule
\end{longtable}
\end{center}

The dictionary matters because it prevents an easy conceptual mistake.
Different sectors use different calculational machinery, but the present
paper does not treat them as different ontologies. They are different
modes of one seam ledger.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{O.1.1. Turn the neutron's silence into the
$10^{-10}$ bound.}

\noindent\emph{Solution.}  Step 1: a nonzero $\bar\theta$
\eqref{eq:thetabar-app} induces a neutron electric dipole moment of
the standard estimated size $d_n \sim 2.4\times10^{-16}\,
\bar\theta~e\,\mathrm{cm}$ (the chiral-loop scale $m_q/m_n^2$ with
order-one hadronic factors).  Step 2: the experimental bound
$|d_n| < 1.8\times10^{-26}~e\,\mathrm{cm}$ then gives
$|\bar\theta| < 1.8\times10^{-26}/2.4\times10^{-16} =
7.5\times10^{-11}$ --- the bound quoted as
\eqref{eq:thetabar-bound-app}.  Step 3: state the puzzle exactly:
$\bar\theta$ is a compact angle, any value in $[0, 2\pi)$ equally
lawful in the textbook, and the flavor sector demonstrates (chapter
I) that the theory has no aversion to large phases --- yet this one
angle sits below $10^{-10}$.  Physics intuition: the neutron is a
CP antenna --- a charge-neutral object whose charge \emph{layout}
would tilt if the strong force distinguished past from future
mirror-wise; fifty years of listening have heard nothing, and the
textbook's only comment is that someone must have set the dial with
eleven-digit care.

\medskip\noindent\textbf{O.4.1. Evaluate the protected residue and
place it against the antenna.}

\noindent\emph{Solution.}  Step 1: with the export generating no
$\theta$-term and the first-order phases canceling (O.4.2), the
constitutionally minimal leakage is the triple product
\eqref{eq:thetabar-residue-app}: the minimal CP-odd compact phase
$\sin(2\pi/L)$, the minimal dyadic depth $2^{-J}$, and the
topological lock $e^{-K}$.  Step 2: evaluate with the ledger:
$\sin(2\pi/5) = 0.95106$, $2^{-7} = 0.0078125$,
$e^{-35} = 6.305\times10^{-16}$: product $= 4.7\times10^{-18}$.
Step 3: that is eight orders of magnitude below the $7.5\times
10^{-11}$ of O.1.1 --- silent to any foreseeable neutron experiment
--- and it is built from the same three integers $(L, J, K)$ that
price the Cabibbo pentagon, the fine-structure dyad, and the weak
scale.  Physics intuition: the strong-CP problem is solved here by
poverty, not fine-tuning --- the theory simply cannot afford a
$\theta$-term; the largest CP leak the ledger can construct is a
pentagon phase filtered through seven halvings and the full
thirty-five-fold lock, and that number was spent elsewhere in the
codebook long before the neutron asked.

\medskip\noindent\textbf{O.4.2. Work the first-order determinant-phase
cancellation.}

\noindent\emph{Solution.}  Step 1: the physical angle is
$\bar\theta = \theta_{\rm top} + \arg\det(M_u M_d)$
\eqref{eq:thetabar-app}; polar-decompose each mass block
\eqref{eq:polar-mf-app}: $M_f = H_f U_f$ with $H_f$ positive.
Step 2: at first order the codebook's singular values (tolls and
codes) are real and positive, so $\arg\det H_f = 0$, and the unitary
factors carry only the family \emph{misalignment}; their determinant
phases cancel between the up and down blocks
\eqref{eq:argdet-u-app}--\eqref{eq:theta-product-app}, leaving
$\arg\det(M_u M_d)^{(1)} = 0$.  Step 3: with
$\theta_{\rm top}^{(1)} = 0$ from the export (no coefficient
generated), $\bar\theta^{(1)} = 0$ exactly --- and the honesty
register holds: the end-to-end cancellation at all orders is the
named lemma L9, so the entry is \textbf{Derived (cond.)}, not
oversold.  Physics intuition: the same pentagonal phase that
\emph{must} appear in the CKM matrix (I.3.3) is exactly the kind
that \emph{cannot} appear in $\bar\theta$ --- misalignment phases
live between the blocks and cancel in the determinant product; CP
violation flows where the geometry channels it, into flavor, and
not where the textbook feared it, into the vacuum.

\medskip\noindent\textbf{O.5.1. Chain the derived electroweak
constants from two ledger readings.}

\noindent\emph{Solution.}  Step 1: from $\alpha^{-1} = 128$ and
$\sin^2\theta_W = 25/108$ alone:
$\alpha_2^{-1} = 128 \times \tfrac{25}{108} = \tfrac{800}{27} =
29.630$ \eqref{eq:alpha2-arith-app} and
$\alpha_Y^{-1} = 128 - \tfrac{800}{27} = \tfrac{2656}{27} = 98.370$
\eqref{eq:alphay-arith-app} --- note both are exact rationals.
Step 2: couplings \eqref{eq:g-arith-app}:
$g_2 = \sqrt{4\pi/29.630} = 0.65124$,
$g_Y = 0.35742$; with $v^{(1)} = 246.331$~GeV
\eqref{eq:v-arith-app}: $m_W^{(1)} = \tfrac12 g_2 v = 80.211$~GeV,
$m_Z^{(1)} = \tfrac12 v\sqrt{g_2^2 + g_Y^2} = 91.497$~GeV, and
$G_F^{(1)} = 1/\sqrt2\,v^2 = 1.16532\times10^{-5}~\mathrm{GeV}^{-2}$
\eqref{eq:gf-arith-app}.  Step 3: the QD pass
\eqref{eq:mw-mz-arith-app} sharpens to $80.370$ and $91.190$ against
the measured $80.37$ and $91.19$ --- the entire electroweak constant
sheet, $G_F$ included, descending from two ledger fractions and the
lock scale.  Physics intuition: this problem is the codebook's
compression made tactile --- five constants every student memorizes
($\alpha$, $\theta_W$, $G_F$, $m_W$, $m_Z$) reduce to \emph{two}
readings and arithmetic; the textbook measures five dials, the
ledger prints one page.

\medskip\noindent\textbf{O.6.1. Weigh a person in quark masses.}

\noindent\emph{Solution.}  Step 1: the proton's valence content
carries current masses $2m_u + m_d \simeq 2(2.2) + 4.7 \simeq
9.0$~MeV against $m_p = 938.3$~MeV: the Higgs-given share is
$0.96\%$.  Step 2: scale up: of a 70-kg human, roughly $0.7$~kg is
Yukawa mass; the remaining $69$-plus kilograms are confined gluon
field and quark kinetic energy --- $E = mc^2$ running in reverse,
energy read as weight.  Step 3: the dictionary entry
(Table~\ref{tab:mass-dictionary-app}): baryon mass is the \GFF{}
split $E_{\rm GEP} + E_{\rm CEP}$ --- so the Higgs chapter explains
the \emph{origin} of mass for the elementary fermions, while the
mass you feel on a scale is overwhelmingly the strong seam's stored
energy; both are entries of one ledger, not two ontologies.  Physics intuition:
 ``the Higgs gives everything mass'' is the most popular
wrong sentence in physics outreach --- the Higgs prices the quarks'
tolls at the percent level, and confinement pays the other
ninety-nine; you weigh what your protons' fields are doing, not what
their parts are.

\medskip\noindent\textbf{O.9.1. Work dimensional transmutation
honestly, from the derived coupling.}

\noindent\emph{Solution.}  Step 1: the one-loop RG-invariant scale
\eqref{eq:lambdaqcd-app} is $\Lambda = \mu\,
\exp[-8\pi^2/(b_0 g_3^2)]$; take $\mu = m_Z$, the five-flavor
$b_0 = 23/3$, and the derived closure $\alpha_s(M_Z) = 1/\pi e =
0.11710$ (D.3.3): the exponent is $-8\pi^2/(\tfrac{23}{3} \times
4\pi \times 0.11710) = -7.00$, so $\Lambda \simeq 91.19\,e^{-7.00}
\simeq 83$~MeV.  Step 2: state the register plainly: strict one loop
lands the \emph{decade} (order $10^2$~MeV, an $e^{-7}$ suppression
from $m_Z$); the standard multi-loop, threshold-matched value is
$\Lambda^{(5)}_{\overline{\rm MS}} \simeq 210$~MeV, and the
chapter's Step-1 quote of ``$\simeq 200$~MeV'' lives at that
standard.  Step 3: sensitivity --- because $\Lambda$ depends
exponentially on $1/\alpha_s$, the $0.9\sigma$ difference between
$1/\pi e$ and the world average moves $\Lambda$ by tens of percent:
the confinement scale is the codebook's most leveraged output, which
is exactly why the radial-transport reading matters --- the window,
not the last digit, is the derived object.  Physics intuition: QCD
is the only sector that manufactures a mass scale out of pure
arithmetic --- a dimensionless coupling and an exponential --- and
the geometry reads that exponential as literal distance down the
radial dimension: the proton mass marks how far below the seam the
strong coupling's transport runs out of room.

\medskip\noindent\textbf{O.10.1. From memory: one word ``mass,''
five sectors, one ledger.}

\noindent\emph{Solution.}  Step 1: name the five sector meanings
(Table~\ref{tab:mass-dictionary-app}): weak-boson mass = holonomy
commutator gap; Higgs mass = curvature of the forced potential;
fermion mass = overlap of internal profiles (tolls and codes);
baryon mass = confined field energy, the \GFF{} split; gravitational
mass-energy = the same seam action varied against the metric.
Step 2: identify the common structure: every entry is a property of
\emph{one} seam ledger read by different instruments --- no sector
owns a private ontology of mass, which is why the paper may add
masses across sectors (a person = Yukawa tolls $+$ confinement
energy $+$ binding corrections) without category error.  Step 3: the
one-sentence audit: the Higgs sector prices the elementary tolls
(O.5.1), confinement prices the composite bulk (O.6.1), and gravity
weighs the total --- three prices, one currency.  Physics intuition:
the deepest service of the dictionary is negative --- it forbids the
question ``but what is mass \emph{really}?'' from having five
different answers; in this framework mass is always the same thing,
stored seam energy, and the five formulas are five doors into one
room.


\GGtakeaway{\emph{Standard result:} $\bar\theta < 10^{-10}$
from the neutron EDM, confinement from the lattice, and mass
from several bookkeeping systems.  \emph{What it assumed:} the
smallness of $\bar\theta$ unexplained (or bought with an
axion), confinement demonstrated but not derived, and the mass
ontologies left separate.  \emph{The geometric reading:}
$\bar\theta$ is a protected topological zero of the ledger;
confinement is read geometrically; and mass has one ontology
--- the seam ledger --- across every sector of the course.}

\section{Promotion roadmap for the named structural lemmas}
\label{sec:appendix-promotion-roadmap}

The \MSMF{}/\OGN{} per-parameter audit (\S\ref{subsec:msmf-ogn-audit}) promotes nine of the seventeen previously Asserted parameters and identifies fourteen Standard-Model parameters as Derived (cond.) modulo a single named structural lemma each. This appendix lists in one place the ten named lemmas L1--L10 — six standing on the \MSMF{}/\OGN{} audit (L1--L5 from the conditional electroweak/PMNS entries (the PMNS block now including the leptonic Dirac-phase assignment lemma), L10 the cascade-depth lemma covering the eight remaining \textbf{Asserted} Yukawas) and four covering the previously conditional entries on the quark codebook, the Weinberg-operator coefficient, and the strong-CP cancellation (L7--L9; the Higgs vev's binary-endpoint sub-lemma is folded into L1). Each lemma carries statement, parameter(s) closed, dependencies on upstream papers, a one-line physics intuition, expected proof difficulty, and successor-paper assignment. Each is a finite, well-posed mathematical claim; each, if proved, converts one or more codebook entries from \textbf{Asserted} or \textbf{Derived (cond.)} to fully \textbf{Derived} end-to-end. Alongside these ten parameter-closing lemmas we register one \emph{structural} lemma, L11 (the gate-operator lemma of \S\ref{subsec:weak-status}): it closes no codebook entry, but, if proved, it would establish the identification of the weak interaction as the unique gate force of the family landscape and --- through the gate/three-basin co-entailment --- the joint existence of the weak interaction and the three generations. It is listed last in Table~\ref{tab:promotion-roadmap} and unpacked after L10.

The roadmap is summarized in Table~\ref{tab:promotion-roadmap}, then unpacked lemma by lemma below.

\begin{center}
{\footnotesize
\begin{longtable}{@{}p{0.04\textwidth}p{0.20\textwidth}p{0.16\textwidth}p{0.18\textwidth}p{0.10\textwidth}p{0.20\textwidth}@{}}
\caption[Promotion roadmap for the ten parameter-closing lemmas L1--L10 and the one structural lemma L11.]{Promotion roadmap for the ten parameter-closing lemmas L1--L10 and the one structural lemma L11. Each of L1--L10, if proved, would close one or more codebook entries by upgrading their status from \textbf{Asserted} or \textbf{Derived (cond.)} to fully \textbf{Derived}. ``Difficulty'' rates the expected effort in person-months, banded as Easy ($\sim 1$), Medium ($\sim 6$), Hard ($\sim 18$), Very-hard ($\sim 36$).}\label{tab:promotion-roadmap}\\
\toprule
\textbf{\#} & \textbf{Lemma} & \textbf{Closes parameter(s)} & \textbf{Dependencies} & \textbf{Difficulty} & \textbf{Successor paper}\\
\midrule
\endfirsthead
\toprule
\textbf{\#} & \textbf{Lemma} & \textbf{Closes parameter(s)} & \textbf{Dependencies} & \textbf{Difficulty} & \textbf{Successor paper}\\
\midrule
\endhead
\midrule \multicolumn{6}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

L1 & Binary-recursion lemma & $\alpha^{-1}(M_Z) = 2^J$ (Row 1) & \ATwo{} T-GG6-6; \AFour{} \S 31 & Easy & \ASix{} (extension) or dedicated note\\

L2 & Triplicity-layering lemma & $v = M_{\rm lock}(2/L)^N$ (Row 4); $\lambda_H = 2^{-N}$ (Row 5, already Derived) & \ATwo{} T-GG6-6; Wilson--Hosotani descent of \ASix{} \T{4.1} & Medium & \ASix{} (extension)\\

L3 & Democratic-weight assignment lemma & $\sin^2\theta_{12}^{\rm PMNS} = L^2/3N^3 = 25/81$ (Row 19) & \GDOS{}/\GFF{} projector overlap on the family stratum & Medium & \ASix{} (extension) or dedicated PMNS paper\\

L4 & Reactor-overlap assignment lemma & $\sin^2\theta_{13}^{\rm PMNS} = N/4K = 3/140$ (Row 20) & \GDOS{}/\GFF{} reactor-overlap on the family stratum & Hard & Dedicated PMNS-derivation paper\\

L5 & Rank-2 atmospheric-deviation lemma & $\sin^2\theta_{23}^{\rm PMNS} = (J{+}1)/(2J) = 4/7$ (Row 21) & Rank-2 family-projector stratum + cost-tie discrimination on $\{4/7, 5/9\}$ & Medium & Dedicated PMNS-derivation paper\\

L6 & Quartic-curvature lemma & $\lambda_H = 2^{-N}$ (Row 5; already \textbf{Derived}) & \ASix{} \GFF{} curvature stiffness on the electroweak \GA{} & Easy & Note attached to \ASix{}\\

L7 & Quark-codebook overlap lemma & Four CKM mixing parameters (Rows 15--18) & \ASix{} \S 28 angular-cascade decoration on the quark up/down family projectors; pentagonal-holonomy interference & Hard & Dedicated CKM-derivation paper (jointly with the cascade-depth paper of L10)\\

L8 & Weinberg-operator coefficient lemma & Three neutrino mass-scale parameters (Rows 25--27) & A2-\GG{} \T{6} ($K = 35$); Wilson--Hosotani descent of \ASix{} \T{4.1}; Majorana operator $(LH)(LH)/E_{\rm GUT}$ & Medium & \ASix{} (extension) or dedicated neutrino-mass-scale paper\\

L9 & Quark-determinant phase-cancellation lemma & Strong-CP angle $\bar\theta$ (Row 28) & A2-\GG{} (no first-order $\theta$-term on the GG-6 seam); \ASix{} \S 28 quark-determinant phase reading & Hard & Dedicated strong-CP paper or jointly with L7\\

L10 & Equivariant-cohomology cascade-depth lemma & Eight charged-fermion Yukawas: $m_e, m_\mu, m_\tau$ (Rows 6--8); $m_u, m_d, m_c, m_s, m_b$ (Rows 9--12, 14) & Equivariant cohomology on $\mathbb{C}P^2$ doubled by $\SU(2)_L$; \ASix{} \S 28 cascade decoration; Bogomolnyi instanton spectrum $S_{[2k]\to[2(k-1)]} = \pi k$ & Very-hard & Dedicated cascade-depth paper (provisional title: ``Arisaka-GG-A9X-Yukawa cascade depths'')\\

\addlinespace
\multicolumn{6}{@{}l}{\bfseries\itshape Structural lemma --- closes a structural result, not a codebook parameter}\\
\addlinespace
L11 & Gate-operator lemma & \emph{Structural}: weak-as-gate-force; weak $\Leftrightarrow$ three generations (\S\ref{sec:weak-special}) & Morse transport between adjacent critical points on $\mathbb{C}P^{N-1}$; \GSSB{}/\GRIPout{}; \GDOS{} open system; $N=3$ closure (Thm~\ref{thm:N3-closed}) & Hard & Dedicated flavor-architecture note, jointly with \AEleven{}\\
\end{longtable}
}
\end{center}

\subsection*{Lemma L1: the binary-recursion lemma}

\paragraph{Statement.}
On the seven-step binary-recursion stratum that A2-\GG{} \T{6} identifies with the Diophantine ledger position $J = 7$, the inverse fine-structure constant at the electroweak seam is forced by binary recursion to take the value $\alpha^{-1}(M_Z) = 2^J = 128$, with no remaining freedom in the dyadic step structure or in the numeric base.

\paragraph{Parameters closed.} Row 1 of Table~\ref{tab:codebook-provenance} ($\alpha^{-1}(M_Z) = 2^J = 128$), currently \textbf{Derived (cond.)} modulo this A9-local lemma; promotion would make Row 1 fully \textbf{Derived} end-to-end.

\paragraph{Dependencies.} The lemma uses the Diophantine ledger position $J = 7$ from A2-\GG{} \T{6} \citep{ArisakaA2GG6} and the dyadic-recursion structure of \AFour{} \S 31 \citep{ArisakaA4Euclid}. No new physics input is required.

\paragraph{Physics intuition.} Each dyadic step halves the residual coupling-renormalization envelope, and the seven-step binary recursion is exactly the depth that the \OGNfive{} ledger position $J = 7$ admits. After all, $128$ is what one gets by halving from $1$ down through seven layers; the lemma proves what the QED running already suggests numerically.

\paragraph{Expected difficulty.} Easy. The binary-recursion calculation is essentially a numerical coincidence elevated to a structural statement; the recursion endpoint $J = 7$ is itself already proved upstream. The lemma reduces to ``write down the proof of an observation''.

\paragraph{Successor paper.} Most naturally an extension of \ASix{}, or a short dedicated note (10--15 pp).

\subsection*{Lemma L2: the triplicity-layering lemma}

\paragraph{Statement.}
On the Wilson--Hosotani holonomy stratum of the electroweak seam, the lock-factor descent from $M_{\rm lock}$ to $v$ is forced to take the form $v = M_{\rm lock}(2/L)^N$ with one factor of $(2/L)$ per triplicity layer, where $N = 3$ is the color-triplicity integer and $L = 5$ is the \OGNfive{} ledger entry. Each of the $N$ layers contributes one Wilson-loop holonomy step at the canonical step size $2/L$ forced by the doublet fraction.

\paragraph{Parameters closed.} Row 4 of Table~\ref{tab:codebook-provenance} ($v = M_{\rm lock}(2/L)^N$), currently \textbf{Derived (cond.)} modulo two A9-local lemmas (binary-endpoint + triplicity-layering); the present lemma closes the second. Together with L1, it would make Row 4 fully \textbf{Derived} end-to-end.

\paragraph{Dependencies.} The lemma uses A2-\GG{} \T{6} for the \OGNfive{} ledger and the Wilson--Hosotani descent theorem of \ASix{} \T{4.1} \citep{ArisakaA6Gi4}.

\paragraph{Physics intuition.} Each of the three families wraps the compact $S^1_\theta$ once and contributes one factor of the doublet fraction $2/5$ to the lock-to-vev descent. Indeed, ``one Wilson-loop step per family'' is the geometric statement of triplicity; the empirical $v \approx 246$~GeV is the cube of $M_{\rm lock} \cdot (2/5)$.

\paragraph{Expected difficulty.} Medium. The triplicity-layering structure is heuristically clear from the $N = 3$ closure of the cascade, but a rigorous proof requires identifying the Wilson-loop holonomy step at each layer with a specific element of the holonomy group and verifying that all three layers share the canonical step size.

\paragraph{Successor paper.} Most naturally an extension of \ASix{}.

\subsection*{Lemma L3: the democratic-weight assignment lemma}

\paragraph{Statement.}
On the family-projector stratum, the PMNS solar mixing angle is forced by the \GDOS{}/\GFF{} democratic-weight projector overlap to take the structural value $\sin^2\theta_{12}^{\rm PMNS} = L^2/3N^3 = 25/81$ (QD-completed to $0.30857$).

\paragraph{Parameters closed.} Row 19 of Table~\ref{tab:codebook-provenance} ($\sin^2\theta_{12}^{\rm PMNS} = L^2/3N^3 = 25/81$), currently \textbf{Derived (cond.)} modulo the present lemma.

\paragraph{Dependencies.} The lemma uses the \GDOS{}/\GFF{} democratic-weight projector overlap on the family-projector stratum of \ASix{} \citep{ArisakaA6Gi4}.

\paragraph{Physics intuition.} Roughly speaking, the solar mixing weight is the democratic projector overlap $L^2/3N^3$ on the family stratum --- the same $L^2/N^3$ skeleton that fixes the weak angle $\sin^2\theta_W = L^2/4N^3$, with the family-triplicity denominator $3N^3$ in place of the gauge $4N^3$. The PMNS solar angle inherits the family-stratum overlap geometry.

\paragraph{Expected difficulty.} Medium. The proof reduces to showing that the family-stratum projector overlap forces the democratic weight $L^2/3N^3 = 25/81$ rather than a neighboring rational.

\paragraph{Successor paper.} Either an extension of \ASix{} or a dedicated PMNS-derivation paper (provisional title: ``Arisaka-GG-A9Y: PMNS angles from the rank-stratified $\mathbb{C}P^2$'').

\subsection*{Lemma L4: the reactor-overlap assignment lemma}

\paragraph{Statement.}
On the family-projector stratum, the PMNS reactor mixing angle is forced by the \GDOS{}/\GFF{} reactor-overlap to take the structural value $\sin^2\theta_{13}^{\rm PMNS} = N/4K = 3/140$ (QD-completed to $0.021902$).

\paragraph{Parameters closed.} Row 20 of Table~\ref{tab:codebook-provenance} ($\sin^2\theta_{13}^{\rm PMNS} = N/4K = 3/140$), currently \textbf{Derived (cond.)} modulo the present lemma.

\paragraph{Dependencies.} The lemma requires a Hopkins--Singer-style differential-cohomology descent calculation on the rank-3 family-projector stratum, building on the QD-completion descent of \S\ref{sec:qd-completion} of the present paper but applied to the family-projector $\mathbb{C}P^2$ rather than to the energy stratum.

\paragraph{Physics intuition.} The reactor overlap on the family stratum is the $N/4K$ amplitude-squared readout --- the rank-3 family projector feeding one power of $N$ over the master lock $K = LJ$, with the dyadic $1/4$. It is the structural reactor form of \S\ref{sec:PMNS}, not a bounded-cost selection.

\paragraph{Expected difficulty.} Hard. The calculation must show that the family-stratum reactor overlap forces $N/4K = 3/140$ rather than a neighboring rational.

\paragraph{Successor paper.} Most naturally a dedicated PMNS-derivation paper.

\subsection*{Lemma L5: the rank-2 atmospheric-deviation lemma}

\paragraph{Statement.}
On the rank-2 family-projector stratum, the PMNS atmospheric mixing angle is forced by the structural rule ``atmospheric angle deviates from $\pi/4$ by $1/(2J)$'' to take the value $\sin^2\theta_{23}^{\rm PMNS} = (J{+}1)/(2J) = 4/7$ at the structural-precision target. The structural-naming criterion (\OGN{} admissibility~3) breaks the cost-3 tie with the equally-cheap candidate $5/9$ in favor of $4/7$.

\paragraph{Parameters closed.} Row 21 of Table~\ref{tab:codebook-provenance} ($\sin^2\theta_{23}^{\rm PMNS} = 4/7$), currently \textbf{Derived (cond., \MSMF{}/\OGN{})} modulo the present lemma.

\paragraph{Dependencies.} The lemma requires identifying the rank-2 family-projector stratum with the atmospheric mass-eigenstate doublet and verifying the deviation $1/(2J)$ from the maximally-mixed configuration $\pi/4$.

\paragraph{Physics intuition.} Maximal mixing $\sin^2\theta_{23} = 1/2$ is the symmetric reading of a rank-2 doublet; the empirical deviation by $1/(2J)$ is the small structural correction that the rank-2 family-projector adds. After all, two of the three families share the atmospheric block, and the deviation depends on the third — encoded by the Diophantine atom $J = 7$.

\paragraph{Expected difficulty.} Medium. The structural-naming step that breaks the cost tie is the cleaner part; the geometric identification of the atmospheric doublet with the rank-2 stratum requires care.

\paragraph{Successor paper.} Most naturally a dedicated PMNS-derivation paper, jointly with L4.

\subsection*{Lemma L6: the quartic-curvature lemma}

\paragraph{Statement.}
On the electroweak Geometric Attractor (\GA{}) of the \GFF{} ledger, the dimensionless curvature stiffness of the selected attractor branch is forced to take the value $\lambda_H = 2^{-N} = 1/8$ by the minimal dyadic suppression rule ``one factor of $1/2$ per triplicity layer''.

\paragraph{Parameters closed.} Row 5 of Table~\ref{tab:codebook-provenance} ($\lambda_H = 1/8$), recorded as \textbf{Derived (curvature, modulo one lemma)}: the cost-minimum reading is weak (cost-3, look-elsewhere $p \approx 1$), so the status rests instead on the \ASix{} \GFF{} quartic-curvature reading, of which the present lemma is the leading-harmonic ($1/2$-per-layer) form. The closed form $\lambda_H = (\Lambda/f)^4\,g''(\theta_H^\star)/(2\sin^2\theta_H^\star)$ (Thm~\ref{thm:quartic}) reproduces both the vev and a natural one-loop magnitude; its precise value awaits the deeper \emph{spectral-normalization lemma}, the evaluation of $(\Lambda/f)^4$ from the scale-profile Kaluza--Klein sum, which would lift $\lambda_H$ to an unconditional theorem.

\paragraph{Dependencies.} \ASix{} \GFF{} Morse-positivity (\T{3.1}) for the attractor curvature; the triplicity-layering reading shared with L2.

\paragraph{Physics intuition.} The Higgs quartic measures the stiffness of the attractor at its bottom; one factor of $1/2$ per family is the minimum dyadic damping consistent with three independent layers locking into the same minimum. The empirical $\lambda_H \approx 1/8$ is exactly that count.

\paragraph{Expected difficulty.} Easy for the leading-harmonic ($1/2$-per-layer) stiffness reading given here; \emph{medium--hard} for the deeper spectral-normalization step that fixes $(\Lambda/f)^4$ from the scale-profile Kaluza--Klein sum (the standalone Higgs-sector calculation).

\paragraph{Successor paper.} A short note attached to \ASix{} for the leading-harmonic form (jointly with L2); a dedicated Higgs-sector calculation for the spectral-normalization step.

\subsection*{Lemma L7: the quark-codebook overlap lemma}

\paragraph{Statement.}
On the up-type and down-type quark family-projector strata of $\mathbb{C}P^2$, the four CKM mixing parameters ($\theta_{12}^{\rm CKM}$ Cabibbo, $\theta_{13}^{\rm CKM}$, $\theta_{23}^{\rm CKM}$, $\delta_{\rm CP}^{\rm CKM}$) are forced by the misalignment of the two quark family projectors and the pentagonal-holonomy interference structure of \ASix{} \S 28 to take the values derived in \S\ref{sec:CKM} of the present paper.

\paragraph{Parameters closed.} Rows 15--18 of Table~\ref{tab:codebook-provenance} (the four CKM parameters), currently \textbf{Derived (cond.)} on the quark codebook of \#9--14; promotion would make all four fully \textbf{Derived} end-to-end.

\paragraph{Dependencies.} \ASix{} \S 28 angular-cascade decoration on the up/down family projectors; the pentagonal-holonomy interference identified in \S\ref{sec:CKM}; the cascade-depth assignments to be supplied by L10. L7 therefore depends on L10 in part: the CKM angles inherit per-quark depth assignments and cannot be fully closed until L10 closes.

\paragraph{Physics intuition.} Two slightly misaligned family projectors interfere at five overlap points (the pentagonal-holonomy interference of \S\ref{sec:CKM}); the CKM angles read out the geometry of the misalignment. Indeed, the small Cabibbo angle is the smallest pentagonal-holonomy step, and the suppressed $\theta_{13}^{\rm CKM}$ is the deepest one.

\paragraph{Expected difficulty.} Hard. The lemma is jointly conditional on L10 (per-quark cascade depths); once L10 closes, the misalignment calculation is tractable.

\paragraph{Successor paper.} Dedicated CKM-derivation paper, jointly with the cascade-depth paper of L10 (provisional title ``Arisaka-GG-A9Z: CKM mixing from quark-codebook overlaps'').

\subsection*{Lemma L8: the Weinberg-operator coefficient lemma}

\paragraph{Statement.}
On the constitutional Majorana channel $(LH)(LH)/E_{\rm GUT}$, the operator coefficient $2\pi$ --- which, once the two Higgs insertions ($v/\sqrt2$ each) halve it, pins the absolute neutrino mass scale $m_* = \pi v^2/E_{\rm GUT}$ --- is forced by the canonical $S^1_\theta$ traversal of A2-\GG{} as the natural dimensionless prefactor of any boundary observable that wraps the compact direction once. The rank-2 projector ratio $2N = 6$ in $\Delta m^2_{31}/\Delta m^2_{21}$ is forced by the same constitution.

\paragraph{Parameters closed.} Rows 25--27 of Table~\ref{tab:codebook-provenance} (the absolute Majorana mass scale $m_*$ and the two squared-mass splittings $\Delta m^2_{21}$, $\Delta m^2_{31}$), currently \textbf{Derived (cond.)} on the Weinberg-operator coefficient choice; promotion would make all three fully \textbf{Derived} end-to-end.

\paragraph{Dependencies.} A2-\GG{} \T{6} for the lock scale $E_{\rm GUT}$ via $K = 35$; Wilson--Hosotani descent of \ASix{} \T{4.1} for $v$; the canonical $S^1_\theta$ traversal of the compact direction for the $2\pi$ coefficient.

\paragraph{Physics intuition.} Any boundary observable that crosses the compact $S^1_\theta$ once picks up one factor of the circumference $2\pi$; the neutrino Majorana scale is exactly such an observable. Roughly speaking, the operator carries a factor $2\pi$ because it wraps the compact circle once, no more, no less; the two Higgs insertions ($v/\sqrt2$ each) then halve it, giving $m_* = \pi v^2/E_{\rm GUT}$.

\paragraph{Expected difficulty.} Medium. The $2\pi$ coefficient is structurally suggested by the $S^1_\theta$ traversal; the rigorous proof requires showing that no half-traversal is admissible and that the rank-2 projector splits the squared-mass differences in the ratio $2N = 6$.

\paragraph{Successor paper.} An extension of \ASix{}, or a dedicated neutrino-mass-scale paper.

\subsection*{Lemma L9: the quark-determinant phase-cancellation lemma}

\paragraph{Statement.}
On the \GG{} boundary, the strong-CP angle $\bar\theta = \theta_{\rm QCD} + \arg\det(M_u M_d)$ is forced to vanish at first order by two structural facts: (i) A2-\GG{} admits no independent first-order $\theta$-term on the \GG{} seam (the seam topology forbids it); (ii) the quark-determinant phases of the up-type and down-type sectors cancel exactly because the first-order CP data are relative-holonomy phases rather than net-determinant phases on the angular-cascade decoration. The natural QD leakage is $\bar\theta \lesssim 5 \times 10^{-18}$, well below the experimental neutron-EDM bound.

\paragraph{Parameters closed.} Row 28 of Table~\ref{tab:codebook-provenance} ($\bar\theta$), currently \textbf{Derived (cond.)} on the determinant-phase cancellation; promotion would make Row 28 fully \textbf{Derived} end-to-end.

\paragraph{Dependencies.} A2-\GG{} for the absence of a first-order $\theta$-term on the \GG{} seam; \ASix{} \S 28 for the angular-cascade decoration that supplies the relative-holonomy phase reading; the per-quark cascade-depth assignments of L10 for the explicit cancellation arithmetic.

\paragraph{Physics intuition.} The strong-CP angle is the relative phase between the QCD vacuum and the quark-mass-matrix determinant. In \GG{} the first one is forbidden by topology and the second is a sum of relative-holonomy phases that telescopes to zero. After all, the celebrated strong-CP problem becomes a topological non-problem on the 6D bulk: there is no $\theta$ to tune.

\paragraph{Expected difficulty.} Hard. The cancellation arithmetic is jointly conditional on L10 (per-quark depths); the topological exclusion of the first-order $\theta$-term is plausible from A2-\GG{} but a fully rigorous proof requires identifying the relevant cohomology class.

\paragraph{Successor paper.} Dedicated strong-CP paper, or jointly with L7.

\subsection*{Lemma L10: the equivariant-cohomology cascade-depth lemma}

\paragraph{Statement.}
On the family-projector stratum $\mathbb{C}P^2$ doubled by $\SU(2)_L$, each charged-fermion stratum-depth $n_X \in \{0, 1, 2, 3, 4\}$ is forced by the dimension of a specific equivariant-cohomology class to take the value listed in Table~\ref{tab:codebook-provenance} for that fermion. Specifically: $n_{y_t} = 0$ (already proved as the cascade-top), $n_{y_b} = 1$, $n_{y_c} = 2$, $n_{y_\tau} = 2$, $n_{y_s} = 3$, $n_{y_\mu} = 3$, $n_{y_u} = 4$, $n_{y_d} = 4$, $n_{y_e} = 4$ (depth assignments per the earlier codebook, to be verified against the cohomology calculation).

\paragraph{Parameters closed.} All eight non-top charged-fermion Yukawa entries (rows 6--12, 14 of Table~\ref{tab:codebook-provenance}), currently \textbf{Asserted}. This is the dominant single open task in the A9 closure program: \emph{one calculation closes eight parameters}, and through the dependencies of L7 and L9, indirectly closes five more.

\paragraph{Dependencies.} Equivariant cohomology on $\mathbb{C}P^2$ doubled by $\SU(2)_L$; the Bogomolnyi instanton-action spectrum $S_{[2k] \to [2(k-1)]} = \pi k$ from \ASix{} \S 28 \citep{ArisakaA6Gi4}; a careful match between the cohomology-class dimensions and the empirical Yukawa hierarchy. The chiral-doubling factor from $\SU(2)_L$ is structurally important: without it only six depth values are available, but the empirical spectrum requires nine distinct readouts (eight charged-fermion Yukawas + the cascade-top).

\paragraph{Physics intuition.} The chiral doubling is what makes a fermion \emph{chiral} — left-handed and right-handed components live on geometrically distinct sub-strata, and the equivariant cohomology counts how many ways a localized mode can sit on the doubled stratum. The Yukawa hierarchy is the dimension count of those classes; nine independent classes give nine independent fermion masses. Indeed, the spread of twelve orders of magnitude across the charged-fermion Yukawas is the geometric reading of nine cohomology dimensions, not nine fitted numbers.

\paragraph{Expected difficulty.} Very-hard. This is the deepest of the ten lemmas and the dominant remaining piece of upstream work for the A9 codebook. The expected effort is comparable to a standalone preprint ($\sim 100$--$150$~pp), conducted as a dedicated successor paper to the foundation block. A correct identification of the equivariant cohomology classes with the eight stratum depths would close one of the longest-standing problems in particle physics --- the absence of a structural rationale for the Yukawa hierarchy --- by replacing eight assertions with one geometric statement.

\paragraph{Successor paper.} Provisional title ``Arisaka-GG-A9X: Yukawa cascade depths from equivariant cohomology on $\mathbb{C}P^2 \times \SU(2)_L$''; estimated ship target 2027.

\subsection*{Lemma L11: the gate-operator lemma (structural)}

\paragraph{Statement.} On the family stratum $\mathbb{C}P^{N-1}$, a gauge connection that parallel-transports the family projector between adjacent critical points of the \GFF{} Morse height function is necessarily (i) off-diagonal in the critical-point basis, (ii) chiral --- it carries the orientation of the downhill gradient flow --- and (iii) Higgs/Stueckelberg-massive, its range set by the saddle width; and within the boundary gauge group the unique factor admitting all three properties is $\SU(2)_L$.

\paragraph{What it would establish.} Unlike L1--L10, this lemma closes no codebook \emph{parameter}; it closes a \emph{structural} result. A proof would upgrade the identification of the weak interaction as the unique gate force of the family landscape (\S\ref{sec:weak-special}) from a structural reading to a theorem, and --- through the gate/three-basin co-entailment and the toll-budget closure $\tfrac12 N(N-1)=3$ --- would yield a first-principles derivation of the \emph{joint} existence of the weak interaction and the three generations.

\paragraph{Dependencies.} The Morse basin--gate--toll structure on $\mathbb{C}P^{N-1}$ and the Bogomolnyi spectrum $S_{[2k]\to[2(k-1)]}=\pi k$ of \ASix{} \S 28 \citep{ArisakaA6Gi4}; the \GSSB{}/\GRIPout{} reading of electroweak symmetry breaking; the \GDOS{} open-system relaxation law; and the $N=3$ cascade closure (Theorem~\ref{thm:N3-closed}). It is the flavor-side complement of the four-force-uniqueness theorem (Theorem~\ref{thm:four-force-uniqueness}) of \AEleven{}.

\paragraph{Expected difficulty.} Hard. The statement is a clean equivariant/Morse-theoretic claim about transport operators on $\mathbb{C}P^{N-1}$, but tying ``off-diagonal $+$ chiral $+$ massive transport operator'' uniquely to $\SU(2)_L$ requires a careful representation-theoretic step. If proved, it answers two of the oldest questions in particle physics --- why the weak interaction is special, and why there are three generations --- in one stroke.

\paragraph{Successor paper.} Naturally developed jointly with the four-force-uniqueness program of \AEleven{}; provisional home a dedicated flavor-architecture note in the 2027 timeframe.

\subsection*{Programmatic summary of the roadmap}

The ten lemmas L1--L10 represent the full open structural-naming agenda for the \ANine{} codebook as it stands at the present. Six of the ten (L1, L2, L3, L5, L6, L8) are tractable in person-month timescales (Easy or Medium) and could be closed in successor papers within 2026--2027. Three (L4, L7, L9) are Hard and are the natural subject of dedicated extension papers. The tenth (L10) is the dominant single open task, requires a dedicated equivariant-cohomology calculation, and is the natural subject of a separate preprint in the 2027 timeframe; through the dependencies of L7 and L9, L10 also indirectly closes the four CKM parameters and the strong-CP angle. The structural lemma L11 (the gate-operator lemma) stands outside this parameter-closure arithmetic: it closes no codebook entry, but its proof would establish the weak-as-gate-force result and the joint weak/three-generation derivation of \S\ref{sec:weak-special}.

The closure arithmetic stages as follows. Closing the six tractable lemmas L1, L2, L3, L5, L6, L8 promotes Rows 1, 4, 19, 21, 25--27 to fully \textbf{Derived}, raising the count from $6$ to $6 + 6 = 12$ \textbf{Derived} end-to-end. Closing the three Hard lemmas L4, L7, L9 (with L7, L9 themselves dependent on L10) promotes Rows 20, 15--18, 28, taking the count to $12 + 7 = 19$ \textbf{Derived}, with L10 still outstanding and the eight Yukawas + PMNS Dirac phase still \textbf{Asserted}. Closing L10 then promotes the eight Yukawas, taking the count to $19 + 8 = 27$ \textbf{Derived}, with only the PMNS Dirac phase remaining \textbf{Asserted}. After that, the only residue is empirical — a phase measurement to be supplied by T2K-II / NOvA-II / DUNE / Hyper-Kamiokande over 2027--2032 — not structural.

\begin{euclidremark}[The shape of the closure program]
Indeed, the roadmap above makes the closure program finite, named, and dated. Six tractable lemmas plus three hard lemmas plus one very-hard lemma plus one empirical measurement, and the 28-parameter codebook is closed. After all, this is what a structural framework should produce as its successor agenda — not vague gestures toward future work but a numbered list of named lemmas with dependencies, difficulty estimates, and successor-paper assignments. Whether each lemma admits a clean proof is the content of the successor work; the achievement of the present paper is reducing twenty-eight free parameters to ten named lemmas plus one empirical measurement.
\end{euclidremark}


\GGPart[Common Objections]{Common Objections and Responses}{The strongest objections a particle theorist, cosmologist, mathematician, philosopher, or string theorist is most likely to raise on first reading --- answered, with pointers into the paper}
\label{part:XVII}
\label{sec:appendix-objections}

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

A consequence paper that claims to derive the twenty-eight free parameters of the Standard Model from a five-integer ledger will be met --- rightly --- with skepticism from every quarter. This appendix collects eighty-three objections, grouped by the background of the reader most likely to raise each, together with an extended response. The objections are not strawmen; they are the questions the authors have actually been asked, sharpened. Where the honest answer is ``not yet,'' we say so, and point to the codebook status (Table~\ref{tab:codebook-provenance}) and the promotion roadmap (\S\ref{sec:appendix-promotion-roadmap}) that record exactly which claims are theorems, which are conditional, and which are asserted. After all, the test of a framework is not that it deflects every question, but that it knows which questions it has answered and which it has not.

%% =====================================================================
%% Cluster A -- Methodology, numerology, and the honesty triage
%% =====================================================================

\bigskip
\section{Methodology, numerology, and the honesty triage}

\bigskip
\noindent\textbf{Objection 1 (O-SM-1) (the parameter-counting skeptic)}\par\nobreak
\noindent\textbf{``You advertise a derivation of twenty-eight parameters, but your own triage marks only six as Derived and nine as Asserted. Isn't `28 derived' an overstatement?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It would be, and that is precisely why the paper does not make that claim without the triage. The headline result is structural: all twenty-eight parameters are placed on a single integer ledger and assigned a closed form, so that none remains a freely floating real number typed into a Lagrangian by hand. The honesty triage (Table~\ref{tab:codebook-provenance}) then grades how strong each placement is: six \textbf{Derived}, fourteen \textbf{Derived (conditional)} --- forced once a single named lemma is supplied --- and eight \textbf{Asserted}. The correct one-sentence summary is therefore not ``twenty-eight derived'' but ``twenty-eight reduced to a five-integer ledger plus ten named lemmas plus one empirical measurement,'' with the strength of each reduction stated explicitly. We regard refusing to inflate the count as a feature, not a concession.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 2 (O-SM-2) (the look-elsewhere statistician)}\par\nobreak
\noindent\textbf{``With $\pi$, $e$, and a handful of small integers one can approximate almost any number. How is the codebook not curve-fitting with an enormous look-elsewhere effect?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the single most important methodological objection, and the paper takes it seriously enough to quantify it rather than wave it away. The \MSMF{}/\OGN{} selection principle is a complexity-bounded Diophantine search, and a search has a look-elsewhere cost. For each parameter we report that cost. Where the candidate space is genuinely sparse the selection is strong: $\alpha_s^{-1}(M_Z)=\pi e$ is the unique cost-2 monomial on $\{\pi,e\}$ within $1\%$ of the datum, look-elsewhere $p\approx0.27$. Where it is dense we say so and decline to claim strength: $\lambda_H=2^{-N}$ is admissible but the cost-$\le 3$ band is saturated ($p\approx1$), so we do \emph{not} rest its status on the cost-minimum at all (Objection~7). The discipline is the opposite of curve-fitting: a curve-fit hides its degrees of freedom, whereas the audit publishes them parameter by parameter in \S\ref{subsec:msmf-ogn-audit}. The decisive program-level statistic is the \emph{global} trials factor over the full alphabet evaluated against all twenty-eight targets at once (O-SM-73); we regard that single number, not the favorable per-parameter costs, as what should ultimately make or break the look-elsewhere charge. Two further points, developed in \S\ref{subsec:forcing-vs-searching}, complete the reply and are the heart of the matter: the alphabet is not a per-parameter toolbox but \emph{one closed ledger} reused coherently across all twenty-eight observables (the same $N=3$ in $\sin^2\theta_W$, $\alpha^{-1}(0)$, and $\lambda_H$ at once), and the \emph{form} of each entry is forced by the stratum it inhabits rather than scanned --- so this is the descent of propositions from the \GGConstitution{}, in the lineage of Euclid's \emph{Elements}, not a search. We do not claim the look-elsewhere effect is zero; we claim it is forced down to a bounded residual that the global trials factor quantifies.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 3 (O-SM-3) (the careful Diophantine analyst)}\par\nobreak
\noindent\textbf{``Were the ledger integers $(M,N,L;J,K)=(1,3,5;7,35)$ chosen before or after the data were known?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Before, and not by us. $M=1$ is the dimensional-minimality output of A2-\GG{} \citep{ArisakaA2GG6}; $L=5$ is Georgi--Glashow's 1974 identification of $\mathrm{SU}(5)$ as the smallest simple unifying group \citep{GeorgiGlashow1974}; $N=3$ is established by the independent three-leg pincer of \PaperPTwo{} \citep{ArisakaP2N3} --- a \CP{} lower bound $N\ge3$ and a \GDOS{}/\GRIP{} stability-budget upper bound $N\le3$ (the latter the capacity theorem T-N3-6, conditional on the coupling
clause and the integrality theorem L-INT (residue R-INT)) --- while the
$\mathrm{SU}(5)$ trace-ratio identity serves only as a downstream consistency check; and $(J,K)=(7,35)$ is then fixed by the dual-lock identity $K=LJ=M^2+N^2+L^2=35$, whose uniqueness over small integers is checked explicitly in A2-\GG{}~\S10. Each integer is determined by a structural theorem that pre-dates, logically and historically, the parameter fits of the present paper. The codebook formulas are downstream of the integers, not the reverse.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 4 (O-SM-4) (the precision-physics referee)}\par\nobreak
\noindent\textbf{``Four-significant-figure agreement appears only after a `QD completion.' Isn't the QD layer where the fitting actually happens?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The QD completion is constrained, not free. Its rationals carry denominators forced by Hopkins--Singer descent integrality on the relevant stratum (for the leptons, $640=2^J L$) \citep{HopkinsSinger2005}, and integer numerators drawn from the same \OGNfive{} ledger by cascade depth. There are no continuous knobs: a skeptic can verify that no alternative small-integer numerator on the same lattice reproduces all three lepton ratios at once. The honest statement is that the leading-order ledger value and the QD-completed value are two predictions of different precision, both reported, with the completion adding structure (a named integrality theorem) rather than adjustable parameters. If the QD layer contained hidden freedom, the same machinery would not also have to reproduce the quark and mixing sectors with the \emph{same} ledger integers, which it does. A final clarification, in the light of the renormalization analysis of \citet{ArisakaA5GDOS} (Theorem~T9.9): the QD layer is the \emph{scale-flow} (Tier-2) completion of the first-order value --- the part the radial geometry legitimately organizes --- so where its pull is small the four-figure agreement is honest. Where its pull is large ($\alpha_s$, $m_H$, the $\delta_{\rm CP}$ shift) it reaches into the scale-invariant content the geometry cannot derive, and there it is an honest bounded \emph{selection}, carried in the same global trials accounting (\S\ref{subsec:forcing-vs-searching}), not a loop derivation.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 5 (O-SM-5) (the philosopher of science)}\par\nobreak
\noindent\textbf{``If each value can be assigned a status after the fact, is the triage falsifiable, or is it flexible bookkeeping that always survives?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The triage is falsifiable because the status labels carry commitments that data can break. A \textbf{Derived} entry asserts a parameter-free value; a measured deviation beyond the stated tolerance falsifies it directly (e.g.\ $\sin^2\theta_W=25/108$ at $>0.5\%$, or any fourth chiral family). A \textbf{Derived (conditional)} entry names a specific mathematical lemma whose failure to be provable would demote it. An \textbf{Asserted} entry makes no theorem claim and so cannot be ``rescued'' by relabeling --- it is simply open. What the triage forbids is the move the objection fears: silently upgrading an entry without either a new theorem or a passing datum. The $\lambda_H$ reclassification of the present version is the test case --- it moved only because an explicit curvature calculation with a passing vev cross-check was carried out, and it carries its remaining open lemma in the open (Objection~7).
\end{objqabody}

%% =====================================================================
%% Cluster B -- The individual derivations
%% =====================================================================

\bigskip
\section{The individual derivations}

\bigskip
\noindent\textbf{Objection 6 (O-SM-6) (the electroweak theorist)}\par\nobreak
\noindent\textbf{``$\sin^2\theta_W=25/108$ is your one genuine theorem, yet you now concede it rests on two unproven lemmas. So is anything derived end-to-end?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The honest answer is: one parameter is derived end-to-end \emph{modulo two named structural steps}, and we label it exactly that way (Theorem~\ref{thm:sw2}). The upstream half is airtight: the trace ratio $\Tr(B_0^2)/\Tr(Y_0^2)=2/5$ is elementary $\mathrm{SU}(5)$ representation theory, giving $\Tr(\hat Y_{\rm split}^2)=5/6$. The assembly $\sin^2\theta_W=(1/N)[\Tr(\hat Y^2)]^2=25/108$ then rests on two physically-motivated but \GG{}-internal steps --- the quadratic neutral-partition lemma (the squaring) and the color-blind averaging lemma (the factor $1/N$). We name them rather than bury them. ``Nothing derived end-to-end'' overstates the deficit: the value uses \emph{no empirical input}, reproduces the measured angle, and its two open steps are ordinary structural assignments, not hidden fits. It is the most-derived of the twenty-eight, and we say precisely how derived it is.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 7 (O-SM-7) (the Higgs/gauge-Higgs-unification specialist)}\par\nobreak
\noindent\textbf{``You relabeled $\lambda_H$ from a `selection' to a `curvature' after the cost-minimum argument failed. Isn't that moving the goalposts?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The relabeling followed a calculation, not a rhetorical retreat, and it was made only after the cost-minimum was found wanting. In gauge-Higgs unification the Higgs is the Wilson-line phase and $\lambda_H$ is, by the attractor-coordinate theorem (Theorem~\ref{thm:attractor}), the quartic curvature of the periodic Wilson--Hosotani holonomy potential at its minimum, with the closed form $\lambda_H=(\Lambda/f)^4 g''(\theta_H^\star)/(2\sin^2\theta_H^\star)$. Evaluated at the minimum fixed by $\sin\theta_H^\star=(2/L)^N$ that \emph{also} fixes the vacuum value $v$, this reproduces the vev as an internal cross-check and yields $\lambda_H=\mathcal{O}(0.1)$ at natural one-loop size, comfortably consistent with $2^{-N}=1/8$. The status moved from ``selected (degenerate)'' to ``curvature, modulo one lemma'' --- a stronger and more honest basis. The goalpost did not move; a better-anchored post replaced a weak one, and the one remaining open step (the spectral normalization) is stated openly.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 8 (O-SM-8) (the precision referee, on $\lambda_H$)}\par\nobreak
\noindent\textbf{``If the curvature calculation only gives an order of magnitude, why call $\lambda_H$ derived at all?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because ``derived'' in the triage means ``fixed by a structural object, not a free real number,'' and the curvature is exactly such an object --- a fourth derivative of a potential whose shape is fixed by the geometry, not a coefficient tuned to the Higgs mass. What is \emph{not} yet fixed is the precise spectral normalization $(\Lambda/f)^4$, which requires the scale-profile Kaluza--Klein sum over the \GG{} field content. We isolate that single missing quantity as the spectral-normalization lemma and label $\lambda_H$ ``Derived (curvature, modulo one lemma)'' accordingly. The residual is small and quantified: the dyadic $1/8$ and the measured $0.129$ require the same normalization to within $3\%$. Calling it ``derived modulo one lemma'' is the accurate description; calling it either ``fully derived'' or ``merely asserted'' would be the dishonest options.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 9 (O-SM-9) (the GUT theorist)}\par\nobreak
\noindent\textbf{``Textbook $\mathrm{SU}(5)$ gives $\sin^2\theta_W=3/8$ at unification, evolved down by running. You quote $25/108\approx0.231$ at $M_Z$ with no running. How can both be defensible?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They describe different objects. The $3/8$ is the group-theoretic ratio of gauge generators at the unification scale, which must then be run down through the desert; the $25/108$ is the value of the boundary observable at $M_Z$, read directly off the split-generator partition of the electroweak attractor. In \GGTheory{} the radial coordinate \emph{is} the renormalization-group axis, so ``running'' and ``descent along the throat'' are the same operation, and the boundary readout already incorporates it. The non-trivial claim is that the geometry delivers the low-energy value without a fitted threshold between $M_Z$ and $E_{\rm GUT}$. That is a falsifiable statement: a measured $\sin^2\theta_W$ departing from $25/108$ at the half-percent level would break it, independently of any GUT-scale bookkeeping.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 10 (O-SM-10) (the flavor physicist)}\par\nobreak
\noindent\textbf{``The mass hierarchy via $e^{-n\pi}$ looks like Froggatt--Nielsen with the Froggatt--Nielsen charge renamed `cascade depth.' What is new?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The resemblance is real and we welcome it, but the exponent is fixed where Froggatt--Nielsen leaves it free. In Froggatt--Nielsen the suppression is $\epsilon^{q}$ with both the small parameter $\epsilon$ and the charges $q$ chosen to fit. Here the suppression is the Bogomolnyi instanton action $S_{[2k]\to[2(k-1)]}=\pi k$ on the family-projector moduli $\mathbb{C}P^{N-1}$ \citep{Witten1982,ArisakaP2N3}: the base $e^{-\pi}$ is not adjustable and the integer depths are the only freedom. That residual freedom is exactly what the triage flags --- the eight charged-fermion depths are the eight \textbf{Asserted} entries (Objection~11). So the honest claim is narrow and testable: \GG{} fixes the \emph{base} of the hierarchy and the doubling structure, and isolates the per-fermion integers as the open problem, rather than fitting a continuous $\epsilon$.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 11 (O-SM-11) (the flavor physicist, continued)}\par\nobreak
\noindent\textbf{``All eight of your Asserted parameters are the charged-fermion cascade depths --- precisely the hard part of flavor. Haven't you simply not solved flavor?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and the paper states it plainly: the charged-fermion Yukawa sector is where the framework is weakest, and the eight depths $n_X\in\{0,1,2,3,4\}$ remain \textbf{Asserted}. What the paper claims is more limited and, we think, still substantial: the spectrum lives on a fixed cascade with a fixed base, the up/down interleaving is structural, and the entire residual ignorance is compressed into one well-posed lemma --- an equivariant-cohomology calculation on the family-projector stratum that would fix each depth as the dimension of a specific cohomology class (the cascade-depth lemma, $L10$). ``Flavor is unsolved'' and ``flavor is reduced to one named calculation'' are both true; the second is the contribution. We do not claim the first is the second.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 12 (O-SM-12) (the mixing-matrix referee)}\par\nobreak
\noindent\textbf{``You now mark the PMNS Dirac phase `Derived (conditional)' at $196^\circ$, yet it is barely measured. Isn't deriving an as-yet-unmeasured number an unfalsifiable bet?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is the opposite of unfalsifiable. The structural rule fixes $\delta_{\rm CP}^{\rm PMNS} = \pi + 2\arcsin(1/7) = 196^\circ$ from the same odd integer $J = 2N+1 = 7$ that orders the atmospheric angle, \emph{before} the measurement --- a forward commitment, not a fit. DUNE and Hyper-Kamiokande will measure $\delta_{\rm CP}$ to $\sim\pm 15^\circ$ over 2027--2032; a central value far from $196^\circ$ falsifies the assignment (falsifier F-Origin-6 of \AEleven{}). The phase is \textbf{Derived (conditional)}, not \textbf{Asserted}, because the structural rule selects a unique value: a flat-monomial cost search does surface a subdominant near-degeneracy $\{\pi, 3\pi/2\}$, but the derivation breaks it by tying the phase to the same $J = 7$ that fixes the atmospheric sector. It is `conditional' only in that it rests on one named lemma --- the leptonic Dirac-phase assignment lemma --- exactly as the three PMNS angles do. A derived value whose experimental test is named, dated, and pending is the strongest form of empirical exposure, not the weakest.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 13 (O-SM-13) (the strong-CP referee)}\par\nobreak
\noindent\textbf{``How is $\bar\theta$ handled --- is the strong-CP angle derived, or quietly set to zero?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is set by a structural cancellation rather than tuned, and it is labeled conditional, not derived end-to-end. The vanishing of the effective $\bar\theta$ follows from the same Abelian structure that gauges baryon number and from the holonomy bookkeeping of the bulk; the residual is a single named cancellation lemma rather than a Peccei--Quinn field inserted by hand \citep{PecceiQuinn1977}. The associated light-axion phenomenology --- the Kaxion --- is a genuine prediction, but it is deferred to the companion paper C1-Kaxion \citep{ArisakaC1Kaxion} rather than claimed here. Within A9 the honest statement is that strong-CP smallness is structural and conditional on one lemma, and its experimental exposure lives in the dedicated axion preprint.
\end{objqabody}

%% =====================================================================
%% Cluster C -- Comparison with other approaches
%% =====================================================================

\bigskip
\section{Comparison with other approaches}

\bigskip
\noindent\textbf{Objection 14 (O-SM-14) (the string theorist)}\par\nobreak
\noindent\textbf{``String theory offers a UV-complete quantum gravity. \GG{} posits a six-dimensional bulk without proving UV completeness. Isn't this just an effective model dressed as a first-principles theory?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} \GG{} makes a different trade than string theory and is explicit about it. String theory secures UV completeness at the cost of an unconstrained moduli landscape ($\sim10^{500}$ vacua), which is exactly the unforced continuous freedom that the parsimony axiom \AX{4} forbids. \GG{} instead fixes the bulk uniquely --- six dimensions, one composite group, one tensor sector, one five-integer ledger --- and pays for it by not yet exhibiting a string-theoretic UV completion. The two programs are not in zero-sum competition: \GG{} uses string-theoretic results as inputs (the Green--Schwarz mechanism appears as the four-component \BIsix{} statement \citep{GreenSchwarz1984,ArisakaA3BI6}) while rejecting the open moduli space. Whether the unique \GG{} bulk admits a string realization is an open and, to us, interesting question --- but the predictive content of the codebook does not wait on its answer.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 15 (O-SM-15) (the supersymmetry advocate)}\par\nobreak
\noindent\textbf{``Without supersymmetry, what stabilizes the electroweak hierarchy, and why is the weak scale not dragged up to $E_{\rm GUT}$?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The weak scale is not a tuned residual here; it is a derived geometric quantity, $v=M_{\rm lock}(2/L)^N$, set by the lock scale and the scale/triplicity factor (Theorem~\ref{thm:vev}). The little-hierarchy smallness $\xi=(v/f)^2\approx(2/L)^{2N}$ is therefore geometric rather than fine-tuned --- the same scale profile that fixes the vev supplies the suppression that, in composite-Higgs language, is usually put in by hand. Supersymmetry is not invoked, and the Coleman--Mandula constraints that motivate it are respected because the extra structure is Abelian gauge symmetry, not a spacetime-symmetry extension. The cost is that the Higgs-mass closure rests on the same spectral-normalization lemma as $\lambda_H$ (Objection~8); we do not claim a complete hierarchy proof, but we do replace ``tune the weak scale'' with ``read it off the geometry.''
\end{objqabody}

\bigskip
\noindent\textbf{Objection 16 (O-SM-16) (the neutrino-anarchy advocate)}\par\nobreak
\noindent\textbf{``Large, seemingly random PMNS mixing is well described by anarchy with no structure. Why prefer specific rationals over the simpler anarchic null hypothesis?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because anarchy and \GG{} make different, testable commitments, and the difference is becoming measurable. Anarchy predicts distributions, not values, and is comfortable with essentially any mixing pattern; \GG{} predicts specific structural forms for the PMNS angles (conditional on named lemmas) and, crucially, a definite reactor and atmospheric structure rather than a flat prior. The discriminator is precision: as $\theta_{13}$, $\theta_{23}$, and eventually $\delta_{\rm CP}$ tighten, anarchy can only ever be ``consistent,'' whereas the \GG{} forms can be falsified by a measured value outside their tolerance. We do not claim anarchy is excluded today; we claim \GG{} stakes a sharper position and is therefore the more useful hypothesis to test.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 17 (O-SM-17) (the lepton-spectrum specialist)}\par\nobreak
\noindent\textbf{``Does \GG{} reproduce or explain the Koide relation, which already ties the charged-lepton masses with a single empirical formula?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The charged-lepton masses in \GG{} come from the cascade plus the QD-completion rationals on the lepton stratum, and the Koide combination is a derived consequence of that spectrum rather than an independent input. The framework's claim is stronger than Koide in ambition --- it assigns each mass a ledger value, not merely a relation among the three --- but weaker in present rigor for the depths (Objection~11). We regard the Koide relation as a useful consistency check: any structural lepton spectrum must reproduce its near-saturation, and the \GG{} spectrum does, without the relation having been imposed. We do not claim to have a first-principles proof of Koide saturation; we note its compatibility and flag the deeper explanation as part of the same open cascade-depth problem.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 18 (O-SM-18) (the renormalization-group referee)}\par\nobreak
\noindent\textbf{``You reinterpret the fifth dimension as the renormalization-group axis rather than a Randall--Sundrum infrared region. Isn't that an \emph{ad hoc} relabeling to dodge Kaluza--Klein graviton bounds?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is a substantive identification with consequences, not a cosmetic relabeling, and it is shared across the cohort. Treating the radial coordinate as the RG scale, $r=\ln(E_{\rm GUT}/\mu)$, means the boundary couplings are RG flows rather than modes of a literal extra dimension, so the spectrum of light Randall--Sundrum gravitons that precision and collider bounds constrain is simply not predicted. What \emph{is} predicted is the lock-cluster structure at $M_{\rm lock}\approx3.85$~TeV (Objection~19). The reframing is testable precisely because it changes what should and should not appear: no tower of sub-TeV KK gravitons, but a specific lock-cluster signature. A relabeling with no observational consequence would be idle; this one reorganizes the predictions.
\end{objqabody}

%% =====================================================================
%% Cluster D -- Phenomenology, falsifiability, and proton decay
%% =====================================================================

\bigskip
\section{Phenomenology, falsifiability, and proton decay}

\bigskip
\noindent\textbf{Objection 19 (O-SM-19) (the LHC phenomenologist)}\par\nobreak
\noindent\textbf{``Your lock cluster at $\approx3.85$~TeV --- why has the LHC not seen it, and what exactly would falsify it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The detailed lock-cluster phenomenology --- the resonance pattern, the branching ratios, and the collider reach --- is deferred to the dedicated preprint P1-KK \citep{ArisakaP1KK}, and A9 imports only the scale. At $3.85$~TeV the states sit near or beyond current sensitivity for the relevant channels, so non-observation to date is expected, not anomalous. The falsifier is sharp and pre-declared: the cluster predicts a specific pattern of nearly-degenerate states at that scale; the absence of any such structure within the predicted window at HL-LHC energies, or a clean resonance at a markedly different scale, would break the lock mechanism. We state the scale here so that the prediction is on the record before the data arrive, and point to P1-KK for the line shapes.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 20 (O-SM-20) (the proton-decay veteran)}\par\nobreak
\noindent\textbf{``$\mathrm{SU}(5)$ is famous for predicting proton decay that Super-Kamiokande has not seen. You assert protons are `absolutely stable.' Isn't that equally exposed --- falsified the moment a single decay is seen?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Yes, and we embrace that exposure: proton stability is a primary falsifier of \GG{}, stated as such. The mechanism is specific. \GG{} does not stop at $\mathrm{SU}(5)$; the anomaly structure \BIsix{} forces an additional gauged $\UoneB{}$ \citep{ArisakaA3BI6}. The dangerous dimension-six $X,Y$ operators that drive decay in ordinary $\mathrm{SU}(5)$ carry net baryon number and are therefore forbidden by exact baryon-number gauge invariance, even though the $\UoneB{}$ boson is itself heavy. The prediction is thus the opposite of minimal $\mathrm{SU}(5)$: not a lifetime near $10^{31}$ years, but absolute stability --- conditional on the one undischarged step (H2), the explicit descent of the boundary gauge symmetry as component-resolved baryon number, flagged open in the proton-stability appendix. A single confirmed $p\to e^+\pi^0$ event would falsify \GG{} as decisively as continued non-observation falsified minimal $\mathrm{SU}(5)$. We regard turning $\mathrm{SU}(5)$'s most famous failure into a sharp, opposite prediction as a strength, and we welcome the next-generation searches.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 21 (O-SM-21) (the gauge-theory specialist)}\par\nobreak
\noindent\textbf{``A gauged $\UoneB{}$ made massive by a St\"uckelberg mechanism still leaves only \emph{global} baryon number. Doesn't the massive boson's longitudinal mode reintroduce baryon-number violation?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No, and the distinction is the crux. The St\"uckelberg field shifts under $\UoneB{}$ so that the combination remaining in the Lagrangian is exactly baryon-number-conserving; the longitudinal mode is the eaten St\"uckelberg scalar, whose couplings are derivative and baryon-number-symmetric, not a source of net $\Delta B\ne0$ vertices. What is broken is the \emph{local} gauge redundancy (the boson is massive); what survives is an \emph{exact global} baryon number, which is all that proton stability requires. This is why the prediction is ``exact global baryon number, residual to a St\"uckelberg-massive $\UoneB{}$'' rather than ``unbroken local $\UoneB{}$.'' The dangerous operators are forbidden by the surviving global symmetry regardless of the boson mass, exactly as a massive photon would still conserve electric charge.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 22 (O-SM-22) (the gauge-coupling-unification referee)}\par\nobreak
\noindent\textbf{``Non-supersymmetric $\mathrm{SU}(5)$ famously fails to unify the three couplings. What makes the \GG{} unification work without superpartners?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The claim is precise and falsifiable: the three couplings meet at $E_{\rm GUT}\approx2.28\times10^{16}$~GeV with no fitted parameter inserted between $M_Z$ and $E_{\rm GUT}$. The mechanism is not added matter tuned to fix the slopes but the geometric identification of the radial/RG axis together with the lock-scale threshold at $M_{\rm lock}\approx3.85$~TeV, which modifies the running in a structurally-fixed way rather than a fitted one. The honest test is the meeting itself: the three-line convergence is the headline empirical signature, and a demonstrated failure of the couplings to meet within the predicted geometry --- given the measured low-energy inputs --- would falsify the construction. We do not claim ordinary non-SUSY $\mathrm{SU}(5)$ unifies; we claim the \GG{} runtime does, and we expose that claim to the data.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 23 (O-SM-23) (the four-generation hunter)}\par\nobreak
\noindent\textbf{``How firmly is $N=3$ established? Could a fourth, heavier family be hiding?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} $N=3$ is among the firmest results in the chain, and a fourth chiral family is a clean falsifier. The family number is fixed three independent ways that must agree: the $\mathrm{SU}(5)$ trace-ratio identity, the anomaly bookkeeping of \BIsix{}, and the topological three-leg pincer proved in \PaperPTwo{} \citep{ArisakaP2N3}. It is not a fitted multiplicity but a forced output of the same ledger that fixes everything else --- indeed $N=3$ enters $\sin^2\theta_W$, $\lambda_H=2^{-N}$, and the vev. A discovered fourth chiral generation would not merely add a row to the codebook; it would break the ledger and falsify the framework. The triage records $N=3$ as derived, and we regard the absence of a fourth family as a prediction rather than an accident.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 24 (O-SM-24) (the cosmology referee)}\par\nobreak
\noindent\textbf{``Deriving the vev from $M_{\rm lock}$ and the scale profile seems to commit you to cosmological outputs --- dark matter, the cosmological constant. Don't those expose the whole edifice to cosmological falsification?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They do, and we accept the exposure while keeping the bookkeeping clean. The cosmological inventory --- $\Omega_m$, $\Omega_\Lambda$, the dark sector, the equation of state --- is derived in the companion paper A10-Cosmos \citep{ArisakaA10Cosmos}, not in A9, precisely so that each claim has a single owner. The relevant point for A9 is consistency: the same lock scale and ledger that fix the electroweak sector also feed the cosmological predictions, so a cosmological falsification (say, a dark-matter mass or an $\Omega$ value outside the predicted structure) would propagate back and threaten the shared inputs. Far from insulating the particle codebook, the shared ledger means the cohort stands or falls together --- which is why we keep the cross-paper coherence map explicit.
\end{objqabody}

%% =====================================================================
%% Cluster E -- Foundations, scope, and the meta-question
%% =====================================================================

\bigskip
\section{Foundations, scope, and the meta-question}

\bigskip
\noindent\textbf{Objection 25 (O-SM-25) (the quantum-field-theory referee)}\par\nobreak
\noindent\textbf{``Is the boundary Standard Model you derive an internally consistent quantum field theory, or only a pattern of numbers that happens to match?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is the ordinary Standard Model as a quantum field theory, with the parameters supplied by the ledger rather than by measurement. \GG{} does not modify the renormalizable $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ dynamics on the boundary; it derives the values of the couplings, masses, and mixings that the Standard Model leaves free. Anomaly cancellation, unitarity, and renormalizability are inherited from the established structure (and, upstream, are tied to the bulk anomaly theorem \BIsix{}). So the answer is both/and: it is a genuine QFT, and the contribution is the parameter pattern. The novelty is not a new low-energy dynamics but a derivation of the constants the low-energy dynamics needs.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 26 (O-SM-26) (the reproducibility referee)}\par\nobreak
\noindent\textbf{``Can an independent reader regenerate every number from scratch, without your code and without tuned constants?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That is a design goal of the paper, and the means are provided. Every codebook entry is a closed-form expression in the ledger integers and the named anchors; Table~\ref{tab:codebook-provenance} gives the formula, the status, and the owning theorem for each, and the fermion-arithmetic appendix walks the explicit numbers. There are no continuous fitted constants to reproduce --- only the integers $(1,3,5;7,35)$, the structural anchors, and arithmetic. A reader can recompute $\alpha^{-1}=2^J+N^2$, $\sin^2\theta_W=25/108$, $v=M_{\rm lock}(2/L)^N$, and the cascade spectrum by hand. We regard ``checkable with a pencil'' as a higher standard than ``checkable with our code,'' and the codebook is written to meet it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 27 (O-SM-27) (the foundational skeptic)}\par\nobreak
\noindent\textbf{``You take $\{c,G_N,\hbar;k_B,Q_B\}$ as anchors. Why should the Standard Model need a \emph{energy--information} constant $Q_B$ at all?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} $Q_B$ does no work in the particle-physics codebook, and we are explicit that the twenty-eight Standard-Model parameters are fixed by the ledger plus the four standard constants $\{c,G_N,\hbar,k_B\}$. $Q_B$ enters only as the boundary anchor of the optional information sector $\UoneQi{}$, which is adopted on empirical grounds elsewhere in the cohort (\AEleven{}, \ASeven{}) and is not required for any number in this paper. We list it among the anchors for cohort-wide uniformity, not because the electroweak or flavor sectors depend on it. A reader who rejects the $\UoneQi{}$ sector entirely loses nothing in the Standard-Model derivation; the codebook is untouched.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 28 (O-SM-28) (the Occam referee)}\par\nobreak
\noindent\textbf{``Is a twenty-two-paper program that derives twenty-eight numbers actually \emph{simpler}, by any reasonable measure, than the Standard Model it claims to explain?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} By the measure that matters for parsimony --- the count of unexplained continuous inputs --- yes. The Standard Model takes twenty-eight real numbers as free inputs; \GG{} replaces them with five integers, a handful of standard constants, and a list of named lemmas. The page count of the program is large because it shows its work, not because the underlying structure is complicated: the generative content is the five-integer ledger, which is far smaller in information than twenty-eight independent reals. Length of exposition and economy of assumptions are different axes. The honest comparison is integers-and-lemmas versus twenty-eight reals, and on that comparison the framework is the more economical hypothesis --- with the lemmas as the explicit IOUs.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 29 (O-SM-29) (the predictive-power referee)}\par\nobreak
\noindent\textbf{``What does \GG{} predict that the Standard Model does not, and which could be tested soon?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Several sharp, near-term exposures, each pre-declared. Absolute proton stability rather than $\mathrm{SU}(5)$-scale decay (Objection~20). No fourth chiral family (Objection~23). $\sin^2\theta_W=25/108$ at the half-percent level with no fitted threshold (Objection~9). A lock cluster near $3.85$~TeV with a specific pattern rather than a generic resonance (Objection~19). A Kaxion at $\sim8~\mu$eV in the companion axion paper. The Standard Model predicts none of these --- it accommodates the measured parameters but forecasts no particular value. Each \GG{} item can move from ``consistent'' to ``falsified'' with foreseeable data, which is the property the objection rightly demands. We collect them in the predictions-and-falsifiability section so that the framework's neck is visibly on the block.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 30 (O-SM-30) (the closing meta-question)}\par\nobreak
\noindent\textbf{``Suppose several of your Asserted entries are never derived. What survives, and why is the paper still worth publishing?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Even in that pessimistic case, the durable contribution is the reframing: the twenty-eight Standard-Model parameters are organized onto a single five-integer ledger, with a published, parameter-by-parameter honesty triage that says exactly how strong each placement is and names the precise mathematical step that would settle each open one. The strong empirical hits --- the integer $137$, $\sin^2\theta_W=25/108$, the $246$~GeV vev, the gauge-coupling meeting, the four-significant-figure codebook --- stand on their own as falsifiable claims regardless of the fate of the cascade-depth lemmas. And the open problems are not vague gestures but a numbered roadmap (\S\ref{sec:appendix-promotion-roadmap}). A framework that reduces twenty-eight free reals to a short list of integers plus an explicit, falsifiable to-do list has changed the question, even if it has not closed every line of it. That, in our view, is what a foundational paper should do: not pretend to finish the subject, but make the remaining questions sharp.
\end{objqabody}


%% ===================================================================
%% Cluster F -- Sharper methodology and statistics (added v11.25)
%% ===================================================================

\bigskip
\section{Sharper methodology and statistics}

\bigskip
\noindent\textbf{Objection 31 (O-SM-31) (the two-readings referee)}\par\nobreak
\noindent\textbf{``You report \emph{two} numbers for most parameters --- a `first-order' ledger value and a `QD-completed' value. With two shots per parameter, can you ever miss?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The two readings are a single prediction at two stated precisions, not two independent shots, and the rule for which applies is fixed in advance, not chosen after seeing the datum. The first-order value is the leading ledger monomial; the QD completion is the \emph{same} value corrected by Hopkins--Singer integrality rationals whose denominators are forced by the stratum (Objection~4). A reader is owed both because they answer different questions --- ``what does the bare ledger give?'' and ``what does the ledger give once the named integrality theorem is applied?'' --- and the honesty triage (Table~\ref{tab:codebook-provenance}) reports the status of the completed value, never the more permissive of the two. The discipline that forbids the hedge is that the QD rationals are the \emph{same} integers across the lepton, quark, and mixing sectors; one cannot tune them for one parameter without breaking another.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 32 (O-SM-32) (the Bayesian)}\par\nobreak
\noindent\textbf{``Your `complexity cost' assigns the integer ledger essentially zero prior penalty. A skeptic would assign the entire six-dimensional bulk, the composite group, and the tensor sector a large penalty. Isn't the whole parsimony claim prior-dependent?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is prior-dependent, and we state the prior rather than smuggle it. The complexity measure is explicitly the \MSMF{}/\OGN{} $\ell^1$ cost on the integer ledger (\S\ref{sec:naming-convention} cites the selection principle, Theorem~\ref{thm:msmf-ogn-selection}), and the look-elsewhere numbers of Objection~2 are computed \emph{within} that prior. A skeptic is free to assign the bulk machinery a large structural penalty; the point we defend is narrower and prior-independent --- that \emph{given} the constitutional bulk, the residual continuous freedom in the parameter sector is zero, so the Bayesian model-complexity debate is about the one-time cost of the framework, not about per-parameter fitting. The honest division is: the bulk is an expensive but fixed prior, paid once; the codebook adds no further adjustable cost. We argue the first is worth it only because the second is genuinely free.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 33 (O-SM-33) (the version-history auditor)}\par\nobreak
\noindent\textbf{``Across versions you have \emph{reclassified} entries --- $\lambda_H$ moved from `selected' to `curvature'. Each reclassification is a hidden degree of freedom. How many revisions before the triage is allowed to stabilize?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Reclassifications are permitted only in the direction that data and proof allow, and each one is logged with its trigger, so the version history is an audit trail rather than a hidden knob. $\lambda_H$ moved because an explicit curvature calculation with a passing vev cross-check was performed (Objection~7), not because a fit improved; the triage's own rule (Objection~5) forbids upgrading an entry without a new theorem or a passing datum. A genuine degree of freedom would let a value be re-tuned to track a shifting measurement; the triage instead pins the value and lets only the \emph{status label} change, monotonically, as proofs arrive. If a future revision had to \emph{downgrade} a Derived entry, that too would be logged --- and would itself be the falsification the objection is right to demand.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 34 (O-SM-34) (the marketing skeptic)}\par\nobreak
\noindent\textbf{```Twenty-eight to zero' is the slogan, but your own triage has fourteen Conditional and eight Asserted --- twenty-two of twenty-eight are not derived end-to-end. Isn't `28-to-zero' overselling?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} `28-to-zero' is a statement about \emph{continuous freedom}, not about completed proofs, and on that axis it is exact: not one of the twenty-eight is a free real number once the ledger is fixed. The fourteen Conditional and eight Asserted entries are not free parameters --- each is pinned to a ledger value and an owning lemma --- so the count of \emph{adjustable reals} is genuinely zero even though the count of \emph{fully proved} entries is six. We agree the slogan can mislead if read as ``twenty-eight proved'', which is exactly why the paper never states it without the triage (Objection~1) and why the accurate phrasing is ``twenty-eight reduced to five integers plus named lemmas plus one measurement.'' The zero refers to knobs, not to theorems; both numbers are reported.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 35 (O-SM-35) (the correlated-evidence statistician)}\par\nobreak
\noindent\textbf{``The integer $N=3$ enters $\sin^2\theta_W$, $\lambda_H=2^{-N}$, and the vev. These are not independent confirmations --- they share one input, so your tally of `hits' overstates the evidence.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct that they are correlated, and the paper neither hides nor double-counts the correlation --- it is the point. $N=3$ is a single forced integer (Theorem~\ref{thm:N3-closed}); the fact that it then appears in the weak angle, the quartic, and the vev is not three independent coincidences but \emph{one} structural commitment with three exposed consequences. That makes the evidence \emph{more} dangerous to the framework, not less: a single measured failure in any of the three would impugn the shared $N=3$ and hence the others, so the correlation concentrates the falsification risk rather than inflating a success count. We do not claim three independent confirmations of $N=3$; we claim one input with three falsifiable outputs, which is the honest and the more exposed statement.
\end{objqabody}

%% ===================================================================
%% Cluster G -- Sharper technical derivations (added v11.25)
%% ===================================================================

\bigskip
\section{Sharper technical derivations}

\bigskip
\noindent\textbf{Objection 36 (O-SM-36) (the QED-precision referee)}\par\nobreak
\noindent\textbf{``$\alpha^{-1}(0)=2^J+N^2=128+9=137$, but the measured value is $137.035999\ldots$. You call the $0.026\%$ residual `higher-order QED'. Isn't that hand-waving --- why should the structural value be exactly the integer part?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The structural object is the bare \OGN{} normalization $\alpha^{-1}=2^J+N^2$, an integer because it counts a dyadic seam depth plus a triplicity square (Theorem~\ref{thm:alpha0}); the measured $137.036$ is that bare value \emph{dressed} by the ordinary low-energy QED vacuum polarization, which is computed in the Standard Model, not invented here. The claim is therefore falsifiable and specific: the integer part is structural and parameter-free, and the residual must match the known QED correction --- not an adjustable offset. If the measured integer part of $\alpha^{-1}(0)$ were ever revised away from $137$, the prediction breaks (Objection~29). ``Higher-order QED'' is not a fudge but a named, independently-calculable quantity; the test is that it has the right sign and magnitude, which it does.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 37 (O-SM-37) (the integrality skeptic)}\par\nobreak
\noindent\textbf{``Your QD denominators differ by sector --- $640=2^J L$ for leptons, a different rational for quarks. If Hopkins--Singer integrality really forced them, why are they not the same everywhere?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the denominator is the integrality modulus of the \emph{specific stratum} a sector lives on, and the lepton and quark family-projectors sit on different strata, so different moduli are not a freedom but a prediction of the descent (Objection~4). $640=2^J L$ is the lepton-stratum modulus; the quark strata carry their own forced denominators from the same Hopkins--Singer mechanism \citep{HopkinsSinger2005} applied to a different cohomology. The skeptic's test is the right one and we invite it: the denominators are not chosen to fit each sector but are read off the stratum's integral structure, and \emph{no} alternative small-integer numerator on the resulting lattice reproduces all the ratios in a sector simultaneously. Same mechanism, stratum-dependent modulus --- that is what an integrality theorem looks like, not a per-sector knob.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 38 (O-SM-38) (the top-quark specialist)}\par\nobreak
\noindent\textbf{``The top sits at cascade depth zero, i.e.\ unsuppressed, so it lands near the vev `for free'. Isn't a near-vev top mass imposed by the depth-zero assignment rather than predicted?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The depth-zero \emph{assignment} is the one non-trivial claim, and it is exposed, not free. Given depth zero the top mass is $\sim v$ with no suppression --- that part is arithmetic --- but \emph{why} the top is the unique unsuppressed family member is the structural content: it is the apex of the family cascade, the only fermion whose overlap integral is order one. That assignment is among the eight \textbf{Asserted} cascade depths (Objection~11), so the paper does \emph{not} claim it as derived; it claims that \emph{if} the cascade-depth lemma assigns the top depth zero --- the natural reading of the apex --- then $m_t\approx v$ follows. A measured top mass far from the vev scale would already be in tension; that it sits where the unsuppressed slot predicts is a consistency the framework gets right but does not yet prove.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 39 (O-SM-39) (the gauge-boson-mass referee)}\par\nobreak
\noindent\textbf{``$m_W$, $m_Z$, and $m_H$ all follow from the vev and the gauge couplings. Counting them as three separate `theorems' (T-SM-33.2, 33.3, 33.4) inflates the score --- they are not independent.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They are not independent, and the paper's logic depends on their \emph{not} being independent --- that is the closure claim, not a padded tally. Once the vev (Theorem~\ref{thm:vev}) and the gauge couplings are fixed by the ledger, $m_W$, $m_Z$, and $m_H$ are \emph{forced} by the ordinary tree-level relations $m_W=\tfrac12 g v$, $m_Z=m_W/\cos\theta_W$, $m_H^2=2\lambda_H v^2$; we state them as separate theorems because each is a separate \emph{checkable} number against the PDG, not because each is a separate assumption. The honest reading is the opposite of inflation: three masses are predicted from \emph{zero} additional inputs beyond $v$, the couplings, and $\lambda_H$, so the three agreements are one structural success exposed three times --- and a single mass off its predicted value would break the shared inputs (cf.\ Objection~35).
\end{objqabody}

\bigskip
\noindent\textbf{Objection 40 (O-SM-40) (the seam-monomial referee)}\par\nobreak
\noindent\textbf{``Each parameter is a monomial in $\pi$, $e$, and the ledger integers. What fixes \emph{which} monomial --- the exponents --- for a given parameter, beyond searching until one lands near the datum?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The exponents are fixed by the physical role of the constant in the seam expression, and the \MSMF{}/\OGN{} principle then forbids importing any exponent not already demanded by that role (Theorem~\ref{thm:msmf-ogn-selection}). $\pi$ enters only as the angular holonomy of the compact $S^1_\theta$ (so it appears with the cascade/instanton action $\pi k$), and $e$ only as the base of the radial redshift along the RG axis (so it appears in scale/overlap factors); a monomial that used $\pi$ where a redshift belongs, or $e$ where a holonomy belongs, is inadmissible regardless of how well it fits. That is why the look-elsewhere counts of Objection~2 are small for the sparse sectors: the admissible monomial space is constrained by the geometry of where each constant can appear, not by a free exponent search. Where the geometry does \emph{not} pin the exponent, the entry is Asserted and we say so.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 41 (O-SM-41) (the neutrino-mass-scale referee)}\par\nobreak
\noindent\textbf{``The absolute neutrino mass scale and the two Majorana phases are essentially unconstrained in your scheme. In the sector where data are weakest, the framework predicts nothing testable. Isn't that telling?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is telling, and the triage tells it: the neutrino sector is, with the charged-fermion depths, where the framework is weakest, and the absolute scale and Majorana phases carry different grades that this response had not kept up
with. The codebook and the \MSMF{}/\OGN{} promotion audit both grade
$\alpha_{21} = 0$ \textbf{Derived (convention)} and $\alpha_{31} = \pi$
\textbf{Derived (structural)} --- a discrete opposite-parity datum of the
rank-two block, not a fit --- and the absolute scale $m_\star$
\textbf{Derived (cond.)}. None of the three is Asserted; the Asserted
class is enumerated exhaustively as the eight charged-fermion Yukawas,
which excludes the Majorana sector by construction. What the scheme does fix conditionally is the \emph{pattern} --- the PMNS angles as projector misalignments on the lepton stratum (Objection~12) and a normal-ordering preference --- while the absolute scale awaits the same cascade-depth lemma that governs the charged sector. We make two honest statements rather than one dishonest one: the framework is not silent (it forecasts mixing structure and ordering, falsifiable as precision improves, Objection~16), but it is genuinely open on the absolute scale and Majorana phases, and a measured neutrinoless-double-beta rate (Objection~49) bears directly on the latter. Weakest-sector honesty is exactly what the triage exists to enforce.
\end{objqabody}

%% ===================================================================
%% Cluster H -- Dependence on the cohort (added v11.25)
%% ===================================================================

\bigskip
\section{Dependence on the rest of the cohort}

\bigskip
\noindent\textbf{Objection 42 (O-SM-42) (the import-auditor)}\par\nobreak
\noindent\textbf{``Strip out what A9 imports --- the ledger from A2, $N=3$ from P2, the cascade from A6, the boundary law from A5 --- and what theorem is \emph{A9's own}?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A9's own content is the boundary parameter map: the theorems that take the imported ledger and bulk and \emph{compute the Standard-Model constants} from them. The weak angle (Theorem~\ref{thm:sw2}), the electroweak attractor coordinates (Theorem~\ref{thm:attractor}), the vev (Theorem~\ref{thm:vev}), the boundary $\alpha^{-1}$ (Theorem~\ref{thm:alpha}), the mass theorems (T-SM-33.x), the conditional proton-stability theorem (Theorem~\ref{thm:proton-stability-a9}), the four-force uniqueness (Theorem~\ref{thm:four-force-uniqueness}), and the \MSMF{}/\OGN{} selection on the boundary (Theorem~\ref{thm:msmf-ogn-selection}) are stated and proved here, not imported. A9 is explicitly a \emph{consequence} paper --- its job is to carry the upstream structure to the 28 numbers --- so heavy importing is by design, and the cross-paper coherence map names every borrowed result. The contribution is the map from a fixed ledger to the measured constants; that map is A9's.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 43 (O-SM-43) (the conditional-inheritance referee)}\par\nobreak
\noindent\textbf{``A6's \GRIP{} cascade carries open problems M9/M10/M11. Your mass derivations use that cascade. Don't your `Derived (conditional)' entries inherit \emph{A6's} conditionals on top of your own --- conditional on conditionals?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Yes, and the triage is built to make that inheritance explicit rather than to launder it. An A9 entry labeled \textbf{Derived (conditional)} on a cascade lemma is, transitively, conditional on whatever that cascade rests on upstream --- including A6's M9 (the \MSMF{} norm), M10 (the \BIsix{} budget that fixes $N\le3$), and M11 (the compensator monodromy) \citep{ArisakaA6Gi4}. We regard this as a feature of an honest cohort: the conditionals do not multiply silently because each is \emph{named} and traceable through the coherence map, so a reader can see that, e.g., the $N\le3$ half of the family count ultimately rests on M10. The alternative --- presenting A9's results as unconditional while the upstream remains open --- is exactly the dishonesty the triage forbids. A9 is no more secure than its imports, and it says so.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 44 (O-SM-44) (the parameter-count auditor)}\par\nobreak
\noindent\textbf{``A3 speaks of \emph{twenty-six} Standard-Model parameters, A9 of \emph{twenty-eight}. Which is it, and does the discrepancy signal loose bookkeeping?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Both are right under stated conventions, and the two extra are the neutrino Majorana phases. The count is nineteen non-neutrino parameters (nine charged-fermion masses, four CKM, three gauge couplings, two Higgs, and $\bar\theta$) plus the neutrino sector. With \emph{Dirac} neutrinos that sector adds seven (three masses, three angles, one phase) for twenty-six; with \emph{Majorana} neutrinos it adds nine (the same plus two Majorana phases) for twenty-eight. A3's earlier statement used the Dirac count; A9, which treats the neutrinos as Majorana, uses twenty-eight. This is a convention difference, not a bookkeeping error, and we will harmonize the wording so both papers quote ``twenty-six (Dirac) / twenty-eight (Majorana)'' to remove the apparent conflict. The codebook itself is unaffected: the two Majorana phases are graded \textbf{Derived} --- $\alpha_{21}$ by convention, $\alpha_{31}$
structurally --- so the two extra parameters are counted but not fitted
(Objection~41).
\end{objqabody}

\bigskip
\noindent\textbf{Objection 45 (O-SM-45) (the modular-flavor theorist)}\par\nobreak
\noindent\textbf{``Modular flavor symmetry also generates masses and mixings from a single modular parameter $\tau$. Why is the \GG{} codebook not just another point in that crowded landscape of flavor models?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the \GG{} flavor structure has \emph{no} continuous modulus to vary, which is exactly the freedom modular models retain. In modular flavor symmetry the complex modulus $\tau$ is a continuous input fixed by a vacuum-selection assumption; vary $\tau$ and the spectrum moves, so the framework spans a family of models. In \GG{} the analogue of $\tau$ is frozen --- the family-projector geometry is $\mathbb{C}P^{N-1}$ with $N=3$ forced (Theorem~\ref{thm:N3-closed}) and the cascade base $e^{-\pi}$ fixed (Objection~10) --- so there is a single point, not a landscape, and the residual freedom is integer cascade depths, not a continuous parameter. The two programs share the ambition of structuring flavor; they differ on whether a continuous modulus survives. We claim none does, which is a sharper and more falsifiable position than a modular family.
\end{objqabody}

%% ===================================================================
%% Cluster I -- Further phenomenology (added v11.25)
%% ===================================================================

\bigskip
\section{Further phenomenology and existing limits}

\bigskip
\noindent\textbf{Objection 46 (O-SM-46) (the resonance-search referee)}\par\nobreak
\noindent\textbf{``The LHC has already excluded many TeV-scale resonances up to $4$--$5$~TeV in dilepton and dijet channels. Is a lock cluster at $3.85$~TeV not already in tension with existing limits?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Not for the cluster's actual quantum numbers, and the distinction is why the detailed analysis lives in P1-KK \citep{ArisakaP1KK} rather than being hand-waved here. The strongest existing limits near $4$~TeV are for narrow, strongly-produced, leptonically-decaying $Z'$-like states; the lock cluster is a set of nearly-degenerate composite resonances whose production and decay pattern is \emph{not} that of a canonical sequential $Z'$, so the published exclusions do not map directly onto it (Objection~19). A9 commits only to the scale, $M_{\rm lock}\approx3.85$~TeV; whether the specific cluster is already constrained is a line-shape-and-channel question answered in P1-KK against the actual searches. The honest statement is that the scale is on the record and the channel-by-channel confrontation with HL-LHC limits is the companion paper's burden --- and a clean exclusion of the predicted pattern in its predicted channels would falsify it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 47 (O-SM-47) (the muon-g-2 referee)}\par\nobreak
\noindent\textbf{``Where does \GG{} stand on the muon anomalous magnetic moment? Does it predict the measured anomaly, accommodate it, or stay silent?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} On $a_\mu$ the boundary theory is the ordinary Standard Model (Objection~25), so at the level of A9 the prediction is the Standard-Model value, with no new light states inserted to manufacture an anomaly. That is a genuine exposure rather than a dodge: the framework forbids the unforced extra sectors that beyond-Standard-Model models routinely invoke to fit $a_\mu$, so if the experimental anomaly survives the current lattice-versus-data-driven debate on the hadronic vacuum polarization and demands genuinely new physics at the relevant scale, that would be evidence \emph{against} the minimal \GG{} boundary content. We do not claim a $\GG{}$ contribution to $a_\mu$; we note that the framework's rigidity means it cannot quietly absorb such an anomaly, which is the kind of vulnerability the predictive-power objection (Objection~29) rightly wants stated.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 48 (O-SM-48) (the FCNC referee)}\par\nobreak
\noindent\textbf{``You introduce a gauged $\UoneB{}$ and an optional $\UoneQi{}$. New neutral gauge bosons generically induce flavor-changing neutral currents tightly bounded by kaon and $B$-meson data. Are yours safe?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They are, because both extra Abelian currents are \emph{flavor-universal} at the boundary, which is precisely the property that suppresses dangerous FCNC. $\UoneB{}$ couples to baryon number, a flavor-diagonal charge identical for all three generations, so it generates no tree-level flavor-changing vertex; the same holds for the information charge of $\UoneQi{}$. FCNC constraints bite on \emph{non-universal} $Z'$ couplings, which the \GG{} charges are not. The boson masses are additionally large (the $\UoneB{}$ is St\"uckelberg-massive, Objection~21), further suppressing any loop-level effect. So the kaon and $B$ bounds that exclude generic $Z'$ models do not constrain these currents at leading order; a measured flavor-changing signal attributable to a universal current would, conversely, be a surprise the framework would have to answer.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 49 (O-SM-49) (the double-beta-decay referee)}\par\nobreak
\noindent\textbf{``If the neutrinos are Majorana, neutrinoless double beta decay should occur. Does \GG{} predict a rate, and is it within reach of the next-generation experiments?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The framework commits to the \emph{Majorana character} --- hence to the existence of the process --- but not yet to a sharp rate, because the rate depends on the absolute mass scale and the Majorana phases, which are graded \textbf{Derived} but not yet \emph{measured} --- which is the
constraint that matters here, since a derived-but-unmeasured phase still
leaves the rate to be predicted rather than checked (Objection~41). So the honest prediction is qualitative-but-real: $0\nu\beta\beta$ must occur at some level, and a definitive demonstration that neutrinos are \emph{Dirac} (e.g.\ a robust null result combined with a cosmological mass detection forcing the Dirac interpretation) would contradict the Majorana assignment and count against the scheme. A measured effective Majorana mass would, conversely, feed back and help fix the currently-open phases, turning an Asserted entry conditional. We neither claim a rate the structure does not yet give nor hide that the sector's openness is exactly what a $0\nu\beta\beta$ measurement would help close.
\end{objqabody}

%% ===================================================================
%% Cluster J -- Rigor, scope, and standing (added v11.25)
%% ===================================================================

\bigskip
\section{Rigor, scope, and standing}

\bigskip
\noindent\textbf{Objection 50 (O-SM-50) (the dynamical-systems referee)}\par\nobreak
\noindent\textbf{``Your gauge and Higgs values are read off `attractor coordinates' of the boundary RG flow. Is the attractor structure --- existence, basin, uniqueness --- actually proved, or assumed because it gives the right numbers?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The attractor-coordinate theorem (Theorem~\ref{thm:attractor}) establishes the structure it uses, and where the proof is partial the entries that depend on it are labeled conditional, not derived. What is established is that the boundary Wilson--Hosotani / RG flow possesses fixed points whose coordinates are the split-generator partitions delivering $\sin^2\theta_W$ and the gauge couplings; what is \emph{not} claimed in full generality is global uniqueness of the basin for every sector, which is among the named lemmas behind the \textbf{Derived (conditional)} entries. So the answer is calibrated: the attractor existence and the relevant coordinates are theorems, the global basin-uniqueness is a stated open step, and no value rests on an unproved attractor while being advertised as fully derived. The objection's worry --- ``assumed because it fits'' --- is exactly what the conditional label is there to prevent.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 51 (O-SM-51) (the boundary-emergence referee)}\par\nobreak
\noindent\textbf{``A5/\GDOS{} gives a CPTP boundary projection. Is the boundary theory \emph{proved} to be the Standard Model, or is `the boundary is the Standard Model' the thing you assume and then decorate with numbers?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The boundary gauge group and chiral content are forced upstream, not assumed here: the four-factor bulk group and its anomaly-free chiral spectrum come from \BIsix{} \citep{ArisakaA3BI6} and the \GDOS{} projection \citep{ArisakaA5GDOS}, and the Standard-Model $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ is what survives the projection of that fixed bulk. A9 does not posit ``the boundary is the SM'' and fit constants to it; it inherits a boundary theory whose group, representations, and anomaly structure are already determined, and then computes the \emph{parameters} that the inherited theory leaves free. The genuinely open part is not the identity of the boundary theory but the rigor of some parameter lemmas (the Conditional and Asserted entries). So ``which theory'' is upstream and forced; ``which constants'' is A9's, and exposed.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 52 (O-SM-52) (the tuned-constant skeptic)}\par\nobreak
\noindent\textbf{``The $\mathcal H_1{+}\mathcal H_2$ reduction carries an explicit $c_R=1$. An $\mathcal O(1)$ constant that happens to equal exactly one looks tuned. Is it derived, or set to the convenient value?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} $c_R=1$ is a normalization forced by the reduction, not a coefficient chosen to land a number, and it is flagged as one of the named steps rather than slipped in. In the $\mathcal H_1{+}\mathcal H_2$ reduction imported from \PaperPTwo{} \citep{ArisakaP2N3}, $c_R$ is the ratio of two integrally-normalized cohomology pairings on the same stratum, so it is a ratio of integers that the construction sets to unity by the matching of the two legs --- not a continuous dial. The honest exposure is that the \emph{value} $c_R=1$ depends on the same descent integrality (Objection~37) that fixes the QD denominators; were that integrality to fail, $c_R$ would not be free to drift to a fitted value but the reduction itself would break. So the proper worry is not ``why exactly one?'' --- the construction answers that --- but ``is the integrality airtight?'', which is the open lemma the triage already names.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 53 (O-SM-53) (the peer-review realist)}\par\nobreak
\noindent\textbf{``The entire cohort is single-author, unrefereed preprints. Why should the community accept twenty-eight derivations, none of which has cleared peer review?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It should not \emph{accept} them on authority --- it should \emph{check} them, which is why the paper is written to be checkable without trusting the author. Every codebook entry is a closed-form expression in named integers with its owning theorem and status given (Table~\ref{tab:codebook-provenance}, Objection~26), so a referee can recompute $\alpha^{-1}=2^J+N^2$, $\sin^2\theta_W=25/108$, and the vev with a pencil and confront each against the PDG independently of any code or claim of authority. Peer review is the right next step and we welcome it; the preprint's job is to put the structure, the triage, and the falsifiers on the record in a form that makes review \emph{possible} and the claims independently falsifiable. ``Not yet refereed'' is a true statement about standing; it is not an argument about correctness, which the explicit arithmetic settles either way.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 54 (O-SM-54) (the single-point-of-failure referee)}\par\nobreak
\noindent\textbf{``Name the one result whose collapse would do the most damage. If you had to bet against \GG{}, where is the framework most likely to break first?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The most load-bearing single result is the integer ledger $(1,3,5;7,35)$ with $N=3$: it enters the weak angle, the quartic, the vev, the fine-structure value, and the cascade base, so its failure would propagate through the whole codebook (Objection~35). The most likely \emph{near-term} break, by contrast, is empirical and sharp --- a measured $\sin^2\theta_W$ off $25/108$ at the half-percent level (Objection~9), a confirmed proton decay (Objection~20), or a fourth chiral family (Objection~23) --- any of which would falsify the framework without waiting on the open lemmas. We would bet against the framework, if at all, at the \emph{charged-fermion cascade depths} (Objection~11): they are the weakest, the most Asserted, and the place where ``reduced to a named calculation'' is most likely to turn out to be ``reduced to a calculation that does not close.'' Naming that as the fragile point, rather than the headline numbers, is the honest answer.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 55 (O-SM-55) (the scheme-coherence referee)}\par\nobreak
\noindent\textbf{``Your flagship $\sin^2\theta_W = 25/108$ is labeled the $\overline{\mathrm{MS}}$ angle at $M_Z$, yet you compare it to $0.23155$, which is the \emph{effective leptonic} angle. Against the $\overline{\mathrm{MS}}$ value $0.23122(4)$ it is a $+6.5\sigma$ miss. Which observable do you actually predict?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Fair, and we have stopped letting the label and the value drift apart. The geometric output $L^2/(4N^3)=25/108=0.23148$ is a prediction of the \emph{effective leptonic} mixing angle $\sin^2\theta_{\mathrm{eff}}^{\ell}$, which it matches to $0.03\%$. Against the $\overline{\mathrm{MS}}$ angle $0.23122(4)$ the same number is a $+6.5\sigma$ tension --- and the marquee caption now states exactly that, in the open, rather than quoting the convenient value under the inconvenient label. A reader is entitled to judge the closed form as a statement about the effective angle; the $\overline{\mathrm{MS}}$ comparison is reported honestly alongside, not hidden.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 56 (O-SM-56) (the pull-tautology referee)}\par\nobreak
\noindent\textbf{``Your first-order theory error $\sigma_{\mathrm{th}}$ is \emph{defined} as the gap between the first-order value and the quantum-defect-completed value. That forces $\mathrm{Pull}\approx\pm1$ on every row by construction. The pull column carries no information.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and we no longer ask that column to do evidential work. The first-order $\sigma_{\mathrm{th}}$ is the estimated size of the omitted quantum-defect correction, so by construction the first-order pull sits near unity; that is a statement about the size of the next term, not about agreement with data. The honest measures are the two we now report alongside it: the relative deviation $\mathrm{Dev}=(\mathrm{Pred}-\mathrm{Obs})/|\mathrm{Obs}|$, and the experimental-only significance $(\mathrm{Pred}-\mathrm{Obs})/\sigma_{\mathrm{obs}}$, which is frankly large for the most precisely measured constants because those rows are post-dictions of long-known inputs. We would rather show that plainly than dress a manufactured pull as a fit quality.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 57 (O-SM-57) (the transcendental-mixing referee)}\par\nobreak
\noindent\textbf{``$\alpha_s^{-1}(M_Z)=\pi e$ multiplies an angular constant by the base of the natural logarithm. Mixing $\pi$ and $e$ in one coupling is the textbook fingerprint of numerology.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The two transcendentals are not thrown together; they label two different primitives of the color-difference channel --- $\pi$ the angular (holonomy) primitive and $e$ the radial (renormalization-group) primitive --- which is exactly why this channel, unlike the electromagnetic one, retains both. That said, we do not over-sell it: $\pi e$ is reported as an \MSMF{}/\OGN{} cost-2 \emph{selection} within $1\%$ of the datum, look-elsewhere $p\approx0.27$, with the empirical value openly acknowledged to enter the filter. It is listed as a selection, not as a derived theorem, and the decisive test is the global trials figure (O-SM-2, O-SM-73), not the cleanliness of the form.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 58 (O-SM-58) (the bulk-construction referee)}\par\nobreak
\noindent\textbf{``You have dropped the words `holographic' and `warped.' Good --- but then what \emph{is} your six-dimensional bulk? There is no bulk action, no boundary CFT, no central charge, no compactification manifold. Without those it is not a higher-dimensional theory at all.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We agree the prestige vocabulary was misleading and have removed it. The 6D bulk is not advanced as a string background or an AdS/CFT dual, and we claim no central charge or GKP--Witten dictionary. It is defined \emph{axiomatically}: \AX{3} (\OGGEP{}) posits a lawful projection from a six-dimensional bulk --- four spacetime dimensions, a radial direction identified with the renormalization-group scale $r=\ln(E_{\mathrm{GUT}}/\mu)$, and a compact phase circle --- onto the observed boundary. \GGTheory{} is an axiomatic--deductive program (\S\ref{sec:central-claim}, ``the deductive structure in three stages''), to be judged on whether its theorems follow from its axioms and reproduce the Standard Model, not on whether it reproduces a string construction it never claimed. The honest cost of that stance is that the bulk's status is postulated, not derived from a UV-complete theory --- which is precisely what the axiom asserts.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 59 (O-SM-59) (the landscape-inversion referee)}\par\nobreak
\noindent\textbf{``You scorn the string landscape for leaving the vacuum undetermined. But string theory at least \emph{constructs} its $10^{500}$ vacua; you construct none and simply \emph{assert} uniqueness. Asserting uniqueness is the easy half.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A fair correction, and we have softened the contrast accordingly. \GGTheory{} does not build a moduli space and count its minima; it fixes its inputs by axiom --- the \OGNfive{} integer ledger --- and derives the rest. The honest comparison is therefore not ``one vacuum versus $10^{500}$'' but ``an axiomatic--deductive program versus a vacuum-counting construction.'' The burden that falls on us in return is real: the uniqueness of the $(1,3,5;7,35)$ ledger must be argued, not assumed, and we do so in A2-\GG{} and \PaperPTwo{} --- with the one genuinely open link, the \BIsix{} budget bound behind $N\le 3$, flagged as a named conjecture rather than a theorem. We claim the deductive route is cleaner; we do not claim it is finished.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 60 (O-SM-60) (the swampland referee)}\par\nobreak
\noindent\textbf{``A serious higher-dimensional proposal must confront the swampland program --- the Weak Gravity, distance, and de~Sitter conjectures. You engage none of them.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We separate the conjectures rather than plead a blanket exemption. The distance and de~Sitter conjectures police low-energy effective theories descending from a gravitational UV completion of the usual string type, and \GGTheory{} does not present itself as such a compactification, so in their standard form they do not directly bind it; its own consistency conditions are different in kind --- the \BIsix{} anomaly closure of \AThree{} and the positivity and finite-resolution requirements of \GDOS{} (\AFive{}) are the constraints an admissible boundary law must satisfy --- and a genuine swampland engagement awaits an independent gravitational completion of the bulk. The Weak Gravity Conjecture is different: it is argued to bind \emph{any} effective theory with dynamical gravity and a gauge field, so we engage it on its own terms rather than plead exemption. \GG{} carries two abelian factors coupled to dynamical gravity --- the gauged baryon number \UoneB{} and the information charge \UoneQi{}. \UoneB{} is St\"uckelberg-massive (Objection~21), and the sharp electric form of the conjecture --- a state with $m \le g\,q\,M_{\mathrm{Pl}}$ --- is formulated for \emph{massless} $\U(1)$s; its status for a St\"uckelberg-massive $\U(1)$ is itself unsettled, while the magnetic form bounds only the cutoff, $\Lambda \lesssim g_B M_{\mathrm{Pl}}$, which the sub-Planckian bulk scale and any coupling in the \PaperPOne{} L-KK-1 menu ($g_B \gtrsim 0.1$) satisfy with orders of magnitude to spare. For \UoneQi{} the required light charged state is, honestly, an open consistency check we have not discharged rather than a box we claim to tick. The honest position is therefore narrower than ``the swampland does not apply'' and sharper than silence: the magnetic Weak Gravity bound is met, the electric bound is either inapplicable (massive \UoneB{}) or an explicitly open check (\UoneQi{}), and a full swampland analysis with the charged spectrum written out is named future work.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 61 (O-SM-61) (the $N=3$ circularity referee)}\par\nobreak
\noindent\textbf{``Your $N=3$ rests on a trace ratio $2/5=(L-N)/L$ --- but that ratio is computed in the $\mathbf{5}$ of $\mathrm{SU}(5)$ using \emph{three} color entries. You put $N=3$ in to get $2/5$ and read $N=3$ back out.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection correctly identifies a circularity \emph{in the trace-ratio presentation}, and that presentation is not where the derivation lives. The actual argument is the three-leg pincer of \PaperPTwo{}: a lower bound $N\ge 3$ from the requirement of a physical \CP{}-violating phase (independent of any color count), and an upper bound $N\le 3$ from a \GDOS{}/\GRIP{} stability-budget theorem (the cascade cannot fund a fourth family), the latter the capacity theorem T-N3-6, conditional on the coupling
clause and the integrality theorem L-INT (residue R-INT). The two bounds
meet at three. The $\mathrm{SU}(5)$ trace-ratio identity is a downstream consistency check that the same integer reappears, not the proof; and $N=3$ is graded in the codebook as a \emph{conditional} derivation, not an unconditional theorem, precisely so that the open budget bound is visible.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 62 (O-SM-62) (the QD-completion referee)}\par\nobreak
\noindent\textbf{``Four-significant-figure agreement only appears after a `quantum-defect completion' that multiplies each first-order value by a bespoke rational ($637/640$, $159/160$, $641/640$, \dots). That is a second fit with one knob per observable.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The concern is exactly right to raise, and it has a sharp test attached. We claim the QD denominators are not free: they are the Hopkins--Singer descent-integrality coefficients of the boundary $X_4$, with $2^J L=640$ the controlling integer, so the admissible numerators form a fixed small set rather than a continuum. If that claim is correct, the QD layer is a computed completion, not a fit; if it is wrong, the four-figure agreement is manufactured. We regard the decisive evidence as the explicit enumeration showing that no alternative small-integer numerator reproduces all three lepton ratios simultaneously, and we agree that this enumeration --- not the agreement itself --- is what should persuade. The first-order value remains the bare prediction, reported separately and with its own honest deviation.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 63 (O-SM-63) (the blind-prediction referee)}\par\nobreak
\noindent\textbf{``Every marquee number --- $\alpha$, $\sin^2\theta_W$, $\alpha_s$, $v$, $m_H$, $y_t$ --- was measured decades ago. Where is a single blind prediction of a not-yet-measured quantity, with an error bar?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We accept the distinction and now separate it explicitly. The gauge--Higgs and charged-lepton closed forms are post-dictions of long-known constants; their value is internal consistency, not foresight. The genuinely forward, presently-testable stakes are few and we name them as such: the neutrino mass sum $\Sigma m_\nu \approx 59$~meV (the near-term cosmological squeeze, DESI-DR2\,+\,CMB $\sim$2026--2028 --- possibly the first channel to fire), the leptonic Dirac phase $\delta_{\mathrm{PMNS}}\approx 196^\circ$ (DUNE/JUNO/Hyper-K), and absolute proton stability (a null, conditional on the $\mathrm{(H2)}$ step). Two further quantitative falsifiers --- the Kaluza--Klein lock near $3.85$~TeV and the Kaxion near $8.1~\mu$eV --- carry no cross-section or coupling \emph{in this paper} and are deferred to \PaperPOne{} and \PaperCOne{}. That is a thin set of live predictions, and we present it as thin rather than padding it with the post-dictions.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 64 (O-SM-64) (the flavor-hierarchy referee)}\par\nobreak
\noindent\textbf{``The fermion mass hierarchy spanning five orders of magnitude is \emph{the} flavor puzzle. Your own table marks all eight non-top charged-fermion Yukawas `Asserted, chosen post-hoc.' So you have not solved flavor at all.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and we say so without hedging. \GGTheory{} fixes the \emph{base} of the hierarchy --- the top Yukawa reference and the geometric cascade scale --- but the eight remaining charged-fermion Yukawas are an explicit discrete ansatz, graded \textbf{Asserted} in the codebook with the per-particle exponents and prefactors flagged as post-hoc. The abstract and summary now state this in the $6/14/8$ triage rather than implying the masses are an output of geometry. What \GGTheory{} offers in the flavor sector is a structural template (a geometric-series cascade of mass ratios) with the rungs not yet derived; the honest headline is ``the base, not the rungs.'' Closing the cascade-depth lemma that would \emph{predict} each exponent is named as the principal open problem of the sector.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 65 (O-SM-65) (the hierarchy-tuning referee)}\par\nobreak
\noindent\textbf{``$v=\tfrac12 M_{\mathrm{Pl}}\,e^{-K}(2/L)^N$ with $K=35$ has $dv/dK=-v$: one unit of your `derived' integer moves the electroweak scale by a factor $e$. You have relocated the hierarchy problem into the assertion that $K$ is exactly $35$.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The sensitivity is real and we now state it in the body. But the point cuts the other way once it is read carefully: $K$ is a \emph{discrete} integer of the \OGNfive{} ledger, not a continuous dial, so the hierarchy is carried by an integer-valued exponent rather than by tuning a small real number. There are no continuous dimensionless free parameters in the construction at all; the only dimensionful input is Newton's constant $G_N$, the energy$\leftrightarrow$spacetime conversion factor of general relativity, from which both $M_{\mathrm{Pl}}$ and $E_{\mathrm{GUT}}$ descend. Whether $K=35$ is itself \emph{forced} (as the dual-lock identity $K=LJ=M^2+N^2+L^2$ claims) is the proper locus of the question, and we put the burden there --- not on a fine-tuned coefficient, of which there is none.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 66 (O-SM-66) (the Planck-mass referee)}\par\nobreak
\noindent\textbf{``Your $v\approx246$~GeV uses the non-reduced Planck mass. With the reduced $M_{\mathrm{Pl}}$ standard in every effective theory it becomes $\approx 49$~GeV --- a factor of five. Which is it, and why?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The convention is now fixed once, cohort-wide: $M_{\mathrm{lock}}$ and the vev chain use the non-reduced Planck mass $M_{\mathrm{Pl}}=(\hbar c/G_N)^{1/2}$. This is not opportunism but a consequence of the single-input structure --- $G_N$ is the lone dimensionful constant, and both $M_{\mathrm{Pl}}$ and the unification scale are defined from it through the geometric scale chain, with the factor of $\sqrt{8\pi}$ absorbed into the once-stated definition rather than re-inserted per equation. We agree that a floating convention would be a factor-of-five escape hatch; that is exactly why the definition is stated explicitly at the single-input paragraph and used uniformly thereafter.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 67 (O-SM-67) (the oblique-parameter referee)}\par\nobreak
\noindent\textbf{``A gauge--Higgs-unification model with a holonomy Higgs and Kaluza--Klein towers generically shifts the oblique parameters $S$ and $T$, which are measured to $\pm0.1$. You predict neither.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We do not, and we will not pretend otherwise. The present paper does not compute $S$, $T$, or $U$ from the holonomy Higgs and the lock-scale tower, and it therefore does not claim consistency with the global electroweak fit beyond the tree-level inputs it reproduces. This is a genuine gap, not a resolved point: a quantitative oblique-parameter prediction --- and its confrontation with the precision fit --- is named as required follow-on work. We flag it here rather than in a footnote because an electroweak-sector claim that ignored $S$ and $T$ would be incomplete, and we would rather the incompleteness be on the record.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 68 (O-SM-68) (the anomaly-polynomial referee)}\par\nobreak
\noindent\textbf{``You call \BIsix{} `anomaly closure,' but what \ANine{} actually uses is a trace-ratio $|a/c|=|a/d|=5/2$ --- a constraint on $\mathrm{U}(1)$ charges, silent on the non-abelian $\mathrm{SU}(5)$ anomalies. That is not Green--Schwarz factorization.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Agreed on the division of labor, and the text now marks it. Within \ANine{} we consume only the trace-ratio constraint from \BIsix{}; we do not re-derive anomaly cancellation here. The full factorizing six-dimensional anomaly polynomial $I_8=\tfrac12 X_4^2$, the modified Bianchi identity, and the flux quantization are established in \AThree{} \citep{ArisakaA3BI6}, where they are presented honestly as an \emph{existence} construction rather than a uniqueness proof. So the correct reading is: \ANine{} inherits a checked anomaly structure and applies one of its consequences; the claim that the observed chiral spectrum is the \emph{unique} anomaly-free choice is not made, and the gap is A3's to close.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 69 (O-SM-69) (the abstract-versus-triage referee)}\par\nobreak
\noindent\textbf{``Your front matter says the twenty-eight parameters are `outputs of geometry, observables of one projection.' Your own codebook says six Derived, fourteen conditional, eight Asserted. Those cannot both be the headline.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The two are reconciled by keeping apart two claims that are easy to run together. ``The parameters are observables of the projection'' is an \emph{ontological} statement about what they are --- it holds whether or not this paper succeeds in computing each one. ``Six Derived, fourteen conditional, eight Asserted'' is the \emph{computational} status of those attempts. The executive summary and the marquee now carry the $6/14/8$ triage at the point of the strong claim, with the explicit sentence that one parameter ($\sin^2\theta_W$) is derived end-to-end modulo two named lemmas and eight charged-fermion Yukawas remain asserted. The vision is intact; the bookkeeping is honest; and a reader of the front matter alone is no longer misled about what was achieved.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 70 (O-SM-70) (the conditional-lemma referee)}\par\nobreak
\noindent\textbf{``The fourteen `Derived (conditional)' entries are where the fitting hides: each is `forced once a single named lemma is supplied,' and several lemmas look reverse-engineered to land on the answer.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the right place to press, and the honest answer distinguishes two kinds of named lemma. Some are genuinely upstream --- fixed by \AThree{}, \ASix{}, or \AFive{} before any Standard-Model number is consulted --- and for those the conditional grade is conservative. Others are \emph{A9-local structural assignments} (the squaring step and the color-blind average behind $\sin^2\theta_W$ are the clearest examples) that we do not claim to have proved from the upstream variational principle; the codebook says so in each comment. The proper standard is a held-out test: a lemma fixed by structure before the observable is known is a derivation, one chosen to match is a fit, and we mark which is which rather than blurring them. Promoting the A9-local lemmas to theorems of the \GFF{} variational route is named as open work.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 71 (O-SM-71) (the axiom-shield skeptic)}\par\nobreak
\noindent\textbf{``Calling \OGGEP{} and \MSMF{} `axioms' looks like a shield: anything you cannot derive you can simply postulate. Why is that not a way of immunizing the theory against criticism?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because declaring something an axiom \emph{raises} the bar on everything downstream rather than lowering it. Once \OGGEP{} and \MSMF{} are fixed, every later step must be a genuine deduction: blind to the data, well-defined in scheme and scale, and free of circularity --- and the paper is built to expose exactly those failures where they occur. The honesty triage publishes which outputs are theorems and which are not; the \MSMF{}/\OGN{} audit publishes the look-elsewhere cost of each selection; the conditional grades flag every undischarged lemma. An axiom that quietly consulted the data to fix a downstream number would violate \MSMF{} itself, and we treat such cases as defects to be marked, not licenses. The axioms constrain us; they do not excuse us. The legitimate criticism is not ``you postulated axioms'' but ``did your deductions honor them'' --- and that is the criticism the apparatus invites.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 72 (O-SM-72) (the top-Yukawa referee)}\par\nobreak
\noindent\textbf{``$y_t=1$ is quoted as a marquee hit, but the running top Yukawa is $\approx0.93$--$0.94$ at the scales where it is measured. At $m_t$ that is many standard deviations from one.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The headline now reads ``$y_t=1$ at the matching scale,'' which is where the cascade-top reference is defined; we no longer present it as an unqualified equality. The physical, testable statement is $m_t=v/\sqrt2$, a parameter-free relation that lands within about one percent of the measured pole mass --- itself a post-diction of a quantity known since 1995, and labeled as such. The running of $y_t$ down to $m_t$ is ordinary Standard-Model physics and is not in dispute; the only correction we have made is to attach the scale to the marquee statement so that ``$y_t=1$'' cannot be read as a claim at $m_t$.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 73 (O-SM-73) (the Eddington referee)}\par\nobreak
\noindent\textbf{``$\alpha^{-1}(0)=2^7+3^2=137$ is Eddington redux. With a small alphabet of integers and transcendentals one reaches most $O(1)$ targets. How is this different?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The difference Eddington never supplied is a quantified look-elsewhere accounting, and that is exactly the test we accept as decisive. Per parameter we already publish the trials cost (sparse where the selection is strong, $p\approx1$ where it is not, and we decline to claim strength in the latter case). The honest, program-level number is the \emph{global} trials factor over the full declared alphabet and operation set, evaluated against all twenty-eight targets at once; we regard that single figure --- not the cleanliness of $2^7+3^2$ --- as what should make or break the numerological charge. If the global $p$ is large, the program fails by its own stated standard, and we would report it as such. We will not pretend the per-parameter picture already settles the matter: a rough back-of-the-envelope pass over the declared alphabet suggests that, of the dozen-or-so closures that look individually clean, an order-half could plausibly be reached by chance --- so the favorable single-parameter costs are suggestive, not probative, and only the global figure, computed once over all twenty-eight targets together, can discharge the charge. Committing to that test, rather than admiring individual coincidences, is the line between this and Eddington. The structural complement to this statistical answer --- why the codebook is \emph{forced} rather than searched, one closed ledger reused across all twenty-eight observables in the lineage of Euclid's \emph{Elements} --- is set out in \S\ref{subsec:forcing-vs-searching}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 74 (O-SM-74) (the single-input auditor)}\par\nobreak
\noindent\textbf{``You advertise `one fixed input,' but the starting point is a six-dimensional bulk \emph{plus} a five-integer ledger \emph{plus} five dimensional constants. That is not one input.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} ``One input'' means one continuous dimensionful free parameter, and on that precise reading the count holds. Of the five dimensional constants, $c$, $\hbar$, and $k_B$ are unit conventions and $Q_B$ is a fixed conversion ratio, leaving Newton's constant $G_N$ as the single dimensionful scale --- and there are no continuous \emph{dimensionless} free parameters at all. The \OGNfive{} ledger is discrete, and its values are argued to be forced (conditionally, in the case of $N\le 3$), not chosen; the bulk itself is posited by \AX{3} rather than fitted. So the accurate phrasing, which the text now uses, is ``one continuous dimensionful input ($G_N$), a discrete integer ledger argued to be unique, and no tunable dimensionless numbers'' --- not the looser ``one input,'' which we agree overstates the economy if read literally.
\end{objqabody}

\section{The limits of geometrization and the standing of the forcing argument}

\bigskip
\noindent\textbf{Objection 75 (O-SM-75) (the renormalization referee)}\par\nobreak
\noindent\textbf{``You geometrize the renormalization group as a radial fifth dimension $r=\ln(E_{\rm GUT}/\mu)$. If so, you should be able to shortcut the brutal high-order loop calculations --- the muon $g-2$ at five loops, say. Can you?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No, and the reason is structural rather than a matter of effort; it is set out in full in the companion \citet{ArisakaA5GDOS} (Theorem~T9.9). The radial coordinate is the variable conjugate to scale, so it geometrizes exactly the \emph{scale-dependent} (logarithmic, RG-resummable) content of an amplitude --- the running couplings, the mass-ratio logs --- and is, by construction, blind to the \emph{scale-invariant} finite coefficients in which the labor of $g-2$ actually resides. Schwinger's $\alpha/2\pi$ and the higher coefficients, carrying $\zeta(3)$, $\pi^2\ln 2$, and worse, are pure numbers with no scale dependence, and no geometry of the scale direction can manufacture them. So \GGTheory{} does not replace the Feynman expansion. What it does instead is compute the \emph{inputs} that expansion takes as given --- the value of $\alpha$, the mass ratios --- a different and complementary task. Perturbation theory computes the corrections to nature's constants; \GGTheory{} computes the constants.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 76 (O-SM-76) (the hadronic / lattice referee)}\par\nobreak
\noindent\textbf{``Grant the perturbative periods. But the strong-coupling regime is where holography helps. Surely there GG-6 can derive the hadronic contributions --- vacuum polarization, light-by-light --- that dominate the $g-2$ uncertainty?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Here the honest answer is `structure and order of magnitude, but not precision,' analyzed in \citet{ArisakaA5GDOS} (T9.9). The hadronic contribution is an irreducible \emph{dynamical} correlator of the boundary QCD --- the full spectral function $R(s)$ --- not an input and not a scale-logarithm; deriving it means solving QCD. A holographic geometrization, GG-6's radial dimension included, genuinely helps at strong coupling, but only in the controlled limit of large $N_c$ and large 't~Hooft coupling, whereas real QCD has $N_c=3$ and coupling of order one, where the corrections are themselves of order one. The GG-6 bulk, being parameter-free, could in principle yield a structural, fitted-dial-free estimate --- a genuine advantage over tuned AdS/QCD --- but at the $\sim10$--$30\%$ level, not the sub-percent the muon anomaly demands. That the data-driven and lattice determinations presently disagree is itself evidence that no geometric shortcut exists. Consistent with this paper's discipline, we make \emph{no} claim to predict the hadronic $g-2$.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 77 (O-SM-77) (the reverse-engineering referee)}\par\nobreak
\noindent\textbf{``Your strongest move is `the form is forced by the stratum' --- $\alpha_s^{-1}=\pi e$ because the color channel carries both the angular and radial primitives. But was that assignment rule fixed \emph{before} you knew the answer, or reverse-engineered to land on $\pi e$? If the latter, the forcing is circular.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the sharpest form of the numerology charge, and it presses in the right place. The defense is provenance, not rhetoric, and it is honest about its own limits. The assignment rules that carry the most weight --- the dyadic seam-depth giving $2^J$, the sector/family ratio giving $\sin^2\theta_W=L^2/(4N^3)$, the Bogomolnyi instanton action $\pi k$ giving the Yukawa cascade --- are fixed by constructions (the \GGConstitution{}, \BIsix{}, the \GFF{} Morse cascade) that exist independently of the numerical targets and are applied identically in sectors with very different data. Where an assignment is \emph{not} independently forced --- and $\alpha_s^{-1}=\pi e$ is exactly such a case, a complexity \emph{selection} on $\{\pi,e\}$ in which the empirical value enters the filter --- we say so and grade it a selection, not a theorem (the six/fourteen/eight triage; \S\ref{subsec:forcing-vs-searching}). So the answer divides cleanly: the load-bearing forms are pre-registered by geometry, the selected forms are flagged as such, and the residual circularity of the selected set is precisely what the global trials factor exists to bound. We do not claim every form is forced; we claim the forced ones are forced and the selected ones are labeled.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 78 (O-SM-78) (the promissory-note referee)}\par\nobreak
\noindent\textbf{``Your entire anti-numerology defense rests on a `global trials factor' you have not computed. You have written pages explaining why I should withhold judgment until a number appears that does not yet exist. Why should I?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The charge is fair, and we will not dodge it: the global trials factor is the decisive computation, it is not yet done, and until it is, the cohort's closures are an honest \emph{ansatz}, not a proof. What we claim in the meantime is bounded and, we think, defensible --- that the construction is \emph{forced or rule-selected} rather than freely searched (\S\ref{subsec:forcing-vs-searching}), that the per-parameter costs are already published (\S\ref{subsec:msmf-ogn-audit}), and that the single number which would settle the matter is \emph{named and accepted as decisive}, the precise commitment Eddington never made. A reader is entitled to withhold final judgment until that number is reported; what the reader is not entitled to do, having seen the over-constraint and the published costs, is assert that the freedom is \emph{hidden}. The computation belongs to a dedicated statistics note and the journal-length version, and we regard producing it as the most important single task ahead of the program --- precisely because, by our own stated standard, a large global $p$ would sink it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 79 (O-SM-79) (the Euclid referee)}\par\nobreak
\noindent\textbf{``You liken the codebook to Euclid's \emph{Elements}. But Euclid \emph{proved} every proposition. You have eight \textbf{Asserted} entries and fourteen conditional ones. A triage is not the \emph{Elements}.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and the analogy is meant narrowly: it concerns the \emph{method}, not the state of completion. The Euclidean point is that the parameters are presented as \emph{propositions descending from a fixed constitution}, not as expressions assembled from a free toolbox --- which is what separates a deductive architecture from a fit, and which holds whether a given proposition is yet fully proved or not. The triage is the honest ledger of \emph{how far} each descent has been carried: six proved end-to-end, fourteen forced modulo one named lemma each, eight still asserted. We would rather publish that ledger than claim a completeness we have not reached --- which is itself the Euclidean virtue, since the \emph{Elements} is admired not for asserting its propositions but for proving them and stating which postulates each rests on. The analogy commits us to finishing the proofs, not to having finished them.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 80 (O-SM-80) (the self-consistency referee)}\par\nobreak
\noindent\textbf{``Suppose the geometry is internally self-consistent. So what? A self-consistent mathematical structure can describe nothing in nature. Consistency is not correctness, and it is not uniqueness.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Entirely agreed, and we are careful not to conflate the three. Internal consistency is necessary but not sufficient: it earns the framework a fair hearing, not assent. \emph{Correctness} is decided empirically, and the program stakes it on falsifiable forward commitments --- the $3.85$~TeV lock cluster, the $8.1~\mu$eV Kaxion, $\delta_{\rm CP}^{\rm PMNS}=196^\circ$, $\sum m_\nu\approx59$~meV --- any of which can break it within the decade. \emph{Uniqueness} --- that this is \emph{the} geometry rather than \emph{a} geometry --- is the weakest of the three claims and is carried explicitly as a conjecture, the spectrum-level uniqueness theorem being open. So the standing we assert is precise: a self-consistent geometry with zero continuous freedom and a dated falsification program, not a proof of nature. The objection names the right hierarchy, and the paper's own triage and falsifier table are built to respect it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 81 (O-SM-81) (the discrete-symmetry referee)}\par\nobreak
\noindent\textbf{``Your CP story presupposes the standard C, P, T apparatus --- unitary this, antiunitary that --- imported wholesale from quantum field theory. You derive parameters while assuming the symmetry table.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The table is now derived, not assumed (T-SM-46.1). In the real presentation the three discrete operations are orientation reversals of the three real factors of the bulk --- fiber, space, clock --- and their unitary/antiunitary character follows: reversal of the fiber commutes with the complex structure because the signed charge reverses with the generator; reversal of the clock anticommutes with it; hence a symmetry is antiunitary exactly when it contains $T$. $CPT$ is total orientation reversal and a symmetry of any real texture; $CP$ fails exactly when the texture is chiral on the circle, which is what the lattice angle $2\pi/5$ measures. The apparatus and the parameters come from the same geometry.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 82 (O-SM-82) (the flavor-provenance referee)}\par\nobreak
\noindent\textbf{``A `derived' CKM phase is meaningless: phases are convention. You can put $\delta$ anywhere you like by rephasing, so deriving $2\pi/5$ derives nothing.''}\par\nobreak
\begin{objqabody}
\emph{Response.} Precisely because phases are convention, the paper derives the invariant, not the convention (T-SM-46.2). Rephasing is per-family dial re-zeroing; a spanning-tree gauge nulls every removable angle; what survives are loop holonomies --- for three families, one --- and the quoted $\delta^{(1)} = 2\pi/5$ is the standard-parameterization name of that single closed-loop angle, whose convention-free content is the Jarlskog invariant the paper checks against PDG. The claim is exactly as rephasing-invariant as an Aharonov--Bohm phase is gauge-invariant, and for the same holonomy reason.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 83 (O-SM-83) (the neutrino-nature referee)}\par\nobreak
\noindent\textbf{``Your neutrino block bets the house on Majorana neutrinos. If the light neutrinos turn out Dirac, the floors, the phases, and the downstream baryogenesis all collapse --- and Majorana-ness was your choice, not your derivation.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The direction of the bet is a theorem and the exposure is stated. T-SM-44.1 proves the Majorana channel is the unique $\Delta L = 2$ structure the fiber admits --- open only at weight zero, where the reality condition \emph{is} the Majorana condition --- and the $LHN$ portal is the unique fiber-neutral door; a Dirac-only sector would require leaving a constitutionally permitted, uniquely minimal term unused, an unforced suppression the framework's own economy clause forbids. So the framework does not merely prefer Majorana: it stakes itself on it, and the falsifier is the standard one --- an established Dirac nature for the light neutrinos (e.g.\ a decisive null program in neutrinoless double beta decay at the relevant masses) retires the neutrino floors as derived and the leptogenesis module with them. The bet is forced, sized, and on the table.
\end{objqabody}

\GGPart[References]{References}{Cited works, each with a relevance note linking it to the present paper}
\label{part:XVIII}

\noindent Unlike a conventional reference list, every entry below carries a short \emph{Relevance} note: a sentence or two saying what the cited work contributes and where in the present paper it bears. The aim is to turn the bibliography from a bare list of sources into a navigable companion --- a reader can move through the references and see, at a glance, which theorem, table, or section each one underwrites.

\noindent The notes are written for two readers at once. For the newcomer, they trace the intellectual lineage of the parameter derivation: from the foundational work on gauge theory, grand unification, the chiral anomaly, and the renormalization group, through to the twenty-two-paper \GGTheory{} cohort whose Standard-Model results the present paper completes. For the specialist, they make the paper's debts explicit and checkable --- each note names the specific result it supports, so that no citation is merely decorative.

\noindent The references are organized in two groups --- first the \GGTheory{} papers by the author (Section~A below), then all other works (Section~B), each in alphabetical order by first author. The cohort papers (\AOne{}--\AEleven{}, with the C- and P-series) are cited under the canonical keys fixed by the cross-paper coherence map of the \GGTheory{} program.

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

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

\begin{thebibliography}{99}
\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{Arisaka, K.\.} (2026). \textit{The \GGConstitution{}: Four Axioms and Seven Postulates of the \GGTheory{} Program, From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} The present manuscript. It serves as the citeable constitutional foundation for the subsequent technical \GG{} and \GDOS{} papers. It fixes the vocabulary (axiom/postulate/theorem), the operational meaning of measurement (\OGGEP), and the parsimony discipline (\MSMF/\OGN) that constrains all later derivations. First cited in \S\ref{sec:phen-intro}.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{Arisaka, K.\.} (2026). \textit{\GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{})}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} This companion paper develops the concrete six-dimensional geometric stage suggested by the axioms (especially \GGEP{} and anomaly inflow requirements). In the present constitutional paper we cite it whenever the 6D bulk, the four-component \BIsix{} anomaly polynomial, the \OGNfive{} integer ledger, or the $\Spinone\times\SU(5)\times\U(1)_B\times\U(1)_{Qi}$ composite gauge group is invoked. First cited in \S\ref{sec:phen-intro}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{Arisaka, K.\.} (2026). \textit{Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{})}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} This companion paper supplies the rigorous Hopkins-Singer differential-cohomology proof that the four-component anomaly polynomial $X_4 = \tr(R\wedge R) - \tr(F_{\SU(5)}\wedge F_{\SU(5)}) - \tfrac{2}{5} F_B \wedge F_B - \tfrac{2}{5} F_{Qi}\wedge F_{Qi}$ is exact, establishing $|a/c|=|a/d|=5/2$ as the geometry-gauge trace-ratio lock that POS-\AF{} demands. First cited in \S\ref{sec:phen-intro}.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{Arisaka, K.\.} (2026). \textit{From Euclid to \GG{}: The Realization of Euclid's Elements in Six Dimensions --- \GUfive{}, \OGNfive{}, and a Parameter-Free Reconstruction of Modern Physics}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} The historical companion to the GG6 Trio. Pedagogical commentary on the long chain from Euclid through Newton, Riemann, Maxwell, and Einstein to the \GG{} stage. We cite this paper whenever the historical positioning of the 6D bulk against its predecessors is invoked. First cited in \S\ref{sec:cohort-A9}.

\bibitem[Arisaka(2026, A5-GDOS)]{ArisakaA5GDOS}
\textbf{Arisaka, K.\.} (2026). \textit{\GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} This companion paper develops the open-system ``runtime'' implied by \OGGEP{} and the postulates (positivity, RG-compatibility, falsifiability). We cite it whenever the boundary law $\cL_{\GDOS}$, the Hilbert-space embedding, the Born rule, the Schr\"odinger limit, the modified TOV equation, or the BRST/Slavnov-Taylor identities are invoked, since their proofs live there as the eleven \GDOS{} Principal Theorems \T{1}--\T{11}. First cited in \S\ref{sec:phen-intro}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{Arisaka, K.\.} (2026). \textit{The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} This companion paper formalizes the four Geometric-Invariance pillars descending from \OGGEP{}: the Total Geometric-Gauge object \TGG, the Covariant Conservation Identity \CCI, the Geometric Free-Energy Functional \GFF, and the Gauge-Runtime Invariance Pipeline \GRIP{} (\DGI{} $\to$ \GSSB{} $\to$ \GA). We cite it when the geometric-side consequences of the \GGConstitution{} are invoked. First cited in \S\ref{sec:phen-intro}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
\textbf{Arisaka, K.\.} (2026). \textit{The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} This companion paper formalizes the four Quantum-Information pillars descending from \OGGEP{}: \QiQUA{} (qua-state preparation), \QiEP{} (quantum equivalence principle), \QiCON{} (decoherence/contraction), and \QiHAL{} (holographic alphabet). We cite it when the quantum-information-side consequences of the \GGConstitution{} are invoked. First cited in \S\ref{subsec:gg6-bulk-gauge-group}.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
\textbf{Arisaka, K.\.} (2026). \textit{The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} This companion paper derives the cosmological parameters $\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$, $\Omega_{\rm DM} = 1/\pi - 1/(2\pi^2)$ from the \GGConstitution{} and the 6D bulk. We cite it whenever the constitution's cosmological-prediction column is invoked. First cited in \S\ref{sec:open-problems}.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{Arisaka, K.\.} (2026). \textit{The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program}. In preparation; to be posted at https://arisaka-gg.org/ on completion. No result of the present paper depends on it.
\par\noindent\textbf{Relevance.} The capstone of the program. Defines the Ground Geometric Attractor (\GGA{}) and proves \textsf{T-GGA-as-Ground-State}, the ground-state refinement on which \S\ref{sec:universal-identity}'s answer to ``why is every electron the same?'' now rests; its eighteen-layer cascade is the recursive extension of the seven-layer survey given in that section. First cited in \S\ref{subsec:gga-refinement}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
\textbf{Arisaka, K.\.} (2026). \textit{The \GG{} Lock Cluster: A Non-Spatial Kaluza--Klein Resonance near 3.85~TeV as the Standing LHC Falsifier of \GGTheory{}}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} The first paper of the Predictions Branch. Specific phenomenology of the lock cluster at $\Mlock = (1/2)\,\MPl\,\e^{-35} \approx 3.85\,\TeV$, with explicit LHC and future-collider falsifiers. Any verified collider deviation from the predicted Kaluza-Klein resonance pattern at $5\sigma$ falsifies $\T{4}$ + $\T{6}$ jointly. First cited in \S\ref{subsec:open-problems-L6}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}
\textbf{Arisaka, K.\.} (2026). \textit{Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} The second paper of the Predictions Branch. The dedicated proof that the fermion family number $N=3$ is uniquely forced within \GGTheory{} by three logically independent legs: (L1) the MSMF lower bound $N \geq 3$ from Kobayashi--Maskawa CP-violation combinatorics; (L2) the \GDOS{} Morse upper bound $N \leq 3$ from family-projector stability on $\mathbb{C}P^{N-1}$ via the cascade-closure constraint $S_{\rm tot}(N) = \pi N(N-1)/2 \leq S_{\rm BI}^{\max} = 3\pi$; (L3) the constructive codebook self-consistency check from the present paper (\ANine{}), which reproduces all twelve charged-fermion masses to four-significant-figure precision uniquely at $N=3$. Its capacity architecture dissolves the would-be soft steps: the bulk-only derivation of the descended class is closed in the negative (its rigidity lemma), the transport geometry is supplied at the Born theorem's own grade by its adiabatic theorems T-AD-1--T-AD-3, strict Morse-positivity is off the critical path (holding as a theorem at the admissible value), and the integrality statement L-INT is a derived theorem with a
single named topological residue R-INT --- the same appendix chain proving the no-go for the naive formulation, the literal scale-profile metric pullback of $\tr R\wedge R$ vanishes identically --- so that the coefficient is fixed by the Fubini--Study normalization of its Appendix~F conventions, the Berry connection being constructed rather than awaited. The instanton budget $S_{\rm BI}^{\max} = 3\pi$ is the capacity theorem T-N3-6 there (extensive form $\pi N$; the residue R-INT is discharged on the
benchmark background in that paper's Appendix~H). Any discovery of a fourth standard family, a precision deviation in $m_\tau/m_\mu$ from $16.817$ by more than $0.5\%$, or a theoretical inconsistency between BI-6 and \GDOS{} Morse-positivity would falsify the chain. Cited in the present paper (\ANine{}) as the canonical owner of the $N=3$ theorem; \ANine{} imports the result without re-derivation. First cited in \S\ref{subsec:gg6-N3}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}
\textbf{Arisaka, K.\.} (2026). \textit{The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} The third paper of the Predictions Branch. The dedicated review-article treatment of the six oldest electron questions --- identity (with the values $y_e$, $m_e$, $e$, spin-$\tfrac12$ derived or imported from the present paper's codebook), absolute stability (no exit gate from the \GGA{} basin), the excited copies (importing \PaperPTwo{} with its fences), the positron (the conjugate readout of the same basin, with the origin of the twin imported at theorem grade from \PaperPFive{}), pointlikeness (structure entirely along the non-spatial directions; readout locality derived from AX-1 of A1; the geometric pointlikeness theorem, with $F \equiv 1$ as its laboratory-facing corollary, held within the declared axiom system), and field response derived as composite geodesic motion on the AX-2 connection --- Newton and Coulomb in parallel from one bulk, reproducing A5's Part VI --- with the two-sector open-system reading (the phase sector moves the electron, the scale sector transmutes it; radiation is the dissipative channel of motion) --- assembled as one theorem-complex about ground-state-ness, with claim-status ledger and pre-registered kill criteria. Consumes the present section's geometric-origin-of-identity reading and the lepton cascade-depth assignment without re-derivation. First cited in \S\ref{subsec:three-electron-questions}.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
\textbf{Arisaka, K.\.} (2026). \textit{The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy --- from Cardano's Cubics to Twistor Space, via the Sixth Dimension of \GGTheory{}}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} The origin-of-$i$ member paper of the Foundations Branch: the standalone, full-length development of the real presentation this paper uses in T-SM-44.1, T-SM-46.1, and T-SM-46.2 --- the two real sources of the derived complex structure, the holomorphy bridge at its Hardy, tube, and twistor levels, and the open items stated as open. For the present paper it supplies the origin of the complex phase structure in which the ledger's valued phases live. First cited in \S\ref{sec:cohort-A9}.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}
\textbf{Arisaka, K.\.} (2026). \textit{The Mirrors of Matter: Why Every Particle Has a Twin, Why Nature's Mirrors Break, and Why the Universe Is Made of Matter}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} The CPT member paper of the Foundations Branch: antiparticles from the handshake of the circle and the clock, C/P/T as orientation reversals of the three real bulk factors, CPT at quasi-free grade from the Wick-quartet analytic fact, the chiral-pentagon CP phases, and the matter--antimatter asymmetry through the weight-zero door. It is the \ANine{}--\ATen{} interface of the cohort map: the ledger phases this paper derives --- the CKM and PMNS Dirac phases and the Majorana parities, each at the grade recorded in the codebook --- reach the baryon-asymmetry chain of \ATen{} through it. First cited in \S\ref{sec:cohort-A9}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}
\textbf{Arisaka, K.\.} (2026). \textit{The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} The Predictions-Branch member that runs the
ground-attractor method of \PaperPThree{} on the baryonic stratum, and the
one companion this paper is in debt to rather than the other way round.
\ANine{} states proton stability as a theorem conditional on the (H2)
baryon-descent step and defers that step to a successor; \PaperPSix{}
discharges it, with a lock theorem that realizes the descended gauged
$\UoneB{}$ and makes exact global baryon number a consequence rather than an
assumption. The selection rule this paper cites in the leptogenesis appendix
--- the one that makes every $\Delta B = \pm 1$ channel unwritable --- is
proved there. Nothing in the twenty-eight-parameter derivation depends on
it, and the proton-stability statement of \S\ref{subsec:fsm4} is unchanged
by it; what changes is that a step this paper left open is now closed
elsewhere in the cohort. First cited in \S\ref{sec:cohort-A9}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
\textbf{Arisaka, K.\.} (2026). \textit{The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}}. Preprint, available at https://arisaka-gg.org/.
\par\noindent\textbf{Relevance.} The cosmology phenomenology companion to \ATen{}. The kaxion field of mass set by the bulk lock-length integer $K = 35$ produces a dark-matter oscillation at fractional frequency $\Delta\nu/\nu = 1/35$ in the $\mu$eV range; explicit haloscope and microwave-cavity falsifiers. Any verified discovery at $\Delta\nu/\nu \neq 1/35$ falsifies the $\OGNfive{}$ ledger and so the constitutional clauses $\AX{4}~\MSMF{}$ and $\POS{6}~\OGN{}$ upstream. First cited in \S\ref{sec:open-problems}.

\bibitem[Arisaka(planned, BookI)]{ArisakaBookI}
\textbf{Arisaka, K.\.} (planned). \textit{\textit{The Geometric Inevitability of Cosmos, Life, and Mind --- A \GGTheory{} Foundation Book}}. Capstone volume of the \GGTheory{} program, in preparation.
\par\noindent\textbf{Relevance.} The planned capstone volume that synthesizes the cohort papers into a single treatment. \ANine{} cites it as the natural successor work for the higher-layer uniqueness theorems of the structure-formation cascade (hadrons, atoms, molecules, single-cell life, and the consciousness layer) beyond the lowest layer closed at theorem level here. First cited in \S\ref{subsec:identity-cascade}.
\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
\textbf{Arisaka, K.~(2026).}
\emph{Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary},
Arisaka-GG-A8-QM.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The quantum-mechanics stratum of the Consequence Trio; recasts textbook QM via \GDOS{} and \OGGEP{}. First cited in \S\ref{sec:cohort-A9}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{Arisaka, K.~(2026).}
\emph{The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry},
Arisaka-GG-A9-SM.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The present paper; listed for completeness as the \ANine{} node of the cohort map. First cited in \S\ref{sec:cohort-A9}.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}
\textbf{Arisaka, K.~(2026).}
\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.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Cosmology and Astrophysics Branch: reads the Hubble tension as one six-dimensional geometric corridor. First cited in \S\ref{sec:cohort-A9}.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}
\textbf{Arisaka, K.~(2026).}
\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.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Cosmology and Astrophysics Branch: black-hole entropy with the horizon as an \OGGEP{} port. First cited in \S\ref{sec:cohort-A9}.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}
\textbf{Arisaka, K.~(2026).}
\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.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The cohort's symmetry review. Its
ladder ends at protection without symmetry, which is the register in
which \S\ref{sec:deep-hierarchy}'s tier-2 argument is stated, and its
reading of symmetry breaking as the one-way \emph{reading} of holonomy
data is the sentence \S\ref{ssec:higgs-cascade} uses to relabel the
Higgs mechanism honestly. First cited in \S\ref{ssec:higgs-cascade}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}
\textbf{Arisaka, K.~(2026).}
\emph{The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology},
Arisaka-GG-W2-Holonomy.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The treatise on the fork this
sector's account runs on --- holonomy requires closed loops, and an
interval has none --- proving the group law for circle transport and
the semigroup law for interval transport, and carrying the
stage-custody correspondence that \S\ref{ssec:higgs-custody} applies
to the electroweak sheet. Its readability theorem is the source of the
Elitzur reading. First cited in \S\ref{ssec:higgs-custody}.

\end{thebibliography}

\bigskip
\section*{B.~Other references}

\begin{thebibliography}{99}
\bibitem[Abachi et al.(1995)]{Abachi1995}
\textbf{Abachi, S. et al. (D0 Collaboration).} (1995). \textit{Observation of the top quark}. Physical Review Letters, \textit{74}, 2632--2637. \url{https://doi.org/10.1103/PhysRevLett.74.2632}
\par\noindent\textbf{Relevance.} The D0 co-discovery of the top quark at the Tevatron, confirming the heaviest Standard-Model fermion at the mass anticipated by precision electroweak fits. Cited in the introduction as part of the framework's record of prediction confirmed. First cited in \S\ref{sec:intro}.

\bibitem[Abe et al.(1995)]{Abe1995}
\textbf{Abe, F. et al. (CDF Collaboration).} (1995). \textit{Observation of top quark production in $\bar p p$ collisions with the Collider Detector at Fermilab}. Physical Review Letters, \textit{74}, 2626--2631. \url{https://doi.org/10.1103/PhysRevLett.74.2626}
\par\noindent\textbf{Relevance.} The CDF discovery of the top quark at $\sim 173$~GeV, the value indirectly predicted beforehand by electroweak radiative corrections. Cited in the introduction as a triumph of Standard-Model predictivity. First cited in \S\ref{sec:intro}.

\bibitem[Adler(1969)]{Adler1969}
\textbf{Adler, S.~L.} (1969). \textit{Axial-vector vertex in spinor electrodynamics}. Physical Review, \textit{177}, 2426--2438. \url{https://doi.org/10.1103/PhysRev.177.2426}
\par\noindent\textbf{Relevance.} With Bell and Jackiw, the discovery of the chiral (triangle) anomaly whose family-by-family cancellation constrains the Standard-Model charge assignments. Cited in the introduction's account of the theory's internal consistency. First cited in \S\ref{sec:intro}.

\bibitem[ADMX Collaboration(2025)]{ADMX2025}
\textbf{ADMX Collaboration (C.~Bartram et al.).} (2025). \textit{Search for axion dark matter from 1.1 to 1.3~GHz with ADMX}. Phys.\ Rev.\ Lett. arXiv:2504.07279.
\par\noindent\textbf{Relevance.} DFSZ-sensitive microwave-cavity haloscope; its Generation-2 band ($0.6$--$2$~GHz) reaches the Kaxion line at $1.95$~GHz. First cited in \S\ref{subsec:fsm2}.

\bibitem[Ahmad et al.(2002)]{Ahmad2002}
\textbf{Ahmad, Q.~R. et al. (SNO Collaboration).} (2002). \textit{Direct evidence for neutrino flavor transformation from neutral-current interactions in the Sudbury Neutrino Observatory}. Physical Review Letters, \textit{89}, 011301. \url{https://doi.org/10.1103/PhysRevLett.89.011301}
\par\noindent\textbf{Relevance.} SNO's neutral-current measurement established solar-neutrino flavor transformation, completing (with Super-Kamiokande) the proof that neutrinos are massive. Cited in the introduction as the empirical origin of the neutrino sector. First cited in \S\ref{sec:intro}.

\bibitem[Arnison et al.(1983)]{Arnison1983}
\textbf{Arnison, G. et al. (UA1 Collaboration).} (1983). \textit{Experimental observation of isolated large transverse energy electrons with associated missing energy at $\sqrt{s}=540$~GeV}. Physics Letters B, \textit{122}, 103--116. \url{https://doi.org/10.1016/0370-2693(83)91177-2}
\par\noindent\textbf{Relevance.} The UA1 discovery of the $W$ boson at CERN, confirming the electroweak prediction of massive charged mediators. Cited in the introduction as a triumph of Standard-Model predictivity. First cited in \S\ref{sec:intro}.

\bibitem[ATLAS Collaboration(2012)]{ATLAS2012}
\textbf{ATLAS Collaboration (Aad, G. et al.).} (2012). \textit{Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC}. Physics Letters B, \textit{716}, 1--29. \url{https://doi.org/10.1016/j.physletb.2012.08.020}
\par\noindent\textbf{Relevance.} The ATLAS observation of the $\sim 125$~GeV Higgs boson, confirming electroweak symmetry breaking. Cited in the introduction and in the discussion of the Higgs sector. First cited in \S\ref{sec:intro}.

\bibitem[Barger et al.(2012)]{BargerMarfatiaWhisnant2012}
\textbf{Barger, V., D.~Marfatia and K.~Whisnant.} (2012). \textit{The Physics of Neutrinos}. Princeton University Press.
\par\noindent\textbf{Relevance.} The shortest complete route from a graduate quantum-field-theory background to the current neutrino literature. Recommended in the course's textbook review as the concise survey; reviewed there for what it covers and where it stops. First cited in \S\ref{app:textbook-review}.

\bibitem[Bell \& Jackiw(1969)]{BellJackiw1969}
\textbf{Bell, J.~S., \& Jackiw, R.} (1969). \textit{A {PCAC} puzzle: $\pi^0\to\gamma\gamma$ in the $\sigma$-model}. Il Nuovo Cimento A, \textit{60}, 47--61. \url{https://doi.org/10.1007/BF02823296}
\par\noindent\textbf{Relevance.} With Adler, the discovery of the chiral anomaly. Cited in the introduction in the discussion of anomaly cancellation as an internal-consistency constraint. First cited in \S\ref{sec:intro}.

\bibitem[Bethe(1947)]{Bethe1947}
\textbf{Bethe, H. A.} (1947). \textit{The electromagnetic shift of energy levels}. Physical Review, \textit{72}, 339--341. \url{https://doi.org/10.1103/PhysRev.72.339}
\par\noindent\textbf{Relevance.} Bethe's 1947 non-relativistic estimate of the Lamb shift was the first successful renormalized QED calculation. We cite Bethe in the historical opening of the IR Abelian transport theorem as the founding moment of low-energy QED transport, extended by Schwinger and Tomonaga to the full $g{-}2$ program. First cited in \S\ref{thm:mH}.

\bibitem[Bilenky(2018)]{Bilenky2018}
\textbf{Bilenky, S.~M.} (2018). \textit{Introduction to the Physics of Massive and Mixed Neutrinos} (2nd ed.). Lecture Notes in Physics 947, Springer.
\par\noindent\textbf{Relevance.} The careful general-$n$ counting of physical angles and phases for Dirac and Majorana neutrinos, and the proof that the Majorana phases cancel identically in every oscillation probability. The counting of \eqref{eq:phasecount-app} and problems~K.1.4 and~K.1.5 follow it. First cited in \S\ref{app:majorana}.

\bibitem[Bouchiat et al.(1972)]{BouchiatIliopoulosMeyer1972}
\textbf{Bouchiat, C., Iliopoulos, J., \& Meyer, P.} (1972). \textit{An anomaly-free version of Weinberg's model}. Physics Letters B, \textit{38}, 519--523. \url{https://doi.org/10.1016/0370-2693(72)90532-1}
\par\noindent\textbf{Relevance.} Showed that the electroweak gauge anomalies cancel between quarks and leptons precisely when each family is complete, locking the quark and lepton charge assignments together. Cited in the introduction's account of internal consistency. First cited in \S\ref{sec:intro}.

\bibitem[Buchm\"uller et al.(2005)]{BuchmullerDiBariPlumacher2005}
\textbf{W.~Buchm\"uller, P.~Di~Bari, and M.~Pl\"umacher.} (2005). \textit{Leptogenesis for pedestrians}. Annals of Physics, \textit{315}, 305--351. hep-ph/0401240. (See also Nuclear Physics B, \textit{643}, 367--390 (2002), Erratum \textit{793}, 362 (2008).)
\par\noindent\textbf{Relevance.} The field-defining review, and the source of the strong-washout efficiency fit \eqref{eq:lg-bdp} used in Part~\ref{part:lepto}. First cited in \S\ref{sec:lg-boltzmann}.

\bibitem[Casas \& Ibarra(2001)]{CasasIbarra2001}
\textbf{J.~A.~Casas and A.~Ibarra.} (2001). \textit{Oscillating neutrinos and $\mu \to e\gamma$}. Nuclear Physics B, \textit{618}, 171--204. hep-ph/0103065.
\par\noindent\textbf{Relevance.} The parameterization in which the neutrino Yukawa matrix is decomposed into light masses, the PMNS matrix and an orthogonal matrix $R$. It is what makes the unflavored \CP{} invariant manifestly $U_{\rm PMNS}$-independent, and hence what forces the chain of Part~\ref{part:lepto} to be flavored. First cited in \S\ref{sec:lg-cp}.

\bibitem[Cabibbo(1963)]{Cabibbo1963}
\textbf{Cabibbo, N.} (1963). \textit{Unitary symmetry and leptonic decays}. Physical Review Letters, \textit{10}, 531--533. \url{https://doi.org/10.1103/PhysRevLett.10.531}
\par\noindent\textbf{Relevance.} Cabibbo's 1963 introduction of the rotation angle that mixes strange and non-strange decays is the original quark-mixing matrix. We cite Cabibbo in the historical opening of the CKM section as the inventor of the structural object that Kobayashi--Maskawa later extended to three families with one CP phase. First cited in \S\ref{sec:CKM}.

\bibitem[CMS Collaboration(2012)]{CMS2012}
\textbf{CMS Collaboration (Chatrchyan, S. et al.).} (2012). \textit{Observation of a new boson at a mass of 125~GeV with the CMS experiment at the LHC}. Physics Letters B, \textit{716}, 30--61. \url{https://doi.org/10.1016/j.physletb.2012.08.021}
\par\noindent\textbf{Relevance.} The CMS observation of the $\sim 125$~GeV Higgs boson, simultaneous with ATLAS. Cited in the introduction and in the discussion of the Higgs sector. First cited in \S\ref{sec:intro}.

\bibitem[Cheng \& Li(1984)]{ChengLi1984}
\textbf{Cheng, T.-P. and L.-F.~Li.} (1984). \textit{Gauge Theory of Elementary Particle Physics}. Oxford University Press.
\par\noindent\textbf{Relevance.} A classic graduate text on the gauge structure of the Standard Model. Recommended in the technical-appendix graduate course for in-depth study of the gauge sector, the $\SU(5)$ embedding, and anomaly cancellation. First cited in \S\ref{app:gauge-step1}.

\bibitem[CMS Collaboration(2019)]{CMSttbar2019}
\textbf{CMS Collaboration (A.~M.~Sirunyan et al.).} (2019). \textit{Search for resonant $t\bar t$ production in proton--proton collisions at $\sqrt{s} = 13$~TeV}. JHEP, \textit{04}, 031. arXiv:1810.05905.
\par\noindent\textbf{Relevance.} The $t\bar t$ resonance search whose Randall--Sundrum Kaluza--Klein-gluon exclusion reaches $4.55$~TeV. Cited in F-SM-1 to explain why that limit, set on a broad template with $\Gamma/M \sim 15$--$30\%$ and $\mathrm{BR}(t\bar t)$ near unity, does not transfer to the narrow lock-cluster member. First cited in \S\ref{subsec:fsm1}.

\bibitem[CMS Collaboration(2020)]{CMSDijet2020}
\textbf{CMS Collaboration (A.~M.~Sirunyan et al.).} (2020). \textit{Search for high mass dijet resonances with a new background prediction method in $pp$ collisions at $\sqrt{s}=13$~TeV}. J.\ High Energy Phys., \textit{05}, 033. arXiv:1911.03947.
\par\noindent\textbf{Relevance.} Run-2 dijet exclusions (leptophobic $Z'$ to $\sim4$~TeV; color-octet scalars to $\sim3.7$~TeV) bracketing the predicted $3.85$~TeV cluster. First cited in \S\ref{subsec:fsm1}.

\bibitem[Calabrese et al.(2025)]{ACTDR62025}
\textbf{ACT Collaboration (E.~Calabrese et al.).} (2025). \textit{The Atacama Cosmology Telescope: DR6 constraints on extended cosmological models}. JCAP, \textit{11}, 063. arXiv:2503.14454.
\par\noindent\textbf{Relevance.} Independent CMB constraint on the neutrino mass sum, $\sum m_\nu<0.089$~eV ($95\%$, $\Lambda$CDM), quoted beside the DESI bound in F-SM-3 so that the model dependence of the squeeze is visible rather than hidden. First cited in \S\ref{subsec:fsm3}.

\bibitem[Capozzi et al.(2025)]{Capozzi2025}
\textbf{F.~Capozzi, E.~Lisi, A.~Marrone, and A.~Palazzo.} (2025). \textit{Neutrino masses and mixing: entering the era of subpercent precision}. Physical Review D, \textit{111}, 093006. arXiv:2503.07752.
\par\noindent\textbf{Relevance.} The independent Bari global fit, corroborating the ordering preference and the subpercent determination of the atmospheric splitting; cited so that the octant discussion does not rest on a single group's analysis. First cited in \S\ref{sec:theta23-split}.

\bibitem[CUORE Collaboration(2025)]{CUORE2025}
\textbf{CUORE Collaboration (D.~Q.~Adams et al.).} (2025). \textit{With or without $\nu$? Hunting for the seed of the matter-antimatter asymmetry}. Science, \textit{390}, 1029. arXiv:2404.04453.
\par\noindent\textbf{Relevance.} The $^{130}$Te $0\nu\beta\beta$ limit, $T^{0\nu}_{1/2}>3.5\times10^{25}$~yr, completing the three-isotope picture against which the predicted $m_{\beta\beta}\simeq1.8$~meV is scored. First cited in \S\ref{subsec:fsm3}.

\bibitem[Coleman \& Weinberg(1973)]{ColemanWeinberg1973}
\textbf{Coleman, S. R. and Weinberg, E.} (1973). \textit{Radiative corrections as the origin of spontaneous symmetry breaking}. Physical Review D, \textit{7}, 1888--1910. \url{https://doi.org/10.1103/PhysRevD.7.1888}
\par\noindent\textbf{Relevance.} Coleman and Weinberg's 1973 paper showed that radiative corrections can generate spontaneous symmetry breaking even when the tree-level potential has no such feature, fixing the dimensionless quartic coupling at one-loop order. We cite Coleman--Weinberg in the historical opening of the quartic-curvature theorem as the inventors of the radiative-quartic mechanism that the geometric-attractor picture replaces with a tree-level branch normal form. First cited in \S\ref{thm:vev}.

\bibitem[Davidson \& Ibarra(2002)]{DavidsonIbarra2002}
\textbf{S.~Davidson and A.~Ibarra.} (2002). \textit{A lower bound on the right-handed neutrino mass from leptogenesis}. Physics Letters B, \textit{535}, 25--32. hep-ph/0202239.
\par\noindent\textbf{Relevance.} The bound \eqref{eq:lg-di} that forces standard hierarchical thermal leptogenesis above $10^{9}$~GeV, and therefore the obstacle that the lock-scale mechanism of Part~\ref{part:lepto} must evade. First cited in \S\ref{sec:lg-di}.

\bibitem[Davidson et al.(2008)]{DavidsonNardiNir2008}
\textbf{S.~Davidson, E.~Nardi and Y.~Nir.} (2008). \textit{Leptogenesis}. Physics Reports, \textit{466}, 105--177. arXiv:0802.2962.
\par\noindent\textbf{Relevance.} The definitive review of the subject: the Boltzmann system, the flavor effects, the bound structure, and the efficiency parametrizations. Where the leptogenesis chapter of this course departs from the standard treatment it departs from this reference, and a reader should be able to locate exactly where. First cited in \S\ref{app:textbook-review}.

\bibitem[DESI Collaboration(2024)]{DESI2024}
\textbf{DESI Collaboration (A.~G.~Adame et al.).} (2024). \textit{DESI 2024 VI: Cosmological constraints from the measurements of baryon acoustic oscillations}. arXiv:2404.03002.
\par\noindent\textbf{Relevance.} DESI\,$+$\,CMB neutrino-mass bound $\sum m_\nu<0.072$~eV, pressing on the predicted $\simeq59$~meV. First cited in \S\ref{subsec:fsm3}.

\bibitem[DESI Collaboration(2025)]{DESI2025}
\textbf{DESI Collaboration (M.~Abdul-Karim et al.).} (2025). \textit{Constraints on neutrino physics from DESI DR2 BAO and DR1 full shape}. Physical Review D, \textit{112}, 083513.
\par\noindent\textbf{Relevance.} The current tightest cosmological bound on the neutrino mass sum: $<0.0642$~eV ($95\%$, $\Lambda$CDM), $<0.053$~eV in the Feldman--Cousins variant, $<0.163$~eV under $w_0w_a$CDM. Because this paper's F-SM-6 pins $w=-1$ two-sidedly, the $\Lambda$CDM row is the one that applies, and the predicted $59.17$~meV sits eight per cent beneath it. First cited in \S\ref{subsec:fsm3}.

\bibitem[Dodelson \& Schmidt(2020)]{DodelsonSchmidt2020}
\textbf{Dodelson, S. and F.~Schmidt.} (2020). \textit{Modern Cosmology} (2nd ed.). Academic Press.
\par\noindent\textbf{Relevance.} The modern data pipeline from a CMB power spectrum or galaxy survey to a parameter constraint --- the background a student needs to read the $\Summnu{}$ bounds of \S\ref{app:pmns-absolute} critically rather than as quoted numbers. First cited in \S\ref{app:textbook-review}.

\bibitem[Ellis \& Sakurai(2016)]{EllisSakurai2016}
\textbf{J.~Ellis and K.~Sakurai.} (2016). \textit{Search for sphalerons in proton--proton collisions}. Journal of High Energy Physics, \textit{04}, 086. arXiv:1601.03654.
\par\noindent\textbf{Relevance.} The quantitative collider phenomenology of sphaleron-induced $\Delta B = \Delta L = 3$ events --- final states, rates under stated suppression assumptions, and LHC constraints. With \citet{TyeWong2015} it brackets the contested near-threshold question on which the FCC-hh rate of \S\ref{sec:lg-accelerator} rides. First cited in \S\ref{sec:lg-accelerator}.

\bibitem[Englert \& Brout(1964)]{EnglertBrout1964}
\textbf{Englert, F. and Brout, R.} (1964). \textit{Broken symmetry and the mass of gauge vector mesons}. Physical Review Letters, \textit{13}, 321--323. \url{https://doi.org/10.1103/PhysRevLett.13.321}
\par\noindent\textbf{Relevance.} Englert and Brout's 1964 paper, simultaneous with Higgs, gave the first complete relativistic gauge-invariant mechanism for vector-boson mass generation by spontaneous symmetry breaking. Cited in the historical opening of the Wilson--Hosotani Higgs-vev theorem and the quartic-curvature theorem as one of the two co-original sources. First cited in \S\ref{sec:vev}.

\bibitem[Esteban et al.(2024)]{NuFIT60}
\textbf{I.~Esteban, M.~C.~Gonzalez-Garcia, M.~Maltoni, I.~Martinez-Soler, and T.~Schwetz.} (2024). \textit{NuFit-6.0: updated global analysis of three-flavor neutrino oscillations}. J.\ High Energy Phys., \textit{12}, 216. arXiv:2410.05380.
\par\noindent\textbf{Relevance.} Global three-flavor oscillation fit; normal ordering favored, with a $\delta_{\mathrm{CP}}$ band consistent with the predicted $196^\circ$. First cited in \S\ref{subsec:fsm3}.

\bibitem[Esteban et al.(2025)]{NuFIT61}
\textbf{I.~Esteban, M.~C.~Gonzalez-Garcia, M.~Maltoni, I.~Martinez-Soler, J.~P.~Pinheiro, and T.~Schwetz.} (2025). \textit{NuFIT 6.1: three-neutrino fit based on data available in November 2025}. Release at \texttt{nu-fit.org}; methodology as in NuFit-6.0, JHEP \textit{12}, 216.
\par\noindent\textbf{Relevance.} The global three-flavor fit used as the primary oscillation benchmark throughout this paper. Its atmospheric best fit lies in the lower octant, against this paper's derived upper-octant value and against the accelerator joint fit; the disagreement is the subject of \S\ref{sec:theta23-split}. First cited in \S\ref{sec:pmns-benchmark}.

\bibitem[Esteban et al.(2026)]{EstebanJUNO2026}
\textbf{I.~Esteban, M.~C.~Gonzalez-Garcia, M.~Maltoni, I.~Martinez-Soler, J.~P.~Pinheiro, and T.~Schwetz.} (2026). \textit{Lessons from the first JUNO results}. JHEP, \textit{04}, 089. arXiv:2601.09791.
\par\noindent\textbf{Relevance.} The post-JUNO global analysis: $\Delta\chi^2(\mathrm{IO}-\mathrm{NO})=4.6$ without atmospheric data and $9.4$ with, roughly doubling the discrimination and placing normal ordering --- this paper's prediction --- at about $2.2$--$2.3\sigma$. First cited in \S\ref{subsec:fsm3}.

\bibitem[Fermi(1934)]{Fermi1934}
\textbf{Fermi, E.} (1934). \textit{Versuch einer {T}heorie der $\beta$-{S}trahlen. {I}}. Zeitschrift f\"ur Physik, \textit{88}, 161--177. \url{https://doi.org/10.1007/BF01351864}
\par\noindent\textbf{Relevance.} Fermi's 1934 four-fermion theory of beta decay introduced the dimensionful coupling $G_F$ that now defines the electroweak scale. We cite Fermi in the historical opening of the boson-mass and Fermi-constant theorem package as the originator of the structural constant whose value the present paper derives from the OGN-5 ledger and the scale chain. First cited in \S\ref{eq:vector-id}.

\bibitem[Fukugita \& Yanagida(2003)]{FukugitaYanagida2003}
\textbf{Fukugita, M. and T.~Yanagida.} (2003). \textit{Physics of Neutrinos and Applications to Astrophysics}. Springer.
\par\noindent\textbf{Relevance.} The bridge from the neutrino mass matrix to cosmology, written by the authors of the leptogenesis proposal itself; the natural book-length companion to App.~\ref{app:leptogenesis}. Reviewed in the course's textbook survey, where its stopping point --- the heavy-neutrino scale and the Dirac Yukawa remain model inputs --- is stated. First cited in \S\ref{app:textbook-review}.

\bibitem[Fukugita \& Yanagida(1986)]{FukugitaYanagida1986}
\textbf{M.~Fukugita and T.~Yanagida.} (1986). \textit{Baryogenesis without grand unification}. Physics Letters B, \textit{174}, 45--47.
\par\noindent\textbf{Relevance.} The paper that proposed leptogenesis: make a lepton asymmetry in heavy-neutrino decay and let the electroweak sphalerons convert it. Part~\ref{part:lepto} is a realization of this mechanism in which the heavy scale is an output rather than an input. First cited in \S\ref{sec:lg-sakharov}.

\bibitem[Fritzsch \& Gell-Mann(1972)]{FritzschGellMann1972}
\textbf{Fritzsch, H. and Gell-Mann, M.} (1972). \textit{Current algebra: Quarks and what else?}. In \textit{Proceedings of the {XVI} International Conference on High Energy Physics}. Chicago/Batavia.
\par\noindent\textbf{Relevance.} Fritzsch and Gell-Mann's 1972 conference paper proposed three color charges (the original color--gauge identification) as the resolution of the $\Delta^{++}$ statistics paradox. We cite Fritzsch--Gell-Mann in the historical opening of the strong angle--scale theorem as the inventors of the color-gauge identification that this paper now derives geometrically. First cited in \S\ref{eq:g2-general}.

\bibitem[Fukuda et al.(1998)]{Fukuda1998}
\textbf{Fukuda, Y. et al. (Super-Kamiokande Collaboration).} (1998). \textit{Evidence for oscillation of atmospheric neutrinos}. Physical Review Letters, \textit{81}, 1562--1567. \url{https://doi.org/10.1103/PhysRevLett.81.1562}
\par\noindent\textbf{Relevance.} The Super-Kamiokande discovery of atmospheric-neutrino oscillation, the first proof that neutrinos have mass and the one empirical fact that forced the Standard Model to grow. Cited in the introduction as the origin of the neutrino sector. First cited in \S\ref{sec:intro}.

\bibitem[Gell-Mann(1962)]{GellMann1962}
\textbf{Gell-Mann, M.} (1962). \textit{Symmetries of baryons and mesons}. Physical Review, \textit{125}, 1067--1084. \url{https://doi.org/10.1103/PhysRev.125.1067}
\par\noindent\textbf{Relevance.} Gell-Mann's 1962 paper introduced the $SU(3)$ flavor symmetry now called the Eightfold Way, the structural antecedent of $SU(3)_c$ as the color-confinement gauge group. We cite Gell-Mann in the historical opening of the strong angle--scale theorem as the originator of the structural identification of the strong sector with an $SU(3)$ algebra. First cited in \S\ref{eq:g2-general}.

\bibitem[Georgi(1999)]{Georgi1999}
\textbf{Georgi, H.} (1999). \textit{Lie Algebras in Particle Physics} (2nd ed.). Perseus Books.
\par\noindent\textbf{Relevance.} Representation theory as a working tool: the $\SU(3)$ chapters, the tensor methods, and the embedding of the Standard-Model group in $\SU(5)$ are the technical backing for the branching arguments of the ledger chapter. First cited in \S\ref{app:textbook-review}.

\bibitem[Georgi \& Glashow(1974)]{GeorgiGlashow1974}
\textbf{Georgi, H., \& Glashow, S.~L.} (1974). \textit{Unity of all elementary-particle forces}. Physical Review Letters, \textit{32}, 438--441.
\par\noindent\textbf{Relevance.} The original $\mathrm{SU}(5)$ grand-unified theory paper. Georgi and Glashow embed $\mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ into a single simple group whose adjoint $\mathbf{24}$ contains the Standard-Model gauge bosons plus heavy $X, Y$ bosons mediating proton decay; the matter content of one family fits into $\bar{\mathbf{5}} \oplus \mathbf{10}$. We cite Georgi--Glashow 1974 in §\ref{sec:classic-su5} as the historical stepping-stone whose non-Abelian spine \GGTheory{} retains and embeds inside the 6D bulk. First cited in \S\ref{sec:classic-su5}.

\bibitem[Gell-Mann et al.(1979)]{GellMannRamondSlansky1979}
\textbf{M.~Gell-Mann, P.~Ramond, and R.~Slansky.} (1979). \textit{Complex spinors and unified theories}. In \textit{Supergravity} (Stony Brook Workshop), Conf.\ Proc.\ C790927, 315--321. arXiv:1306.4669.
\par\noindent\textbf{Relevance.} One of the three independent 1977--1980 statements of the seesaw mechanism. Cited beside Minkowski and Yanagida in Part~\ref{part:lepto} as the historical origin of the heavy-Majorana route that the constitutional door of \S\ref{sec:lg-door} arrives at without freedom. First cited in \S\ref{sec:lg-door}.

\bibitem[Harvey \& Turner(1990)]{HarveyTurner1990}
\textbf{J.~A.~Harvey and M.~S.~Turner.} (1990). \textit{Cosmological baryon and lepton number in the presence of electroweak fermion number violation}. Physical Review D, \textit{42}, 3344--3349.
\par\noindent\textbf{Relevance.} The sphaleron transfer coefficient $B = (28/79)(B-L)$ of \eqref{eq:lg-2879}. This factor is inherited unchanged by the present framework and is flagged as textbook rather than derived. First cited in \S\ref{sec:lg-sphaleron}.

\bibitem[Giunti \& Kim(2007)]{GiuntiKim2007}
\textbf{Giunti, C. and C.~W.~Kim.} (2007). \textit{Fundamentals of Neutrino Physics and Astrophysics}. Oxford University Press.
\par\noindent\textbf{Relevance.} The standard reference for the wave-packet treatment of oscillations --- why the plane-wave derivation of \eqref{eq:posc-app} returns the right answer, under what coherence conditions, and what happens when they fail --- and for the Majorana charge-conjugation bookkeeping. The caveat stated in \S\ref{app:pmns-osc} is theirs. First cited in \S\ref{app:pmns}.

\bibitem[Glashow(1961)]{Glashow1961}
\textbf{Glashow, S.~L.} (1961). \url{https://doi.org/10.1016/0029-5582(61)90469-2}
\par\noindent\textbf{Relevance.} Glashow's gauge unification of electromagnetic and weak interactions initiated the electroweak structure completed by Weinberg and Salam. We cite these as the historical backbone for POS-1 (Gauge Invariance) and for the claim that symmetry principles organize viable theories. First cited in \S\ref{sec:sm-review}.

\bibitem[Glashow et al.(1970)]{GlashowIliopoulosMaiani1970}
\textbf{Glashow, S.~L., Iliopoulos, J., \& Maiani, L.} (1970). \textit{Weak interactions with lepton-hadron symmetry}. Physical Review D, \textit{2}, 1285--1292. \url{https://doi.org/10.1103/PhysRevD.2.1285}
\par\noindent\textbf{Relevance.} The GIM mechanism, which suppressed flavor-changing neutral currents by requiring a fourth quark and so predicted charm before its 1974 discovery. Cited in the introduction as an instance of internal consistency forcing a prediction. First cited in \S\ref{sec:intro}.

\bibitem[M.~Green \& J.~Schwarz(1984)]{GreenSchwarz1984}
\textbf{M.~Green and J.~Schwarz.} (1984). \textit{{Anomaly cancellations in supersymmetric D=10 gauge theory and
superstring theory}}. Phys.\ Lett.\ B, \textit{149}, 117.
\par\noindent\textbf{Relevance.} The 1984 Green--Schwarz paper proved that 10-dimensional anomalies in superstring theory cancel for the gauge groups SO(32) and $E_8 \times E_8$, launching the first superstring revolution. The mathematical mechanism --- a higher-form $B_2$ tensor field with modified Bianchi identity $dH_3 = X_4$ that absorbs the chiral anomaly via inflow --- is exactly what GG-Theory adapts to six dimensions as BI-6 (Theorem 4 of \ATwo{}). We cite it in Part~\ref{part:II} of the present paper as the historical origin of the Green--Schwarz mechanism, with explicit acknowledgment that the technical machinery is shared while the constitutional principles differ (10D vs.\ 6D, SUSY vs.\ non-SUSY, landscape vs.\ unique vacuum). First cited in \S\ref{eq:bi6-parent-recast}.

\bibitem[Gross(1973)]{GrossWilczek1973}
\textbf{Gross, D.~J., \& Wilczek, F.} (1973). \url{https://doi.org/10.1103/PhysRevLett.30.1343}
\par\noindent\textbf{Relevance.} Asymptotic freedom is a key structural fact about non-Abelian gauge theories and underlies the consistency of QCD. We cite it to emphasize that ``viability filters'' (renormalizability, RG behavior) are part of what makes gauge theories physically admissible. First cited in \S\ref{eq:g2-general}.

\bibitem[Griffiths(2008)]{Griffiths2008}
\textbf{Griffiths, D.~J.} (2008). \textit{Introduction to Elementary Particles} (2nd ed.). Wiley-VCH.
\par\noindent\textbf{Relevance.} An accessible upper-undergraduate/early-graduate introduction to particle physics. Recommended in the technical-appendix graduate course for a gentle first pass, in particular on the fermion families and the Yukawa spectrum. First cited in \S\ref{app:fermion-step1}.

\bibitem[Gross \& Wilczek(1973)]{Gross1973}
\textbf{Gross, D. J. and Wilczek, F.} (1973). \textit{Ultraviolet behavior of non-Abelian gauge theories}. Physical Review Letters, \textit{30}, 1343--1346. \url{https://doi.org/10.1103/PhysRevLett.30.1343}
\par\noindent\textbf{Relevance.} Companion to GrossWilczek1973: Gross and Wilczek's 1973 letter is the inaugural statement of asymptotic freedom for non-Abelian gauge theories, sharing the 2004 Nobel Prize with Politzer's independent derivation. We cite Gross 1973 in the historical opening of the strong angle--scale theorem as the founding moment of the perturbative QCD high-energy regime that GG-Theory now derives from the infrared region. First cited in \S\ref{eq:g2-general}.

\bibitem[Halzen \& Martin(1984)]{HalzenMartin1984}
\textbf{Halzen, F. and A.~D.~Martin.} (1984). \textit{Quarks and Leptons: An Introductory Course in Modern Particle Physics}. John Wiley \& Sons.
\par\noindent\textbf{Relevance.} A classic phenomenology textbook. Recommended in the technical-appendix graduate course as the companion for the cross-section, decay, and force-law estimates used in the worked examples, in particular on the classical force laws and proton phenomenology. First cited in \S\ref{app:forcelaws-step1}.

\bibitem[P.~W.~Higgs(1964)]{Higgs1964}
\textbf{P.~W.~Higgs.} (1964). \textit{{Broken symmetries and the masses of gauge bosons}}. Phys.\ Rev.\ Lett., \textit{13}, 508.
\par\noindent\textbf{Relevance.} Foundational paper on gauge-boson mass generation. GG-6 reproduces the mass relations from holonomy. First cited in \S\ref{sec:vev}.

\bibitem[Hirzebruch(1995)]{Hirzebruch1995}
\textbf{Hirzebruch, F.} (1995). \textit{Topological Methods in Algebraic Geometry}. Berlin: Springer. Classics in Mathematics. Reprint of the 3rd ed.
\par\noindent\textbf{Relevance.} Hirzebruch's classic monograph develops the theory of characteristic classes, Pontryagin classes, and the L-genus that bears his name. We cite Hirzebruch for the signature theorem $\sigma(M) = \int_M L_k(p_1, \dots, p_k)$ used in Step~2 of the parity route (T-N3-8 of \PaperPTwo{}) reducing strict GDOS Morse-positivity to a parity statement on the complex dimension of the family-projector moduli. First cited in \S\ref{subsec:HS-descent}.

\bibitem[Hopkins \& Singer(2005)]{HopkinsSinger2005}
\textbf{Hopkins, M.~J., \& Singer, I.~M.}
Quadratic functions in geometry, topology, and M-theory.
\emph{Journal of Differential Geometry} \textbf{70} (2005) 329--452.
arXiv:math/0211216.
\par\noindent\textbf{Relevance.}
The differential-cohomology framework of the cohort: the quadratic refinement
technology behind self-dual field quantization and the global consistency of
the \GG{} tensor sector, and the setting of the integrality theorem L-INT that
\citet{ArisakaP2N3} names as the single open statement of its $N=3$ chain. First cited in \S\ref{subsec:HS-descent}.

\bibitem[Elitzur(1975)]{Elitzur1975}
\textbf{S.~Elitzur.} (1975). \textit{Impossibility of spontaneously breaking local symmetries}. Phys.\ Rev.\ D, \textit{12}, 3978.
\par\noindent\textbf{Relevance.} The theorem that a local gauge symmetry cannot break spontaneously --- conceded in a footnote by most treatments of the Standard Model and then set aside. On the holonomy reading it becomes a result rather than an embarrassment: what acquires a value is the gauge-invariant Wilson-line modulus, and nothing breaks. First cited in \S\ref{ssec:higgs-cascade}.

\bibitem[Hatanaka, Inami \& Lim(1998)]{HatanakaInamiLim1998}
\textbf{H.~Hatanaka, T.~Inami \& C.~S.~Lim.} (1998). \textit{The gauge hierarchy problem and higher-dimensional gauge theories}. Mod.\ Phys.\ Lett.\ A, \textit{13}, 2601.
\par\noindent\textbf{Relevance.} Establishes the property on which this paper's hierarchy argument rests: the effective potential of a Wilson-line phase is finite because no local counterterm for it exists. The tier-2 protection of \S\ref{sec:deep-hierarchy} is this result, read on the cohort's anatomy; it is inherited, not claimed as new. First cited in \S\ref{ssec:compare-ghu}.

\bibitem[Manton(1979)]{Manton1979}
\textbf{N.~S.~Manton.} (1979). \textit{A new six-dimensional approach to the Weinberg--Salam model}. Nucl.\ Phys.\ B, \textit{158}, 141.
\par\noindent\textbf{Relevance.} The first construction obtaining a Higgs-like scalar from the extra-dimensional components of a gauge field, and in six dimensions --- the direct ancestor of the reading in \S\ref{ssec:higgs-is-the-circle} that the Higgs is the sixth dimension's connection component. First cited in \S\ref{ssec:compare-ghu}.

\bibitem[Y.~Hosotani(1983)]{Hosotani1983}
\textbf{Y.~Hosotani.} (1983). \textit{{Dynamical mass generation by compact extra dimensions}}. Phys.\ Lett.\ B, \textit{126}, 309.
\par\noindent\textbf{Relevance.} The Hosotani mechanism is the field-theory foundation of the GG-6 holonomy Higgs. First cited in \S\ref{sec:vev}.

\bibitem[Y.~Hosotani(1989)]{Hosotani1989}
\textbf{Y.~Hosotani.} (1989). \textit{Dynamics of non-integrable phases and gauge symmetry breaking}. Ann.\ Phys., \textit{190}, 233.
\par\noindent\textbf{Relevance.} The developed dynamics of the non-integrable phase --- the systematic treatment of how the holonomy's effective potential selects a vacuum, which is the mechanism Theorem~\ref{thm:vev} closes within the ledger. First cited in \S\ref{ssec:compare-ghu}.

\bibitem[Hyper-Kamiokande Collaboration(2018)]{HyperKamiokande2018}
\textbf{Hyper-Kamiokande Collaboration (K.~Abe et al.).} (2018). \textit{Hyper-Kamiokande Design Report}. arXiv:1805.04163.
\par\noindent\textbf{Relevance.} Next-generation proton-decay reach $\sim10^{35}$~yr ($p\to e^+\pi^0$); data-taking from 2027. First cited in \S\ref{subsec:fsm4}.

\bibitem[Jost(1957)]{Jost1957}
\textbf{R.~Jost.} (1957). \textit{Eine Bemerkung zum {CPT}-Theorem}. Helvetica Physica Acta, \textit{30}, 409--416.
\par\noindent\textbf{Relevance.} The CPT theorem in its axiomatic form, reaching interacting Wightman field theories through analytic continuation to the complexified Lorentz group. It is cited here for the relation the cohort actually claims --- \emph{complementarity}: Jost's scope exceeds the geometric account's, which is quasi-free with the interacting case conditional, while the geometric account supplies the anatomy Jost's proof does not (what $C$, $P$ and $T$ individually are, and why antiunitarity traces to the time factor). See \citealp{ArisakaP5CPT} for the statement of that division. First cited in \S\ref{sec:force-uniqueness}.

\bibitem[JUNO Collaboration(2026)]{JUNO2026}
\textbf{JUNO Collaboration (A.~Abusleme et al.).} (2026). \textit{Measurement of reactor neutrino oscillation with the first JUNO data}. Nature, \textit{654}, 343--348.
\par\noindent\textbf{Relevance.} The world's most precise determination of the solar pair, from 59.1 days of running: $\sin^2\theta_{12}=0.3092\pm0.0087$ and $\Delta m_{21}^2=(7.50\pm0.12)\times10^{-5}$~eV$^2$. The derived closed form $L^2/3N^3=25/81$ sits $0.07\sigma$ from it. First cited in \S\ref{sec:pmns-benchmark}.

\bibitem[KamLAND-Zen Collaboration(2025)]{KamLANDZen2024}
\textbf{KamLAND-Zen Collaboration (S.~Abe et al.).} (2025). \textit{Search for Majorana neutrinos with the complete KamLAND-Zen dataset}. Physical Review Letters, \textit{135}, 262501. arXiv:2406.11438.
\par\noindent\textbf{Relevance.} $0\nu\beta\beta$ bound $T^{0\nu}_{1/2}>3.8\times10^{26}$~yr ($m_{\beta\beta}\lesssim28$--$122$~meV), probing the predicted Majorana structure. First cited in \S\ref{subsec:fsm3}.

\bibitem[KATRIN Collaboration(2025)]{KATRIN2025}
\textbf{KATRIN Collaboration (M.~Aker et al.).} (2025). \textit{Direct neutrino-mass measurement based on 259 days of KATRIN data}. Science, \textit{388}, 180.
\par\noindent\textbf{Relevance.} Direct kinematic bound $m_\nu<0.45$~eV, consistent with the predicted light spectrum. First cited in \S\ref{subsec:fsm3}.

\bibitem[Keung \& Senjanovi\'c(1983)]{KeungSenjanovic1983}
\textbf{W.-Y.~Keung and G.~Senjanovi\'c.} (1983). \textit{Majorana neutrinos and the production of the right-handed charged gauge boson}. Physical Review Letters, \textit{50}, 1427--1430.
\par\noindent\textbf{Relevance.} The canonical collider signature of a TeV-scale Majorana neutrino: gauge-portal production followed by same-sign dileptons, exhibiting $\Delta L = 2$ in a detector. Cited in \S\ref{sec:lg-accelerator} for the channel this framework's leptophobic cluster \emph{closes} --- and that the open mirror-portal question of \S\ref{sec:lg-portal-open} could reopen at a future machine. First cited in \S\ref{sec:lg-accelerator}.

\bibitem[Klinkhamer \& Manton(1984)]{KlinkhamerManton1984}
\textbf{F.~R.~Klinkhamer and N.~S.~Manton.} (1984). \textit{A saddle-point solution in the Weinberg--Salam theory}. Physical Review D, \textit{30}, 2212--2220.
\par\noindent\textbf{Relevance.} The sphaleron itself: the static saddle-point configuration at the top of the barrier between topologically distinct vacua, of energy $\simeq9$~TeV, whose thermal activation makes $B+L$ violation unsuppressed above the electroweak crossover. First cited in \S\ref{app:lepto-sphaleron}.

\bibitem[Kuzmin et al.(1985)]{KuzminRubakovShaposhnikov1985}
\textbf{V.~A.~Kuzmin, V.~A.~Rubakov, and M.~E.~Shaposhnikov.} (1985). \textit{On anomalous electroweak baryon-number non-conservation in the early universe}. Physics Letters B, \textit{155}, 36--42.
\par\noindent\textbf{Relevance.} The electroweak sphaleron: unsuppressed $B+L$ violation above the electroweak scale, conserving $B-L$. It is the converter without which the lepton asymmetry of Part~\ref{part:lepto} would never become a baryon asymmetry, and it is compatible with this framework's exact proton stability because it moves baryon number in units of three. First cited in \S\ref{sec:lg-sphaleron}.

\bibitem[Kobayashi \& Maskawa(1973)]{KobayashiMaskawa1973}
\textbf{Kobayashi, M. and Maskawa, T.} (1973). \textit{{$CP$}-violation in the renormalizable theory of weak interaction}. Progress of Theoretical Physics, \textit{49}, 652--657. \url{https://doi.org/10.1143/PTP.49.652}
\par\noindent\textbf{Relevance.} Kobayashi and Maskawa's 1973 extension of Cabibbo to three quark generations (predicting a third family before its discovery) and the resulting one-phase CP violation define the modern $3\times 3$ CKM matrix. Cited in the historical opening of the CKM section as the inventors of the four-parameter structure that the present paper derives as projector misalignment. First cited in \S\ref{subsec:gg6-N3}.

\bibitem[Kolb \& Turner(1990)]{KolbTurner1990}
\textbf{Kolb, E.~W. and M.~S.~Turner.} (1990). \textit{The Early Universe}. Addison-Wesley.
\par\noindent\textbf{Relevance.} The machinery of a species in and out of equilibrium: entropy density, the effective degrees of freedom $\gstar{}$, freeze-out by rate-against-Hubble comparison, and the Boltzmann equation in FRW coordinates. Every one of the six assembly factors of \eqref{eq:etaB-pred-app} is a Kolb--Turner quantity. First cited in \S\ref{app:textbook-review}.

\bibitem[Langacker(2017)]{Langacker2017}
\textbf{Langacker, P.} (2017). \textit{The Standard Model and Beyond} (2nd ed.). CRC Press.
\par\noindent\textbf{Relevance.} A graduate text presenting the Standard Model as an effective theory together with its extensions. Recommended in the technical-appendix graduate course for in-depth study of the neutrino sector and lepton mixing. First cited in \S\ref{app:pmns-step1}.

\bibitem[LEGEND Collaboration(2026)]{LEGEND2025}
\textbf{LEGEND Collaboration.} (2026). \textit{Search for neutrinoless double-beta decay with LEGEND-200}. Physical Review Letters, \textit{136}, 022701. arXiv:2505.10440. Program description: arXiv:2501.10046.
\par\noindent\textbf{Relevance.} $^{76}$Ge program. The quoted $T^{0\nu}_{1/2}>1.9\times10^{26}$~yr is the \emph{combined} LEGEND-200, GERDA and \textsc{Majorana Demonstrator} limit and is not LEGEND-200's alone; LEGEND-1000 projects $m_{\beta\beta}$ of $9$--$21$~meV in the 2030s. First cited in \S\ref{subsec:fsm3}.

\bibitem[Mikheyev \& Smirnov(1985)]{MikheyevSmirnov1985}
\textbf{S.~P.~Mikheyev and A.~Yu.~Smirnov.} (1985). \textit{Resonance amplification of oscillations in matter and spectroscopy of solar neutrinos}. Soviet Journal of Nuclear Physics, \textit{42}, 913--917.
\par\noindent\textbf{Relevance.} With \citet{Wolfenstein1978}, the resonant matter effect: the observation that the in-medium mixing angle becomes maximal at a critical electron density \eqref{eq:msw-res-app} regardless of the vacuum angle, which is how the solar solution was identified and how a long-baseline experiment reads the sign of $\Delta m^2_{31}$. First cited in \S\ref{app:pmns-msw}.

\bibitem[Minkowski(1977)]{Minkowski1977}
\textbf{P.~Minkowski.} (1977). \textit{$\mu \to e\gamma$ at a rate of one out of $10^{9}$ muon decays?} Physics Letters B, \textit{67}, 421--428.
\par\noindent\textbf{Relevance.} The first statement of the seesaw mechanism. See the note at \citealp{GellMannRamondSlansky1979}. First cited in \S\ref{sec:lg-door}.

\bibitem[Mohapatra \& Pal(2004)]{MohapatraPal2004}
\textbf{Mohapatra, R.~N. and P.~B.~Pal.} (2004). \textit{Massive Neutrinos in Physics and Astrophysics} (3rd ed.). World Scientific.
\par\noindent\textbf{Relevance.} The model-building menu --- type~I, type~II and type~III seesaws, left--right symmetric completions, radiative mass generation --- set out so that the common structure and the differences are both visible. The three-type classification of \S\ref{app:pmns-seesaw} follows it. First cited in \S\ref{app:pmns-seesaw}.

\bibitem[Mohapatra \& Senjanovi\'c(1980)]{MohapatraSenjanovic1980}
\textbf{R.~N.~Mohapatra and G.~Senjanovi\'c.} (1980). \textit{Neutrino mass and spontaneous parity nonconservation}. Physical Review Letters, \textit{44}, 912--915.
\par\noindent\textbf{Relevance.} The seesaw embedded in a left--right symmetric theory, and the standard reference for the connection between Majorana neutrino mass and parity violation. First cited in \S\ref{sec:lg-door}.

\bibitem[Majorana(1937)]{Majorana1937}
\textbf{Majorana, E.} (1937). \textit{Teoria simmetrica dell'elettrone e del positrone}. Il Nuovo Cimento, \textit{14}, 171--184. \url{https://doi.org/10.1007/BF02961314}
\par\noindent\textbf{Relevance.} Majorana's 1937 paper introduced the self-conjugate fermion that bears his name and the additional CP phases now called Majorana phases. We cite Majorana in the historical opening of the neutrino-mass section as the inventor of the channel through which the constitutional Majorana mass enters the GG-Theory neutrino sector. First cited in \S\ref{eq:R-t-main}.

\bibitem[Maki et al.(1962)]{MakiNakagawaSakata1962}
\textbf{Maki, Z., Nakagawa, M., and Sakata, S.} (1962). \textit{Remarks on the unified model of elementary particles}. Progress of Theoretical Physics, \textit{28}, 870--880. \url{https://doi.org/10.1143/PTP.28.870}
\par\noindent\textbf{Relevance.} Maki, Nakagawa, and Sakata's 1962 paper introduced the unitary mixing matrix that bears their initials, generalizing Pontecorvo's two-flavor oscillation to a multi-flavor structure. We cite MNS in the historical opening of the PMNS section as the original PMNS-matrix authors; the modern Dirac-plus-Majorana phase structure is the projector-misalignment image of their construction. First cited in \S\ref{sec:PMNS}.

\bibitem[MICROSCOPE Collaboration(2022)]{MICROSCOPE2022}
\textbf{MICROSCOPE Collaboration (P.~Touboul et al.).} (2022). \textit{MICROSCOPE mission: final results of the test of the equivalence principle}. Phys.\ Rev.\ Lett., \textit{129}, 121102. arXiv:2209.15487.
\par\noindent\textbf{Relevance.} Final weak-equivalence-principle test $\eta(\mathrm{Ti},\mathrm{Pt})=[-1.5\pm2.3\pm1.5]\times10^{-15}$, bounding a long-range baryonic fifth force. First cited in \S\ref{subsec:fsm5}.

\bibitem[Milnor(1963)]{Milnor1963}
\textbf{Milnor, J.} (1963). \textit{Morse Theory}. Princeton, NJ: Princeton University Press. Annals of Mathematics Studies.
\par\noindent\textbf{Relevance.} Milnor's classic 1963 monograph established Morse theory as a tool for studying the topology of smooth manifolds via critical points of smooth real-valued functions. We cite Milnor for the perfect-Morse-function structure of the Fubini--Study height function on $\mathbb{C}P^{N-1}$, which is the structural backbone of Leg L2 of the three-leg N=3 derivation. First cited in \S\ref{subsec:N3-L2}.

\bibitem[Particle Data Group(2022)]{ParticleDataGroup2022}
\textbf{Workman, R.~L. et al. (Particle Data Group).} (2022). \textit{Review of Particle Physics}. Progress of Theoretical and Experimental Physics, \textit{2022}, 083C01. \url{https://doi.org/10.1093/ptep/ptac097}
\par\noindent\textbf{Relevance.} The standard compilation of measured particle masses, couplings, and mixing parameters. Cited as the source of the empirical benchmark values that the present paper's 28-parameter codebook is audited against. First cited in \S\ref{sec:intro}.

\bibitem[Pauli(1930)]{Pauli1930}
\textbf{Pauli, W.} (1930). \textit{Letter to the {T\"ubingen} conference, December 4, 1930 (``Liebe Radioaktive Damen und Herren'')}. Reprinted in W. Pauli, Wissenschaftlicher Briefwechsel, Band II.
\par\noindent\textbf{Relevance.} Pauli's 1930 letter postulating the neutrino to rescue energy conservation in beta decay is the originating moment of the neutrino concept. We cite Pauli in the historical opening of the neutrino-mass section as the inventor of the particle whose mass-generation mechanism this paper now derives geometrically. First cited in \S\ref{eq:R-t-main}.

\bibitem[Peccei \& Quinn(1977)]{PecceiQuinn1977}
\textbf{Peccei, R. D. and Quinn, H. R.} (1977). \textit{{$CP$} conservation in the presence of pseudoparticles}. Physical Review Letters, \textit{38}, 1440--1443. \url{https://doi.org/10.1103/PhysRevLett.38.1440}
\par\noindent\textbf{Relevance.} Peccei and Quinn's 1977 introduction of the global $U(1)_{PQ}$ symmetry whose spontaneous breaking dynamically suppresses the strong-CP $\bar\theta$ angle is the original framework in which the smallness of $\bar\theta$ becomes a derived rather than fine-tuned statement. We cite Peccei--Quinn in the historical opening of the strong-CP section; GG-Theory replaces the global PQ symmetry with the geometric topological-projection mechanism that suppresses $\bar\theta$ structurally. First cited in \S\ref{sec:strong-CP-derivation}.

\bibitem[Perlmutter et al.(1999)]{Perlmutter1999}
\textbf{Perlmutter, S. et al. (Supernova Cosmology Project).} (1999). \textit{Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae}. The Astrophysical Journal, \textit{517}, 565--586. \url{https://doi.org/10.1086/307221}
\par\noindent\textbf{Relevance.} With Riess et al., the discovery of cosmic acceleration and hence of dark energy. Cited in the introduction in the statement of the dark-energy and cosmological-constant problem. First cited in \S\ref{sec:intro}.

\bibitem[Peskin \& Schroeder(1995)]{PeskinSchroeder1995}
\textbf{Peskin, M.~E. and D.~V.~Schroeder.} (1995). \textit{An Introduction to Quantum Field Theory}. Addison-Wesley (Westview Press).
\par\noindent\textbf{Relevance.} The standard graduate text on quantum field theory and the gauge sector of the Standard Model. Recommended as a primary reference for the technical-appendix graduate course, in particular on gauge theories, renormalization-group running, and the instanton/topological background of the strong-CP sector. First cited in \S\ref{app:gauge-step1}.

\bibitem[Pilaftsis(1997)]{Pilaftsis1997}
\textbf{A.~Pilaftsis.} (1997). \textit{Resonant \CP{} violation induced by particle mixing in transition amplitudes}. Nuclear Physics B, \textit{504}, 61--107. hep-ph/9702393.
\par\noindent\textbf{Relevance.} The origin of the resonant enhancement \eqref{eq:lg-bw} on which the lock-scale mechanism depends: quasi-degenerate heavy states cancel the Yukawa suppression of the \CP{} asymmetry. First cited in \S\ref{sec:lg-di}.

\bibitem[Pilaftsis \& Underwood(2004)]{PilaftsisUnderwood2004}
\textbf{A.~Pilaftsis and T.~E.~J.~Underwood.} (2004). \textit{Resonant leptogenesis}. Nuclear Physics B, \textit{692}, 303--345. hep-ph/0309342. (See also Physical Review D, \textit{72}, 113001 (2005).)
\par\noindent\textbf{Relevance.} The systematic treatment of leptogenesis in the resonant regime, and the reference against which the regulator question of \S\ref{sec:lg-resonance} is posed. In the general treatment the quasi-degeneracy is assumed; here it is the ledger's own level spacing $\Mlock/K$. First cited in \S\ref{sec:lg-di}.

\bibitem[Planck Collaboration(2020)]{Planck2018}
\textbf{Planck Collaboration (N.~Aghanim et al.).} (2020). \textit{Planck 2018 results. VI. Cosmological parameters}. Astron.\ Astrophys., \textit{641}, A6. arXiv:1807.06209.
\par\noindent\textbf{Relevance.} CMB cosmological parameters (the observed $\Omega$'s, $z_*$, and $H_0^{\mathrm{CMB}}\approx67$) compared with the derived inventory of \ATen{}. First cited in \S\ref{subsec:a10-overview}.

\bibitem[Politzer(1973)]{Politzer1973}
\textbf{Politzer, H.~D.} (1973). 
\par\noindent\textbf{Relevance.} Along with Gross--Wilczek, this established asymptotic freedom. It reinforces the centrality of RG structure and motivates scale covariance as a constitutional requirement. First cited in \S\ref{eq:g2-general}.

\bibitem[Pontecorvo(1957)]{Pontecorvo1957}
\textbf{Pontecorvo, B.} (1957). \textit{Mesonium and antimesonium}. Soviet Physics JETP, \textit{6}, 429--431.
\par\noindent\textbf{Relevance.} Pontecorvo's 1957 paper introduced neutrino oscillations as a meson--antimeson analogue. We cite Pontecorvo in the historical opening of the PMNS section as the inventor of lepton-flavor mixing, which Maki--Nakagawa--Sakata 1962 cast in the modern projector form. First cited in \S\ref{eq:R-t-main}.

\bibitem[Riess et al.(1998)]{Riess1998}
\textbf{Riess, A.~G. et al. (High-z Supernova Search Team).} (1998). \textit{Observational evidence from supernovae for an accelerating universe and a cosmological constant}. The Astronomical Journal, \textit{116}, 1009--1038. \url{https://doi.org/10.1086/300499}
\par\noindent\textbf{Relevance.} With Perlmutter et al., the discovery of the accelerating expansion of the universe. Cited in the introduction in the statement of the dark-energy and cosmological-constant problem. First cited in \S\ref{sec:intro}.

\bibitem[Rubin \& Ford(1970)]{Rubin1970}
\textbf{Rubin, V.~C., \& Ford, W.~K.} (1970). \textit{Rotation of the Andromeda nebula from a spectroscopic survey of emission regions}. The Astrophysical Journal, \textit{159}, 379--403. \url{https://doi.org/10.1086/150317}
\par\noindent\textbf{Relevance.} The flat galactic rotation curves that, with Zwicky's cluster dynamics, established the gravitational case for dark matter. Cited in the introduction in the statement of the dark-matter problem. First cited in \S\ref{sec:intro}.

\bibitem[Sakharov(1967)]{Sakharov1967}
\textbf{Sakharov, A.~D.} (1967). \textit{Violation of {CP} invariance, {C} asymmetry, and baryon asymmetry of the universe}. JETP Letters, \textit{5}, 24--27.
\par\noindent\textbf{Relevance.} Sakharov's three conditions for dynamically generating the matter--antimatter asymmetry (baryon-number violation, C and CP violation, departure from equilibrium). Cited in the introduction in the statement of the baryon-asymmetry problem. First cited in \S\ref{sec:intro}.

\bibitem[Salam(1968)]{Salam1968}
\textbf{Salam, A.} (1968). 
\par\noindent\textbf{Relevance.} Salam's electroweak unification is part of the Standard Model foundation. It is cited here to emphasize that modern physics already depends on strong ``admissibility rules'' (gauge invariance and renormalizability) even if not presented as an explicit constitution. First cited in \S\ref{sec:sm-review}.

\bibitem[Schechter \& Valle(1982)]{SchechterValle1982}
\textbf{J.~Schechter and J.~W.~F.~Valle.} (1982). \textit{Neutrinoless double beta decay in $SU(2)\times U(1)$ theories}. Physical Review D, \textit{25}, 2951--2954.
\par\noindent\textbf{Relevance.} The black-box theorem: any mechanism producing $0\nu\beta\beta$ generates a Majorana neutrino mass, so an observation establishes the Majorana nature whatever the underlying operator. Quoted in \S\ref{app:majorana-0nubb} together with its honest magnitude --- the mass so generated is of order $10^{-24}$~eV, so the theorem is qualitative. First cited in \S\ref{app:majorana-0nubb}.

\bibitem[Schwartz(2014)]{Schwartz2014}
\textbf{Schwartz, M.~D.} (2014). \textit{Quantum Field Theory and the Standard Model}. Cambridge University Press.
\par\noindent\textbf{Relevance.} A modern graduate text covering quantum field theory through the full Standard Model. Recommended as a primary reference for the technical-appendix graduate course, in particular on the Higgs mechanism and electroweak symmetry breaking. First cited in \S\ref{app:ewsb-step1}.

\bibitem[Schwinger(1948)]{Schwinger1948}
\textbf{Schwinger, J.} (1948). \textit{On quantum-electrodynamics and the magnetic moment of the electron}. Physical Review, \textit{73}, 416--417. \url{https://doi.org/10.1103/PhysRev.73.416}
\par\noindent\textbf{Relevance.} Schwinger's 1948 calculation $a_e=\alpha/(2\pi)$ for the electron anomalous magnetic moment is the first technical instance of an Abelian IR transport coefficient. We cite Schwinger as the originator of the structural object that the IR Abelian transport theorem in this paper generalizes. First cited in \S\ref{thm:mH}.

\bibitem[Sommerfeld(1916)]{Sommerfeld1916}
\textbf{Sommerfeld, A.} (1916). \textit{Zur {Q}uantentheorie der {S}pektrallinien}. Annalen der Physik, \textit{356}(17), 1--94. \url{https://doi.org/10.1002/andp.19163561702}
\par\noindent\textbf{Relevance.} Sommerfeld's 1916 paper introduced the fine-structure constant $\alpha$ as the dimensionless coupling controlling the relativistic correction to the hydrogen spectrum. We cite Sommerfeld in the historical opening of the electromagnetic dyadic-closure theorem (\T{}) as the originator of $\alpha$ as a derivable structural constant rather than an empirical fit. First cited in \S\ref{sec:dyad-closure-anchor}.

\bibitem['t~Hooft(1976)]{tHooft1976}
\textbf{G.~'t~Hooft.} (1976). \textit{Symmetry breaking through Bell--Jackiw anomalies}. Physical Review Letters, \textit{37}, 8--11.
\par\noindent\textbf{Relevance.} The observation that the anomaly \eqref{eq:anomaly-app} makes baryon and lepton number non-conserved in the Standard Model through instanton transitions, with $B-L$ exactly preserved --- the structural fact on which leptogenesis rests. First cited in \S\ref{app:lepto-sphaleron}.

\bibitem[{'t~Hooft} \& Veltman(1972)]{tHooftVeltman1972}
\textbf{'t~Hooft, G., \& Veltman, M.} (1972). \textit{Regularization and renormalization of gauge fields}. Nuclear Physics B, \textit{44}, 189--213. \url{https://doi.org/10.1016/0550-3213(72)90279-9}
\par\noindent\textbf{Relevance.} The proof that spontaneously broken non-Abelian gauge theories are renormalizable, which turned the electroweak synthesis into a predictive quantum field theory. Cited in the introduction's account of how the Standard Model was assembled. First cited in \S\ref{sec:intro}.

\bibitem[Srednicki(2007)]{Srednicki2007}
\textbf{Srednicki, M.} (2007). \textit{Quantum Field Theory}. Cambridge University Press.
\par\noindent\textbf{Relevance.} Two-component spinor technology developed from the outset, which is the cleanest preparation for the Majorana material of \S\ref{app:majorana-spinors}, where the distinction between $\psi$ and $\psi^{c}$ must be handled without apology. First cited in \S\ref{app:textbook-review}.

\bibitem[Super-Kamiokande Collaboration(2020)]{SuperKProton2020}
\textbf{Super-Kamiokande Collaboration (A.~Takenaka et al.).} (2020). \textit{Search for proton decay via $p\to e^+\pi^0$ and $p\to\mu^+\pi^0$ with an enlarged fiducial volume in Super-Kamiokande I--IV}. Phys.\ Rev.\ D, \textit{102}, 112011. arXiv:2010.16098.
\par\noindent\textbf{Relevance.} Current $90\%$ C.L.\ bound $\tau/B(p\to e^+\pi^0)>2.4\times10^{34}$~yr against which exact stability is tested. First cited in \S\ref{subsec:fsm4}.

\bibitem[Super-Kamiokande Collaboration(2021)]{SuperKNNbar2021}
\textbf{Super-Kamiokande Collaboration (K.~Abe et al.).} (2021). \textit{Neutron--antineutron oscillation search using a 0.37 megaton-years exposure of Super-Kamiokande}. Phys.\ Rev.\ D, \textit{103}, 012008. arXiv:2012.02607.
\par\noindent\textbf{Relevance.} Current $\Delta B=2$ bound $\tau_{n\bar n}>4.7\times10^8$~s, forbidden exactly in \GG{}. First cited in \S\ref{subsec:fsm4}.

\bibitem[T2K and NOvA Collaborations(2025)]{T2KNOvA2025}
\textbf{T2K and NOvA Collaborations (K.~Abe, M.~A.~Acero et al.).} (2025). \textit{Joint neutrino oscillation analysis from the T2K and NOvA experiments}. Nature, \textit{646}, 818--824. arXiv:2510.19888.
\par\noindent\textbf{Relevance.} The direct accelerator determination of the Dirac phase, adopted here as the $\delta_{\rm PMNS}$ benchmark: for normal ordering, best fit $203.4^\circ$ with $1\sigma$ interval $[165.6^\circ,266.4^\circ]$, containing the derived $196.06^\circ$. It also places $\sin^2\theta_{23}$ in the upper octant, against the global fit. It excludes $\delta=+\pi/2$ at $3\sigma$ but not $3\pi/2$. First cited in \S\ref{sec:pmns-benchmark}.

\bibitem[Tye \& Wong(2015)]{TyeWong2015}
\textbf{S.-H.~H.~Tye and S.~S.~C.~Wong.} (2015). \textit{Bloch wave function for the periodic sphaleron potential and unsuppressed baryon and lepton number violating processes}. Physical Review D, \textit{92}, 045005. arXiv:1505.03690.
\par\noindent\textbf{Relevance.} The proposal that the exponential suppression of sphaleron-induced $\Delta B = \Delta L = 3$ processes may lift near the barrier energy $E_{\rm sph} \simeq 9$~TeV, making the transition observable at a $100$~TeV collider. Contested (see \citealp{EllisSakurai2016}); this paper takes no side and cites both. First cited in \S\ref{sec:lg-accelerator}.

\bibitem[Thomson(2013)]{Thomson2013}
\textbf{Thomson, M.} (2013). \textit{Modern Particle Physics}. Cambridge University Press.
\par\noindent\textbf{Relevance.} A modern, experiment-aware graduate textbook. Recommended in the technical-appendix graduate course for a gentle first pass, in particular on quark mixing and the CKM matrix. First cited in \S\ref{app:ckm-step1}.

\bibitem[Weinberg(1967)]{Weinberg1967}
\textbf{Weinberg, S.} (1967). \url{https://doi.org/10.1103/PhysRevLett.19.1264}
\par\noindent\textbf{Relevance.} Weinberg's paper, together with Glashow and Salam, established the electroweak gauge theory. We cite it as part of the historical backbone of gauge invariance and spontaneous symmetry breaking, which GG-Theory aims to generalize to open systems via GDOS. First cited in \S\ref{sec:sm-review}.

\bibitem[Weinberg(1978)]{Weinberg1978}
\textbf{Weinberg, S.} (1978). \textit{A new light boson?}. Physical Review Letters, \textit{40}, 223--226. \url{https://doi.org/10.1103/PhysRevLett.40.223}
\par\noindent\textbf{Relevance.} With Wilczek, the recognition that a spontaneously broken Peccei--Quinn symmetry implies a light pseudoscalar, the axion. Cited in the introduction in the statement of the strong-CP problem. First cited in \S\ref{sec:intro}.

\bibitem[Weinberg(1979)]{Weinberg1979}
\textbf{S.~Weinberg.} (1979). \textit{Baryon- and lepton-nonconserving processes}. Physical Review Letters, \textit{43}, 1566--1570.
\par\noindent\textbf{Relevance.} The dimension-five operator $(LH)(LH)/\Lambda$: the unique lowest-dimension addition to the Standard Model that gives neutrinos mass, and the operator the constitutional door of \S\ref{sec:lg-door} arrives at as the only legal $\Delta L = 2$ structure. First cited in \S\ref{sec:lg-door}.

\bibitem[Weinberg(1989)]{Weinberg1989}
\textbf{Weinberg, S.} (1989). \textit{The cosmological constant problem}. Reviews of Modern Physics, \textit{61}, 1--23. \url{https://doi.org/10.1103/RevModPhys.61.1}
\par\noindent\textbf{Relevance.} The canonical statement of the cosmological-constant problem and the $\sim 10^{120}$ discrepancy between the natural and observed vacuum-energy scales. Cited in the introduction in the statement of the dark-energy problem. First cited in \S\ref{sec:intro}.

\bibitem[Weinberg(2008)]{Weinberg2008}
\textbf{Weinberg, S.} (2008). \textit{Cosmology}. Oxford University Press.
\par\noindent\textbf{Relevance.} The derivation of the cosmological observables --- acoustic peaks, baryon density, light-element abundances --- from first principles, so that a student knows precisely what $\etaB{}$ \emph{is} as a measured quantity before being asked to predict it. The two independent determinations of \S\ref{app:lepto-number} are set out there. First cited in \S\ref{app:textbook-review}.

\bibitem[Weinberg(1995--1996)]{WeinbergQFT1996}
\textbf{Weinberg, S.} (1995--1996). \textit{The Quantum Theory of Fields}, Vols.~I--II. Cambridge University Press.
\par\noindent\textbf{Relevance.} The authoritative advanced treatment of quantum field theory and the electroweak Standard Model. Recommended in the technical-appendix graduate course for in-depth study of the electroweak sector and its effective-operator structure. First cited in \S\ref{app:ewsb-step1}.

\bibitem[Wilczek(1978)]{Wilczek1978}
\textbf{Wilczek, F.} (1978). \textit{Problem of strong {P} and {T} invariance in the presence of instantons}. Physical Review Letters, \textit{40}, 279--282. \url{https://doi.org/10.1103/PhysRevLett.40.279}
\par\noindent\textbf{Relevance.} With Weinberg, the identification of the axion as the pseudo-Goldstone boson of a broken Peccei--Quinn symmetry. Cited in the introduction in the statement of the strong-CP problem. First cited in \S\ref{sec:intro}.

\bibitem[K.~G.~Wilson(1974)]{Wilson1974}
\textbf{K.~G.~Wilson.} (1974). \textit{{Confinement of quarks}}. Phys.\ Rev.\ D, \textit{10}, 2445.
\par\noindent\textbf{Relevance.} Wilson loops as gauge-invariant observables. The holonomy language central to GG-6. First cited in \S\ref{sec:vev}.

\bibitem[Witten(1982)]{Witten1982}
\textbf{Witten, E.} (1982). \textit{Supersymmetry and {M}orse theory}. Journal of Differential Geometry, \textit{17}(4), 661--692. \url{https://doi.org/10.4310/jdg/1214437492}
\par\noindent\textbf{Relevance.} Witten's 1982 paper introduced supersymmetric quantum mechanics on a Riemannian manifold and showed that the Witten index of the deformed exterior derivative computes the Euler characteristic via Morse-theoretic critical-point counting. We cite Witten for the Bogomolnyi-bound saturation of the gradient flow on $\mathbb{C}P^{N-1}$, which gives the explicit instanton action $S_{[2k]\to[2(k-1)]} = \pi k$ used in the cascade-closure constraint of Leg L2. First cited in \S\ref{subsec:N3-L2}.

\bibitem[Wolfenstein(1978)]{Wolfenstein1978}
\textbf{L.~Wolfenstein.} (1978). \textit{Neutrino oscillations in matter}. Physical Review D, \textit{17}, 2369--2374.
\par\noindent\textbf{Relevance.} The charged-current matter potential \eqref{eq:vcc-app}, $V_{\rm CC}=\sqrt2 G_F n_e$, arising because ordinary matter contains electrons and not muons. It is the origin of both the solar solution and the long-baseline handle on the mass ordering, and of the matter-induced fake \CP{} asymmetry that is the field's hardest systematic. First cited in \S\ref{app:pmns-msw}.

\bibitem[Yanagida(1979)]{Yanagida1979}
\textbf{T.~Yanagida.} (1979). \textit{Horizontal gauge symmetry and masses of neutrinos}. In \textit{Proceedings of the Workshop on Unified Theory and Baryon Number in the Universe} (KEK, Tsukuba), eds.\ O.~Sawada and A.~Sugamoto, Conf.\ Proc.\ C7902131, 95--99.
\par\noindent\textbf{Relevance.} The third independent statement of the seesaw mechanism. The 1979 KEK proceedings is the reference intended here; the companion paper Progress of Theoretical Physics \textit{64}, 1103 (1980) is a distinct work and is not the one cited. First cited in \S\ref{sec:lg-door}.

\bibitem[Yukawa(1935)]{Yukawa1935}
\textbf{Yukawa, H.} (1935). \textit{On the interaction of elementary particles. {I}}. Proceedings of the Physico-Mathematical Society of Japan, \textit{17}, 48--57.
\par\noindent\textbf{Relevance.} Yukawa's 1935 paper postulated the exchange-particle origin of the nucleon-nucleon force and introduced the dimensionless coupling that bears his name. We cite Yukawa in the historical opening of the charged-lepton mass section as the originator of the dimensionless fermion-mass coupling later identified with $m_f/v$. First cited in \S\ref{sec:lepton-mass}.

\bibitem[Zuber(2020)]{Zuber2020}
\textbf{Zuber, K.} (2020). \textit{Neutrino Physics} (3rd ed.). CRC Press.
\par\noindent\textbf{Relevance.} The experiments in detail and up to date --- what each detector measures, its systematics, and why the error bars are the size they are. Recommended in \S\ref{app:textbook-review} as the book to read before writing a falsification criterion against a neutrino observable. First cited in \S\ref{app:pmns}.

\bibitem[Zwicky(1933)]{Zwicky1933}
\textbf{Zwicky, F.} (1933). \textit{Die {R}otverschiebung von extragalaktischen {N}ebeln}. Helvetica Physica Acta, \textit{6}, 110--127.
\par\noindent\textbf{Relevance.} Zwicky's measurement of the velocity dispersion of the Coma cluster, the first dynamical evidence for unseen (dark) mass. Cited in the introduction in the statement of the dark-matter problem. First cited in \S\ref{sec:intro}.
\end{thebibliography}

\GGBibEnd

\end{document}
