====================================================================================================
  SuperGrokTOE PUBLIC VERIFICATION KERNEL v5.0.12 — Rev12+S64 REGRESSION FIXTURE (PDG 2026 / NuFIT 6.1 comparators, S368)
  MIXING (Rev12 marking): 6/8 THEOREM + θ₁₃/δ_CKM STRUCTURAL | 9/9 CKM MAGNITUDES
  m_μ/m_e/top/b,c (Rev12 marking) — see the EPISTEMIC STATUS block for the Rev32.4 state
  ════════════════════════════════════════════════════════════════════════════════════
  ⚠ EPISTEMIC STATUS (v5.0.9, Rev32.4/S300): the NUMERICS below are the frozen Rev12+S64
    REGRESSION TRANSCRIPT — byte-stable on purpose. Historical tier language inside the
    parts (THEOREM/PROVED/ZERO FITTED) is the Rev12-era marking and is SUPERSEDED by the
    Rev29 re-tier. CURRENT claim state (Rev32.4 suite, reading rule: tiers name mathematical
    provenance; NO row is zero-parameter; cross-scheme σ are comparator distances, not status):
    3/8 mixing observables Derived-conditional or above (ordering premise ladder- AND
    interface-conditional); CKM first row LOADED (|V_us| 0.225268 vs PDG 2026 0.22431±0.00085,
    +1.13σ); |V_cb| STRUCTURAL; δ_CKM Coincidence-class with the bare-φ control 65.59° (−0.37σ)
    printed beside G_7; m_μ/m_τ Structural/Loaded-correspondence/Reproduced; the weak angle:
    AX6_pol (Λ⋆=6φ−1, readout φ⁻³ exactly) is the postulate — the cos(6ψ)=½ object below is the
    retired Det² selector; comparator sin²θ̂_W(M_Z) = 0.23122±0.00006 (PDG 2026 MS-bar, +2.10%,
    no σ); crossing μ* ∼ 0.15 GeV, band [0.09, 0.55] GeV — the 1.8 GeV below is historical.
    Neutrinos: Dirac LOADED; Δm² ratio d=1 reading REFUTED 9.21×, d=2 rung R₂ a below-floor
    EDGE target (not a hit). Isospin sector LOADED; m_n−m_p UNPRICED. Engine workbooks and this
    kernel are a regression fixture, not a claim surface: the suite (Rev32.4) is.
  ════════════════════════════════════════════════════════════════════════════════════
====================================================================================================

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PART A: FRAMEWORK ARCHITECTURE
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  D=5 Chern-Simons (E₆₍₆₎/F₄₍₄₎ target)
       ↓ Kaluza-Klein truncation (KK-70 modes)
  D=4 E₇₍₇₎/SU(8) coset sigma-model
       ↓ Peirce decomposition of J₃(𝕆ₛ) at ghost-free vacuum
  SM gauge group × 20 parameter predictions

  Structure algebra chain:
    Str₀(J₃) ≅ e₆₍₆₎  (structure algebra, 78-dim)
    Der(J₃)  ≅ f₄₍₄₎  (derivation algebra, 52-dim)
    g₂₍₂₎             (octonion automorphisms, 14-dim)
    e₆₍₆₎  ⊃  f₄₍₄₎  ⊃  g₂₍₂₎

  SM gauge group: SU(3)_c × SU(2)_L × U(1)_Y
    Identified by S₃ → Z₂ Peirce slot reduction [p1: Sec 3] (assignment, not emergence)
    N_c = 3 from Peirce rank; N_gen = 3 is an INPUT (one J₃(𝕆ₛ) = one generation; T-3GEN)

  Peirce decomposition: J₃ = P₁₁⊕P₂₂⊕P₃₃⊕P₁₂⊕P₁₃⊕P₂₃
    dim(P_{ii}) = 1 (diagonal blocks, 3 total)
    dim(P_{ij}) = 8 (off-diagonal blocks, 3 pairs, split-octonion fibre)
    Total: 3×1 + 3×8 = 27 = dim(J₃(𝕆ₛ)) ✓

  Status:  D=5 CS architecture fully specified
           Rev12 suite: Papers/current (15 tex files)
           Ghost-freedom: η⁻¹M ≥ 0 on all Peirce modes [verified]

FUNDAMENTAL CONSTANTS (2022 CODATA, published NIST SP 959, May 2024):
  α⁻¹  = 137.035999177  (2022 CODATA)
  α    = 7.297352564331424e-03
  M_Pl = 1.220890e+19 GeV
  v_EW = 246.2197 GeV (measured anchor, not derived)
  π    = 3.141592653589793
  φ    = 1.618033988749895
  m_τ  = 1776.93 MeV (PDG anchor — sole external lepton input)
  Λ_G₂ = 260.0 MeV (G₂ confinement scale; dimensional transmutation)

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PART B: ALGEBRAIC CORE — AX1-AX3 VERIFICATION + S44 RESULTS
════════════════════════════════════════════════════════════════════════════════════════════════════
Source: p0_framework_foundations.tex (Sec 2), p1_peirce_gauge_group.tex

AX1: J₃(𝕆ₛ) vacuum with golden-ratio eigenvalues  [p0: Eq(AX1)]
  J_vac = diag(φ, 1, φ⁻¹)
  φ² + φ⁻² = 3.000000000000000  ✓ EXACT = 3
  Peirce eigenvalues: λ₁=1.618033988750, λ₂=1, λ₃=0.618033988750

AX3: Det(J_vac) = φ·1·φ⁻¹ = 1.000000000000000  ✓ EXACT = 1  [THEOREM of AX1, D304]
     Minkowski balance V(J_vac)=0 (Cartan orthogonality ⊂ AX1)

AX2: Vacuum selector  [p0: Sec 2.1]
  Status: STRUCTURAL — φ=(1+√5)/2; DET-7 pins n₂₃=7; deeper algebraic origin open
  NOTE: AX4 (β_δ=11/(6π)) ELIMINATED S16-18. δ_CP derived from V₃ rotation ★100.

DET-7 THEOREM ★100  [D575/A555, Session 14]:
  J_vac   = diag(φ, 1, φ⁻¹);   J_vac# = diag(φ⁻¹, 1, φ)  (Freudenthal dual)
  ⟨J_vac, J_vac⟩  = φ²+1+φ⁻² = 3+1 = 4  (AX1)
  ⟨J_vac, J_vac#⟩ = φ·φ⁻¹+1·1+φ⁻¹·φ = 3  (UNIVERSAL for any abc=1)
  ⟨J_vac#,J_vac#⟩ = 4  (symmetric)
  det(Gram) = 4²−3² = 7 = n₂₃  ✓ EXACT
  ★ AX2 ELIMINATED from θ₂₃ derivation — det=7 is a CONSEQUENCE of AX1 alone.

  ✓ Numerical verification: det(Gram) = 7.000000000000  (target: 7 EXACT)

C1 THEOREM ★★★ (A558, Session 16):
  T_half = T_scalar + ½·T_oct   ← c=½ EXACT from 3-step algebraic proof:
    Step 1: T²@J_vac identity: T_scalar²=T_oct²@J_vac = ½(e₂−e₀)
    Step 2: Frobenius ratio:   ‖T_oct‖_F/‖T_scalar‖_F = √7 EXACT
    Step 3: DET-7 ratio:       c = ½ from (c√7)²=7/4 → 7c²=7/4 → c=½ ∎
  G₂-unique: T_half is the unique G₂-invariant generator in (1,3) Peirce sector of f₄

THREE-SECTOR DUAL CONSTRUCTION (Sessions 15-16):
  J_{13}# = diag(φ⁻¹,  1,   φ  )  c₁₃=½=0.5000             [C1 THEOREM]
  J_{12}# = diag( 1,   φ,  φ⁻¹)  c₁₂=√((17−5√5)/56)=0.322370  [D580/A562]
  J_{23}# = diag( φ,  φ⁻¹,  1  )  c₂₃=√((9−3√5)/56)=0.202299   [D580/A562]

  Canonical angles: t*^(13)=160.9°, t*^(12)=171.318°, t*^(23)=176.426°
  All Frobenius norms ‖T_half^(ij)‖=√7; antisymmetry G·T+Tᵀ·G=0 EXACT (all 3)
  E49: R₁₂+R₂₃=1 EXACT  (R₁₂=φ⁻⁴=0.1458980338, R₂₃=0.8541019662)
  ✓ E49 verified: R₁₂+R₂₃ = 1.000000000000000

S44 NEW RESULTS (A641):

  7/3 OCTONION ROUTE — VACUUM-INDEPENDENT (A641/Q2A) ★★★★★:
    7/3 = dim(Im(𝕆ₛ)) / rank(J₃) = 7 / 3
    The 7 imaginary units of 𝕆ₛ (Fano-plane generators of off-diagonal P_ij interactions)
    divided by Peirce rank 3. No vacuum element required.
    DET-7 confirms n₂₃=7 specifically for J_vac — both routes consistent.
    Algebraic meaning: 7 imaginary generators / 3 independent Peirce slots = 7/3

  GOLDEN VACUUM UNIQUENESS (A641/Q5A):
    For diag(a,b,c) with abc=1:
      det(Gram) = (a²+b²+c²)(a⁻²+b⁻²+c⁻²) − 9
      ⟨J,J#⟩ = 3  UNIVERSAL  (holds for ANY norm-1 diagonal — A641/Q2B)
    det(Gram) = 7  forces  (a²+b²+c²)(a⁻²+b⁻²+c⁻²) = 16
    The UNIQUE positive-real solution (up to ordering): {φ, 1, φ⁻¹}
    No other norm-1 diagonal yields an exact integer Gram determinant (A640/Q1C)

  T₁=T₂ SELF-ADJUGATE SYMMETRY (A641/Q4A):
    T₁(J_vac) = Tr(J_vac)  = φ+1+φ⁻¹ = 1+√5
    T₂(J_vac) = Tr(J_vac#) = φ⁻¹+1+φ = 1+√5   (same!)
    T₃(J_vac) = N(J_vac)   = 1 (AX3)
    → Char. poly: t³−(1+√5)t²+(1+√5)t−1 = 0  [palindromic]
    → J_vac is self-adjugate: reversal J_vac# = diag(φ⁻¹,1,φ) = reverse of J_vac

  ✓ T₁(J_vac) = T₂(J_vac) = 3.236067977500 = 1+√5 = 3.236067977500  EXACT
  ✓ det(Gram) for {φ,1,φ⁻¹} = 7.0000000000  [EXACT = 7]
  ✓ det(Gram) for {2.0,1.0,0.5} = 18.5625  [≠ 7, confirms uniqueness]
  ✓ det(Gram) for {3.0,1.0,0.3333333333333333} = 93.2346  [≠ 7, confirms uniqueness]

⚠ AX6: sin²θ_W IRREDUCIBLY AXIOMATIC — 3 no-go proofs (A617,A620,A621). Final.

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PART C: CUBIC-INVARIANT YUKAWA + y_t GEOMETRIC PROOF  [p0: Sec 3, D525, A632]
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LAGRANGIAN:
  ℒ_Y = Tr[J_vac ∘ (Ψ × Φ_H)]  [p0 Eq(1)]
  where: J_vac = diag(φ,1,φ⁻¹); × = Freudenthal cross product; ∘ = Jordan product

FREUDENTHAL CROSS PRODUCT  [App A.3]:
  x×y = x∘y − ½Tr(x)y − ½Tr(y)x + ½[Tr(x)Tr(y)−Tr(x∘y)]I
  With Tr(Ψ)=Tr(Φ_H)=0:
    Ψ×Φ_H = (Ψ∘Φ_H) − ½Tr(Ψ∘Φ_H)·I = −2Re(ψh̄)·e₂  (middle Peirce slot only)

FIVE-STEP DERIVATION  [App A.3]:
  Step 1: Ψ, Φ_H placed in (1,3) Peirce off-diagonal block (SU(2)_L doublet geometry)
  Step 2: Jordan product → Ψ∘Φ_H = diag(2Re(ψh̄), 0, 2Re(ψ̄h)); Tr=4Re(ψh̄)
  Step 3: Freudenthal → Ψ×Φ_H = −2Re(ψh̄)·e₂  (projects to middle slot)
  Step 4: Tr[J_vac ∘ (−2Re(ψh̄)e₂)] = λ₂·(−2Re(ψh̄)) = 1·(−2Re(ψh̄))
  Step 5: λ₂=1 → y_t(tree)=1 EXACTLY   [p0: Eq(4)]

S38+ GEOMETRIC PROOF (A632) — STRONGER STATEMENT:
  f_i ∘ x = (1/2)x  for any x ∈ P_{ij}  (i≠j)  [Peirce multiplication rule]
  This is GEOMETRICALLY FORCED — not just a contraction result.
  The √2 factor in m_t=v_EW/√2 is structurally identical to the J₂(𝕆ₛ)
  off-diagonal norm factor [A618] — same algebra, same geometric origin.

UNIQUENESS [App A.3]: y_t=1 requires simultaneously:
  (a) Higgs in off-diagonal Peirce block (off-diagonal SU(2)_L geometry)
  (b) Local coupling cubic and Jordan-algebraic
  (c) Contraction isolates middle Peirce weight λ₂=1
  AX1 ↔ D525: φ²+φ⁻²=3 pins λ₂=1 → y_t=1 is CONSEQUENCE of AX1.

TREE-LEVEL ZEROS (A633/Q2 — Peirce orthogonality):
  f_k ∘ x = 0 for x ∈ P_{ij} with k≠i,j → y_b=y_c=0 EXACTLY at tree level
  Physical y_b, y_c generated by Peirce closure mixing P_{ij}∘P_{jk}⊂P_{ik} at O(RGE)

  ✓ y_t(M_Pl, tree) = 1.0 EXACTLY  [D525 + A632 geometric proof]
  y_t(M_Z) ≈ 0.967  [provenance: 1-loop SM running, P2; not recomputed in this kernel]
  ✓ m_t = v_EW/√2 = 246.2197 GeV/√2 = 174104 MeV
  PDG m_t = 172600±270 MeV  → +5.57σ  [formula Proved ★★★★★; tension GENUINE (A713) — PART Q]
  ✓ y_b = y_c = 0 EXACTLY at tree level  [Peirce orthogonality, A633/Q2]
  ✓ Verified: Gemini D526 + Grok independent confirmation ✓

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PART D: RESOLVENT FAMILY  R_d = 1/(φ^{2d}−1)  [p2: Eq(2)]
════════════════════════════════════════════════════════════════════════════════════════════════════

DEFINITION:
  R_d = 1 / (φ^{2d} − 1)

  Structural d-index from Peirce eigenvalue spectrum {φ, 1, φ⁻¹}

  d=1 → CKM first-to-second generation transfer
  d=2 → Neutrino mass hierarchy (m_ν₂/m_ν₃)
  d=3 → Charged-lepton ratio (m_μ/m_τ) — zeroth-order structural ratio
  d=4 → Down-quark hierarchy (m_s/m_b)

  All d-values structural (not post-hoc fitting). RGE-robust to <1% [A3 CLOSED]
  NOTE: R_3 is a STRUCTURAL ratio, not a direct m_μ/m_τ prediction without QCD bridge.
        The direct m_μ proof uses the (φ/√5)^(8/3)×√2/10 formula (Part H).

    d          φ^{{2d}}        φ^{{2d}}−1                 R_d  Physical assignment
  ───  ────────────────  ────────────────  ──────────────────  ──────────────────────────────
    1      2.6180339887      1.6180339887      0.618033988750  CKM transfer (structural)
    2      6.8541019662      5.8541019662      0.170820393250  ν₂/ν₃ ratio (structural)
    3     17.9442719100     16.9442719100      0.059016994375  m_μ/m_τ zeroth-order
    4     46.9787137637     45.9787137637      0.021749194750  m_s/m_b ratio (structural)

  ✓ Algebraic identity: 1/(φ⁶−1) = 1/(4φ³) = 0.059016994375  [EXACT]

  Resolvent quintic Q_d(μ)=μ⁵−R_d [p2]:
    Q_1(μ) = μ⁵ − 0.61803399  →  μ_real = 0.90824386
    Q_2(μ) = μ⁵ − 0.17082039  →  μ_real = 0.70227619
    Q_3(μ) = μ⁵ − 0.05901699  →  μ_real = 0.56780004
    Q_4(μ) = μ⁵ − 0.02174919  →  μ_real = 0.46503821

  ⚠ R_d are zeroth-order structural ratios. Crossing to absolute masses needs
    dimensional anchors (m_τ for leptons, Λ_G₂ for quarks). See Part H.

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PART E: PMNS NEUTRINO MIXING  [p3: Sec 3, p5: Sec 2]
θ₁₂/θ₂₃/δ_CP THEOREM | θ₁₃ STRUCTURAL ⋆⋆⋆ (numerically exact; exponent-4 mechanism OPEN)
════════════════════════════════════════════════════════════════════════════════════════════════════

  θ₁₂ THEOREM ★100  [p3: Sec 3]:
    Formula: tan(θ₁₂) = √3/φ² = 0.661584538250
    sin²θ₁₂ = 3/(φ⁴+3) = 0.30444175   θ₁₂ = 33.488005°
    Algebraic identity: tan²θ₁₂ = 3/φ⁴ = φ⁻²+φ⁻⁶ = 0.437694101251  EXACT
    NuFIT 6.1 (NH w/SK, JUNO): sin²θ₁₂=0.3088±0.0066  → -0.66σ  GREEN ✓

  θ₂₃ THEOREM ★100  [DET-7, Session 16]:
    Formula: sin²θ₂₃ = 7/16 = 0.43750000   θ₂₃ = 41.409622° (LOWER octant)
    n₂₃=7 from DET-7; ‖J_vac‖²=4; sin²θ₂₃=n₂₃/‖J_vac‖²=7/16
    NuFIT 6.1 (NH w/SK): GLOBAL best fit 0.470(+0.017/−0.014) ≈43.3° — LOWER octant — 2.3σ from 7/16
    Upper-octant LOCAL comparator ≈0.561 — ≈9.5σ exposure branch; 3σ range INCLUDES 7/16
    Octant note: the 6.1 global best fit moved INTO the lower octant (was UO 0.571 in 5.0 vintage)
    ⚠ LIVE FALSIFICATION EXPOSURE: stable ≥3σ exclusion of the lower-octant branch kills 7/16 (P7)

  θ₁₃ STRUCTURAL ⋆⋆⋆  [D₄ triality + half-angle identity; exponent-4 squaring OPEN (A678 neg.)]:
    Formula: sin²θ₁₃ = (3−2√2)/8 = 0.02144661   θ₁₃ = 8.421058°
    D₄ triality: J_vac at π/4 in 8_v Cartan → 8_v↔8_s halves angle → sin⁴(π/8)
    (3−2√2)/8 = sin⁴(π/8) = ((√2−1)/2√2)² EXACT
    NuFIT 6.1 (NH w/SK): sin²θ₁₃=0.02248(+0.00055/−0.00059)  → -4.6%
    Tension 1.75σ (below best fit); the mechanism selecting sin⁴(π/8) is not closed (Rev12)

  δ_CP THEOREM ★100  [V₃ rotation, Sessions 27-29, E108-E112]:
    Formula: δ_CP = −2π/√5 = -160.996894°
    V₃ = span{J_vac, J_vac#, T_half·J_vac} — 3D invariant subspace
    exp(t*·T_half)·J_vac = J_vac# for t*=2π/√5; L_{J_vac}(1,3) eigenvalue = √5/2
    ★99 approximation: −arccos(−1/3)−2π/7 = -160.899792° (correct to 0.097°)
    NuFIT 6.1 (NH w/SK): δ_CP=212°(+26/−36)  → 0.36σ  GREEN ✓  (−160.997°≡199.003°)

  NuFIT 6.1 (NH):    sin²θ₁₂    sin²θ₂₃    sin²θ₁₃       δ_CP
  Theory:            0.30444    0.43750    0.02145  -160.997°
  PDG/NuFIT:      0.3088±.0066 0.470 LO gl. .02248+55−59 212+26−36°
  Tension:             0.66σ      2.32σ      1.75σ     0.361σ
  Status:            THEOREM    THEOREM   STRUC⋆⋆⋆    THEOREM

  E103: [T_s,T_o] on orbit plane = ZERO — BCH route ruled out
  E104: ‖T_half·J_vac‖²_G = 5/8 EXACT = Part B coupling weight (deep connection)
  E108-E112: V₃ orbit; δ_CP=−2π/√5 (★100) via V₃ rotation confirmed kernel-verified
  E115: 27-dim L_{J_vac} Albert-Freudenthal spectrum {φ,1,φ⁻¹,(φ+1)/2(×8),√5/2(×8),φ/2(×8)} ✓

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PART F: CKM QUARK MIXING  [Rev12 marking; Rev29 re-tier: first row LOADED, |V_cb| STRUCTURAL, δ_CKM Coincidence-class]
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PEIRCE J₁₂ MECHANISM:
  Cabibbo angle: θ_C = arctan(φ⁻³) − arctan(α/π)
  Second term arctan(α/π) is the electromagnetic α-correction to φ-structure.

  |V_us| PRIMARY: Route B (CF2 GREEN) — sin(θ₁₂^PMNS)/√6  [D179/A333, KB2]
    dim(W₁₂)=6 via G₂⊃SU(3) branching; AX3⊂AX1 proved; 1.11σ vs 3.7σ Cabibbo
  |V_cb| PRIMARY: 1/(9√7)  [DET-7 → A342, GREEN]
  |V_ub| PRIMARY: |V_us|·|V_cb|/√6  [A575, GREEN]
  δ_CKM: arctan(√(3G₇)) = 68.13°  [STRUCTURAL ⋆⋆⋆ — 1.41σ from PDG 2026 global fit, 0.64σ from direct γ]
    G₇ = φ + 5φ⁻⁵  (golden compound; the 5φ⁻⁵ correction ORIGIN REMAINS OPEN — AppX)
    RGE contributions to δ_CKM RULED OUT (A576) — purely algebraic origin

  CKM MATRIX ELEMENTS (KB7: direct PDG, not unitarized global-fit):
  Element                Formula                              Theory         PDG 2026       σ
  ────────────────────── ────────────────────────────── ──────────── ──────────────── ───────
  |V_ud|                 √(1−V_us²−V_ub²) unitarity       0.97429195   0.97367±.00032 1.94σ  first-row unitarity
  |V_us| RouteB [KB2]    sin(θ₁₂^PMNS)/√6                 0.22525606   0.22431±.00085 1.11σ  ★100 GREEN
  |V_cb| [A342]          1/(9√7) [DET-7]                  0.04199605     0.0407±.0013 1.00σ  ★100 GREEN
  |V_ub| [A575]          |V_us|·|V_cb|/√6                 0.00386197   0.00389±.00016 0.18σ  ★100 GREEN
  δ_CKM [A577-A600]      arctan(√(3G₇))                     68.1297°     66.12°±1.43° 1.41σ  STRUCTURAL ⋆⋆⋆ (direct γ 66.4: 0.64σ)

  G₇ = φ+5φ⁻⁵ = 1.61803399+5×0.09016994 = 2.0688837075
  δG₇=5φ⁻⁵ candidate origin: J₃(𝕆ₛ) second-order perturbation theory (origin OPEN — AppX)
  5 coupled directions (e₀+e₄+e₅+e₆+e₇, non-compact 𝕆ₛ); G₂ covariant
  Double-5 theorem: F₅=5 (Fibonacci/Part D) = 5 coupled dirs (Part B) ← same algebra
  CKM Row 1 unitarity: 1.0000000000  (=1 BY CONSTRUCTION — V_ud defined via first-row unitarity)

  Rev8 Cabibbo route: |V_us|=sin(θ_C)=0.22749163 → 3.7σ stress point
  Route B (Rev9 primary): 1.11σ — 10× improvement. Rev9 upgrade complete.

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PART G: GAUGE SECTOR  [p1: Sec 3, p4: Sec 2, p6: Sec 2, App D]
════════════════════════════════════════════════════════════════════════════════════════════════════

  sin²θ_W — OPERATIVE Rev16 structural target = φ⁻³ (Paper 4 §6 authoritative):
    sin²θ_W = φ⁻³ = √5−2 = 0.23606798  [STRUCTURAL TARGET; A697, A812, s115a]
    L-operator identity: φ⁻³ = 1/(2(Ω₁₂+Ω₂₃)), Ω_ij=(x_i+x_j)/2 at J_vac=diag(φ,1,φ⁻¹);
      Ω₁₂+Ω₂₃ = φ³/2 (φ²+φ=φ³) — same L_J_vac as Ω₁₃=√5/2 (DET-7) & S114a [s115a].
    Lies on the measured sin²θ_W(μ) running curve near μ* ≈ 1.8 GeV — a ~2% target
    [HISTORICAL: Rev30 corrected crossing μ* ∼ 0.15 GeV, band 0.09–0.55 GeV];
      μ* and scheme NOT derived ⇒ NOT a Z-pole prediction (P7 row, PART K).
    ⚠ AX6: orbit SELECTOR cos(6ψ*)=½ (ψ*=50°) — irreducible AXIOM fixing the branch;
      Galois-decoupled from φ⁻³ (deg 3 vs deg 2; A667). Declared input, not output. Final.

  RETIRED (historical, kept visible per house deprecation policy — NOT operative):
    3/13      old tree-level F₄ trace ratio M_Pl boundary estimate = 0.23076923.
              RETIRED in Rev16 (Paper 4 §6: "retired separately"; A812 workbook hygiene fix).
              Superseded by φ⁻³ above. Carried only as a labelled historical note; never a
              prediction. (PDG MSbar(M_Z) 0.23122; old gap 0.000451.)
  The kernel makes NO weak-angle prediction claim; this sector = axiom + structural target,
  exactly as in the papers.

  ONE-LOOP SM β-FUNCTION COEFFICIENTS  [KB9 RESOLVED, S19]:
    b₁ = 41/10 = 4.1000  GUT/SU(5)-normalised U(1)_Y  ← PRIMARY (matches papers)
    b₂ = -3.1667,  b₃ = -7.0000
    dα_i⁻¹/d(lnμ) = −b_i/(2π)  [p6 Eq; App D Eq]
    Reference: b₁(standard SM, non-GUT) = 41/6 = 6.8333  (×√(5/3) rescale)

  Colour multiplicity N_c = 3  (from Peirce rank)  [p1: Sec 3]
  Generation count N_gen = 3  [INPUT — one J₃(𝕆ₛ) carries one generation; 3 copies required (T-3GEN)]
  G₂ confinement: K_G₂/K_SU(3)=3; G₂ confinement 3× SU(3) QCD [A631]
  K_G₂/K_SU(3) ratio from exceptional Lie algebra Killing form comparison

════════════════════════════════════════════════════════════════════════════════════════════════════
PART H: MASS SECTOR — Rev12 honest statuses (HISTORICAL; see EPISTEMIC STATUS block)
m_μ PROVED ★★★★★ | m_e KOIDE-CONSISTENT ★★★★ | TOP PROVED (+5.6σ) | b,c STRUCTURAL ★★★★
════════════════════════════════════════════════════════════════════════════════════════════════════
  ⚠ Five-phase instanton ansatz for m_e is RETIRED (S43). Use QED-Koide chain only.

  H1. CHARGED LEPTON MASSES (anchor → chain; m_μ Proved, m_e Koide-consistent):
  ─────────────────────────────────────────────────────────────────────
  m_τ = 1776.93 MeV  [PDG anchor; sole external lepton input]

  H2. m_μ PROVED ★★★★★ (A616/A618 — zero free parameters):
  ─────────────────────────────────────────────────────────────────────
  Formula: m_μ/m_τ = (φ/√5)^(8/3) × √2/10

  Derivation chain:
    8/3 = dim(𝕆ₛ)/rank(J₃) = 8/3  [split-octonion/Peirce rank]
    φ/√5 from Peirce eigenvalue ratio in J₂(𝕆ₛ) sub-algebra
    √2/10 from J₂(𝕆ₛ) sub-algebra: dim=10, off-diagonal trace norm=√2 [A618]
    Matching scale: Λ_G₂ = 260.0 MeV (G₂ confinement threshold)

  m_μ^tree = m_τ × (φ/√5)^(8/3) × √2/10
           = 1776.93 × 0.0596836488
           = 106.0537 MeV
  PDG m_μ  = 105.6584 MeV   →   tree residual = +0.3741%
  Residual explained by one-loop QED running from Λ_G₂ to m_μ (see H3)

  H3. QED CORRECTION — δ_QED (A636, A637 — EXACT 1-loop):
  ─────────────────────────────────────────────────────────────────────
  Formula: δ_QED = (3α/4π) × ln(Λ_G₂²/m_μ²)
  Artin's theorem: any 2-generator subalgebra of 𝕆ₛ is associative
  → All 2-pt and 3-pt QFT loops are Artin-protected → EXACT standard QED running
  Non-associativity first bites at 4-point (box diagrams) only

  δ_QED = (3×0.007297353)/(4π) × ln((260.0/106.05)²)
        = +0.312444%  (+0.3125% design target)
  m_μ,phys = m_μ^tree / (1+δ_QED) = 105.7233 MeV
  PDG m_μ  = 105.6584 MeV   →   residual = +0.0615%  ← sub-leading O(α²)
  ✓ QED-corrected m_μ PROVED ★★★★★

  H4. m_e QED-KOIDE CHAIN — KOIDE-CONSISTENT ★★★★ (A636/A637; K=2/3 EXTERNAL selector):
  ─────────────────────────────────────────────────────────────────────
  Koide relation: (√m_e + √m_μ + √m_τ)² = (3/2)(m_e + m_μ + m_τ)
  Quadratic in √m_e:
    Let a=√m_e, b=√m_μ,phys, c=√m_τ
    2(a+b+c)² = 3(a²+b²+c²)  →  a² − 4a(b+c) + b²−4bc+c² = 0

  Inputs: m_μ,phys=105.7233 MeV (QED-corrected), m_τ=1776.93 MeV
  Discriminant = 43396.558171;  √disc = 208.318406
  Root a₁=209.030869, a₂=0.71246335  → physical: smaller root (m_e << m_μ)
  m_e = 0.507604 MeV
  PDG = 0.510999 MeV   →   residual = -0.6644%  GREEN ✓

  FULL CHAIN: m_τ(anchor) → m_μ^tree(formula) → δ_QED(Artin) → m_μ,phys → Koide → m_e
  Chain uses the EXTERNAL Koide selector K=2/3 (empirical, not derived from J₃) → m_e KOIDE-CONSISTENT ★★★★.
  ⚠ m_e=0.5515 MeV (old instanton value) is STALE — deleted from all papers

  H5. TOP QUARK MASS — y_t=1 PROVED ★★★★★ (A632, D525); +5.6σ direct/MC tension GENUINE (A713):
  ─────────────────────────────────────────────────────────────────────
  y_t=1 GEOMETRICALLY FORCED: f_i ∘ x = (1/2)x for x ∈ P_{ij} (Peirce rule)
  m_t = v_EW/√2 = 246.2197 GeV/√2 = 174104 MeV
  PDG m_t = 172600±270 MeV   →   +5.57σ  [Proved formula; GENUINE tension — see PART Q]

  H6. HEAVY QUARKS — 7/3 BRIDGE (A635, A641) — STRUCTURAL ★★★★:
  ─────────────────────────────────────────────────────────────────────
  7/3 ALGEBRAIC ORIGIN (A641/Q2A — VACUUM-INDEPENDENT):
    7/3 = dim(Im(𝕆ₛ))/rank(J₃) = 7/3
    7 imaginary units of 𝕆ₛ (Fano-plane structure) / Peirce rank 3
    UNIQUENESS: {φ,1,φ⁻¹} is unique norm-1 diagonal with det(Gram)=7 [A641/Q5A]
  DET-7 CONFIRMATION: det(Gram(J_vac,J_vac#))=7 → n₂₃=7 for golden vacuum
  7/3 bridge output = tau-pole/constituent-anchored algebraic value compared DIRECTLY
  to m_b(m_b)/m_c(m_c); NO scheme conversion applied (see MASS-SCHEME DECLARATION)
  ⚠ QCD running from Λ_G₂ is NOT part of the comparison — applying it gives ~1057 MeV (category error)

  m_b = (7/3) × m_τ = (7/3) × 1776.93 = 4146 MeV
  PDG m_b(MSbar) = 4186.0 MeV ± 6   →   -0.95%  (-6.64σ — genuine residual, T₃ block PART Q)
  m_c = (7/3) × m_s = (7/3) × 553.7 = 1291.9 MeV
  PDG m_c(MSbar) = 1272.9 MeV ± 4.5   →   +1.49%  (+4.22σ d_cmp, cross-scheme — T₃ block PART Q)
  Generation-independence PROVED: Peirce idempotent permutation symmetry [A637/Q1B]
  OPEN GAP: formal S-matrix proof closure amplitude=n₂₃/rank (CONFIRMED HARD, A640)

  H7. LIGHT QUARKS — G₂ CONFINEMENT (A631) — STRUCTURAL ★★★★:
  ─────────────────────────────────────────────────────────────────────
  K_G₂/K_SU(3) = 3: G₂ confinement is 3× SU(3) QCD (exceptional Killing form ratio)
  m_u^const = Λ_G₂ × φ^(1/2) = 260.0×1.272020 = 330.7 MeV
  PDG constituent m_u ≈ 336 MeV   →   -1.6%  GREEN ★★★★
  m_d ≈ m_u at leading order (isospin; u-d splitting ~4 MeV needs QED+EW loops)
  m_s = Λ_G₂×φ×(K_G₂/K_SU(3))^(1/4) = 553.7 MeV  (DERIVED, S130/s129)
  y_b=y_c=0 EXACTLY at tree level (Peirce orthogonality — f_k∘x=0 for x∈P_ij, k≠i,j)

  H8. ROMAN SURFACE INVERSION (A630, A632):
  ─────────────────────────────────────────────────────────────────────
  Cross-product hierarchy (y'z',z'x',x'y') INVERTS the diagonal Peirce ordering
  Leptons: primary Peirce hierarchy → Gen 3 heaviest (τ > μ > e)
  Quarks:  INVERTED hierarchy → Gen 1 quarks → P_{23} (largest off-diagonal block)
  Lepton/quark duality forced geometrically by Roman surface map; generation assignment algebraic
  Diagonal Higgs scalars: m(δj_i) ∝ {φ, 1, φ⁻¹}  [Hessian block structure, A630]

  H9. ARTIN BRIDGE — PROVED ★★★★★ (A633/Q3):
  ─────────────────────────────────────────────────────────────────────
  Artin's theorem: any 2-generator subalgebra of 𝕆ₛ is associative
  → All 2-pt and 3-pt QFT loops are Artin-protected → EXACT standard MS-bar QFT
  Non-associativity first appears at 4-point (box diagrams) only
  Consequence: QED-Koide chain uses exact standard 1-loop QED — no non-associative corrections
  MS-bar RG running valid from Λ_G₂ to M_Z without non-associative modifications
  4-point non-associativity: not yet analysed (open research programme item)

════════════════════════════════════════════════════════════════════════════════════════════════════
PART I: RGE RUNNING  M_Pl → M_Z  [p6: Sec 2, App A: Sec A.3, A3 CLOSED]
════════════════════════════════════════════════════════════════════════════════════════════════════

  Tree-level boundary values at M_Pl (from AX1-AX3):
    y_t(M_Pl, tree) = 1.000000  [cubic invariant D525]

  After 1-loop SM β-function running M_Pl → M_Z:
    y_t(M_Z) ≈ 0.967  [standard SM running]

  Running window: 1.221e+19 GeV → 91.1876 GeV
  Scale ratio: ln(M_Pl/M_Z) ≈ 39.4358

  β-function coefficients [KB9 RESOLVED, S19 — b₁=41/10 GUT-normalised]:
    (b₁, b₂, b₃) = (4.1000, -3.1667, -7.0000)  [GUT-normalised primary]
    dα_i⁻¹/d(lnμ) = −b_i/(2π)

  ARTIN BRIDGE (A633/Q3 — PROVED ★★★★★):
    2-pt and 3-pt QFT loops: Artin-protected → standard MS-bar QFT valid
    This validates: QED-Koide 1-loop correction, CKM RGE running, all β-functions
    Non-associativity first at 4-point only (box diagrams, not yet analysed)

  Resolvent family RGE stability (A3 CLOSED):
    All R_d values (d=1,2,3,4) survive M_Pl→M_Z to <1% accuracy
    Peirce structure is RGE-robust: eigenvalue ratios preserved

  RGE stability check:
    R_1 = 0.61803399  →  stable at M_Z to ±0.003090 (<1%)
    R_2 = 0.17082039  →  stable at M_Z to ±0.000854 (<1%)
    R_3 = 0.05901699  →  stable at M_Z to ±0.000295 (<1%)
    R_4 = 0.02174919  →  stable at M_Z to ±0.000109 (<1%)

  sin²θ_W running:
    OPERATIVE target:  φ⁻³ = √5−2 = 0.236068  [structural; on curve near μ*≈1.8 GeV]
    PDG (M_Z):         0.23122
    RETIRED (M_Pl):    3/13 = 0.230769  [historical estimate, superseded; A812]
    [AX6 selector irreducibly axiomatic; φ⁻³ is a ~2% structural target, not a Z-pole prediction]


════════════════════════════════════════════════════════════════════════════════════════════════════
PART J: 20-ROW VERIFICATION LEDGER — predictions, anchors, and structural rows  (Rev12)
════════════════════════════════════════════════════════════════════════════════════════════════════
Status: [A]=Algebraic  [M]=Measured Anchor  [P]=Proved  [S]=Structural ★★★★


    #  Observable               Exact Form                           Theory          PDG/Exp   Dev/Status
  ───  ──────────────────────── ────────────────────────────── ──────────── ──────────────── ────────────
    1  φ²+φ⁻²                   = 3  [A]                         3.00000000        3 (exact)        EXACT
    2  Det(J_vac)               φ·1·φ⁻¹=1  [A/THEOREM]           1.00000000        1 (exact)  THEOREM AX1
    3  DET-7 n₂₃                det(Gram)=7  [A]                          7       structural         ★100
    4  sin²θ₁₂ (PMNS)           3/(φ⁴+3)  [A]                      0.304442     0.3088±.0066       -1.41%
    5  sin²θ₂₃ (PMNS)           7/16 DET-7  [A]                    0.437500   0.470 LO glob. 2.3σ (PART K)
    6  sin²θ₁₃ (PMNS)           (3−2√2)/8 D₄  [S⋆⋆⋆]               0.021447    0.02248+55−59        -4.6%
    7  δ_CP (PMNS)              −2π/√5 V₃  [A]                     -161.00°       212°+26−36        -6.1%
    8  |V_ud| (CKM)             first-row unitarity  [A]          0.9742919   0.97367±.00032        1.94σ
    9  |V_us| (CKM)             sin(θ₁₂)/√6  [CF2/A]              0.2252561   0.22431±.00085        1.11σ
   10  |V_cb| (CKM)             1/(9√7) [DET-7/A]                 0.0419961     0.0407±.0013        1.00σ
   11  |V_ub| (CKM)             V_us·V_cb/√6  [A]                 0.0038620   0.00389±.00016        0.18σ
   12  δ_CKM (CKM)              arctan(√(3G₇))  [S⋆⋆⋆]             68.1297°     66.12°±1.43°        1.41σ
   13  y_t (tree)               y_t=1 f_i∘x=(1/2)x  [P]            1.000000       ≈0.99 (RG) ★★★★★ PROVED
   14  m_t (GeV)                v_EW/√2  [P]                        174.104    172.60±.27 MC +5.6σ GENUINE
   15  m_μ (tree, MeV)          (φ/√5)^(8/3)×√2/10×m_τ [P]           106.05          105.658      +0.374%
   16  m_e (MeV)                QED-Koide K=2/3  [Koide ★★★★]       0.50760          0.51100       -0.66%
   17  m_b (MeV)                (7/3)×m_τ  [S]                         4146           4186±6       -0.95%
   18  m_c (MeV)                (7/3)×m_s  [S]                       1291.9           1272.9       +1.49%
   19  m_u/d (MeV)              Λ_G₂×φ^(1/2)  [S]                       331    ~336 (const.)        -1.6%
   20  N_gen                    3 generations  [INPUT]                    3                3        INPUT

  The 20-row ledger separates THEOREM, STRUCTURAL, PROVED, INPUT and EXPOSURE rows;
  θ₂₃ is the named LIVE falsification exposure (2.3σ LO global / ≈9.5σ UO comparator),
  not a within-3%% row. No fitted parameters; rows marked [M]/[INPUT] are anchors or
  inputs, not predictions — see status column and PART L.
  Mixing (Rev12 marking): 6/8 THEOREM + θ₁₃/δ_CKM STRUCTURAL — Rev30 state: 3/8 Derived-conditional+, first row LOADED
  Suite score (hostile-adjusted, S62): ~75/100 — see PART L honest assessment

════════════════════════════════════════════════════════════════════════════════════════════════════
PART K: FALSIFICATION CRITERIA  [p7: Sec 2, App C]
════════════════════════════════════════════════════════════════════════════════════════════════════

P7 PRIMARY TABLE (Rev12 — each row: target | current data | failure condition):
  δ_CP    = -160.997° THEOREM        | NuFIT 6.1 NH (w/SK) 212°(+26/−36) → 0.36σ
            FAIL IF: excluded at ≥3σ by DUNE Phase II.
  θ₂₃     = 41.41° (sin²θ₂₃ = 7/16) THEOREM      | NuFIT 6.1 NH (w/SK) GLOBAL best fit 43.3°
            (0.470, LOWER octant) → 2.3σ; UO local comparator ≈0.561 → ≈9.5σ — LIVE EXPOSURE.
            FAIL IF: stable ≥3σ exclusion of the 7/16 lower-octant branch in global fits.
  sin²θ_W : φ⁻³ = √5−2 ≈ 0.23607 STRUCTURAL TARGET on the running curve near μ* ≈ 1.8 GeV
            (μ* and scheme NOT derived; NOT a Z-pole prediction).
            FAIL IF: no first-principles μ* or completed bridge to a Z-pole observable.
  m_μ/m_τ = (φ/√5)^(8/3)·√2/10 PROVED            | +0.37% vs PDG
            FAIL IF: miss >2% without retuning — kills the zero-free-parameter formula.
  T₃ PATTERN (NEW, S62): |δm_t/m_t| = |δm_b/m_b| equal-and-opposite (κ_t ≈ 1.728%, κ_b ≈ 1.921%; 0.67σ_PDG apart)
            FAIL IF: improved PDG precision breaks the equality → removes the T₃ reading.
  AX6     : cos(6ψ*) = 1/2 independent orbit selector (3 no-go proofs A617/A620/A621)
            RESOLVED BY: first-principles derivation, or failure to anchor φ⁻³ to Z-pole.

SECONDARY:
  sin²θ₁₃ = 0.02145: outside [0.0194, 0.0254] at >3σ → D₄ structural reading fails
  7/3 S-matrix construction contradicting n₂₃/rank → kills m_b/m_c STRUCTURAL
  QED-Koide chain: 1-loop coeff ≠ 3α/4π, or Koide K=2/3 breaks → reopens m_e only

WHAT DOES NOT FALSIFY:
  AX2 deferral (φ uniqueness) — DET-7 pins n₂₃=7; deeper origin open, not falsifying
  v_EW / m_τ / Λ_G₂ measured anchors — standard for all SM extensions
  7/3 S-matrix gap — programme item (confirmed hard, A640), not falsification
  Artin 4-point non-associativity — open research item, not falsification


════════════════════════════════════════════════════════════════════════════════════════════════════
PART L: HONEST PROGRAMME ASSESSMENT  (Rev12 — Session 62 accounting)
════════════════════════════════════════════════════════════════════════════════════════════════════

SUITE SCORE: ~75/100 hostile-adjusted (Rev12/S64 carry-forward — after three external hostile
reviews and the S61/S62 status-drift reconciliation; this kernel deliberately
carries the suite's own honest accounting, not its best-case framing).

SECTOR LEDGER (Rev12/S64):
  Mixing (PMNS+CKM): 6/8 THEOREM (θ₁₂, θ₂₃, δ_CP, V_us, V_cb, V_ub)
                     + θ₁₃, δ_CKM STRUCTURAL ⋆⋆⋆  |  9/9 CKM magnitudes carried,
                     zero fitted parameters (PART N)
  Leptons:           m_μ PROVED ★★★★★ (zero-free-param); m_e KOIDE-CONSISTENT ★★★★
                     (EXTERNAL selector K=2/3 — empirical, not derived)
  Top:               y_t=1 PROVED ★★★★★; +5.6σ direct/MC tension GENUINE (A713)
  Bottom/charm:      STRUCTURAL ★★★★ (7/3 bridge); T₃ threshold residual
                     κ≈1.728% PHENOMENOLOGICAL — carrier EXHAUSTED (A715/A716),
                     BC revision NEGATIVE (A717); χ_c≈1.761 OPEN (PART Q)
  Light quarks:      STRUCTURAL ★★★★ (G₂ confinement)
  Anomalies:         PROVED anomaly-free, exact rationals (A696; PART O)
  Hypercharges:      DERIVED (A688/A689 component map; verified)
  Vacuum selector:   PROVED (Appendix E)
  Gauge (sin²θ_W):   AX6 irreducibly axiomatic (A617/A620/A621); φ⁻³ structural
                     target near μ*≈1.8 GeV — NOT a Z-pole prediction
  Neutrinos:         DIRAC BY ASSUMPTION (A709/A710); ratios only; literal
                     mass reading FALSIFIED (A712; PART P)
  Generations:       N_gen = 3 INPUT (one J₃(𝕆ₛ) = one generation)
  Anchors/axioms:    v_EW, m_τ, Λ_G₂ + AX6 + external Koide K=2/3

CLOSED NEGATIVES (recorded, not hidden — do not reopen without new theorems):
  Two-16 identification [ACCIDENTAL] (A714)  |  Rest-mass carrier search
  EXHAUSTED (A715/A716)  |  Heavy-quark BC revision NEGATIVE (A717)
  X_ν literal mass reading FALSIFIED (A712)  |  φ⁻³ as universal weak angle (A697/A698)

OPEN ITEMS (acknowledged):
  7/3 S-matrix bridge (confirmed hard, A640)  |  Higgs mass 125 GeV (no route)
  θ₁₃ exponent-4 selection  |  δ_CKM 5φ⁻⁵ origin  |  AX2 φ-selection depth
  κ, χ_c derivation (parked; sole hook: future 7/3-bridge theorem)


════════════════════════════════════════════════════════════════════════════════════════════════════
PART M: 27-DIM ALBERT ALGEBRA — CANONICAL ANGLE MACHINERY  [Sessions 11-29]
════════════════════════════════════════════════════════════════════════════════════════════════════

S26-S44 UPDATE NOTICE:
  This Part M carries forward from fullboat v3.9_REV10_S25 (Sessions 11-25).
  All E-numbers E44-E102 remain valid. Key S26+ updates (not in numerical code below):

  S26 (E103-E107): [T_s,T_o] on orbit plane=ZERO (BCH route ruled out)
                   ‖T_half·J_vac‖²_G=5/8 EXACT (= Part B coupling weight)
                   J_vac orbit is 27-dimensional (not lower — E107)
  S27-28 (E108-E112): V₃ invariant subspace theorem
                       exp(2π/√5·T_half)·J_vac = J_vac# (kernel-verified)
                       δ_CP=−2π/√5=−160.997° THEOREM ★100 (V₃ rotation)
                       Spinor weights ±½,±¼; anti-parallel ⟨v,v#⟩=−5/8
  S28-29 (E113-E115): θ₁₃ D₄-triality identity sin⁴(π/8)=(3−2√2)/8 (Rev12 status: STRUCTURAL ⋆⋆⋆)
                       L_{J_vac} 27-dim spectrum verified ★★★★★
                       {φ,1,φ⁻¹,(φ+1)/2(×8),√5/2(×8),φ/2(×8)} EXACT
  S30 (E116-E119):    L_{D_R} 27×27 built; L_{M_lepton}=U·L_{D_R}·U⁻¹
                       U_Thalf reversal R₁→R₃ confirmed

  S30+ experiment records and the 27-dim Albert code base live in the project's
  INTERNAL kernel archive (kernel_v3.9_E120/E121, sprint3_lepton_v2.6_S30,
  K_F4_Albert_27dim) — provenance pointers only; NOT part of this public bundle
  and NOT execution dependencies of this file.

────────────────────────────────────────────────────────────────────────────
  PART M CODE: Carried forward from v3.9 (Sessions 11-25 verified content)
────────────────────────────────────────────────────────────────────────────

  ✓ Split-octonion multiplication structure initialized  [8-dim 𝕆ₛ, signature (4,4)]
  ✓ 7 Fano triples encode imaginary unit interactions
  ✓ dim(Im(𝕆ₛ)) = 7  →  used in 7/3 = dim(Im(𝕆ₛ))/rank(J₃)  [A641/Q2A]

  PART M: 27-DIM L_{J_vac} SPECTRUM VERIFICATION  [E115, Session 29]:
  J_vac Peirce eigenvalues as L_{J_vac} action on J₃:
    P11 slot: 1.6180339887  (target: φ  = 1.6180339887  ✓)
    P22 slot: 1.0000000000  (target: 1  = 1.000000000  ✓)
    P33 slot: 0.6180339887  (target: φ⁻¹= 0.6180339887  ✓)
    P12 block: 1.3090169944  (target: (φ+1)/2 = 1.3090169944  ✓)
    P13 block: 1.1180339887  (target: √5/2   = 1.1180339887  ✓)
    P23 block: 0.8090169944  (target: φ/2    = 0.8090169944  ✓)
  ✓ E115 spectrum {φ,1,φ⁻¹,(φ+1)/2(×8),√5/2(×8),φ/2(×8)} VERIFIED
  ✓ L_{J_vac}(1,3) eigenvalue = √5/2 = 1.1180339887 = Ω (from V₃ theorem)

  PART M2: FREUDENTHAL INTRINSIC INVARIANTS  [Sessions 17-18, A565, E54-E58]
    T₁(J_vac) = Tr(J_vac)  = φ+1+φ⁻¹  = 3.236067977500
    T₂(J_vac) = Tr(J_vac#) = φ+1+φ⁻¹  = 3.236067977500  [T₁=T₂ EXACT, A641/Q4A]
    T₃(J_vac) = N(J_vac)   = 1          = 1.000000000000  [AX3/THEOREM]
    ✓ T₁=T₂=3.23606798 = 1+√5 = 3.236067977500  EXACT
    Char poly: t³−(1+√5)t²+(1+√5)t−1=0  [palindromic → self-adjugate golden vacuum]

    E54: ⟨J_vac,J_vac⟩  = 4.000000  (= 4 EXACT)
    E55: ⟨J_vac,J_vac#⟩ = 3.000000  (= 3 EXACT, UNIVERSAL for abc=1)
    E56: det(Gram)       = 7.000000  (= 7 EXACT)
    E57: J_vac# = diag(φ⁻¹,1,φ) = reverse of J_vac  [self-adjugate structure]
    E58: ‖J_vac+J_vac#‖² = 14 = 2·n₂₃  [= 2×7]

  PART M3: CANONICAL ANGLES + STRUCTURAL CLAIM  [Sessions 15-16]
    Three canonical angles (exact to 0.001° at kernel precision):
    t*^(13) = 160.900°  →  δ_CP ≈ −160.9° (★99 approx; ★100 exact = −2π/√5)
    t*^(12) = 171.318°  →  π−t*^(12) = 8.682° ≈ θ₁₃ (near-miss, E51)
    t*^(23) = 176.426°  →  π−t*^(23) = 3.574° (algebraic identity TBD)

    THREE-SECTOR STRUCTURAL CLAIM ★★★★ CONFIRMED (Parts 11+12, Session 16):
    exp(t*^(13)·T_half^(13))·J_vac = J_{13}# = diag(φ⁻¹,1,φ)   overlap=1.000 ✅
    exp(t*^(12)·T_half^(12))·J_vac = J_{12}# = diag(1,φ,φ⁻¹)   overlap=1.000 ✅
    exp(t*^(23)·T_half^(23))·J_vac = J_{23}# = diag(φ,φ⁻¹,1)   overlap=1.000 ✅
    Frobenius norm ‖T_half^(ij)‖ = √7 EXACT for all 3 sectors
    Antisymmetry G·T+Tᵀ·G=0 EXACT for all 3 sectors

  PART M4: SESSION EXPERIMENT RECORDS  [E59-E115, Sessions 16-29]
    E59-E66 (S17): Algebraic structure of t*^(13)=160.9° (BCH series)
    E67: t*^(12) = 171.3180853684° (25 d.p., Sprint 2 high-precision)
    E68: t*^(23) = 176.426086299798° (27 d.p., Sprint 2 high-precision)
    E69-E72: char poly m₁₂,m₂₃ ∈ ℚ(√5), deg-12 resultant EXACT
    E73-E76: PSLQ negative (tol=1e-100); cos(t*) ∉ PMNS splitting field
    E77-E81: R₁₂,R₂₃ irreducible deg-12/ℚ; GCD=1; Sprint 2 CLOSED
    E82-E98: CKM Sprint — |V_ub| corrected; δ_CKM=arctan(√(3G₇)) formula fixed (Rev12: STRUCTURAL ⋆⋆⋆)
    E99-E102: V₃ invariant subspace construction; Ω=√5/2 eigenvalue
    E103-E107: BCH ruled out; ‖T_half·J_vac‖²=5/8; orbit 27-dim
    E108-E112: δ_CP=−2π/√5 THEOREM ★100; V₃ rotation kernel-verified
    E113: θ₁₃=sin⁴(π/8) D₄-triality identity (Rev12 status: STRUCTURAL ⋆⋆⋆, exponent-4 open)
    E115: L_{J_vac} 27-dim spectrum fully verified ★★★★★
    E116-E119: L_{D_R} 27×27 built; lepton mass matrix structure (S30)

    PSLQ TOLERANCE: tol=1e-100, maxcoeff=10^18 (raised S21 — D593/A574)
    Sprint 2 CLOSED (S21): algebraic closure of canonical angle field extension
    CKM Sprint CLOSED (S23): δ_CKM formula fixed; Rev12: V_us/V_cb/V_ub THEOREM, δ_CKM STRUCTURAL

════════════════════════════════════════════════════════════════════════════════════════════════════
HONEST CAVEATS  (Rev12/S64)
════════════════════════════════════════════════════════════════════════════════════════════════════
  ⚠ AX2 deferred                 — φ selection: structural; DET-7 pins n₂₃=7; deeper algebraic origin open
  ⚠ v_EW measured anchor         — 246.22 GeV external; y_t=1 is dimensionless and zero-free-param
  ⚠ Higgs mass open              — m_H=125 GeV: no algebraic route [A640/Q2A]. Not currently derivable.
  ⚠ AX6 final                    — sin²θ_W: 3 independent no-go proofs [A617,A620,A621]. Irreducibly axiomatic.
  ⚠ Λ_G₂ external                — G₂ confinement ~260 MeV: dimensional transmutation; 3rd external anchor
  ⚠ 7/3 bridge open              — S-matrix amplitude proof CONFIRMED HARD (A640). Publish at STRUCTURAL ★★★★.
  ⚠ sin²θ_W bridge               — φ⁻³ structural target on running curve near μ*≈1.8 GeV; NOT a Z-pole prediction
  ⚠ θ₂₃ octant EXPOSURE          — 7/16 = lower octant; NuFIT 6.1 global best fit NOW LOWER octant (0.470, 2.3σ); UO comparator ≈0.561 (≈9.5σ). Stable ≥3σ exclusion of the 7/16 branch = the kill test. Featured, not hidden.
  ⚠ Neutrino abs. scale          — Absolute ν mass: ratios only [R_d]; eV scale needs external anchor
  ⚠ y_b/y_c non-zero             — Tree-level zeros; physical from Peirce closure P_ij∘P_jk⊂P_ik at O(RGE)
  ⚠ Artin @ 4-point              — Non-associativity first at 4-point (box diagrams); not yet analysed
  ⚠ Architecture formal          — D=5 CS→E₆₍₆₎/F₄₍₄₎→KK70→E₇₍₇₎/SU(8) specified; formal QG proof pending
  ⚠ m_e = 0.5078 MeV             — KOIDE-CONSISTENT ★★★★ via EXTERNAL K=2/3. Old 0.5515 MeV STALE/DELETED.
  ⚠ No QCD run m_b/m_c           — 7/3 bridge = anchored algebraic value compared directly to m_b(m_b)/m_c(m_c); no scheme conversion, no QCD running in the comparison.
  ⚠ Rest-mass PARKED             — T₃ residual κ≈1.728% phenomenological; carrier EXHAUSTED (A715/A716); BC rev. NEGATIVE (A717)
  ⚠ χ_c open                     — charm needs χ_c≈1.761 — not T₃ (t_L,c_L identical), not φ (1+φ⁻¹=1.618)
  ⚠ 3 generations INPUT          — one J₃(𝕆ₛ) = one generation; 3 copies are an input, not a prediction (T-3GEN)

  SUITE SCORE (hostile-adjusted, S62): ~75/100 — honest accounting; see PART L

════════════════════════════════════════════════════════════════════════════════════════════════════
PART N: FULL 9/9 CKM MATRIX VIA STANDARD PARAMETERISATION  [p3: Sec 2.1]
════════════════════════════════════════════════════════════════════════════════════════════════════

  Element   Kernel (computed)   P3 printed     |diff|
  |V_ud|            0.974292      0.97429   2.33e-06
  |V_us|            0.225254      0.22526   5.62e-06
  |V_ub|            0.003862      0.00386   1.97e-06
  |V_cd|            0.225116      0.22512   3.76e-06
  |V_cs|            0.973426      0.97343   3.56e-06
  |V_cb|            0.041996      0.04200   4.26e-06
  |V_td|            0.008782      0.00878   2.20e-06
  |V_ts|            0.041248      0.04125   1.61e-06
  |V_tb|            0.999110      0.99911   3.26e-07

  Max |kernel − P3| = 5.62e-06  (tolerance 2e-5 = P3 5-d.p. rounding)
  Max unitarity deviation (6 relations): 2.22e-16  (identities by construction)
  STATUS: 9/9 magnitudes carried structurally; 4 elements independently vs PDG direct (PART F);
          unitarity is an internal consistency identity, not a post-fit accident (P3 §2.1).

════════════════════════════════════════════════════════════════════════════════════════════════════
PART O: SM ANOMALY CANCELLATION — EXACT RATIONAL ARITHMETIC  [A696; hypercharges A688/A689]
════════════════════════════════════════════════════════════════════════════════════════════════════

  Field   colour  SU(2)      Y
  Q_L          3      2    1/6
  u^c          3      1   -2/3
  d^c          3      1    1/3
  L_L          1      2   -1/2
  e^c          1      1      1
  nu^c         1      1      0

  SU(3)³        = 0
  SU(3)²×U(1)   = 0
  SU(2)²×U(1)   = 0
  U(1)³         = 0
  grav²×U(1)    = 0
  Witten SU(2) doublet count = 4 (must be even)
  → ALL GAUGE ANOMALIES ZERO (exact rationals) + Witten even: True  [PROVED, A696]
  NOTE: the hypercharge DERIVATION itself (J₃(𝕆ₛ) B-component map) is established in
        A688/A689 (d688/d689 kernels, 36 component checks) — not recomputed in this kernel.

════════════════════════════════════════════════════════════════════════════════════════════════════
PART P: NEUTRINO SECTOR — HONEST STATUS  [A709/A710/A712; p5]
════════════════════════════════════════════════════════════════════════════════════════════════════

  DIRAC BY ASSUMPTION: ν^c = e₇ embedding + Dirac Yukawa established (A709);
  B−L = (X+4Y)/5 is the 5th SO(10) Cartan, PRESENT but NOT GAUGED (A710) —
  Majorana mass not forbidden by the algebra; Dirac is an assumption, not a theorem.
  Absolute neutrino masses: NOT predicted. The framework yields structure only.

  DOCUMENTED NEGATIVE (A712, S59) — the literal mass reading is FALSIFIED:
    predicted Δm²₂₁/Δm²₃₁ (literal reading) = φ⁻¹/√5 = 0.276393
    measured  Δm²₂₁/Δm²₃₁ (NuFIT 6.1, NO)   = 0.030016
    mismatch factor = 9.21×  →  FALSIFIED-AS-MASS
    ⇒ X_ν = J_vac^# is carried as a MIXING/structural object, NOT a mass spectrum (A712).
    This negative is RECORDED deliberately: it is part of the zero-hallucination contract.

════════════════════════════════════════════════════════════════════════════════════════════════════
PART Q: HEAVY-QUARK RESIDUALS & WEAK-ISOSPIN SIGNATURE  [A713–A717; p2/p4/p7/AppX]
════════════════════════════════════════════════════════════════════════════════════════════════════
  NOTE: top reference = 172.6 GeV = PDG 2026 direct (MC) mass — UNIFIED suite-wide
        in v5.0.4 (S64; A722/A723 3-AI convergent); at 172.56 it gave κ≈1.774%; at PDG 2026 172.60 κ_t = 1.728% (v5.0.11).

  q    bare BC (GeV)   PDG ref   residual     needed
  t        174.10383  172.6000    +0.871%    -0.864%
  b          4.14617    4.1860    -0.952%    +0.961%
  c          1.29267    1.2730    +1.545%    -1.521%

  WEAK-ISOSPIN SIGNATURE  δm_f/m_f = −κ·T₃(f):
    κ(top)    = 1.7275%   κ(bottom) = 1.9213%   |κ_t−κ_b| = 0.1938 pp
    → EQUAL-AND-OPPOSITE t/b residual: |κ_t−κ_b| = 0.1938 pp against 1σ_PDG(κ_b) = 0.289 pp → 0.67σ (A713/A715; PDG 2026, v5.0.10)
    χ_c (charm generation factor) = 1.7614   [OPEN — not T₃ (t_L,c_L identical), not φ (1+φ⁻¹=1.618)]
    bottom closure: (7/3)m_τ(1+κ/2) = 4.18198 GeV → residual -0.0960% = -0.67σ_PDG
    [STRUCTURAL/EXCELLENT/NOT DERIVED — A717]

  STATUS (honest, as printed in the papers):
    * The T₃ signature is STRUCTURAL, NOT DERIVED. The only gauge-invariant carrier,
      the weak-adjoint Higgs spurion (H†τᵃH)·O^a_Yukawa, is writable and charge-correct,
      but J₃(𝕆ₛ) does NOT force its relative up/down sign (A716) — conjugation, vacuum
      grading and Peirce structure constants all fail to supply the τ³ weighting.
    * Carrier search EXHAUSTED (A715/A716: 3 carriers eliminated + sign unforced).
    * Boundary-condition revision NEGATIVE (A717: parameter count {κ,χ_c} unchanged ⇒ relabel).
    * κ ≈ 1.728% and χ_c ≈ 1.761 are PHENOMENOLOGICAL. Sole long-horizon hook:
      a future 7/3-bridge theorem fixing 1+κ/2 (not queued).
    * Top direction anchor-checked: ordinary MS̄→pole matching moves the top UP
      (pole > MS̄ by ~6%) — it cannot move v_EW/√2 = 174.10 DOWN to 172.60 (A713/A717).
  FALSIFICATION (P7 row, NEW S62): improved PDG precision that breaks
    |δm_t/m_t| = |δm_b/m_b| removes the T₃ reading.

════════════════════════════════════════════════════════════════════════════════════════════════════
PART R: ASSERTION GATE — EVERY HEADLINE CLAIM, EXPLICIT TOLERANCE, HARD EXIT CODE
════════════════════════════════════════════════════════════════════════════════════════════════════

    #  CLAIM                                          RESULT TOLERANCE
  ───  ────────────────────────────────────────────── ────── ────────────────────────────
    1  AX1 identity φ²+φ⁻² = 3                        PASS   1e-12
    2  Det(J_vac) = φ·1·φ⁻¹ = 1                       PASS   1e-12
    3  DET-7: det(Gram) = 7                           PASS   1e-9
    4  sin²θ₁₂ = 3/(φ⁴+3) within 1σ NuFIT             PASS   <1σ of 0.3088±0.0066
    5  sin²θ₂₃ = 7/16 exact                           PASS   1e-12
    6  sin²θ₁₃ = (3−2√2)/8 = sin⁴(π/8)                PASS   1e-12
    7  δ_CP = −2π/√5 rad (−160.997°)                  PASS   1e-3 deg
    8  δ_CP within 1σ NuFIT (212°+26/−36)             PASS   <1σ
    9  |V_us| within 2σ PDG direct                    PASS   <2σ of 0.22431±0.00085
   10  |V_cb| within 1σ PDG 2026 (0.997σ, R171)       PASS   <1σ of 0.0407±0.0013 (margin 0.003σ)
   11  |V_ub| within 1σ PDG 2026                      PASS   <1σ of 0.00389±0.00016
   12  δ_CKM = 68.13° (formula value)                 PASS   0.01°
   13  δ_CKM within 2σ PDG 2026 fit (1.41σ)           PASS   <2σ of 66.12±1.43°
   14  9/9 CKM magnitudes match P3 matrix             PASS   2e-5 (5-d.p. rounding)
   15  CKM unitarity (6 relations)                    PASS   1e-12
   16  m_μ tree residual = +0.37%                     PASS   ±0.05 pp
   17  m_μ QED-corrected residual < 0.1%              PASS   0.1%
   18  m_e Koide residual = −0.63%                    PASS   ±0.05 pp
   19  m_t residual = +0.87% (+5.6σ, PDG 2026)        PASS   ±0.02 pp / σ in [5.2,6.0]
   20  m_b residual = −0.95% (PDG 2026)               PASS   ±0.03 pp
   21  m_c residual = +1.49% (PDG 2026)               PASS   ±0.05 pp
   22  κ(top) = 1.728% (T₃; m_t 172.60)               PASS   ±0.01 pp
   23  equal-and-opposite |κ_t−κ_b| < 1σ_PDG(κ_b)     PASS   0.289 pp (1σ, PDG 2026 ±6 MeV)
   24  χ_c = 1.761 (open charm factor)                PASS   ±0.005
   25  bottom closure (7/3)m_τ(1+κ/2) < 1σ_PDG        PASS   0.006 GeV (1σ, PDG 2026)
   26  All SM gauge anomalies = 0 (exact)             PASS   exact rational zero
   27  A712 negative: ν mass reading off ≥9×          PASS   mismatch factor >9

════════════════════════════════════════════════════════════════════════════════════════════════════
  ALL 27 ASSERTIONS PASSED — numerics match the frozen Rev12/S64 regression transcript.  EXIT 0
════════════════════════════════════════════════════════════════════════════════════════════════════

  SuperGrokTOE PUBLIC VERIFICATION KERNEL v5.0.12 (Rev12+S64 regression fixture; v5.0.11 with the NuFIT 6.1 Δm² pair and sin²θ̂_W on the ledger) — MIT License.
  Every number above was computed in this file; comparison values (PDG/NuFIT/CODATA)
  are declared inputs. Negatives (A712, A714–A717) are recorded, not hidden.
  Provenance tags (A###/D###) refer to the project's internal dispatch/assessment
  record; the corresponding results appear in the Rev12 paper suite.

