s1133 — Peirce category / null-side typing build
  structure constants: sgtoe_kernel_v5.0.8.py :: oct_product  sha256 bc8c1f658942b641…  (expected bc8c1f6589…)
  [PASS] v5.0.8 table sha matches the registered kernel (COMPUTED)

PART A — signature arithmetic on the A689 dictionary
  N(e_i) = [1, 1, 1, 1, -1, -1, -1, -1]
  [PASS] N(e₀..e₃) = +1, N(e₄..e₇) = −1 (the (4,4) split signature, computed from the table)
  Higgs/neutral (e₀,e₇): N(e_a)=+1 N(e_b)=-1 → plane signature (1,1); N(e_a±e_b) = 0.0e+00, 0.0e+00
  [PASS] Higgs/neutral (e₀,e₇) is a HYPERBOLIC (1,1) plane with two null lines (|N| ≤ 1e-12 absolute)
  colour 1 (e₁,−e₆): N(e_a)=+1 N(e_b)=-1 → plane signature (1,1); N(e_a±e_b) = 0.0e+00, 0.0e+00
  [PASS] colour 1 (e₁,−e₆) is a HYPERBOLIC (1,1) plane with two null lines (|N| ≤ 1e-12 absolute)
  colour 2 (e₂,e₅): N(e_a)=+1 N(e_b)=-1 → plane signature (1,1); N(e_a±e_b) = 0.0e+00, 0.0e+00
  [PASS] colour 2 (e₂,e₅) is a HYPERBOLIC (1,1) plane with two null lines (|N| ≤ 1e-12 absolute)
  colour 3 (e₃,e₄): N(e_a)=+1 N(e_b)=-1 → plane signature (1,1); N(e_a±e_b) = 0.0e+00, 0.0e+00
  [PASS] colour 3 (e₃,e₄) is a HYPERBOLIC (1,1) plane with two null lines (|N| ≤ 1e-12 absolute)
  [PASS] σ(e₀+e₇) = e₀−e₇ and σ(e₀−e₇) = e₀+e₇: σ SWAPS the two null lines of the Higgs plane (H_u ↔ H_d is a null-line exchange)
  [PASS] σ fixes every colour line pointwise (it is the identity on e₁..e₆)

PART B — J₃(𝕆_s) built from the table alone (no Jordan-product formula imported)
  [PASS] X∘Y is hermitian for random hermitian X, Y (closure of the Jordan product built from the table)
  [PASS] J_vac ∘ J_vac = diag(φ², 1, φ⁻²) (sanity; relative 1e-12)
  [PASS] (X∘X)∘X = X∘(X∘X) on a random X (power-associativity; a table/sign error would fail here)
  [PASS] Jordan identity (X²∘Y)∘X = X²∘(Y∘X) on random X, Y (relative 1e-10)

PART C — the Peirce category 𝔾: objects f₁,f₂,f₃; Hom(f_i,f_j) = P_ij; composition = Jordan product
  worst leak outside the Peirce target block over 50 random trials × all (i,j,k): 0.00e+00
  [PASS] P_ij ∘ P_jk ⊂ P_ik and P_ij ∘ P_ij ⊂ ℝf_i ⊕ ℝf_j EXACTLY (absolute ≤ 1e-12): the Peirce closure of Paper 2 :353, computed
  composition map identification (which octonion product each c_ijk is, in the block coordinates of P(i,j,·)):
    c_123(x,y) = ['½ x·y']
    c_132(x,y) = ['½ x·y']
    c_213(x,y) = ['½ x·y']
    c_231(x,y) = ['½ ȳ·x̄']
    c_312(x,y) = ['½ ȳ·x̄']
    c_321(x,y) = ['½ ȳ·x̄']
  [PASS] every c_ijk is EXACTLY ½ × one octonion product of (x or x̄) with (y or ȳ) — the Hom-composition is the octonion product with a fixed conjugation pattern
  functor 𝔾 → (one generation): weights w_ij ∈ P_ij (24 real dims); closure w_13 := c(w_12, w_23) removes 8 → 16-dim family
  [PASS] round trip c(c(x,y), ȳ) = ¼ N(y) x for a generic weight (alternativity; relative 1e-12)
  with a NULL weight y = e₀+e₇: N(y) = 0.0e+00; |c(c(x,y),ȳ)| = 0.0e+00; |c(x,y)| = 0.982 (nonzero)
  [PASS] with a NULL weight the round trip is ZERO (absolute ≤ 1e-12) while c(x,y) itself is not: functoriality on 𝔾 degenerates exactly on the null cone

PART D — replication along the DISCRETE category vs the framed lift of the Peirce swaps
  [PASS] discrete Kan extension (T⊕T⊕T): Y_u and Y_d commute and their generation-block structure is scalar ⇒ any induced frame is I (Paper 3's theorem, one level down)
  [PASS] X ↦ R X Rᵀ (R ∈ SO(3) real) is a Jordan automorphism of the table-built J₃ (random X,Y; relative 1e-11)
  [PASS] R₁₂(π/2) swaps the idempotents f₁ ↔ f₂ (it covers the Peirce transposition)
  R₁₂(π/2)² = R₁₂(π) = diag(−1,−1,+1): on P₁₂ → +1, on P₁₃ → −1, on P₂₃ → −1, on f₁,f₂,f₃ → +1
  ★ HOUSE EXPECTATION FALSIFIED (kept, S294 rule): the draft expected the half-twist on the SWAPPED block P₁₂; it sits on the two ADJACENT blocks P₁₃, P₂₃.
  [PASS] R₁₂(π/2)² acts as −1 on P₁₃ and P₂₃, +1 on P₁₂ and on every f_i (a HALF-TWIST on Hom-objects; invisible on the diagonal)
  braid relation on SO(3): |R₁₂R₂₃R₁₂ − R₂₃R₁₂R₂₃| = 1.22e-16; R₁₂⁴ = I: True; R₁₂² = I: False
  [PASS] quarter-turns satisfy the BRAID relation and have order 4, not 2: the Peirce swaps lift to a B₃ representation inside Aut J₃ (a framed lift, not S₃)
  group generated: order 24 (rotational octahedral); its image on the DIAGONAL: 6 permutations; commutant dim on the diagonal = 2 (control: on ℝ³ as vectors = 1, irreducible)
  [PASS] the framed lift acts on the GENERATION DIAGONAL through its S₃ quotient (6 permutations, commutant dim 2 = span{I,J}): c1119's no-go is unchanged there — the framing lives ONLY on the Hom-objects
  [PASS] control: the same group is IRREDUCIBLE on ℝ³ as vectors (commutant dim 1) — the diagonal action is a quotient, not the vector action
  [PASS] R₁₂² = diag(−1,−1,+1) is the identity on the diagonal idempotents and −1 on P₁₃, P₂₃: the half-twist is invisible to a diagonal-only (generation-endomorphism) search

VERDICT (typing, no number): (A) σ is a NULL-SIDE object — it exchanges the null lines of the Higgs plane and fixes the colour
  lines; every A689 line is hyperbolic, so A1548 reopen (2) 'positive complex structure' = a null-cone side choice. (B) J₃(𝕆_s)
  from the table alone reproduces the Peirce closure exactly; the Hom-composition is the octonion product. (C) a functor on the
  Peirce category has a 16-dim choice space and collapses on null weights. (D) the Peirce swaps lift to a braid-relation pair
  with a half-twist on the adjacent P_ik, P_jk; on the diagonal it is still S₃ (span{I,J}). Mixing data cannot live on the diagonal; it must enter
  through Hom(f_i,f_j) — the coend/Kan route of the lens review §3(c). Nothing here is a mass or an angle.

RESULT: ALL GATES PASS
