s1159 — ladder-centralizer census (house-computed; A1568 §0 gates)
── carrier 27: J3(O_s), g = str_0(J) ──
  [PASS] G1.27 representation: dim Der = 52, dim str_0 = 78, brackets close  Der 52, e6 78, closure 1.3e-15
  [PASS] G1.27b ambient trace form on str_0: signature (42,36,0) ⇒ split real form e6(6) (maximal compact 36 = sp(4))  sig (42, 36, 0)
  [PASS] G2.27 S_27 = L_vac − (tr/27)I lies in str_0 (residual to the span)  res 2.1e-15
  [PASS] G2.27b spectrum of L_vac on the 27 = {φ,1,φ⁻¹ ×1; φ²/2, √5/2, φ/2 ×8} (the golden ladder of App I :342 lives on the three 8-blocks)  {0.618034: 1, 0.809017: 8, 1.0: 1, 1.118034: 8, 1.309017: 8, 1.618034: 1}
  [PASS] G3.27 centralizer C(L_vac) ⊂ e6(6): nullspace dim printed; every basis element commutes; basis independent  dim 30, max‖[X,L]‖ 6.7e-16
  [PASS] G4.27 C(L_vac) closes under brackets  res 9.9e-16
  [PASS] G5.27 real form: ambient trace form on C(L_vac) has signature EXACTLY (18,12,0), centre EXACTLY 2, derived algebra [C,C] of dim 28 ⇒ so(4,4) [(16,12)] ⊕ ℝ² (name from bracket structure + A_SUB_010 certificate)  sig (18, 12, 0), centre 2, dim[C,C] 28, compact part 12
  [PASS] G6.27 theory-selected elements Y=L_diag(1,1,−2) (App I :342) and L_K, K=diag(−1,0,1) (App I :560) lie in C(L_vac) and are NONCOMPACT (Tr X² > 0)  resY 1.3e-15 TrY² 18.000; resK 1.3e-15 TrK² 6.000
  [PASS] G7.27 candidate exhibited: a compact J_L ∈ C(L_vac) with J_L² = −I EXACTLY on an explicit invariant 2-plane inside the φ/2 eigenblock, [J_L,L_vac]=0; a second exhibit on a DIFFERENT plane ⇒ the (plane, J_L) candidate set is not a single point; no registered invariant selects among them (G6: all registered elements noncompact) ⇒ outcome (c) FREE  exhibits 2, planes distinct True; compact SUBALGEBRA dim 12 (a subalgebra dimension, not the candidate-variety dimension)
── carrier 56: certified E7(7) 56, SL(8,R) frame ──
  [PASS] G1.56 representation: 133 independent generators, symplectic (XᵀΩ+ΩX=0), brackets close  rank 133, symp 0.0e+00
  [PASS] G1.56b ambient trace form on e7: signature (70,63,0) ⇒ split E7(7) (v1 V3 reproduced)  sig (70, 63, 0)
  [PASS] G2.56 H = ρ_sl8(diag(1,1,1,1,1,1,−3,−3))/2 ∈ e7 (App H :78–:82) with weights {−3:1, −1:27, +1:27, +3:1} (App H :122)  res 1.5e-15; weights {-3.0: 1, -1.0: 27, 1.0: 27, 3.0: 1}
  [PASS] G3.56 C(H) ⊂ e7(7): dim printed; commutes; independent  dim 79
  [PASS] G4.56 C(H) closes  res 1.8e-16
  [PASS] G5.56 real form: ambient form on C(H) has signature EXACTLY (43,36,0), centre EXACTLY 1 (ℝH), derived algebra [C,C] dim 78 (all pairs) ⇒ e6(6) [(42,36)] ⊕ so(1,1)  sig (43, 36, 0), centre 1, dim[C,C] 78, compact part 36
  [PASS] G6.56 grading spaces g_{±4} = {x: [D_56,x] = ±4x} are 27-dimensional (App H :126–:127)  dim g+4 27, g−4 27
  [PASS] G7.56 Jacobson–Morozov: a regular E ∈ g_{+4} and F ∈ g_{−4} with [E,F] = H (residual); K = E − F is COMPACT (Tr K² < 0, imaginary spectrum) and [H,K] ≠ 0 ⇒ the registered compact clock generator is NOT in C(H)  [E,F]−H res 4.1e-15; TrK² -72.000; ‖[H,K]‖ 52.575
  [PASS] G8.56 compact part of C(H) is positive-dimensional (a 36-dim SUBALGEBRA, the maximal compact of e6(6)); the registered compact element Z_2 (App G, TYPED) lies in C(H) but App G registers its charge grading as INDEPENDENT of the D_KK grading ⇒ a compact line is SELECTED, the relative angle per rung is NOT ⇒ outcome (c) FREE for the corrected reason  compact subalgebra dim 36; Z_2 typed (App G :49–:52, :92, :124–:133)
  [PASS] G9 LOSS GATE (A1567) over ALL SIX carrier rows (2 computed + 4 typed, typed rows are declared inputs with locators): every row in (c)/(d) ⇒ the programme LOSES in the registered class at Rev32.5 — no theory-selected (J_L, α) in any ladder-scaling centralizer. Certification: COMPUTED for the 27 and the 56; TYPED for the other four (this gate does not upgrade a typed row)  rows {'27': '(c)', '56': '(c)', 'V3 plane': '(d)', 'KK tower (App G)': '(c)', 'Bergman/resolvent (p2)': '(c)', 'braid iterate (App H)': '(c)'}
  [PASS] V1 AST scan of this kernel: no check() predicate is a literal, contains a literal conjunct, or compares literal to literal  vacuous predicates found: []
SEMANTIC_SHA256: 27c1b3d15c81ab4b9be54d877d1b7c434e104d481f03e9348cd6e85488eb91e5
RESULT: 20/20 gates pass
