s1213 — B-frame globally over O(4,4)
receipts {'d692': 'f72e724f81f24ca9', 'd708': '792b12b968d39a74'}
  computed  PASS   numpy_octonions_match_d692   [τ(e_i,e_j,e_k) numpy vs d692.tau0, all 512]
  computed  PASS   baseline_and_sigma_valid   [identity and σ against the 76 equations]
  computed  PASS   T1_T2_equation_coverage_all_V6xV6_pairs   [e0-slot pairs 36/36, e7-slot pairs 36/36]
  computed  PASS   T1_identity_tau_uuu_equals_N_Re   [200 random u: τ(u,u,u) = N(u) Re(u)]
  computed  PASS   T1_null_v_has_rank4_left_multiplication_and_Lv2_is_v2   [200 random null imaginary v: rank L_v = 4, L_v² = −N(v)]
  argument  ARG    T1_proof
  computed  PASS   T2_J_paracomplex_antiisometry_null_eigenspaces   [J = L_{ē7}|V6: J² = 1, JᵀηJ = −η, V± 3-dim totally null, η pairs V₊ with V₋ nondegenerately]
  computed  PASS   T2_every_GL3_frame_is_valid   [50 random A ∈ GL(3): isometry, [h, J] = 0, max equation residual 4.3e-14]
  computed  PASS   g2_split_has_dim_14_and_preserves_eta   [dim Der(𝕆ₛ) = 14]
  computed  PASS   T3_orbit_dim_6_and_stabilizer_sl3_traceless_on_Vplus   [dim g2·e7 = 6, dim stab = 8, max |tr on V₊| = 2.2e-16]
  computed  PASS   T3_G2_times_GL3_frames_all_valid   [30 random exp(g2)·GL(3) frames, max equation residual 1.5e-12]
  computed  PASS   T3_G2_transitive_on_N_minus_1_imaginaries_constructive   [40 random u with N(u) = −1: an automorphism g with g e7 = u built from a basic triple (e7, e1, e2); max defect 1.3e-13]
  argument  ARG    T3_transitivity_standard
  computed  PASS   T4_psi_well_defined_on_all_384_and_automorphisms_have_psi_plus1   [ψ over the 384 valid signed permutations: {'-1': 192, '1': 192}; ψ over the 192 automorphisms: {1}]
  computed  PASS   T4_two_components_sigma_in_the_other   [ψ(σ) = -1; components hit by signed permutations: ['-1', '1']]
  computed  PASS   T4_psi_constant_along_identity_component_path   [ψ along exp(tX)·GL3(diag(1+t,1,1)), t ∈ [0,1]: {1}]
  computed  PASS   T5_commutant_is_span_P07_P6_and_valid_part_is_1_and_colour_flip   [commutant dim 2 = span{P_{e0,e7}, P_{V6}}: True; valid (M e0 = e0) diag(a, b·1₆, a): 2]
  computed  PASS   crosslink_compact_su3c_is_not_in_so44   [max |Tᵀη + ηT| over the 8 colour generators = 1.155]
  argument  ARG    crosslink_reading

VERDICT S = {M ∈ O(4,4) keeping the 76 equations} fixes e0 (T1), equals G2(2)·GL(3,R)_{e7}, dim 15 (T2–T3), has two components with ψ(σ) = -1 and signed permutations in both (ψ counts {'-1': 192, '1': 192}) (T4); the compact-gauge commutant is {1, colour flip} over all of S (T5). The App X caveat is discharged; the compact SU(3)_c is not an O(4,4) object.
RESULT_HASH 10ea075f3f172a7c
s1213: 16 computed PASS-able, 3 argument, 0 failed
