s1266 — the two oriented 5D → 4D reductions: O-mirror, free relative sign, so(8) / so(4,4) twins, the eight-member census, the core, the T² signature
  Q1_O_mirror        {"‖O·X − X‖": "<1e-12", "O D56 O⁻¹ + D56": "<1e-12", "O D_sol O⁻¹ + D_sol": "<1e-12", "OᵀΩO + Ω (anti-symplectic)": "<1e-12", "[D56, D_sol]": "<1e-12", "D56: pieces sum − X": "<1e-12", "D56: O·X⁻ − X⁺": "<1e-12", "D_sol: pieces sum − X": "<1e-12", "D_sol: O·X⁻ − X⁺": "<1e-12"}
  Q2_pencils         {"D56 (1,1)": {"closure": "<1e-10", "Ω-constraint": "<1e-10", "(dim, derived, Killing)": [28, 28, [12, 16, 0]], "ad-commutant": 1, "rank": 4, "real form (D4 by Killing)": "so(6,2)"}, "D56 (1,-1)": {"closure": "<1e-10", "Ω-constraint": "<1e-10", "(dim, derived, Killing)": [28, 28, [0, 28, 0]], "ad-commutant": 1, "rank": 4, "real form (D4 by Killing)": "so(8)"}, "D56 (1,2)": {"closure": "<1e-10", "Ω-constraint": "<1e-10", "(dim, derived, Killing)": [28, 28, [12, 16, 0]]}, "D56 (2,-1)": {"closure": "<1e-10", "Ω-constraint": "<1e-10", "(dim, derived, Killing)": [28, 28, [0, 28, 0]]}, "D56 (1,0)": {"closure": "<1e-10", "Ω-constraint": "<1e-10", "(dim, derived, Killing)": [28, 27, [0, 16, 12]]}, "D56 (0,1)": {"closure": "<1e-10", "Ω-constraint": "<1e-10", "(dim, derived, Killing)": [28, 27, [0, 16, 12]]}, "D56 O-parity": {"X⁻ + X⁺ O-even": "<1e-12", "X⁻ − X⁺ O-odd": "<1e-12"}, "D_sol (1,1)": {"closure": "<1e-10", "Ω-constraint": "<1e-10", "(dim, derived, Killing)": [28, 28, [12, 16, 0]], "ad-commutant": 1, "rank": 4, "real form (D4 by Killing)": "so(6,2)"}, "D_sol (1,-1)": {"closure": "<1e-10", "Ω-constraint": "<1e-10", "(dim, derived, Killing)": [28, 28, [16, 12, 0]], "ad-commutant": 1, "rank": 4, "real form (D4 by Killing)": "so(4,4)"}, "D_sol (1,2)": {"closure": "<1e-10", "Ω-constraint": "<1e-10", "(dim, derived, Killing)": [28, 28, [12, 16, 0]]}, "D_sol (2,-1)": {"closure": "<1e-10", "Ω-constraint": "<1e-10", "(dim, derived, Killing)": [28, 28, [16, 12, 0]]}, "D_sol (1,0)": {"closure": "<1e-10", "Ω-constraint": "<1e-10", "(dim, derived, Killing)": [28, 27, [8, 8, 12]]}, "D_sol (0,1)": {"closure": "<1e-10", "Ω-constraint": "<1e-10", "(dim, derived, Killing)": [28, 27, [8, 8, 12]]}, "D_sol O-parity": {"X⁻ + X⁺ O-even": "<1e-12", "X⁻ − X⁺ O-odd": "<1e-12"}}
  Q3_bigrading       {"joint eigenbasis residual": "<1e-12", "56 bigraded multiplicities": [[[-6, 2], 1], [[-2, -2], 10], [[-2, 2], 16], [[-2, 6], 1], [[2, -6], 1], [[2, -2], 16], [[2, 2], 10], [[6, -2], 1]], "X bigraded weights present": [[-6, 2], [-2, -2], [-2, 6], [2, -6], [2, 2], [6, -2]], "X piece norm² fractions": {"(-6, 2)": 0.125, "(6, -2)": 0.125, "(-2, -2)": 0.25, "(2, 2)": 0.25, "(-2, 6)": 0.125, "(2, -6)": 0.125}, "pieces sum − X": "<1e-12", "consistent sign classes (of 32)": 8, "real-form classes": {"so(6,2)": 4, "so(4,4)": 2, "so(8)": 2}, "members": {"(1, 1, 1, 1, 1, 1)": {"(dim, derived, Killing)": [28, 28, [12, 16, 0]], "ad-commutant": 1, "rank": 4, "real form": "so(6,2)", "O-parity": "even"}, "(1, 1, 1, 1, -1, -1)": {"(dim, derived, Killing)": [28, 28, [16, 12, 0]], "ad-commutant": 1, "rank": 4, "real form": "so(4,4)", "O-parity": "even"}, "(1, 1, -1, -1, 1, 1)": {"(dim, derived, Killing)": [28, 28, [12, 16, 0]], "ad-commutant": 1, "rank": 4, "real form": "so(6,2)", "O-parity": "even"}, "(1, 1, -1, -1, -1, -1)": {"(dim, derived, Killing)": [28, 28, [0, 28, 0]], "ad-commutant": 1, "rank": 4, "real form": "so(8)", "O-parity": "even"}, "(1, -1, 1, -1, 1, -1)": {"(dim, derived, Killing)": [28, 28, [0, 28, 0]], "ad-commutant": 1, "rank": 4, "real form": "so(8)", "O-parity": "odd"}, "(1, -1, 1, -1, -1, 1)": {"(dim, derived, Killing)": [28, 28, [12, 16, 0]], "ad-commutant": 1, "rank": 4, "real form": "so(6,2)", "O-parity": "odd"}, "(1, -1, -1, 1, 1, -1)": {"(dim, derived, Killing)": [28, 28, [16, 12, 0]], "ad-commutant": 1, "rank": 4, "real form": "so(4,4)", "O-parity": "odd"}, "(1, -1, -1, 1, -1, 1)": {"(dim, derived, Killing)": [28, 28, [12, 16, 0]], "ad-commutant": 1, "rank": 4, "real form": "so(6,2)", "O-parity": "odd"}}}
  Q4_contractions    {"D56 X⁻": {"dim": 28, "radical dim": 13, "radical abelian": false, "radical is an ideal": true, "Levi Killing inertia": [0, 15, 0]}, "D56 X⁺": {"dim": 28, "radical dim": 13, "radical abelian": false, "radical is an ideal": true, "Levi Killing inertia": [0, 15, 0]}, "D_sol X⁻": {"dim": 28, "radical dim": 13, "radical abelian": false, "radical is an ideal": true, "Levi Killing inertia": [8, 7, 0]}, "D_sol X⁺": {"dim": 28, "radical dim": 13, "radical abelian": false, "radical is an ideal": true, "Levi Killing inertia": [8, 7, 0]}, "(1, 0, 1, 0, 0, 1)": {"dim": 28, "radical dim": 16, "radical abelian": true, "radical is an ideal": true, "Levi Killing inertia": [4, 8, 0], "closure, Ω": ["<1e-10", "<1e-10"]}, "(1, 0, 1, 0, 0, -1)": {"dim": 28, "radical dim": 16, "radical abelian": true, "radical is an ideal": true, "Levi Killing inertia": [0, 12, 0], "closure, Ω": ["<1e-10", "<1e-10"]}}
  Q5_core            {"dim": 8, "closure": "<1e-12", "derived dim": 6, "derived algebra (dim, derived, Killing)": [6, 6, [0, 6, 0]], "centre dim": 2, "centre ⊂ span(anchors)": "<1e-09", "anchors ⊂ centre": "<1e-09", "[core, D56], [core, D_sol]": ["<1e-12", "<1e-12"], "anchors: commute, Frobenius overlap": ["<1e-12", "<1e-12"]}
  Q6_T2_signature    {"centraliser of D56": {"dim": 79, "derived": [78, 78, [42, 36, 0]], "centre dim": 1}, "centraliser of D_sol": {"dim": 79, "derived": [78, 78, [42, 36, 0]], "centre dim": 1}, "centraliser of S": {"dim": 67, "derived": [66, 66, [36, 30, 0]], "centre dim": 1}, "centraliser of {D56, D_sol}": {"dim": 47, "derived": [45, 45, [25, 20, 0]], "centre dim": 2}, "S 56-spectrum": [[-1.0, 12], [0.0, 32], [1.0, 12]], "commutant of that so(5,5)": {"dim": 4, "derived": [3, 3, [2, 1, 0]], "centre dim": 1}, "S in the sl(2) part, Δ in the commutant": ["<1e-09", "<1e-09"], "Δ in the sl(2) part (expected NOT)": 1.0, "X S-weight norm² fractions": {"-1": 0.5, "1": 0.5}, "sl(2): ad_S eigenvalues, [e,f] − h": [[-2.0, 0.0, 2.0], "<1e-10"], "Weyl: W S W⁻¹ + S, W Δ W⁻¹ − Δ": ["<1e-10", "<1e-10"], "Weyl: W D56 W⁻¹ + D_sol, W D_sol W⁻¹ + D56, WᵀΩW − Ω": ["<1e-10", "<1e-10", "<1e-10"]}
  Q7_single_pieces   {"lane CSS comparison basis": [[13, 12, [0, 1, 12]], 16], "(-6, 2)": [[13, 12, [0, 1, 12]], 16], "(6, -2)": [[13, 12, [0, 1, 12]], 16], "(-2, -2)": [[22, 22, [0, 6, 16]], 6], "(2, 2)": [[22, 22, [0, 6, 16]], 6], "(-2, 6)": [[13, 12, [0, 1, 12]], 16], "(2, -6)": [[13, 12, [0, 1, 12]], 16]}
  receipt   PASS filed_d1414_house_frames_npz
  receipt   PASS filed_chatgpt_d1414_maps_npz
  computed  PASS Q1_X_is_the_O_mirror_sum_of_one_oriented_half_in_both_gradings
  computed  PASS Q2_the_relative_sign_is_free_same_sign_so62_opposite_so8_D56_or_so44_Dsol_one_half_CSO
  computed  PASS Q3_eight_of_32_sign_classes_are_consistent_4_so62_2_so44_2_so8_no_so71_no_so53
  computed  PASS Q4_halves_have_radical13_Levi_so6_or_so42_double_contractions_radical16_abelian_Levi_4_8_and_0_12
  computed  PASS Q5_the_core_of_all_eight_members_is_su2_su2_u1_u1_with_centre_spanned_by_the_two_anchors
  computed  PASS Q6_the_two_gradings_carry_the_T2_signature_joint_Levi_so55_commutant_gl2_Weyl_sends_D56_to_minus_Dsol
  computed  PASS Q7_the_four_extreme_bigraded_pieces_have_the_CSS_comparison_invariants_13_12_0_1_12_fixed16
  argument  ARG  what_it_answers
  argument  ARG  what_symmetry_it_implies
  argument  ARG  the_T2_signature
VERDICT: X = X⁻ + O·X⁻ in both gradings; the relative sign of the halves is free (whole pencil consistent): equal → so(6,2), opposite → so(8) (D56) / so(4,4) (D_sol), one half → CSO; the bigraded census gives 8 consistent members (4 so(6,2), 2 so(4,4), 2 so(8)); their common core is su(2) ⊕ su(2) ⊕ u(1) ⊕ u(1) with the two anchors as centre; the two gradings carry the T² signature (joint Levi so(5,5), commutant gl(2), Weyl D56 → −D_sol). Algebraic only.
RESULT_HASH 1979ecc0adb55c53
s1266: 7 computed, 2 receipt, 3 argument, 0 failed
