s1308 — house checks of A1787 (the Gemini Möbius / Spin(4,4) thought)
  Q1_gauge_algebras                  {"X (registered, λ = 1)": {"gauge algebra dim, Killing (+, −)": [28, [12, 16]], "in ρ(sl(8)); O h O in ρ(sl(8))": ["<1e-12", "<1e-12"], "η signature (+, −); η-null residual": [6, 2, "<1e-12"], "g solves φ(A) = gAg⁻¹: second-smallest / largest singular value, max residual": ["gap 0.866", "<1e-12"], "det g > 0; g preserves η up to scale": [true, "<1e-10"], "g ∝ I": true, "sign det g₊₊, sign det g₋₋ (η-blocks)": [1, 1]}, "D56-flipped twin X₊ − X₋": {"gauge algebra dim, Killing (+, −)": [28, [0, 28]], "in ρ(sl(8)); O h O in ρ(sl(8))": ["<1e-12", "<1e-12"], "η signature (+, −); η-null residual": [8, 0, "<1e-12"], "g solves φ(A) = gAg⁻¹: second-smallest / largest singular value, max residual": ["gap 0.866", "<1e-12"], "det g > 0; g preserves η up to scale": [true, "<1e-10"], "g ∝ I": false, "sign det g₊₊, sign det g₋₋ (η-blocks)": [1, 0]}, "D_sol-flipped twin X₊ − X₋": {"gauge algebra dim, Killing (+, −)": [28, [16, 12]], "in ρ(sl(8)); O h O in ρ(sl(8))": ["<1e-12", "<1e-12"], "η signature (+, −); η-null residual": [4, 4, "<1e-12"], "g solves φ(A) = gAg⁻¹: second-smallest / largest singular value, max residual": ["gap 0.866", "<1e-12"], "det g > 0; g preserves η up to scale": [true, "<1e-10"], "g ∝ I": false, "sign det g₊₊, sign det g₋₋ (η-blocks)": [1, 1]}}
  Q2_F0_commutant_and_three_eights   {"dim sl(2)³; ideal dims": [9, [3, 3, 3]], "O Iⱼ O ⊂ Iₖ: residual matrix (rows j, cols k; 0 = contained)": [["<1e-10", "1.000e+00", "1.000e+00"], ["1.000e+00", "<1e-10", "1.000e+00"], ["1.000e+00", "1.000e+00", "<1e-10"]], "centraliser in e7: dim, Killing (+, −); O-stable (residual)": [28, [16, 12], "<1e-10"], "Vₖ dims; O Vⱼ ⊂ Vₖ residual matrix": [[16, 16, 16], [["<1e-10", "1.000e+00", "1.000e+00"], ["1.000e+00", "<1e-10", "1.000e+00"], ["1.000e+00", "1.000e+00", "<1e-10"]]], "Vₖ orthonormal columns; span of the three = 48": ["<1e-10", 48]}
  Q3_O_versus_J0                     {"O Ω + Ω O (antilinear)": "<1e-12", "CONTROL O Ω − Ω O": "2.000e+00", "Ω² + I": "<1e-12"}
  Q4_lambda_to_one_over_lambda       {"D56": {"λ = 2: O·X_λ − λ·X_{1/λ}": "<1e-12", "λ = 0.5: O·X_λ − λ·X_{1/λ}": "<1e-12", "λ = -3: O·X_λ − λ·X_{1/λ}": "<1e-12", "O·X₊₁ − X₊₁; O·X₋₁ + X₋₁": ["<1e-12", "<1e-12"], "CONTROL λ = 2: |O·X_λ − c X_λ| / |X_λ| (best c)": "6.000e-01"}, "D_sol": {"λ = 2: O·X_λ − λ·X_{1/λ}": "<1e-12", "λ = 0.5: O·X_λ − λ·X_{1/λ}": "<1e-12", "λ = -3: O·X_λ − λ·X_{1/λ}": "<1e-12", "O·X₊₁ − X₊₁; O·X₋₁ + X₋₁": ["<1e-12", "<1e-12"], "CONTROL λ = 2: |O·X_λ − c X_λ| / |X_λ| (best c)": "6.000e-01"}}
  receipt   PASS filed_d1414_house_frames_npz
  receipt   PASS filed_chatgpt_d1417_maps_npz
  receipt   PASS s1288_results_machinery_actX_rho
  receipt   PASS s1283_results_machinery_closure_ideal_sl2r_cubed
  computed  PASS Q1_on_X_and_both_flipped_twins_Ad_O_is_conjugation_by_a_unique_det_positive_element_of_the_identity_component_of_O_eta_so_no_half_spin_eight_is_swapped
  computed  PASS Q2_O_preserves_each_sl2_factor_of_F0_and_each_of_the_three_16_dim_pieces_2k_tensor_8k_so_it_permutes_none_of_the_three_eights_of_the_so44_commutant
  computed  PASS Q3_O_anticommutes_with_J0_equal_Omega_so_it_exchanges_the_8_and_the_8bar_of_SU8
  computed  PASS Q4_O_maps_X_lambda_to_lambda_times_X_one_over_lambda_on_both_lines_with_projective_fixed_points_exactly_plus_and_minus_one
  argument  ARG  reading
  argument  ARG  scope
VERDICT: O swaps no half-spin eight of any registered so(4,4) / so(8)-type algebra (identity-component action on the gauge algebras; each eight of F0's so(4,4) preserved); it reverses the SU(8) complex structure (8 ↔ 8̄, handedness); and O·X_λ = λ·X_{1/λ}, fixed projectively exactly at λ = ±1.
RESULT_HASH 2e2cb305e904fa89
s1308: 4 computed, 4 receipt, 2 argument, 0 failed
