s1303 — no symplectic map fixes X and reverses D56
  Q1_gradings_and_pieces                       {"D56: symmetric, max |eigenvalue − integer|": ["<1e-12", "<1e-09"], "D56 on 56 (weight: multiplicity)": {"-6": 1, "-2": 27, "2": 27, "6": 1}, "X weights under D56": [-6, -2, 2, 6], "|X_w|² (D56)": {"-6": 84.0, "-2": 252.0, "2": 252.0, "6": 84.0}, "Σ pieces − X; max eigen-residual act_t(D56, X_w) − w X_w": ["<1e-12", "<1e-10"]}
  Q2_omega_pairing                             {"B(X_w, X_w′) / (|X_w||X_w′|), all pairs": {"-6,-6": 0.0, "-6,-2": 0.0, "-6,2": 0.0, "-6,6": -1.0, "-2,-6": 0.0, "-2,-2": 0.0, "-2,2": 1.0, "-2,6": 0.0, "2,-6": 0.0, "2,-2": -1.0, "2,2": 0.0, "2,6": 0.0, "6,-6": 1.0, "6,-2": 0.0, "6,2": 0.0, "6,6": 0.0}, "random e7 element: |gᵀΩg − Ω|, relative change of B(X2, X−2)": ["<1e-12", "<1e-10"], "CONTROL random GL(56) conjugation: |GᵀΩG − Ω| > 0.1, relative change of B(X2, X−2)": [true, 0.069759], "O: |OᵀΩO + Ω|, (B(O·X2, O·X−2) + B(X2, X−2)) / |B|": ["<1e-12", "<1e-12"]}
  Q3_D56_premises                              {"B(X6, X−6) / |X6|²": 1.0, "B(X2, X−2) / |X2|²": -1.0}
  Q4_Dsol_premises                             {"D_sol: symmetric, max |eigenvalue − integer|": ["<1e-12", "<1e-09"], "D_sol on 56": {"-6": 1, "-2": 27, "2": 27, "6": 1}, "X weights under D_sol": [-6, -2, 2, 6], "|X_w|² (D_sol)": {"-6": 84.0, "-2": 252.0, "2": 252.0, "6": 84.0}, "B(X_w, X_−w) / |X_w|² (D_sol)": {"2": -1.0, "6": 1.0}, "O D_sol O⁻¹ + D_sol": "<1e-12"}
  Q5_the_line                                  {"λ = 1": {"B(X2, λX−2) / |X2|², B(X6, λX−6) / |X6|²": [-1.0, 1.0], "gauge algebra (dim, derived, Killing +,−,0)": [28, 28, [12, 16, 0]]}, "λ = 0.5": {"B(X2, λX−2) / |X2|², B(X6, λX−6) / |X6|²": [-0.5, 0.5], "gauge algebra (dim, derived, Killing +,−,0)": [28, 28, [12, 16, 0]]}, "λ = 2": {"B(X2, λX−2) / |X2|², B(X6, λX−6) / |X6|²": [-2.0, 2.0], "gauge algebra (dim, derived, Killing +,−,0)": [28, 28, [12, 16, 0]]}, "λ = -0.5": {"B(X2, λX−2) / |X2|², B(X6, λX−6) / |X6|²": [0.5, -0.5], "gauge algebra (dim, derived, Killing +,−,0)": [28, 28, [0, 28, 0]]}, "λ = -1": {"B(X2, λX−2) / |X2|², B(X6, λX−6) / |X6|²": [1.0, -1.0], "gauge algebra (dim, derived, Killing +,−,0)": [28, 28, [0, 28, 0]]}, "λ = -2": {"B(X2, λX−2) / |X2|², B(X6, λX−6) / |X6|²": [2.0, -2.0], "gauge algebra (dim, derived, Killing +,−,0)": [28, 28, [0, 28, 0]]}}
  Q6_O_realises_the_anti_symplectic_reversal   {"O·X − X": "<1e-12", "O D56 O⁻¹ + D56": "<1e-12", "max_w |O·X_w − X_−w|": "<1e-12", "OᵀΩO + Ω": "<1e-12", "O² − I": "<1e-12"}
  receipt   PASS filed_d1414_house_frames_npz
  receipt   PASS s1288_results_machinery_actX_act_t_basis_inertia_algebra
  computed  PASS Q1_D56_has_weights_6_2_minus2_minus6_on_X_with_norms_84_252_252_84_and_the_pieces_resolve_X
  computed  PASS Q2_only_opposite_pieces_pair_B_X6_Xm6_is_plus_one_and_B_X2_Xm2_minus_one_in_units_of_the_norms_and_B_is_symplectic_invariant_and_O_odd
  computed  PASS Q2_CONTROL_a_non_symplectic_conjugation_moves_B_so_the_invariance_is_not_vacuous
  computed  PASS Q3_THEOREM_PREMISE_the_opposite_D56_pairings_are_nonzero_so_no_symplectic_map_fixes_X_and_reverses_D56
  computed  PASS Q4_D_sol_too_has_a_nonzero_opposite_pairing_so_no_symplectic_map_fixes_X_and_reverses_D_sol
  computed  PASS Q5_on_the_line_the_pairing_scales_as_lambda_and_the_only_B_allowed_symplectic_reversal_target_X_minus_lambda_has_the_other_real_form
  computed  PASS Q6_O_fixes_X_reverses_D56_exchanges_the_pieces_and_is_an_anti_symplectic_involution
  argument  ARG  theorem
  argument  ARG  reading
  argument  ARG  scope
VERDICT: B(X2, X−2) = −|X2|² and B(X6, X−6) = +|X6|² are non-zero, and a symplectic reversal of D56 fixing X would force them to vanish — so none exists in Sp(56, ℝ), let alone in E7(7); the same for D_sol and along the whole λ-line. The Möbius monodromy is necessarily anti-symplectic (O realises it).
RESULT_HASH 0e0f49335ac4d5c0
s1303: 7 computed, 2 receipt, 3 argument, 0 failed
