s1273 — the sign of the dyonic angle, the quarter-turn and the end flip
  Q1_parity                        {"PᵀΩP + Ω": "<1e-12", "P normalises e7 (residual)": "<1e-12", "I₄(Pv) vs I₄(v) (relative)": "<1e-12", "P·T(η₆₂, ω) − T(η₆₂, −ω) at ω = 0.3, π/4, 1.1": ["<1e-12", "<1e-12", "<1e-12"]}
  Q2_no_quadratic_odd_invariant    {"number of contractions": 12, "max |value| per tensor": {"X": "<1e-09", "ω=0": "<1e-09", "ω=0.3": "<1e-09", "ω=+π/4": "<1e-09", "ω=−π/4": "<1e-09"}}
  Q3_quarter_turn                  {"R(kπ/8) scan (η₆₂)": {"0": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "<1e-10", "I₄ preserved": "<1e-10"}, "1": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "3.464e+00", "I₄ preserved": "1.207e+00"}, "2": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "4.899e+00", "I₄ preserved": "1.542e+00"}, "3": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "3.464e+00", "I₄ preserved": "3.347e-01"}, "4": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "<1e-10", "I₄ preserved": "<1e-10"}, "5": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "3.464e+00", "I₄ preserved": "1.207e+00"}, "6": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "4.899e+00", "I₄ preserved": "1.542e+00"}, "7": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "3.464e+00", "I₄ preserved": "3.347e-01"}, "8": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "<1e-10", "I₄ preserved": "<1e-10"}, "9": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "3.464e+00", "I₄ preserved": "1.207e+00"}, "10": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "4.899e+00", "I₄ preserved": "1.542e+00"}, "11": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "3.464e+00", "I₄ preserved": "3.347e-01"}, "12": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "<1e-10", "I₄ preserved": "<1e-10"}, "13": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "3.464e+00", "I₄ preserved": "1.207e+00"}, "14": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "4.899e+00", "I₄ preserved": "1.542e+00"}, "15": {"symplectic": "<1e-12", "commutes with the 56 gauge generators": "<1e-12", "normalises e7": "3.464e+00", "I₄ preserved": "3.347e-01"}}, "R(π/2)·T(η, ω) − T(η, ω + π/2) at ω = 0.3, 1.1": {"(6,2)": ["<1e-12", "<1e-12"], "(8,0)": ["<1e-12", "<1e-12"], "(4,4)": ["<1e-12", "<1e-12"], "(5,3)": ["<1e-12", "<1e-12"], "(7,1)": ["<1e-12", "<1e-12"]}}
  Q4_plus_minus_pi4_conjugate      {"R(π/2)·T(η, −π/4) − T(η, +π/4)": {"(6,2)": "<1e-12", "(8,0)": "<1e-12", "(4,4)": "<1e-12", "(5,3)": "<1e-12", "(7,1)": "<1e-12"}, "R(π/2)·(−T(η₆₂, −π/4)) − X": "<1e-12"}
  Q5_O_is_parity_times_quarter_turn {"‖O − candidate‖_max": {"R(π/2)·P": 0.0, "−P·R(π/2)": 0.0, "P·R(−π/2)": 0.0, "P·R(π/2)": 2.0, "−R(π/2)·P": 2.0}, "O·T(η₆₂, ω) − T(η₆₂, π/2 − ω) at ω = 0.3, 1.1": ["<1e-12", "<1e-12"], "O·X − X": "<1e-12", "OᵀΩO + Ω": "<1e-12"}
  receipt   PASS filed_d1414_house_frames_npz
  computed  PASS Q1_the_parity_P_is_anti_symplectic_normalises_e7_preserves_I4_and_reverses_the_dyonic_angle
  computed  PASS Q2_every_quadratic_contraction_with_one_inverse_omega_vanishes
  computed  PASS Q3_the_multiplicity_rotation_normalises_e7_and_preserves_I4_exactly_at_quarter_turns_and_shifts_omega_by_pi2_in_every_signature
  computed  PASS Q4_T_minus_pi4_and_T_plus_pi4_are_conjugate_by_the_exhibited_quarter_turn_in_every_signature_and_the_registered_X_is_reached
  computed  PASS Q5_the_end_flip_O_equals_the_quarter_turn_times_parity_acts_as_omega_to_pi2_minus_omega_and_fixes_the_registered_pi4
  argument  ARG  membership
  argument  ARG  what_it_answers
  argument  ARG  scope
VERDICT: the parity P reverses the dyonic angle; the multiplicity quarter-turn R(π/2) is in E7(ℝ) (symplectic, normalises e7, preserves I₄; exactly at quarter-turns) and shifts ω by π/2 in every signature, so T(η, −π/4) and T(η, +π/4) are E7(ℝ)-conjugate — the sign of ω is not an invariant and supplies no parity-odd bit for χ; the end flip is exactly O = R(π/2)·P, acting as ω ↦ π/2 − ω with the registered π/4 its fixed point. Algebraic only.
RESULT_HASH 511c70d02ead59f0
s1273: 5 computed, 1 receipt, 3 argument, 0 failed
