s1270 — the relative sign: the ω = π/4 dyonic SO(6,2), its SU*(8) kernel, and the SL(8)-electric-type twins
  Q1_normal_form                   {"max ρ-residual and fit residual over the 56 generators": "<1e-12", "electric coefficient on ρ(η₆₂K_ab) (value: count)": {"-0.707107": 28}, "magnetic coefficient on ρ(K_abη₆₂) (value: count)": {"-0.707107": 28}, "‖X − (±T(η₆₂, ±π/4))‖_max": {"+T(η₆₂, +π/4)": 1.414214, "+T(η₆₂, −π/4)": 1.414214, "−T(η₆₂, +π/4)": 0.0, "−T(η₆₂, −π/4)": 1.414214}, "‖X electric‖ − ‖X magnetic‖": "<1e-12", "dyonic angle |ω| from norms (rad / π)": 0.25, "quadratic constraints of T(η, ω) at ω = 0, 0.3, π/4, 1.1 (max of both)": {"(6,2) odd 78": ["<1e-12", "<1e-12", "<1e-12", "<1e-12"], "(6,2) odd 12": ["<1e-12", "<1e-12", "<1e-12", "<1e-12"], "(8,0)": ["<1e-12", "<1e-12", "<1e-12", "<1e-12"], "(4,4)": ["<1e-12", "<1e-12", "<1e-12", "<1e-12"], "(5,3)": ["<1e-12", "<1e-12", "<1e-12", "<1e-12"], "(7,1)": ["<1e-12", "<1e-12", "<1e-12", "<1e-12"]}}
  Q2_census_is_the_omega_family    {"consistent classes": 8, "members": {"(1, 1, 1, 1, 1, 1)": {"matches": ["−T((6,2) odd 78, +π/4)"], "kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [27, 36, 0]}, "(1, 1, 1, 1, -1, -1)": {"matches": ["−T((4,4), +π/4)"], "kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "(1, 1, -1, -1, 1, 1)": {"matches": ["+T((6,2) odd 12, +π/4)"], "kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [27, 36, 0]}, "(1, 1, -1, -1, -1, -1)": {"matches": ["+T((8,0), +π/4)"], "kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "(1, -1, 1, -1, 1, -1)": {"matches": ["+T((8,0), −π/4)"], "kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "(1, -1, 1, -1, -1, 1)": {"matches": ["+T((6,2) odd 12, −π/4)"], "kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [27, 36, 0]}, "(1, -1, -1, 1, 1, -1)": {"matches": ["−T((4,4), −π/4)"], "kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "(1, -1, -1, 1, -1, 1)": {"matches": ["−T((6,2) odd 78, −π/4)"], "kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [27, 36, 0]}}, "plane swap ρ((1 7)(2 8)): ‖g·T(62|78, ±π/4) − T(62|12, ±π/4)‖, det, symplectic": [["<1e-12", "<1e-12"], 1.0, "<1e-12"]}
  Q3_kernel_stabiliser_table       {"(6,2): stabiliser dim at ω = kπ/32": {"0": 63, "1": 28, "2": 28, "3": 28, "4": 28, "5": 28, "6": 28, "7": 28, "8": 63, "9": 28, "10": 28, "11": 28, "12": 28, "13": 28, "14": 28, "15": 28, "16": 63}, "table": {"(6,2) odd 78": {"0": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "+π/4": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [27, 36, 0]}, "−π/4": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [27, 36, 0]}, "π/2": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}}, "(8,0)": {"0": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "+π/4": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "−π/4": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "π/2": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}}, "(4,4)": {"0": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "+π/4": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "−π/4": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "π/2": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}}, "(5,3)": {"0": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "+π/4": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 28, "Tr56 inertia": [15, 13, 0]}, "−π/4": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 28, "Tr56 inertia": [15, 13, 0]}, "π/2": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}}, "(7,1)": {"0": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}, "+π/4": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 28, "Tr56 inertia": [7, 21, 0]}, "−π/4": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 28, "Tr56 inertia": [7, 21, 0]}, "π/2": {"kernel dim": 28, "Lagrangian residual": "<1e-12", "stabiliser dim": 63, "Tr56 inertia": [35, 28, 0]}}}}
  Q4_su_star_8                     {"stabiliser (dim, derived, Tr56 inertia)": [63, 63, [27, 36, 0]], "rank (generic centraliser in the stabiliser)": 7, "ad-commutant dim (two generic elements)": 1, "gauge algebra ⊂ stabiliser (residual)": "<1e-10", "intersections with the e6(6) ⊕ ℝ of each grading": {"D56": {"centraliser dim": 79, "∩ stabiliser": [36, 36, [0, 36, 0]], "rank": 4, "ad-commutant dim": 1}, "D_sol": {"centraliser dim": 79, "∩ stabiliser": [36, 36, [16, 20, 0]], "rank": 4, "ad-commutant dim": 1}}}
  receipt   PASS filed_d1414_house_frames_npz
  computed  PASS Q1_X_is_exactly_minus_T_eta62_pi4_the_omega_deformed_dyonic_so62_at_dyonic_angle_pi4_electric_eta_K_magnetic_K_eta
  computed  PASS Q2_the_eight_census_members_are_exactly_plus_minus_T_eta_plus_minus_pi4_so62_kernels_su_star_8_twins_sl8
  computed  PASS Q3_at_pi4_the_kernel_stabiliser_follows_p_minus_q_mod_8_sl8R_for_0_su_star_8_for_4_gauge_only_for_2_and_6_and_omega_0_pi2_are_sl8R
  computed  PASS Q4_the_registered_kernel_stabiliser_is_su_star_8_containing_the_gauge_algebra_meeting_the_D56_e6_in_usp8_and_the_Dsol_e6_in_sp22
  argument  ARG  what_the_relative_sign_is
  argument  ARG  what_separates_and_what_does_not
  argument  ARG  scope
VERDICT: X = −T(η₆₂, π/4) — the ω-deformed SO(6,2) at dyonic angle π/4; the eight census members are ±T(η, ±π/4); the kernel stabiliser (an E7-invariant) is su*(8) for every so(6,2) member and sl(8,ℝ) for the so(8) / so(4,4) twins, so the registered gauging is not E7(ℝ)-conjugate to any SL(8)-electric gauging while the twins pass that test; across signatures the π/4 kernel follows p − q mod 8. The constraints do not select η; the invariant separates it. Algebraic only.
RESULT_HASH a9334cef694a194a
s1270: 4 computed, 1 receipt, 3 argument, 0 failed
