s1272 — A1775 follow-ups: nilpotent orbits across the SL(8)-frame so(p,q), and (r, Z) under Stab(X)
  Q1_twin_gauge_algebra        {"twin constraints": ["<1e-12", "<1e-12"], "twin gauge span vs ρ(so(η₄₄)) (max residual both ways)": "<1e-10", "twin gauge algebra": [28, 28, [16, 12, 0]]}
  Q2_nilpotent_table           {"per signature (signed diagram → invariants)": {"(4,4)": {"71[++]": {"orbit dim": 114, "56-power ranks": [48, 40, 32, 26, 20, 14, 8, 6, 4, 2, 0]}, "71[−−]": {"orbit dim": 114, "56-power ranks": [48, 40, 32, 26, 20, 14, 8, 6, 4, 2, 0]}, "53[++]": {"orbit dim": 106, "56-power ranks": [44, 32, 20, 14, 8, 4, 0]}, "53[−−]": {"orbit dim": 106, "56-power ranks": [44, 32, 20, 14, 8, 4, 0]}, "5111[++−−]": {"orbit dim": 100, "56-power ranks": [40, 30, 20, 12, 4, 2, 0]}, "5111[−++−]": {"orbit dim": 100, "56-power ranks": [40, 30, 20, 12, 4, 2, 0]}, "44": {"orbit dim": 100, "56-power ranks": [40, 30, 20, 12, 4, 2, 0]}, "3311[++++]": {"orbit dim": 84, "56-power ranks": [36, 20, 4, 2, 0]}, "3311[+−+−]": {"orbit dim": 84, "56-power ranks": [36, 20, 4, 2, 0]}, "3311[−−−−]": {"orbit dim": 84, "56-power ranks": [36, 20, 4, 2, 0]}, "3221[++]": {"orbit dim": 82, "56-power ranks": [32, 14, 4, 0]}, "3221[−−]": {"orbit dim": 82, "56-power ranks": [32, 14, 4, 0]}, "311111[++++−−]": {"orbit dim": 66, "56-power ranks": [24, 12, 0]}, "311111[−++−−−]": {"orbit dim": 66, "56-power ranks": [24, 12, 0]}, "2222": {"orbit dim": 66, "56-power ranks": [24, 12, 0]}, "221111[++−−]": {"orbit dim": 52, "56-power ranks": [20, 2, 0]}}, "(6,2)": {"5111[++++]": {"orbit dim": 100, "56-power ranks": [40, 30, 20, 12, 4, 2, 0]}, "3311[−−++]": {"orbit dim": 84, "56-power ranks": [36, 20, 4, 2, 0]}, "311111[++++++]": {"orbit dim": 66, "56-power ranks": [24, 12, 0]}, "311111[−++++−]": {"orbit dim": 66, "56-power ranks": [24, 12, 0]}, "221111[++++]": {"orbit dim": 52, "56-power ranks": [20, 2, 0]}}, "(5,3)": {"71[−+]": {"orbit dim": 114, "56-power ranks": [48, 40, 32, 26, 20, 14, 8, 6, 4, 2, 0]}, "53[+−]": {"orbit dim": 106, "56-power ranks": [44, 32, 20, 14, 8, 4, 0]}, "5111[+++−]": {"orbit dim": 100, "56-power ranks": [40, 30, 20, 12, 4, 2, 0]}, "5111[−+++]": {"orbit dim": 100, "56-power ranks": [40, 30, 20, 12, 4, 2, 0]}, "3311[+−++]": {"orbit dim": 84, "56-power ranks": [36, 20, 4, 2, 0]}, "3311[−−+−]": {"orbit dim": 84, "56-power ranks": [36, 20, 4, 2, 0]}, "3221[−+]": {"orbit dim": 82, "56-power ranks": [32, 14, 4, 0]}, "311111[+++++−]": {"orbit dim": 66, "56-power ranks": [24, 12, 0]}, "311111[−+++−−]": {"orbit dim": 66, "56-power ranks": [24, 12, 0]}, "221111[+++−]": {"orbit dim": 52, "56-power ranks": [20, 2, 0]}}, "(7,1)": {"311111[−+++++]": {"orbit dim": 66, "56-power ranks": [24, 12, 0]}}, "(8,0)": {}}, "distinct orbit dims": {"(4,4)": [52, 66, 82, 84, 100, 106, 114], "(6,2)": [52, 66, 84, 100], "(5,3)": [52, 66, 82, 84, 100, 106, 114], "(7,1)": [66], "(8,0)": []}, "labels (by dimension; see NOT CLAIMED)": {"52": "2A1", "66": "A2", "82": "A2+2A1", "84": "2A2 (A1775 label; dim also A3, A2+3A1)", "100": "A4 (A1775 label; dim also A3+A2+A1)", "106": "D5(a1) or A4+A2", "114": "A6, D5+A1 or D6(a1)"}, "max residuals": {"so(η) residual": "<1e-10", "form transport residual": "<1e-10", "gauge residual": "<1e-10"}, "Jordan types re-read == partitions": true, "triality pairs in (4,4): (2⁴ vs every 31⁵, 4² vs every 51³) same ranks and orbit dim": [true, true], "(6,2) set ⊂ (4,4) set": true}
  Q3_dehn_orbit_absent         {"ρ(E12): ranks, orbit dim": [[12, 0], 34], "lane's e: ranks, orbit dim": [[12, 0], 34], "e in the registered gauge span (relative residual)": 0.707107, "orbit 34 in any signature list": false, "partition 21⁶ realisable in any so(p,q)": false}
  Q4_stab_and_r_orthogonal     {"stab_e7(X) dim and invariants": [28, 28, [12, 16, 0]], "stab vs gauge span (max residual both ways)": "<1e-10", "X-variation of Z, r": ["<1e-12", 1.0], "trace-form projection of r on the gauge algebra (relative)": "<1e-12", "ρ-preimage of r: residual, r − (E68 − E86), rᵀη − ηr": ["<1e-12", "<1e-12", "<1e-12"], "ρ-preimage of Z: residual, Z − (E87 − E78), Zᵀη + ηZ": ["<1e-12", "<1e-12", "<1e-12"]}
  Q5_gauge_rotation_and_wick   {"ρ(E16 − E61) in the gauge span": "<1e-10", "‖g₀·X − X‖/‖X‖": "<1e-10", "g₀Zg₀⁻¹ − Z": "<1e-10", "g₀rg₀⁻¹ = s·ρ(E18 − E81) for s in": [1], "lane's Q: preimage residual, Q − (E18 + E81)": ["<1e-12", "<1e-12"], "W(E18 + E81)W⁻¹ + i(E18 − E81)": "<1e-12", "g₀ symplectic": "<1e-10"}
  Q6_compact_so3               {"dim span{r, Z, [r, Z]}": 3, "algebra": [3, 3, [0, 3, 0]], "dim of its intersection with the gauge algebra": 1, "that intersection ∝ Z (residual)": "<1e-10"}
  receipt   PASS filed_d1414_house_frames_npz
  receipt   PASS filed_chatgpt_d1415_maps_npz
  computed  PASS Q1_the_Dsol_twin_minus_T_eta44_pi4_is_consistent_and_its_gauge_algebra_is_rho_so_eta44
  computed  PASS Q2_so44_meets_seven_complex_e7_orbits_so62_four_so53_seven_so71_one_so8_none_triality_pairs_coincide
  computed  PASS Q3_the_minimal_dehn_orbit_34_meets_no_sl8_frame_so_pq_and_e_is_not_gauge
  computed  PASS Q4_stab_of_X_is_exactly_the_gauge_so62_Z_lies_in_it_and_r_is_trace_orthogonal_to_it_as_the_rotation_of_the_mixed_plane_68
  computed  PASS Q5_a_gauge_rotation_fixes_X_and_Z_and_carries_r_to_the_wick_partner_of_the_lanes_gauge_boost_Q
  computed  PASS Q6_r_and_Z_span_a_compact_so3_whose_only_gauge_direction_is_Z
  argument  ARG  what_a_closes
  argument  ARG  what_b_closes
  argument  ARG  scope
VERDICT: so(4,4) meets seven nonzero complex E7 nilpotent orbits (52, 66, 82, 84, 100, 106, 114), so(6,2) four, so(5,3) seven, so(7,1) one, so(8) none; the minimal Dehn orbit (34) meets no SL(8)-frame so(p,q). stab_e7(X) is the gauge so(6,2); Z is in it, r is trace-orthogonal to it (the rotation of a mixed plane); a gauge rotation fixing X and Z carries r to the Wick partner of the lane's gauge boost Q; r and Z span a compact so(3) meeting the gauge algebra in ⟨Z⟩. No element of Stab(X) relates r and Z. Algebraic only.
RESULT_HASH 3601f8112b099216
s1272: 6 computed, 2 receipt, 3 argument, 0 failed
