s1271 — the T² parabolic and the 6D-origin obstruction
  Q1_t2_parabolic                  {"ad_Δ spectrum on e7 (value: count)": {"-2.0": 10, "-1.0": 32, "0.0": 49, "1.0": 32, "2.0": 10}, "max |Im| of ad_Δ eigenvalues": "<1e-09", "Levi = centraliser of Δ (dim, derived, Tr56 inertia)": [49, 48, [27, 21, 1]], "so(5,5) = derived centraliser of {S, Δ}": [45, 45, [25, 20, 0]], "sl(2) = derived commutant of so(5,5)": [3, 3, [2, 1, 0]], "Δ on the 56 (value: count)": {"-2.0": 2, "-1.0": 16, "0.0": 20, "1.0": 16, "2.0": 2}, "dim of the (Δ = 0, S = +1) space": 10, "so(5,5)-invariance of it (residual)": "<1e-10", "invariant symmetric forms on it": 1, "signature": [5, 5, 0]}
  Q2_css_positive_control          {"CSS basis size": 13, "D56-grade norm² of the CSS basis": {"-4.0": 12.0, "0.0": 1.0, "4.0": 0.0}, "derived series": [13, 12, 0], "invariants": [13, 12, [0, 1, 12]]}
  Q3_commuting_twist_ansatz        {"anchors in so(5,5) (relative residual)": ["<1e-10", "<1e-10"], "[A56, A_sol]": "<1e-12", "dim g₊ (grades +1, +2)": 42, "span{A56, A_sol} ⊕ g₊: dim, closure residual": [44, "<1e-10"], "its derived series": [44, 42, 10, 0]}
  Q4_registered_spans_all_grades   {"registered gauge algebra: derived series": [28, 28], "Δ-grade norm² of an orthonormal gauge basis": {"-2.0": 2.0, "-1.0": 8.0, "0.0": 8.0, "1.0": 8.0, "2.0": 2.0}, "census members: grade mass on each side": {"(6,2) odd 78 +π/4": {"negative grades": 240.0, "positive grades": 240.0}, "(6,2) odd 78 −π/4": {"negative grades": 240.0, "positive grades": 240.0}, "(6,2) odd 12 +π/4": {"negative grades": 240.0, "positive grades": 240.0}, "(6,2) odd 12 −π/4": {"negative grades": 240.0, "positive grades": 240.0}, "(8,0) +π/4": {"negative grades": 240.0, "positive grades": 240.0}, "(8,0) −π/4": {"negative grades": 240.0, "positive grades": 240.0}, "(4,4) +π/4": {"negative grades": 240.0, "positive grades": 240.0}, "(4,4) −π/4": {"negative grades": 240.0, "positive grades": 240.0}}}
  Q5_spinor_reality                {"Clifford residual, chirality anticommutation": ["<1e-12", "<1e-12"], "Weyl halves": {"(8,0)": {"+": {"self-intertwiners": 1, "intertwining residual": "<1e-10", "J² ∝ I residual": "<1e-10", "sign of J²": 1.0, "imag of J²/|J²|": "<1e-10", "type": "real", "Σ leak off the half": "<1e-12"}, "−": {"self-intertwiners": 1, "intertwining residual": "<1e-10", "J² ∝ I residual": "<1e-10", "sign of J²": 1.0, "imag of J²/|J²|": "<1e-10", "type": "real", "Σ leak off the half": "<1e-12"}}, "(7,1)": {"+": {"self-intertwiners": 0, "type": "complex", "Σ leak off the half": "<1e-12"}, "−": {"self-intertwiners": 0, "type": "complex", "Σ leak off the half": "<1e-12"}}, "(6,2)": {"+": {"self-intertwiners": 1, "intertwining residual": "<1e-10", "J² ∝ I residual": "<1e-10", "sign of J²": -1.0, "imag of J²/|J²|": "<1e-10", "type": "quaternionic", "Σ leak off the half": "<1e-12"}, "−": {"self-intertwiners": 1, "intertwining residual": "<1e-10", "J² ∝ I residual": "<1e-10", "sign of J²": -1.0, "imag of J²/|J²|": "<1e-10", "type": "quaternionic", "Σ leak off the half": "<1e-12"}}, "(5,3)": {"+": {"self-intertwiners": 0, "type": "complex", "Σ leak off the half": "<1e-12"}, "−": {"self-intertwiners": 0, "type": "complex", "Σ leak off the half": "<1e-12"}}, "(4,4)": {"+": {"self-intertwiners": 1, "intertwining residual": "<1e-10", "J² ∝ I residual": "<1e-10", "sign of J²": 1.0, "imag of J²/|J²|": "<1e-10", "type": "real", "Σ leak off the half": "<1e-12"}, "−": {"self-intertwiners": 1, "intertwining residual": "<1e-10", "J² ∝ I residual": "<1e-10", "sign of J²": 1.0, "imag of J²/|J²|": "<1e-10", "type": "real", "Σ leak off the half": "<1e-12"}}}}
  Q6_signature_obstruction         {"real 8s and their invariant-form signatures (vector, then real-type spinors)": {"(6,2)": [[6, 2]], "(8,0)": [[8, 0], [8, 0], [8, 0]], "(4,4)": [[4, 4], [4, 4], [4, 4]]}, "can 8 ⊕ 2 carry (5,5)?": {"(6,2)": false, "(8,0)": false, "(4,4)": true}, "the 10 of the Levi so(5,5) (Q1)": [5, 5]}
  receipt   PASS filed_d1414_house_frames_npz
  receipt   PASS filed_chatgpt_d1414_maps_npz
  computed  PASS Q1_the_t2_parabolic_grades_e7_as_10_32_49_32_10_with_levi_so55_sl2_R_and_the_so55_vector_ten_has_signature_5_5
  computed  PASS Q2_positive_control_the_known_5d_to_4d_css_algebra_lies_on_one_side_of_its_grading_and_is_solvable
  computed  PASS Q3_two_commuting_u1_twists_in_so55_plus_the_nilradical_close_on_a_solvable_44_dim_algebra
  computed  PASS Q4_the_registered_so62_is_perfect_and_meets_every_delta_grade_and_every_census_member_meets_both_signs
  computed  PASS Q5_weyl_spinors_of_spin_p_q_are_real_for_8_0_and_4_4_quaternionic_for_6_2_complex_for_7_1_and_5_3
  computed  PASS Q6_so62_and_so8_admit_no_5_5_ten_so_they_lie_in_no_t2_parabolic_so44_is_not_obstructed
  argument  ARG  the_obstruction
  argument  ARG  the_two_u1s_as_twists
  argument  ARG  scope
VERDICT: the T² parabolic of Δ has Levi so(5,5) ⊕ sl(2) ⊕ ℝ and a (5,5) vector 10; two commuting U(1) twists inside it generate only solvable algebras; the registered so(6,2) is perfect and spans every grade, and — its spinors being quaternionic — admits no (5,5) 10, so it lies in no conjugate of the T² parabolic: no torus origin in maximal 6D supergravity (twisted or gauged). The so(8) twin is obstructed too; the so(4,4) twin is not. Algebraic only; sphere and generalised reductions out of scope.
RESULT_HASH 86f3f8d093cfc6ea
s1271: 6 computed, 2 receipt, 3 argument, 0 failed
