s1355 — the s1351 lattice L on the commutant orbit
  receipt   PASS   filed_d1414_house_frames_npz
  receipt   PASS   filed_chatgpt_d1423_maps_npz
  receipt   PASS   s1351_source_whose_construction_is_executed
  receipt   PASS   filed_d1414_house_frames_npz
  receipt   PASS   filed_chatgpt_d1423_maps_npz
  receipt   PASS   s1345_source_whose_hermite_code_is_copied
  receipt   PASS   s1345_results
  receipt   PASS   s1346_results
  computed  PASS   H0_the_Z_algebra_generated_by_C_is_all_of_M56Z   [{"E_a E_-a diagonal with entries in {0,±1}": true, "weights isolated by projector products": "56/56", "root-graph reach from weight 0": 56, "root vectors have unit entries": true} ⇒ every e_ww (projector products) and every e_w′w (± a path of unit root vectors times e_ww) lies in the algebra]
  argument  ARG    H0_consequence
  computed  PASS   H1a_Ad_Gamma_acts_on_C_by_integer_matrices_of_det_1_with_a_15_dim_fixed_space   [coroot HNF rank 7; det Ad(B1, B2, O, C56) on C = [1, 1, 1, 1]; ℚ-fixed dimension 15]
  computed  PASS   H1b_the_plus_minus_one_isotypes_are_15_15_21_15_and_none_with_B1_or_B2_odd   [ℚ-dimensions of the joint (B1,B2,O,C56)-eigenspaces for every ±1 character: nonzero {'++++': 15, '+++-': 15, '++-+': 21, '++--': 15}; total 66 of 133]
  computed  PASS   H1c_mod_3_5_7_the_same_isotypes_and_no_other_character_mod_2_the_fixed_space_is_36   [{"2": {"fixed_dim": 36, "generator_eigenvalues": [[1], [1], [1], [1]], "nontrivial_characters": {}}, "3": {"fixed_dim": 15, "generator_eigenvalues": [[1], [1], [1, 2], [1, 2]], "nontrivial_characters": {"1,1,1,2": 15, "1,1,2,1": 21, "1,1,2,2": 15}}, "5": {"fixed_dim": 15, "generator_eigenvalues": [[1], [1], [1, 4], [1, 4]], "nontrivial_characters": {"1,1,1,4": 15, "1,1,4,1": 21, "1,1,4,4": 15}}, "7": {"fixed_dim": 15, "generator_eigenvalues": [[1], [1], [1, 6], [1, 6]], "nontrivial_characters": {"1,1,1,6": 15, "1,1,6,1": 21, "1,1,6,6": 15}}}]
  computed  PASS   H1d_each_sign_isotype_brackets_into_the_commutant   [max distance of [V_χ, V_χ] from the commutant: {'+++-': '< 1e-9', '++-+': '< 1e-9', '++--': '< 1e-9'}]
  computed  PASS   H2_L_is_a_new_gamma_stable_omega_unimodular_chevalley_admissible_lattice_with_lambda2_8   [{"Y in the (+,+,-,+) isotype": true, "Y^2 = 0": true, "Y/2 integral": false, "g^T Omega g = Omega": true, "det g": 1, "Gamma-stable (max distance of g^-1 gamma g to Z)": "< 1e-12", "g normalises e7 (distance)": "< 1e-9", "stabiliser covolume of Lambda0 / L": ["1/2", "1/2"], "lambda2(Lambda0)": 2.0, "lambda2(L)": 8.0, "on L: sqrt2 X / Z / Gram distance to Z": [0.5, 0.5, 0.5]}]
  computed  PASS   Q1_exp_2pi_R_is_the_identity_on_all_56_the_15_acts_through_SO0_eta_with_no_spinor_rep   [max |exp(2πR) − 1| on the 56: < 1e-8]
  computed  PASS   Q2_sigma_is_an_orthochronous_eta_isometry_from_Z6_onto_L_cap_V6   [σᵀησ = η exactly, det σ = 1, σ₆₆ = 3; |det C6| = 1 (L ∩ V6 has covolume 1 in V6)]
  computed  PASS   Q3_c_in_exp_of_the_15_carries_Lambda0_onto_L_exactly   [{"log real, det(1+σ′) ≠ 0": true, "c within 1e-10 of (½)ℤ": true, "2σ′ integral": true, "C2·F == F·(2σ′) (exact)": true, "[C2, γ] == 0, γ = B1 B2 O C56 (exact)": true, "C2ᵀ Ω C2 == 4Ω (exact)": true, "C2 · (Ω⁻¹ C2ᵀ Ω) == 4·1 (exact)": true, "g⁻¹c integral (exact)": true, "c⁻¹g integral (exact)": true, "|det g⁻¹c|": 1}]
  computed  PASS   Q4_the_lifts_of_the_15_Vinberg_pair_products_of_I51_are_integral_on_Lambda0   [15 / 15 pair products r_i r_j lift into Aut(Λ₀)]
  argument  ARG    reading

VERDICT THE s1351 LATTICE IS ON THE ORBIT: an explicit c = exp(lift(log σ′)) in the identity component of the commutant carries Λ₀ onto L = (1 + Y/2)Λ₀ (exact integer checks); the 15 acts on the 56 through SO⁰(5,1) (no spinors) and the integral Lorentz group lifts into Aut(Λ₀).
RESULT_HASH cd93bf4a829a00b9
s1355: 18 computed/receipt PASS-able, 2 argument, 0 failed
