s1351 — lattices beyond the orbits
  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   H3_every_computed_ZGamma_invariant_of_L_equals_that_of_Lambda0_so_no_new_A1785_class_is_detected   [13 invariants compared; differing: []; Λ₀: {"Smith B1-1": {"0": 44, "1": 12}, "Smith B2-1": {"0": 44, "1": 12}, "Smith O-1": {"0": 28, "1": 28}, "Smith O+1": {"0": 28, "1": 28}, "Smith C56-1": {"0": 24, "2": 32}, "Smith C56+1": {"0": 32, "2": 24}, "Smith B1B2-1": {"0": 32, "1": 24}, "Smith [B1;B2]-1": {"0": 32, "1": 24}, "braid-fixed rank": 32, "F2 rank (O-1) there": 16, "O-split index (log2)": 28, "Gamma-fixed stabiliser order: rank": 15, "… its trace form, Smith": {"24": 15}}]
  argument  ARG    scope
  argument  ARG    reading

VERDICT LATTICES BEYOND THE ORBITS: Chevalley-admissible ∧ Ω-unimodular ⇔ gΛ₀ with g ∈ E7^± (proved: the ℤ-algebra of C is M₅₆(ℤ)). Ad Γ on C has the commutant (15) and three sign isotypes (15, 21, 15). An O-odd nilpotent gives L = (1 + Y/2)Λ₀ ≠ Λ₀, Γ-stable, Ω-unimodular, Chevalley, λ² = 8, with every computed ℤΓ invariant equal to Λ₀'s: no new class detected; whether L lies on the commutant orbit is OPEN.
RESULT_HASH 22c18da4f36df40b
s1351: 12 computed/receipt PASS-able, 3 argument, 0 failed
