s1348 — S374.13(b): the equivalence in s1212's loss condition
  computed  PASS   expectation_1_block_reproduces_A1741   [|Fix(A^4)| = 45, orbits 13, periods {1: 1, 4: 10, 2: 2} (A1741 §2 house table: 45 / 13 / {1:1, 2:2, 4:10})]
  computed  PASS   carrier_F_d_is_det_formula   [|Fix(A^d)| d=1..4 = [1, 5, 16, 45] vs |2 − tr A^d|]
  computed  PASS   normalizer_contains_minus_I_unit_and_reverser   [28 normalizer elements in |M_ij| ≤ 6: 14 centralizer (units with C² = A: [((-2, 1), (-1, 1)), ((2, -1), (1, -1))]), 14 reversers]
  computed  PASS   normalizer_elements_preserve_Fix_A4   [28 elements × 45 points]
  computed  PASS   W4_is_the_U_invariant_subspace   [U_4 W = W and rank W = 13]
  computed  PASS   linking_form_well_defined_A_invariant_F_unitary_commutes   [b well defined on sampled coset shifts True; A-invariant True; F unitary True; [F, U_4] = 0 True]
  receipt   PASS   s1343_is_the_filed_kernel_this_succeeds
  receipt   PASS   s1212_is_the_filed_kernel_whose_lines_59_to_216_are_copied
  receipt   PASS   s1344_is_the_filed_kernel_whose_lines_55_to_215_are_copied
  receipt   PASS   s1344_results_the_span_census_this_extends
  computed  PASS   E3a_Gamma_bar_has_order_8_nonabelian_irreps_1_1_1_1_2   [|Γ̄| = 8, nonabelian True, 5 conjugacy classes, |[Γ̄,Γ̄]| = 2 ⇒ 4 one-dim irreps, remaining irrep dims [2]]
  computed  PASS   E3b_Gamma_bar_isotypic_components_of_W4_are_invariant_and_sum_to_13   [class-sum element symmetric True; 3 isotypic components, dims [4, 4, 5]; max leak under Γ̄ < 1e-12]
  computed  PASS   E3c_F_W_has_order_4_with_one_dim_irreps_and_U4_is_the_identity_on_W4   [F_W⁴ = 1 True; F_W eigenvalue multiplicities {'i^0': 5, 'i^1': 2, 'i^2': 4, 'i^3': 2}; ‖U_4 W − W‖ < 1e-12]
  computed  PASS   E1_left_side_holds_for_every_generic_eight_space   [H_E = 1 − P_E on 5 seeded E: lowest multiplicities [8, 8, 8, 8, 8], distance from V [0.542193, 0.54108, 0.544807, 0.53736, 0.54178], coupled fractions [0.552508, 0.551855, 0.552609, 0.550571, 0.551558]]
  computed  PASS   E2_those_eight_spaces_are_invariant_under_no_nontrivial_registered_element   [min over the 7 nontrivial g ∈ Γ̄ of ‖(1−P_E)(g⊗1)P_E‖: [0.977956, 0.986368, 0.986653, 0.94923, 0.985978]; over F, F², F³: [0.989237, 0.987206, 0.99028, 0.994901, 0.992951]]
  computed  PASS   E4_every_Gamma_bar_invariant_eight_space_is_the_lowest_eigenspace_of_a_coupled_H_outside_V   [4 Γ̄-invariant 8-spaces (−I-odd + isotypic unions); C = seed 2024 Hermitian, V- and uncoupled components removed (dist from V 1.000, coupled fraction 1.000); lowest mults [8, 8, 8, 8]; min gap 0.607983; distances from V [0.215517, 0.215517, 0.218329, 0.218861]; spin-blind [True, True, True, False]]
  argument  ARG    E3d_no_registered_group_acts_on_both_sides
  argument  ARG    reading

E3 types: {"Gamma_bar_order": 8, "Gamma_bar_nonabelian": true, "conjugacy_classes": 5, "irrep_dims": [1, 1, 1, 1, 2], "W4_isotypic_components": [{"dim": 4, "character_on_Gamma_bar": [4, 0, 0, -4, 0, 0, 0, 0]}, {"dim": 4, "character_on_Gamma_bar": [4, 4, 4, 4, -4, -4, -4, -4]}, {"dim": 5, "character_on_Gamma_bar": [5, 5, 5, 5, 5, 5, 5, 5]}], "F_W_eigen_multiplicities": {"i^0": 5, "i^1": 2, "i^2": 4, "i^3": 2}, "U4_on_W4": "identity"}
E4 rows:
   {"E": "minus_I_odd_sector", "dim": 8, "invariant_leak_max": 0.0, "lowest_mult": 8, "gap": 0.632881, "overlap_with_E": 1.0, "dist_from_V": 0.215517, "coupled_fraction": 0.215517, "spin_blind": true}
   {"E": "isotypic_union_[0]_dim4_x_C2", "dim": 8, "invariant_leak_max": 0.0, "lowest_mult": 8, "gap": 0.632881, "overlap_with_E": 1.0, "dist_from_V": 0.215517, "coupled_fraction": 0.215517, "spin_blind": true}
   {"E": "isotypic_union_[1]_dim4_x_C2", "dim": 8, "invariant_leak_max": 0.0, "lowest_mult": 8, "gap": 0.607983, "overlap_with_E": 1.0, "dist_from_V": 0.218329, "coupled_fraction": 0.218827, "spin_blind": true}
   {"E": "isotypic_union_[0, 1]_dim8_x_spin_up_line", "dim": 8, "invariant_leak_max": 0.0, "lowest_mult": 8, "gap": 0.615868, "overlap_with_E": 1.0, "dist_from_V": 0.218861, "coupled_fraction": 0.35974, "spin_blind": false}

VERDICT S374.13(b): the "equivalently" is FALSE AS WRITTEN. Its left side (an unregistered coupling outside V with an 8-dim extreme eigenspace) exists for every 8-space; generic such spaces are invariant under no registered element, so they are images of no intertwiner; the converse holds trivially. Its right side names no Hom-space of a registered group. Struck; replacement in header.
RESULT_HASH 71d2aac828bfdd93
s1348: 16 computed/receipt PASS-able, 2 argument, 0 failed
