W33-Theory/data/PART_W33_PASS5667_5674_Q5_REYE_EQUIVARIANT_ORIENTATION.json at master · wilcompute/W33-Theory · GitHub

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{
  "schema": "w33.pass5667_5674.q5_reye_equivariant_orientation.v1",
  "status": "PASS",
  "check_count": 56,
  "checks_passed": 56,
  "all_pass": true,
  "checks": {
    "pass5417_action_order_576": true,
    "cover_orbits_are_1_plus_12": true,
    "fixed_cover_vertex_is_7": true,
    "moving_cover_action_order_576": true,
    "moving_cover_action_is_T12_165": true,
    "moving_stabilizer_is_48_48": true,
    "moving_subdegrees_are_1_3_8": true,
    "latin_action_order_576": true,
    "latin_action_is_T12_165": true,
    "explicit_cover_to_latin_conjugator": true,
    "explicit_cover_to_library_T165_conjugator": true,
    "outside_vertices_are_312": true,
    "all_design_rows_have_size_6": true,
    "design_point_replication_is_144": true,
    "design_pair_multiplicity_is_60": true,
    "triple_spectrum_is_0_16_24": true,
    "zero_shell_has_16_triples": true,
    "fixed_cover_point_absent_from_zero_shell": true,
    "zero_shell_is_12_4_16_3": true,
    "zero_shell_is_source_invariant": true,
    "zero_shell_levi_has_48_edges": true,
    "zero_shell_levi_aut_order_576": true,
    "levi_point_action_is_T12_165": true,
    "levi_line_action_is_T16_1034": true,
    "source_line_action_is_faithful": true,
    "twelve_action_is_not_even": true,
    "sixteen_action_is_entirely_even": true,
    "twelve_sign_kernel_has_order_288": true,
    "twelve_sign_kernel_is_288_1025": true,
    "cover_conjugator_transports_even_Latin_kernel": true,
    "distinct_design_rows_are_132": true,
    "row_multiplicities_are_120_times_2_plus_12_times_6": true,
    "heavy_shell_has_12_blocks": true,
    "heavy_shell_excludes_fixed_point": true,
    "heavy_shell_point_degree_is_6": true,
    "heavy_pair_intersections_are_2_or_3": true,
    "heavy_action_is_T12_165": true,
    "point_and_heavy_stabilizers_share_type_48_48": true,
    "point_and_heavy_stabilizers_not_source_conjugate": true,
    "two_twelve_images_are_permutation_conjugate": true,
    "carrier_map_lifts_to_source_automorphism": true,
    "carrier_automorphism_witness_has_order_8": true,
    "carrier_outer_coset_has_order_2": true,
    "exactly_48_inner_regaugings_give_outer_involutions": true,
    "carrier_outer_involution_is_order_2_and_outer": true,
    "outer_involution_exchanges_point_and_heavy_stabilizer_classes": true,
    "all_three_carrier_permutation_characters_have_rank_3": true,
    "untwisted_carrier_character_pairings_are_2": true,
    "sign_twisted_point_module_is_orthogonal_to_heavy_module": true,
    "sign_twisted_point_module_is_orthogonal_to_line_module": true,
    "sign_character_twist_is_not_the_outer_carrier_twist": true,
    "natural_seven_group_has_order_168": true,
    "all_natural_seven_embeddings_enumerated": true,
    "relative_placement_orbit_counts_are_53_62_58": true,
    "parity_separates_161_163_from_165": true,
    "every_join_is_A12_A12_S12": true
  },
  "pass_5667_action_gate": {
    "fixed_cover_position_one_based": 1,
    "fixed_cover_vertex": 7,
    "moving_cover_positions_one_based": [ 2, 13, 8, 10, 6, 4, 5, 11, 7, 3, 12, 9 ],
    "moving_cover_vertices": [ 31, 321, 190, 227, 141, 112, 129, 255, 158, 74, 278, 194 ],
    "group_order": 576,
    "transitive_id": 165,
    "stabilizer_id": [ 48, 48 ],
    "subdegrees": [ 1, 3, 8 ],
    "cover_to_latin_conjugator_one_based": [ 1, 9, 4, 8, 12, 7, 10, 2, 5, 3, 6, 11 ],
    "cover_to_library_T165_conjugator_one_based": [ 1, 3, 10, 5, 12, 8, 6, 4, 2, 7, 11, 9 ]
  },
  "pass_5668_design": {
    "parameters": [13,6,60],
    "rows_with_multiplicity": 312,
    "point_replication": 144,
    "triple_containment_spectrum_value_count": [ [ 0, 16 ], [ 16, 30 ], [ 24, 240 ] ],
    "moving_triple_orbits_size_multiset_multiset_distinct_representative": [ [ 12, 16, 8, [ 2, 3, 8 ] ], [ 16, 0, 0, [ 2, 4, 9 ] ], [ 48, 24, 8, [ 2, 4, 5 ] ], [ 144, 24, 10, [ 2, 3, 4 ] ] ]
  },
  "pass_5669_reye_zero_shell": {
    "zero_triples_in_cover_positions": [ [ 2, 4, 9 ], [ 2, 5, 12 ], [ 2, 6, 10 ], [ 2, 7, 13 ], [ 3, 4, 13 ], [ 3, 5, 10 ], [ 3, 6, 12 ], [ 3, 7, 9 ], [ 4, 5, 8 ], [ 4, 6, 11 ], [ 5, 7, 11 ], [ 6, 7, 8 ], [ 8, 9, 12 ], [ 8, 10, 13 ], [ 9, 10, 11 ], [ 11, 12, 13 ] ],
    "zero_triples_on_moving_twelve": [ [ 1, 6, 12 ], [ 1, 7, 11 ], [ 1, 5, 4 ], [ 1, 9, 2 ], [ 10, 6, 2 ], [ 10, 7, 4 ], [ 10, 5, 11 ], [ 10, 9, 12 ], [ 6, 7, 3 ], [ 6, 5, 8 ], [ 7, 9, 8 ], [ 5, 9, 3 ], [ 3, 12, 11 ], [ 3, 4, 2 ], [ 12, 4, 8 ], [ 8, 11, 2 ] ],
    "configuration": [12,4,16,3],
    "levi_edges": 48,
    "levi_aut_order": 576,
    "point_action_transitive_id": 165,
    "line_action_transitive_id": 1034,
    "line_action_subdegrees": [ 1, 6, 9 ]
  },
  "pass_5670_5671_orientation": {
    "twelve_action_even": false,
    "sixteen_action_even": true,
    "twelve_sign_kernel_order": 288,
    "twelve_sign_kernel_id": [ 288, 1025 ],
    "twelve_sign_kernel_structure": "(C2 x C2 x C2 x C2) : (C3 x S3)",
    "latin_even_kernel_is_literal_conjugate": true
  },
  "pass_5672_natural_PSL27_join": {
    "embedding_type": "natural degree 7 plus five fixed points",
    "number_of_S12_conjugates": 23760,
    "comparison_transitive_ids": [ 161, 163, 165 ],
    "comparison_groups_lie_in_A12": [ true, true, false ],
    "relative_placement_orbit_counts": [ 53, 62, 58 ],
    "relative_placement_orbit_size_sets": [ [ 24, 72, 96, 144, 288, 576 ], [ 12, 72, 96, 144, 288, 576 ], [ 24, 48, 72, 96, 144, 192, 288, 576 ] ],
    "join_order_sets": [ [ 239500800 ], [ 239500800 ], [ 479001600 ] ],
    "all_joins_are_A12": [ true, true, false ],
    "all_joins_are_S12": [ false, false, true ]
  },
  "pass_5673_5674_heavy_dual": {
    "distinct_rows": 132,
    "row_multiplicity_value_count": [ [ 2, 120 ], [ 6, 12 ] ],
    "heavy_blocks_in_cover_positions": [ [ 2, 3, 4, 5, 6, 7 ], [ 2, 3, 9, 10, 12, 13 ], [ 2, 4, 5, 10, 11, 13 ], [ 2, 4, 6, 8, 12, 13 ], [ 2, 5, 7, 8, 9, 10 ], [ 2, 6, 7, 9, 11, 12 ], [ 3, 4, 5, 9, 11, 12 ], [ 3, 4, 6, 8, 9, 10 ], [ 3, 5, 7, 8, 12, 13 ], [ 3, 6, 7, 10, 11, 13 ], [ 4, 7, 8, 9, 11, 13 ], [ 5, 6, 8, 10, 11, 12 ] ],
    "heavy_action_transitive_id": 165,
    "heavy_pair_intersection_value_count": [ [ 2, 18 ], [ 3, 48 ] ],
    "point_stabilizer_id": [ 48, 48 ],
    "heavy_stabilizer_id": [ 48, 48 ],
    "stabilizers_conjugate_in_source_group": false,
    "permutation_images_conjugate_in_S12": true,
    "outer_twist_conjugator_one_based": [ 1, 10, 12, 9, 4, 8, 7, 2, 3, 11, 6, 5 ],
    "source_automorphism_witness_order": 8,
    "source_outer_coset_order": 2,
    "inner_regaugings_yielding_outer_involutions": 48,
    "outer_involution_twist_on_cover13_one_based": [ 1, 2, 11, 6, 12, 7, 9, 3, 10, 13, 8, 5, 4 ],
    "outer_involution_order": 2,
    "outer_involution_maps_point_to_heavy_stabilizer_class": true
  },
  "sign_twist_module_firewall": {
    "point_heavy_character_inner_product": 2,
    "point_line_character_inner_product": 2,
    "heavy_line_character_inner_product": 2,
    "sign_twisted_point_heavy_inner_product": 0,
    "sign_twisted_point_line_inner_product": 0,
    "odd_element_point_fixed_count_distribution": [ [ 0, 168 ], [ 2, 108 ], [ 6, 12 ] ],
    "odd_element_heavy_fixed_count_distribution": [ [ 0, 108 ], [ 1, 96 ], [ 2, 72 ], [ 4, 12 ] ],
    "boundary": "Tensoring the point permutation module by its sign character is disjoint from the heavy and line permutation modules; this sign twist is not the source outer involution."
  },
  "prior_ownership": {
    "pass_5300": "owns the even-Latin/Hoffman central-quotient bridge",
    "pass_5596": "owns the F4 short-root-pair to Latin action",
    "pass_5606": "already specified the q5 cover-to-Latin gate; this run activates it",
    "pass_5659": "owns the independent Reye identification as T12_165"
  },
  "boundary": "Exact finite GAP theorem. The 2-design rows are a multiset. The natural PSL(2,7) join theorem is scoped to its degree-7 action with five fixed points. The point and heavy twelve-carriers are not source-equivariantly identified; their permutation images agree only after an outer twist. No continuum, dynamics, particle, gauge-force, or physical unification claim follows."
}