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"evidence/rank8_creative_20261022/unified/verification.json", "evidence/rank8_creative_20261022/unified/composite_verification.json", "evidence/rank8_creative_20261022/unified/prime_gauss_audit.json" ], "limitation": "These restrictions alone establish neither existence nor nonexistence.", "outcome": "Equality in a dimension-character bound forces the cyclic modular representation through its character conductor. Signed prime and composite refinements give exact twist support and spectral capacities, with no twist-order cutoff.", "proof": "evidence/rank8_creative_20261022/unified/SATURATED_CONDUCTOR.md", "scope": "Necessary conditions for a pseudo-unitary Galois form of any complex categorification.", "verifiers": [ "evidence/rank8_creative_20261022/unified/verify_saturated_conductor.py", "evidence/rank8_creative_20261022/unified/verify_composite_support.py", "evidence/rank8_creative_20261022/unified/prime_gauss_audit.py" ] }, { "domain": "complete_signed_conductor_phase_capacity_and_packet_product_refinement", "evidence": [ "evidence/rank8_creative_20261022/r928_r941/verification.json" ], "limitation": "A feasible rational phase witness is not a center, associator, or categorification. Pair products do not complete the center fusion ring.", "outcome": "All ten complete R928 patterns retain exact feasible necessary witnesses after oriented complex STS and individual conductor-fibre capacities.", "proof": "evidence/rank8_creative_20261022/r928_r941/PROOF.md", "scope": "Complete existing pattern/charge domain; necessary center and distinguished-product constraints.", "verifiers": [ "evidence/rank8_creative_20261022/r928_r941/verify.py" ] }, { "domain": "complete_integer_phase_branching_refinement", "evidence": [ "evidence/rank8_creative_20261022/r928_r941/integer_followup/verification.json", "evidence/rank8_creative_20261022/r928_r941/integer_followup/branches.json", "evidence/rank8_creative_20261022/r928_r941/integer_followup/search.json" ], "limitation": "Four necessary patterns survive. These witnesses construct neither complete modular data nor a fusion category; the ring remains unresolved.", "outcome": "Exact integer phase counts eliminate patterns 44, 58, 196, 243, 245 and 248 by complete binary branching trees with 49 rational Farkas leaves. Patterns 40, 49, 186 and 191 have explicit nonnegative integer phase counts and exact rational auxiliary witnesses.", "proof": "evidence/rank8_creative_20261022/r928_r941/integer_followup/PROOF.md", "scope": "All ten oriented complex-STS patterns; phase orbit counts are integral while rotation and endpoint-mixture auxiliaries remain continuous.", "verifiers": [ "evidence/rank8_creative_20261022/r928_r941/integer_followup/verify.py" ] } ], "limitations": [ "No completed search here exhausts all complex associators or all constructions.", "A feasible necessary-condition system is not a categorification.", "Extensively investigated exotic candidate is a research-screening description, not a proof of categorifiability or novelty." ], "new_attempts": [ { "details": { "FP_global_dimension_minimal_polynomial": [ "1", "-576", "11520", "-73728", "147456" ], "general_complex_exclusion_claimed_by_this_audit": false, "invertible_count": 2, "invertibles_abelian": true, "largest_real_root_isolating_interval": [ "9104089/16389", "1111/2" ], "rational_scaling_outcome": "impossible_by_minimal_polynomial_degree", "routes": { "deequivariantization_of_NG": "excluded", "direct_or_Morita_equivalent_to_NG": "excluded", "equivariantization_of_NG": "excluded", "group_extension_of_NG": "excluded", "neutral_component_of_nontrivially_graded_NG": "excluded" } }, "domain": "all_NG_Cn_n_and_five_single_operation_routes_all_positive_n", "evidence": [ "evidence/rank8_frontier_20261021/structural_models/near_group_scaling.json" ], "limitation": "No arbitrary chains, inserted Morita changes, or noncentral condensations are covered.", "outcome": "family_excluded", "proof": "evidence/rank8_frontier_20261021/structural_models/NEAR_GROUP_SCALING.md", "scope": "general_complex_construction_family", "verifiers": [ "evidence/rank8_frontier_20261021/structural_models/near_group_scaling.py" ] }, { "details": { "PSD_Gram_survivors": 24, "bounded_selfdual_vectors_tested": 1920, "entrywise_Gram_survivors": 664, "independent_complete_Python_domain": 1920, "invertible_stabilizer_survivors": 1920, "one_plus_simple_surviving_indices": [ 7 ] }, "domain": "all_connected_separable_algebra_underlying_objects_entrywise_and_PSD_Gram", "evidence": [ "evidence/rank8_frontier_20261021/structural_models/algebra_candidates.json" ], "limitation": "Surviving objects are not algebra structures; no multiplication, associator, model or general exclusion is claimed.", "outcome": "complete_necessary_object_filter_survives", "proof": "evidence/rank8_frontier_20261021/structural_models/ALGEBRA_CANDIDATES.md", "scope": "general_complex_algebra_construction_constraints", "verifiers": [ "evidence/rank8_frontier_20261021/structural_models/algebra_candidates.py" ] }, { "details": { "complete_signed_count": 32, "conclusion": "every categorification is spherical and has a pseudo-unitary Galois conjugate; no existence or exclusion conclusion", "normalized_indicator_domains_used": false }, "domain": "complete_signed_sphericalization_and_general_Galois_PU_reduction", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative_followup/scope_gates.json" ], "limitation": "This proves a scope reduction, not categorifiability or unitarity. Necessary PU constraints apply after Galois conjugation.", "outcome": "Galois_pseudo_unitary_reduction", "proof": "evidence/rank8_frontier_20261021/noncommutative_followup/README.md", "scope": "general_complex_fusion_category", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative_followup/verify_scope_gates.py" ] }, { "details": { "allowed_order_sequence": [ 8, 12, 24 ], "complete_order_candidates": [ 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30 ], "dimension_bounded_product_count": 5, "exponent_box_size": 12, "positive_integer_scaling_matches": [] }, "domain": "all_levels_SU2_adjoint_finite_products_integer_multiplier_chains_and_Morita_duals", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative_followup/su2_all_frontier.json" ], "limitation": "Other Lie types, arbitrary algebra condensation, dimension-decreasing de-equivariantization, or general categorifications.", "outcome": "family_excluded_by_exact_global_dimension", "proof": "evidence/rank8_frontier_20261021/noncommutative_followup/SU2_FAMILY.md", "scope": "general_complex_construction_family_dimension_obstruction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative_followup/su2_all_frontier.py", "evidence/rank8_frontier_20261021/noncommutative_followup/su2_product_family.py" ] }, { "details": { "families": [], "global_dimension_minimal_polynomial": [ "1", "-576", "11520", "-73728", "147456" ], "id": "R8_3_928", "outcome": "family_dimension_excluded_by_degree", "positive_root_interval": [ "9104089/16389", "1111/2" ], "scope": "specific_model_family_only", "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" }, "domain": "all_generalized_near_group_dimension_parameters_with_integer_scalings", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative_followup/generalized_near_group_all_frontier.json" ], "limitation": "This is a necessary dimension-family test; retained parameter triples are not categories, and integral targets lie outside its irrational quadratic parameter theorem.", "outcome": "family_dimension_excluded_by_degree", "proof": "evidence/rank8_frontier_20261021/noncommutative_followup/GENERALIZED_NEAR_GROUP_TABLE.md", "scope": "specific_model_family_only", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative_followup/generalized_near_group_all_frontier.py" ] }, { "details": { "complete_joint_pattern_count": 343, "independent_Python_nodes": 561, "native_search_nodes": 1665 }, "domain": "complete_integer_Gram_and_all_twenty_distinguished_cubic_coefficients", "evidence": [ "evidence/rank8_frontier_20261021/quartic_integer_center/gram_cubic.json", "evidence/rank8_frontier_20261021/quartic_integer_center/source.json" ], "limitation": "All343 patterns are necessary center data; their existence does not construct a modular category.", "outcome": "343_exact_necessary_integer_patterns_survive", "proof": "evidence/rank8_frontier_20261021/quartic_integer_center/PROOF.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/quartic_integer_center/source.py", "evidence/rank8_frontier_20261021/quartic_integer_center/gram_cubic.py" ] }, { "details": { "forced_low_translate_high_indices": [ 40, 41, 42, 43, 57 ], "source_invertible_index": 7 }, "domain": "forced_order_two_central_lift_and_complete_tensor_orbit_graph", "evidence": [ "evidence/rank8_frontier_20261021/review/quartic_central_orbits.json" ], "limitation": "A central lift does not imply a tensor-factor decomposition of the nonbraided source category.", "outcome": "central_invertible_forced_in_all_integer_patterns", "proof": "evidence/rank8_frontier_20261021/review/QUARTIC_CENTRAL_ORBITS.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/review/verify_quartic_central_orbits.py" ] }, { "details": { "complete_case_count": 15, "exact_feasible_count": 10, "exact_infeasible_count": 5 }, "domain": "all_semion_parity_patterns_with_fixed_even_twist_moments", "evidence": [ "evidence/rank8_frontier_20261021/r928_semion_rotation/results.json" ], "limitation": "These are finite necessary moment systems; exact primal survival is not a semionic construction.", "outcome": "exact_feasible_relaxation; exact_infeasible_relaxation", "proof": "evidence/rank8_frontier_20261021/review/QUARTIC_CENTRAL_ORBITS.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/r928_semion_rotation/verify.py" ] }, { "details": { "complete_case_count": 10, "exact_feasible_count": 10, "exact_infeasible_count": 0 }, "domain": "complete_correlated_semion_pair_twist_cones", "evidence": [ "evidence/rank8_frontier_20261021/r928_semion_rotation/pair_rotation.json" ], "limitation": "These are finite necessary moment systems; exact primal survival is not a semionic construction.", "outcome": "exact_feasible_relaxation", "proof": "evidence/rank8_frontier_20261021/review/QUARTIC_CENTRAL_ORBITS.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/r928_semion_rotation/pair_rotation.py" ] }, { "details": { "bounded_integer_search": { "exact_integer_witness_indices": [ 40, 49, 186, 191 ], "numeric_negative_is_exclusion": false, "outcomes": { "exact_nonnegative_integer_witness": 4, "integer_feasibility_unresolved": 6 }, "status": "bounded_attempt_finished", "uncertified_integer_indices": [ 44, 58, 196, 243, 245, 248 ] }, "correlated_semion_pairs": { "outcomes": { "exact_feasible_relaxation": 10 }, "status": "exact_complete" }, "id": "R8_3_928", "joint_spectrum": { "input_patterns": 343, "status": "exact_complete", "surviving_patterns": 255 }, "semion_fixed_moments": { "outcomes": { "exact_feasible_relaxation": 10, "exact_infeasible_relaxation": 5 }, "status": "exact_complete" }, "supported_modular_lattices": { "new_exclusions": 0, "status": "exact_finite_tests_complete", "tested_primes": [ 2, 3, 5, 7, 11, 13 ] } }, "domain": "latest_complete_joint_Galois_spectra_and_correlated_semion_integer_attempts", "evidence": [ "evidence/rank8_frontier_20261021/r928_semion_rotation/final_campaign.json" ], "limitation": "Six bounded R928 integer searches remain unresolved integer-feasibility problems; numerical negative solver outcomes are not exclusions. Exact positive integer witnesses are necessary data only.", "outcome": "documented_finite_domains_finished", "proof": "evidence/rank8_frontier_20261021/review/QUARTIC_CENTRAL_ORBITS.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/r928_semion_rotation/verify_all.py", "evidence/rank8_frontier_20261021/r928_semion_rotation/charged_joint_spectrum.py", "evidence/rank8_frontier_20261021/noncommutative_joint_spectrum/verify.py", "evidence/rank8_frontier_20261021/r928_semion_rotation/final_campaign.py" ] }, { "details": { "low_count": 1, "passing_low_count": 1 }, "domain": "complete_central_image_low_induction_and_Gram_moment_constraints", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/central_support_results.json" ], "limitation": "This PU necessary-condition test applies after the separately proved Galois reduction; surviving data need not lift to a center.", "outcome": "necessary_data_survives", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py", "evidence/rank8_frontier_20261021/noncommutative/verify_independent.py" ] }, { "details": { "central": true, "dual_fixed_basis_count": 2, "id": "R8_3_928", "invertible": 7, "left_action": [ 7, 2, 1, 4, 3, 6, 5, 0 ], "matrix_block_formal_codegree": [ "4" ], "matrix_block_size": 2, "matrix_involution_trace": -2, "matrix_involution_type": "symplectic", "negative_quotient": { "multiplication": [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ], [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, -1, 0 ], [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ] ], "representatives": [ 1, 3, 5 ], "squares": [ [ -1, 0, 0, 0 ], [ -1, 0, 0, 0 ], [ -1, 0, 0, 0 ] ] }, "pointed_bimodule_orbits": [ { "basis": [ 0, 7 ], "cohomology_attempts": [ { "checked_triples": 8, "omega_exponent": 0, "restricted_cocycle_trivial": true }, { "checked_triples": 8, "omega_exponent": 1, "restricted_cocycle_trivial": true } ], "stabilizer": [ [ 0, 0 ], [ 1, 1 ] ] }, { "basis": [ 1, 2 ], "cohomology_attempts": [ { "checked_triples": 8, "omega_exponent": 0, "restricted_cocycle_trivial": true }, { "checked_triples": 8, "omega_exponent": 1, "restricted_cocycle_trivial": true } ], "stabilizer": [ [ 0, 0 ], [ 1, 1 ] ] }, { "basis": [ 3, 4 ], "cohomology_attempts": [ { "checked_triples": 8, "omega_exponent": 0, "restricted_cocycle_trivial": true }, { "checked_triples": 8, "omega_exponent": 1, "restricted_cocycle_trivial": true } ], "stabilizer": [ [ 0, 0 ], [ 1, 1 ] ] }, { "basis": [ 5, 6 ], "cohomology_attempts": [ { "checked_triples": 8, "omega_exponent": 0, "restricted_cocycle_trivial": true }, { "checked_triples": 8, "omega_exponent": 1, "restricted_cocycle_trivial": true } ], "stabilizer": [ [ 0, 0 ], [ 1, 1 ] ] } ], "right_action": [ 7, 2, 1, 4, 3, 6, 5, 0 ], "scalar_dimension": 4, "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" }, "domain": "exact_matrix_block_involution_and_pointed_bimodule_cocycle_constraints", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/matricial_structure.json" ], "limitation": "This restricted domain does not exhaust general categorifications.", "outcome": "necessary_structure_compatible", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "based_ring_and_pointed_coherence_constraints", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "exact_or_numeric_branch_statuses": [ "undecided_numeric" ], "height": 12, "low_index": 0 }, "domain": "finite_order_higher_rotation_LP", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/certified_928_h12_low0.json" ], "limitation": "Individual infeasible branches do not exclude the ring; numeric undecided is not an exact result.", "outcome": "undecided_numeric", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "exact_or_numeric_branch_statuses": [ "exact_feasible_relaxation" ], "height": 6, "low_index": 0 }, "domain": "finite_order_higher_rotation_LP", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/certified_928_h6_low0.json" ], "limitation": "Individual infeasible branches do not exclude the ring; numeric undecided is not an exact result.", "outcome": "exact_feasible_relaxation", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "exact_or_numeric_branch_statuses": [ "exact_feasible_relaxation" ], "height": 6, "low_index": null }, "domain": "joint_rotation_and_partial_STS_LP", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/modular_928_h6.json" ], "limitation": "Individual infeasible branches do not exclude the ring; numeric undecided is not an exact result.", "outcome": "exact_feasible_relaxation", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] } ], "prior_published_attempts": [ { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "scope_inapplicable", "reason": "Not a rank-eight near-group ring.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "near_group", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "rank_two", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "zero_spectrum", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "lower_rank_subring", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "scope_inapplicable", "reason": "No rank-seven closed based subring.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "rank7_certified_subring", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "dnumber", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "noncomm_dnumber", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "scope_inapplicable", "reason": "No integral positive FP character; integral Drinfeld branch only.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "integral_drinfeld", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "scope_detail": "Bounded exact obstruction search; no conclusion is not a proof that all character fields are cyclotomic.", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "cyclotomic", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "scope_inapplicable", "reason": "The inherited module implementation requires an invertible involution with one or two fixed simples.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "involution_module", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "scope_detail": "Search restricted to fusion-matrix irreducible factors of degrees3through6; no conclusion is not a proof of cyclotomicity.", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "field_automorphisms", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "subring_dimension", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "primitive_codegree", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "column_counts": [ 27, 45, 175, 175 ], "completed": true, "id": "R8_3_928", "multiplicities": [ 1, 1, 1, 2 ], "objects": [ { "cases": [ { "passes": true, "principal_minors": [ [ "-1/8", "9", "-26", "16" ], [ "96", "-480", "768", "-384" ], [ "96", "-480", "768", "-384" ], [ "407280", "-2160000", "3521280", "-1775616" ] ], "second_indicator": 0, "signs": [ 1, 1, 1, 1 ] } ], "lower_trivial_twist_mass": [], "object": 1, "total_mass": [ "-1/8", "9", "-26", "16" ] }, { "cases": [ { "passes": true, "principal_minors": [ [ "-1/8", "9", "-26", "16" ], [ "96", "-480", "768", "-384" ], [ "96", "-480", "768", "-384" ], [ "407280", "-2160000", "3521280", "-1775616" ] ], "second_indicator": 0, "signs": [ 1, 1, 1, 1 ] } ], "lower_trivial_twist_mass": [], "object": 2, "total_mass": [ "-1/8", "9", "-26", "16" ] }, { "cases": [ { "passes": true, "principal_minors": [ [ "3/8", "-22", "86", "-80" ], [ "132", "-480", "192", "384" ], [ "132", "-480", "192", "384" ], [ "679824", "-3609600", "5894400", "-2979840" ] ], "second_indicator": 0, "signs": [ 1, 1, 1, 1 ] } ], "lower_trivial_twist_mass": [], "object": 3, "total_mass": [ "3/8", "-22", "86", "-80" ] }, { "cases": [ { "passes": true, "principal_minors": [ [ "3/8", "-22", "86", "-80" ], [ "132", "-480", "192", "384" ], [ "132", "-480", "192", "384" ], [ "679824", "-3609600", "5894400", "-2979840" ] ], "second_indicator": 0, "signs": [ 1, 1, 1, 1 ] } ], "lower_trivial_twist_mass": [], "object": 4, "total_mass": [ "3/8", "-22", "86", "-80" ] }, { "cases": [ { "passes": true, "principal_minors": [ [ "-1/8", "10", "-46", "48" ], [ "228", "-1056", "1344", "-384" ], [ "228", "-1056", "1344", "-384" ], [ "1504560", "-7979520", "13006080", "-6555648" ] ], "second_indicator": 0, "signs": [ 1, 1, 1, 1 ] } ], "lower_trivial_twist_mass": [], "object": 5, "total_mass": [ "-1/8", "10", "-46", "48" ] }, { "cases": [ { "passes": true, "principal_minors": [ [ "-1/8", "10", "-46", "48" ], [ "228", "-1056", "1344", "-384" ], [ "228", "-1056", "1344", "-384" ], [ "1504560", "-7979520", "13006080", "-6555648" ] ], "second_indicator": 0, "signs": [ 1, 1, 1, 1 ] } ], "lower_trivial_twist_mass": [], "object": 6, "total_mass": [ "-1/8", "10", "-46", "48" ] }, { "cases": [ { "passes": true, "principal_minors": [ [ "-1/8", "8", "-30", "32" ], [ "96", "-384", "384" ], [], [] ], "second_indicator": -1, "signs": [ 1, 1, 0, 0 ] }, { "passes": true, "principal_minors": [ [ "-1/8", "8", "-30", "32" ], [ "96", "-384", "384" ], [], [] ], "second_indicator": 1, "signs": [ 1, 1, 0, 0 ] } ], "lower_trivial_twist_mass": [], "object": 7, "total_mass": [ "-1/8", "8", "-30", "32" ] } ], "obstructed_objects": [], "outcome": "no_conclusion", "scope": "pseudo_unitary_only", "source": "evidence/rank8_m3_20261021/noncommutative/central_moments_results.json", "stage": "central_twist_moments", "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "no_conclusion", "report": { "original_components": 1, "scalar_rank": 4 }, "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/noncommutative/results.json", "stage": "noncommutative_scalar_sphericalization", "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" }, { "attempt_kind": "exclusion", "attempts": [], "budget_skipped_objects": 0, "candidate_object_count": 0, "completed_objects": 0, "frontier_ledger_digest": "23837060c2c8186fb971d5c7515620cc0720508d025337e12154eb20b1935270", "id": "R8_3_928", "object_family": "one nontrivial dual-pair atom or sum of two distinct nonunit self-dual orbit atoms", "outcome": "inapplicable_noncommutative", "per_object_case_budget": 100000, "profile_derived_from_tensor": { "commutative": false, "dual": [ 0, 2, 1, 4, 3, 6, 5, 7 ], "fixed_point_involutions": [], "invertibles": [ 0, 7 ] }, "scope": "general_complex_fusion_category", "scope_note": "a modular survivor gives no exclusion; it is not a feasible categorification or an exact module witness", "source": "evidence/rank8_m3_20261021/noncommutative/split_module_object_results.json", "stage": "split_module_object_results", "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "no_conclusion", "report": { "lowest_numerical_eigenvalue": -5.644629513881443e-07, "scope": "numerical discovery only" }, "source": "evidence/rank8_m3_20261021/noncommutative/unitary_primary3_results.json", "stage": "unitary_primary3_results", "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/one_spectrum_m3.json", "stage": "one_spectrum", "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" }, { "attempt_kind": "construction", "id": "R8_3_928", "outcome": "implementation_ineligible", "reason": "Nonintegral noncommutative FP-field quotient not implemented in this diagnostic.", "scope": "weakly_group_theoretical_models_only", "source": "evidence/rank8_m3_20261021/pipeline/frobenius_results.json", "stage": "strong_frobenius_weak_group_theoretical", "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "scope_inapplicable", "reason": "No closed based subring of rank6or7 exists.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/unified_scope_attempts.json", "stage": "expanded_exact_negative_subring_transfer", "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" }, { "attempt_kind": "exclusion", "id": "R8_3_928", "outcome": "scope_inapplicable", "reason": "Regular Casimir polynomial has no irreducible quadratic factor.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/unified_scope_attempts.json", "stage": "multiquadratic_casimir_compression", "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" }, { "attempt_kind": "construction", "family": "72 explicitly constructed rank-eight tensors", "id": "R8_3_928", "matching_test_numbers": [], "outcome": "no_match_in_tested_family", "scope": "negative means no match in this finite list; crossed products are only abstract ring tests unless separately certified", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "72 explicitly constructed rank-eight tensors", "tested_tensors": 72 }, { "attempt_kind": "construction", "family": "ordinary finite-group representation categories", "id": "R8_3_928", "outcome": "family_excluded", "reason": "noncommutative ring", "rejected_by_elementary_filter": true, "scope": "This construction family only; no arbitrary categorification exclusion.", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "ordinary finite-group representation categories" }, { "attempt_kind": "construction", "family": "pointed and ordinary Tambara-Yamagami categories", "id": "R8_3_928", "outcome": "family_excluded", "reason": "their fusion multiplicities are at most1, but this source has maximum3", "rejected": true, "scope": "This construction family only; no arbitrary categorification exclusion.", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "pointed and ordinary Tambara-Yamagami categories" }, { "attempt_kind": "construction", "family": "Morita sources obtained by cyclic-prime extension/equivariantization", "id": "R8_3_928", "outcome": "no_conclusion", "scope": "if applicable, excludes Morita equivalence to a nontrivial cyclic-prime extension or equivariantization; not arbitrary categorification", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "Morita sources obtained by cyclic-prime extension/equivariantization", "structural_filter": { "adjoint_basis": [ 0, 1, 2, 3, 4, 5, 6, 7 ], "center_has_no_nontrivial_invertible_if_categorified": false, "central_invertible_indices": [ 0, 7 ], "invertible_indices": [ 0, 7 ], "scope": "if applicable, excludes Morita equivalence to a nontrivial cyclic-prime extension or equivariantization; not arbitrary categorification" } }, { "attempt_kind": "construction", "family": "bounded classical WZW dimension-ratio search", "id": "R8_3_928", "leads": [], "outcome": "heuristic_search_completed", "scope": "heuristic numerical leads only; no tensor identification or general exclusion", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "bounded classical WZW dimension-ratio search", "tested_models": 1505 }, { "attempt_kind": "construction", "casimir_characteristic_polynomial": [ 1, -608, 30336, -665600, 8114176, -58982400, 254803968, -603979776, 603979776 ], "casimir_matrix": [ [ 8, 8, 8, 8, 8, 12, 12, 0 ], [ 8, 64, 56, 68, 68, 88, 88, 8 ], [ 8, 56, 64, 68, 68, 88, 88, 8 ], [ 8, 68, 68, 88, 80, 104, 104, 8 ], [ 8, 68, 68, 80, 88, 104, 104, 8 ], [ 12, 88, 88, 104, 104, 144, 136, 12 ], [ 12, 88, 88, 104, 104, 136, 144, 12 ], [ 0, 8, 8, 8, 8, 12, 12, 8 ] ], "families": [ { "all_larger_odd_orders": { "first_order": 9, "source_upper_bound": 688, "strict_target_lower_bound": 729 }, "family": "odd_Haagerup_Izumi_and_all_Morita_duals", "outcome": "specific_family_Morita_model_excluded", "tested_odd_orders": [ { "casimir_remainder": [ -226103671089, 740498581755 ], "dimension_polynomial": [ 1, -39, 117 ], "odd_order": 3, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ -2191077832954791, 11361702442419851 ], "dimension_polynomial": [ 1, -145, 725 ], "odd_order": 5, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ -392067593047965885, 2798307055732470115 ], "dimension_polynomial": [ 1, -371, 2597 ], "odd_order": 7, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" } ] }, { "all_larger_odd_orders": { "first_order": 7, "source_upper_bound": 688, "strict_target_lower_bound": 2401 }, "family": "specified_near_group_and_all_Morita_duals", "outcome": "specific_family_Morita_model_excluded", "tested_odd_orders": [ { "casimir_remainder": [ -449749166719707, 4418811550063323 ], "dimension_polynomial": [ 1, -117, 1053 ], "odd_order": 3, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ 16136908813725850149, -418384855372263643349 ], "dimension_polynomial": [ 1, -725, 18125 ], "odd_order": 5, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" } ] } ], "global_dimension_diagonal_lower_bound": 144, "global_dimension_upper_bound": 688, "id": "R8_3_928", "outcome": "no_model_from_either_tested_family", "scope": "specific_model_families_only_not_categorification_exclusion", "source": "evidence/rank8_m3_20261021/models/fresh_202609/odd_hi_dimension_filter.json", "source_dual": [ 0, 2, 1, 4, 3, 6, 5, 7 ], "stage": "odd_hi_and_specified_near_group_Morita_dimension_filter", "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" }, { "AH_dimension_polynomial_remainder": [ -88555211961925632, 359589063046463488 ], "attempt_kind": "construction", "casimir_characteristic_polynomial": [ 1, -608, 30336, -665600, 8114176, -58982400, 254803968, -603979776, 603979776 ], "family_outcome": "different_global_dimension_so_no_AH_Morita_model", "general_categorification_excluded": false, "id": "R8_3_928", "invertibles": [ 0, 7 ], "maximum_multiplicity": 3, "outcome": "family_excluded", "rank": 8, "scope": "known_Asaeda_Haagerup_Morita_family_only", "source": "evidence/rank8_m3_20261021/models/asaeda_haagerup_attempt.json", "stage": "known_Asaeda_Haagerup_Morita_family_filter", "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" } ], "prior_stage_completed": [ { "domain": "integral_proper_subcategories_and_braided_Morita_representatives", "evidence": [ "evidence/rank8_research_20261022/braided_morita/PROOF.md", "evidence/rank8_research_20261022/braided_morita/verification.json", "site/BRAIDED_CATEGORIFICATION.md" ], "limitation": "This restricts braided Morita constructions; arbitrary unbraided categorification remains unresolved for the six surviving rings.", "outcome": "Nonintegral total FP dimension and integral proper fusion subcategories force any braided Morita representative to yield a braiding on the original category. The general braiding exclusions therefore exclude every braided Morita representative.", "scope": "general_complex_fusion_category", "verifiers": [ "evidence/rank8_research_20261022/braided_morita/verify.py", "site/verify_braided_scope.py" ] }, { "domain": "pointed_center_and_rank_square_Morita_obstructions", "evidence": [ "evidence/rank8_research_20261022/r928_r941/PROOF.md", "evidence/rank8_research_20261022/r928_r941/verification.json" ], "limitation": "This excludes braided representatives only and supplies no unbraided categorification.", "outcome": "An independent center argument excludes braided Morita representatives: the R928 center has a semion and no nontrivial boson, while all fourteen R941 charge allocations have centralizer rank 22, incompatible with the rank-square formula for a forced fermionic center.", "scope": "general_complex_Morita_model_obstruction", "verifiers": [ "evidence/rank8_research_20261022/r928_r941/verify.py" ] }, { "domain": "dimension_block_conductor_bound_and_saturation", "evidence": [ "evidence/rank8_research_20261022/pointed_followup/dimension_block/PROOF.md", "evidence/rank8_research_20261022/pointed_followup/dimension_block/verification.json" ], "limitation": "All seven incoming rings pass the scalar screen. No refined cone run, model, or exclusion follows from this criterion; R1795 is separately excluded by C2 Fourier packing.", "outcome": "Every additional twist-one central simple outside the dimension orbit has dimension ratio in Q(sqrt(2)); its coordinates satisfy a+b=3c and a+2b=6d. All forced I(1) summands satisfy the applicable mask.", "scope": "necessary_center_conditions_in_a_pseudo_unitary_Galois_form", "verifiers": [ "evidence/rank8_research_20261022/pointed_followup/dimension_block/verify.py" ] } ], "prior_unitary_obstruction_source": null, "prior_unitary_status": "not_separately_decided", "screening_case_specific_focus": "General Galois-PU reduction and quaternionic rational block; complete joint integer Gram/cubic domain forces a central semion, leaving 15 parity patterns and ten exact correlated height-six moment witnesses; four exact integer witnesses and six explicitly unresolved bounded integer attempts; connected-algebra underlying-object filters and specified unbounded model families tested.", "screening_definition": "Extensively investigated exotic candidate means that the ring survived the documented, case-specific campaign of known and newly developed exclusion tests and explicit model searches, including complete finite domains and specified unbounded construction families. It proves neither existence of a fusion category nor novelty of a future realization. Each case lists exactly what was exhausted, what survived and the remaining hypotheses.", "screening_stage": "extensively_investigated_exotic_candidate", "tensor_sha256": "94dfb90235dbb3ba95864a1be23437943ea3b1316ec82ca44baf419262890614" }, { "categorification_or_novelty_proved": false, "general_status": "unresolved", "id": "R8_3_941", "latest_completed": [ { "domain": "saturated_dimension_character_conductor_and_signed_spectral_capacity", "evidence": [ "evidence/rank8_creative_20261022/unified/verification.json", "evidence/rank8_creative_20261022/unified/composite_verification.json", "evidence/rank8_creative_20261022/unified/prime_gauss_audit.json" ], "limitation": "These restrictions alone establish neither existence nor nonexistence.", "outcome": "Equality in a dimension-character bound forces the cyclic modular representation through its character conductor. Signed prime and composite refinements give exact twist support and spectral capacities, with no twist-order cutoff.", "proof": "evidence/rank8_creative_20261022/unified/SATURATED_CONDUCTOR.md", "scope": "Necessary conditions for a pseudo-unitary Galois form of any complex categorification.", "verifiers": [ "evidence/rank8_creative_20261022/unified/verify_saturated_conductor.py", "evidence/rank8_creative_20261022/unified/verify_composite_support.py", "evidence/rank8_creative_20261022/unified/prime_gauss_audit.py" ] }, { "domain": "complete_signed_conductor_phase_capacity_and_packet_product_refinement", "evidence": [ "evidence/rank8_creative_20261022/r928_r941/verification.json" ], "limitation": "A feasible rational phase witness is not a center, associator, or categorification. Pair products do not complete the center fusion ring.", "outcome": "All fourteen complete R941 charge allocations retain exact feasible necessary witnesses after signed conductor-fibre capacities; 55 exact integral distinguished pair products per allocation are forced.", "proof": "evidence/rank8_creative_20261022/r928_r941/PROOF.md", "scope": "Complete existing pattern/charge domain; necessary center and distinguished-product constraints.", "verifiers": [ "evidence/rank8_creative_20261022/r928_r941/verify.py" ] } ], "limitations": [ "No completed search here exhausts all complex associators or all constructions.", "A feasible necessary-condition system is not a categorification.", "Extensively investigated exotic candidate is a research-screening description, not a proof of categorifiability or novelty." ], "new_attempts": [ { "details": { "FP_global_dimension_minimal_polynomial": [ "1", "-576", "11520", "-73728", "147456" ], "general_complex_exclusion_claimed_by_this_audit": false, "invertible_count": 2, "invertibles_abelian": true, "largest_real_root_isolating_interval": [ "9104089/16389", "1111/2" ], "rational_scaling_outcome": "impossible_by_minimal_polynomial_degree", "routes": { "deequivariantization_of_NG": "excluded", "direct_or_Morita_equivalent_to_NG": "excluded", "equivariantization_of_NG": "excluded", "group_extension_of_NG": "excluded", "neutral_component_of_nontrivially_graded_NG": "excluded" } }, "domain": "all_NG_Cn_n_and_five_single_operation_routes_all_positive_n", "evidence": [ "evidence/rank8_frontier_20261021/structural_models/near_group_scaling.json" ], "limitation": "No arbitrary chains, inserted Morita changes, or noncentral condensations are covered.", "outcome": "family_excluded", "proof": "evidence/rank8_frontier_20261021/structural_models/NEAR_GROUP_SCALING.md", "scope": "general_complex_construction_family", "verifiers": [ "evidence/rank8_frontier_20261021/structural_models/near_group_scaling.py" ] }, { "details": { "PSD_Gram_survivors": 38, "bounded_selfdual_vectors_tested": 184320, "entrywise_Gram_survivors": 11314, "independent_complete_Python_domain": null, "invertible_stabilizer_survivors": 93120, "one_plus_simple_surviving_indices": [ 3, 4, 5, 6, 7 ] }, "domain": "all_connected_separable_algebra_underlying_objects_entrywise_and_PSD_Gram", "evidence": [ "evidence/rank8_frontier_20261021/structural_models/algebra_candidates.json" ], "limitation": "Surviving objects are not algebra structures; no multiplication, associator, model or general exclusion is claimed.", "outcome": "complete_necessary_object_filter_survives", "proof": "evidence/rank8_frontier_20261021/structural_models/ALGEBRA_CANDIDATES.md", "scope": "general_complex_algebra_construction_constraints", "verifiers": [ "evidence/rank8_frontier_20261021/structural_models/algebra_candidates.py" ] }, { "details": { "complete_signed_count": 128, "conclusion": "every categorification is spherical and has a pseudo-unitary Galois conjugate; no existence or exclusion conclusion", "normalized_indicator_domains_used": false }, "domain": "complete_signed_sphericalization_and_general_Galois_PU_reduction", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative_followup/scope_gates.json" ], "limitation": "This proves a scope reduction, not categorifiability or unitarity. Necessary PU constraints apply after Galois conjugation.", "outcome": "Galois_pseudo_unitary_reduction", "proof": "evidence/rank8_frontier_20261021/noncommutative_followup/README.md", "scope": "general_complex_fusion_category", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative_followup/verify_scope_gates.py" ] }, { "details": { "allowed_order_sequence": [ 8, 12, 24 ], "complete_order_candidates": [ 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30 ], "dimension_bounded_product_count": 5, "exponent_box_size": 12, "positive_integer_scaling_matches": [] }, "domain": "all_levels_SU2_adjoint_finite_products_integer_multiplier_chains_and_Morita_duals", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative_followup/su2_all_frontier.json" ], "limitation": "Other Lie types, arbitrary algebra condensation, dimension-decreasing de-equivariantization, or general categorifications.", "outcome": "family_excluded_by_exact_global_dimension", "proof": "evidence/rank8_frontier_20261021/noncommutative_followup/SU2_FAMILY.md", "scope": "general_complex_construction_family_dimension_obstruction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative_followup/su2_all_frontier.py", "evidence/rank8_frontier_20261021/noncommutative_followup/su2_product_family.py" ] }, { "details": { "families": [], "global_dimension_minimal_polynomial": [ "1", "-576", "11520", "-73728", "147456" ], "id": "R8_3_941", "outcome": "family_dimension_excluded_by_degree", "positive_root_interval": [ "9104089/16389", "1111/2" ], "scope": "specific_model_family_only", "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, "domain": "all_generalized_near_group_dimension_parameters_with_integer_scalings", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative_followup/generalized_near_group_all_frontier.json" ], "limitation": "This is a necessary dimension-family test; retained parameter triples are not categories, and integral targets lie outside its irrational quadratic parameter theorem.", "outcome": "family_dimension_excluded_by_degree", "proof": "evidence/rank8_frontier_20261021/noncommutative_followup/GENERALIZED_NEAR_GROUP_TABLE.md", "scope": "specific_model_family_only", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative_followup/generalized_near_group_all_frontier.py" ] }, { "details": { "complete_joint_pattern_count": 343, "independent_Python_nodes": 561, "native_search_nodes": 1665 }, "domain": "complete_integer_Gram_and_all_twenty_distinguished_cubic_coefficients", "evidence": [ "evidence/rank8_frontier_20261021/quartic_integer_center/gram_cubic.json", "evidence/rank8_frontier_20261021/quartic_integer_center/source.json" ], "limitation": "All343 patterns are necessary center data; their existence does not construct a modular category.", "outcome": "343_exact_necessary_integer_patterns_survive", "proof": "evidence/rank8_frontier_20261021/quartic_integer_center/PROOF.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/quartic_integer_center/source.py", "evidence/rank8_frontier_20261021/quartic_integer_center/gram_cubic.py" ] }, { "details": { "forced_low_translate_high_indices": [ 40, 41, 42, 43, 57 ], "source_invertible_index": 3 }, "domain": "forced_order_two_central_lift_and_complete_tensor_orbit_graph", "evidence": [ "evidence/rank8_frontier_20261021/review/quartic_central_orbits.json" ], "limitation": "A central lift does not imply a tensor-factor decomposition of the nonbraided source category.", "outcome": "central_invertible_forced_in_all_integer_patterns", "proof": "evidence/rank8_frontier_20261021/review/QUARTIC_CENTRAL_ORBITS.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/review/verify_quartic_central_orbits.py" ] }, { "details": { "bosonic_branch": { "dependencies": [ { "path": "manuscripts/rank4/papers/companion_sections/14_finite_certification.tex", "sha256": "0ec47c3c3da2c46bfee20b06214b9f7704edea841955cb2c86b7b0522f199e8b" }, { "path": "supplements/rank4/data/historical_32.json.gz", "sha256": "61a5c638e7d20d4e7fa33c9bcb8bb787c3735ebfebb59e024d761c2139be9c51" }, { "path": "supplements/rank4/data/through32_mapping.json.gz", "sha256": "d88e7333de292167d5fce39d5c311efc9614d0f47a89adbf0eda1819a2c4c632" } ], "historical_label": "F_{6,19}", "historical_source_to_quotient_basis_map": [ 0, 3, 1, 2 ], "orbits": [ [ 0, 3 ], [ 1, 2 ], [ 4, 5 ], [ 6, 7 ] ], "outcome": "excluded_by_unrestricted_rank_four_bound", "quotient_global_dimension": "72+48sqrt(2)+40sqrt(3)+28sqrt(6)", "quotient_tensor": [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 1, 0, 0 ], [ 1, 2, 2, 4 ], [ 0, 2, 2, 3 ], [ 0, 4, 3, 4 ] ], [ [ 0, 0, 1, 0 ], [ 0, 2, 2, 3 ], [ 1, 2, 2, 2 ], [ 0, 3, 2, 4 ] ], [ [ 0, 0, 0, 1 ], [ 0, 4, 3, 4 ], [ 0, 3, 2, 4 ], [ 1, 4, 4, 6 ] ] ] }, "fermionic_branch": { "outcome": "necessary_central_lift_has_twist_minus_one", "translated_low_twists": [ { "forced_twist": -1, "high_index": 40, "low_index": 0, "moments_1_through_6": [ -1, 1, -1, 1, -1, 1 ], "monodromy": 1 }, { "forced_twist": -1, "high_index": 41, "low_index": 1, "moments_1_through_6": [ -1, 1, -1, 1, -1, 1 ], "monodromy": 1 }, { "forced_twist": -1, "high_index": 42, "low_index": 2, "moments_1_through_6": [ -1, 1, -1, 1, -1, 1 ], "monodromy": 1 }, { "forced_twist": -1, "high_index": 43, "low_index": 3, "moments_1_through_6": [ -1, 1, -1, 1, -1, 1 ], "monodromy": 1 }, { "forced_twist": 1, "high_index": 57, "low_index": 4, "moments_1_through_6": [ 1, 1, 1, 1, 1, 1 ], "monodromy": -1 } ] }, "free_source_permutation": [ 3, 2, 1, 0, 5, 4, 7, 6 ], "gram_receipt_sha256": "484c887ee83f0584436deba2ab85e1c24cf58c98bae384c254acd4a76779c65a", "id": "R8_3_941", "order_two_object": 3, "scope": "under the independently verified Galois-PU reduction, unique central lift and positive source indicator; no rank-four bound is applied to a fermionic quotient", "source_receipt_sha256": "83edb031f16d7aa1f1c7c4f3578e11f4da3b86adeae6fd3fc88f8f54ad1f92d4", "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, "domain": "complete_bosonic_central_lift_quotient_and_fermionic_alternative", "evidence": [ "evidence/rank8_frontier_20261021/quartic_integer_center/r941_fermionic.json" ], "limitation": "The ordinary rank-four bound applies only to the Tannakian branch; no ordinary rank-four classification is imposed on a fermionic quotient.", "outcome": "bosonic_branch_excluded_fermionic_lift_forced", "proof": "evidence/rank8_frontier_20261021/quartic_integer_center/R941_FERMION.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/quartic_integer_center/r941_fermionic.py" ] }, { "details": { "complete_charge_allocation_count": 93, "independent_full_box_assignments_tested": 41664, "input_Gram_cubic_patterns": 343, "rref_assignments_tested": 446, "surviving_pattern_count": 68 }, "domain": "complete_integer_fermion_monodromy_charge_allocations", "evidence": [ "evidence/rank8_frontier_20261021/quartic_integer_center/r941_charges.json" ], "limitation": "A surviving charge allocation fixes necessary eigenvalue multiplicities only; it is not a center or associator.", "outcome": "exact_charge_allocations_survive", "proof": "evidence/rank8_frontier_20261021/quartic_integer_center/R941_FERMION.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/quartic_integer_center/r941_charges.py" ] }, { "details": { "all_quotient_fusion_subrings": [ [ 0 ], [ 0, 1, 2, 3 ] ], "all_unit_containing_subsets_tested": 8, "direct_tensor_products_of_two_rank_two_supercategories": "excluded_by_all_m_type_and_absence_of_rank_two_subring", "id": "R8_3_941", "limitation": "No classification of general rank-four superfusion categories, arbitrary superalgebra condensations, or mixed construction chains is asserted.", "primary_sources": [ { "locator": "Section4.1.3, with free fermion action giving m-type orbit simples", "url": "https://arxiv.org/abs/1709.01941" }, { "locator": "Section3 and AppendixE; rank-two computations cover stated low multiplicities, completeness beyond them is conjectural", "url": "https://arxiv.org/abs/2208.02757" }, { "locator": "Introduction theorem(a),(b): centers of types C6+6 and C2xC4+8, pointed-factor removal and bosonic condensation", "url": "https://arxiv.org/abs/2305.09108" } ], "quotient_global_dimension_polynomial": [ 1, -288, 2880, -9216, 9216 ], "quotient_rank": 4, "quotient_receipt_sha256": "eef4760f4887b137da99e9cde6950ae662694e7684516899c474bcd7c9cbc015", "quotient_simple_types": [ "m", "m", "m", "m" ], "scope": "specified direct super tensor-product and published near-group-center constructions only", "specified_near_group_center_families": [ { "center_global_dimension": "360*sqrt(15) + 1440", "dimension_field": "Q(sqrt(15))", "outcome": "every_rational_rescaling_of_these_center_dimensions_differs_from_quartic_target", "source_global_dimension": "6*sqrt(15) + 30", "type": "near_group_order_6_multiplicity_6" }, { "center_global_dimension": "1536*sqrt(6) + 3840", "dimension_field": "Q(sqrt(6))", "outcome": "every_rational_rescaling_of_these_center_dimensions_differs_from_quartic_target", "source_global_dimension": "16*sqrt(6) + 48", "type": "near_group_order_8_multiplicity_8" } ], "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, "domain": "all_direct_rank_two_super_products_and_specified_published_near_group_center_routes", "evidence": [ "evidence/rank8_frontier_20261021/quartic_integer_center/r941_super_constructions.json" ], "limitation": "No classification of general rank-four superfusion categories, arbitrary superalgebra condensations, or mixed construction chains is asserted.", "outcome": "specified_super_construction_families_excluded", "proof": "evidence/rank8_frontier_20261021/quartic_integer_center/R941_FERMION.md", "scope": "general_complex_scoped_construction_obstructions", "verifiers": [ "evidence/rank8_frontier_20261021/quartic_integer_center/r941_super_constructions.py" ] }, { "details": { "charge_surviving_pattern_count": 68, "complete_charge_allocation_count": 93, "exact_excluded_count": 77, "exact_feasible_count": 16, "input_integer_pattern_count": 343 }, "domain": "all_fermion_charge_allocations_correlated_height6_rotation_STS_and_fermion_S_row", "evidence": [ "evidence/rank8_frontier_20261021/quartic_integer_center/r941_paired_rotation.json", "evidence/rank8_frontier_20261021/review/r941_fermion_operator_review.json", "evidence/rank8_frontier_20261021/review/r941_fermion_campaign_review.json" ], "limitation": "The complete finite correlated cone is necessary data and does not classify superfusion associators.", "outcome": "exact_feasible_relaxation; exact_infeasible_relaxation", "proof": "evidence/rank8_frontier_20261021/quartic_integer_center/R941_FERMION.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/quartic_integer_center/r941_paired_rotation.py", "evidence/rank8_frontier_20261021/review/verify_r941_fermion_operator.py", "evidence/rank8_frontier_20261021/review/verify_r941_fermion_campaign.py" ] }, { "details": { "exact_integer_witness_count": 12, "exact_modular_exclusion_count": 0, "planned_allocation_indices": [ 4, 8, 19, 21, 31, 34, 43, 45, 56, 60, 72, 73, 77, 79, 85, 86 ], "status": "all_planned_integer_count_attempts_finished", "unresolved_integer_feasibility_count": 4 }, "domain": "all_sixteen_surviving_correlated_fermion_cones_supported_modular_and_bounded_integer_attempts", "evidence": [ "evidence/rank8_frontier_20261021/quartic_integer_center/r941_integer_counts.json", "evidence/rank8_frontier_20261021/review/r941_integer_witness_review.json" ], "limitation": "complete supported modular tests at primes2,3,5,7,11,13 and one30-second/20000-node bounded MIP attempt per surviving cone; numerical infeasibility is not an exact exclusion", "outcome": "finite_attempts_finished_with_exact_witnesses_or_explicit_unresolved_integer_feasibility", "proof": "evidence/rank8_frontier_20261021/quartic_integer_center/R941_FERMION.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/quartic_integer_center/r941_integer_counts.py", "evidence/rank8_frontier_20261021/review/verify_r941_integer_witnesses.py" ] }, { "details": { "actual_center_control": { "actual_fixed_fermion_objects": 3, "center_rank": 9, "mixed_integer_matrix_trace_identities": 56, "model": "Z(Ising)" }, "complete_charge_allocation_count": 93, "mixed_traces_per_allocation": 420, "paired_allocations_removed": [ 4, 86 ], "paired_surviving_allocations": [ 8, 19, 21, 31, 34, 43, 45, 56, 60, 72, 73, 77, 79, 85 ], "retained_exact_integer_witness_allocations": [ 8, 19, 21, 31, 34, 43, 45, 56, 60, 72, 73 ], "retained_integer_feasibility_unresolved": [ 77, 79, 85 ], "status": "exact_complete" }, "domain": "complete_involutive_permutation_fixed_point_bounds_and_420_mixed_fusion_traces_per_charge_allocation", "evidence": [ "evidence/rank8_frontier_20261021/review/r941_fusion_traces.json" ], "limitation": "This final finite necessary-condition gate removes two earlier witnesses; fourteen moment systems remain, with no categorical construction implied.", "outcome": "fourteen_correlated_survivors_eleven_integer_witnesses_three_unresolved_integer_cases", "proof": "evidence/rank8_frontier_20261021/review/R941_FUSION_TRACE_REVIEW.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/review/verify_r941_fusion_traces.py" ] }, { "details": { "counterexample": "explicit unitary Vec^omega(Q8) with a freely acting central fermion but every scalar character positive on its source label" }, "domain": "proposed_negative_scalar_character_obstruction_for_a_central_fermion", "evidence": [ "evidence/rank8_frontier_20261021/review/fermion_no_negative_character.json" ], "limitation": "This counterexample rejects an invalid universal implication; it neither constructs nor excludes R941.", "outcome": "universal_implication_refuted_by_an_actual_unitary_pointed_counterexample", "proof": "evidence/rank8_frontier_20261021/review/FERMION_DIMENSION_CHARACTER_COUNTEREXAMPLE.md", "scope": "scoped_obstruction_validation", "verifiers": [ "evidence/rank8_frontier_20261021/review/fermion_no_negative_character.py" ] }, { "details": { "charged_joint_spectrum": { "input_allocations": 93, "status": "exact_complete", "surviving_allocations": 93, "surviving_patterns": 68 }, "id": "R8_3_941", "joint_spectrum": { "input_patterns": 343, "status": "exact_complete", "surviving_patterns": 255 } }, "domain": "latest_complete_joint_Galois_spectra_and_correlated_semion_integer_attempts", "evidence": [ "evidence/rank8_frontier_20261021/r928_semion_rotation/final_campaign.json" ], "limitation": "Six bounded R928 integer searches remain unresolved integer-feasibility problems; numerical negative solver outcomes are not exclusions. Exact positive integer witnesses are necessary data only.", "outcome": "documented_finite_domains_finished", "proof": "evidence/rank8_frontier_20261021/review/QUARTIC_CENTRAL_ORBITS.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/r928_semion_rotation/verify_all.py", "evidence/rank8_frontier_20261021/r928_semion_rotation/charged_joint_spectrum.py", "evidence/rank8_frontier_20261021/noncommutative_joint_spectrum/verify.py", "evidence/rank8_frontier_20261021/r928_semion_rotation/final_campaign.py" ] }, { "details": { "low_count": 1, "passing_low_count": 1 }, "domain": "complete_central_image_low_induction_and_Gram_moment_constraints", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/central_support_results.json" ], "limitation": "This PU necessary-condition test applies after the separately proved Galois reduction; surviving data need not lift to a center.", "outcome": "necessary_data_survives", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py", "evidence/rank8_frontier_20261021/noncommutative/verify_independent.py" ] }, { "details": { "central": true, "dual_fixed_basis_count": 6, "id": "R8_3_941", "invertible": 3, "left_action": [ 3, 2, 1, 0, 5, 4, 7, 6 ], "matrix_block_formal_codegree": [ "4" ], "matrix_block_size": 2, "matrix_involution_trace": 2, "matrix_involution_type": "orthogonal", "negative_quotient": { "multiplication": [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ], [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, -1, 0 ], [ 0, -1, 0, 0 ], [ 1, 0, 0, 0 ] ], "representatives": [ 1, 4, 6 ], "squares": [ [ -1, 0, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 0, 0, 0 ] ] }, "pointed_bimodule_orbits": [ { "basis": [ 0, 3 ], "cohomology_attempts": [ { "checked_triples": 8, "omega_exponent": 0, "restricted_cocycle_trivial": true }, { "checked_triples": 8, "omega_exponent": 1, "restricted_cocycle_trivial": true } ], "stabilizer": [ [ 0, 0 ], [ 1, 1 ] ] }, { "basis": [ 1, 2 ], "cohomology_attempts": [ { "checked_triples": 8, "omega_exponent": 0, "restricted_cocycle_trivial": true }, { "checked_triples": 8, "omega_exponent": 1, "restricted_cocycle_trivial": true } ], "stabilizer": [ [ 0, 0 ], [ 1, 1 ] ] }, { "basis": [ 4, 5 ], "cohomology_attempts": [ { "checked_triples": 8, "omega_exponent": 0, "restricted_cocycle_trivial": true }, { "checked_triples": 8, "omega_exponent": 1, "restricted_cocycle_trivial": true } ], "stabilizer": [ [ 0, 0 ], [ 1, 1 ] ] }, { "basis": [ 6, 7 ], "cohomology_attempts": [ { "checked_triples": 8, "omega_exponent": 0, "restricted_cocycle_trivial": true }, { "checked_triples": 8, "omega_exponent": 1, "restricted_cocycle_trivial": true } ], "stabilizer": [ [ 0, 0 ], [ 1, 1 ] ] } ], "right_action": [ 3, 2, 1, 0, 5, 4, 7, 6 ], "scalar_dimension": 4, "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, "domain": "exact_matrix_block_involution_and_pointed_bimodule_cocycle_constraints", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/matricial_structure.json" ], "limitation": "This restricted domain does not exhaust general categorifications.", "outcome": "necessary_structure_compatible", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "based_ring_and_pointed_coherence_constraints", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "exact_or_numeric_branch_statuses": [ "undecided_numeric" ], "height": 12, "low_index": 0 }, "domain": "finite_order_higher_rotation_LP", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/certified_941_h12_low0.json" ], "limitation": "Individual infeasible branches do not exclude the ring; numeric undecided is not an exact result.", "outcome": "undecided_numeric", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "exact_or_numeric_branch_statuses": [ "exact_feasible_relaxation" ], "height": 6, "low_index": 0 }, "domain": "finite_order_higher_rotation_LP", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/certified_941_h6_low0.json" ], "limitation": "Individual infeasible branches do not exclude the ring; numeric undecided is not an exact result.", "outcome": "exact_feasible_relaxation", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { 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"exclusion", "id": "R8_3_941", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "dnumber", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_941", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "noncomm_dnumber", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_941", "outcome": "scope_inapplicable", "reason": "No integral positive FP character; integral Drinfeld branch only.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "integral_drinfeld", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_941", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "scope_detail": "Bounded exact obstruction search; no conclusion is not a proof that all character fields are cyclotomic.", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "cyclotomic", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_941", "outcome": "scope_inapplicable", "reason": "The inherited module implementation requires an invertible involution with one or two fixed simples.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "involution_module", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_941", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "scope_detail": "Search restricted to fusion-matrix irreducible factors of degrees3through6; no conclusion is not a proof of cyclotomicity.", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "field_automorphisms", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_941", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "subring_dimension", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_941", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "primitive_codegree", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "column_counts": [ 27, 45, 175, 175 ], "completed": true, "id": "R8_3_941", "multiplicities": [ 1, 1, 1, 2 ], "objects": [ { "cases": [ { "passes": true, "principal_minors": [ [ "3/8", "-22", "86", "-80" ], [ "132", "-480", "192", "384" 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"8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, { "attempt_kind": "exclusion", "attempts": [], "budget_skipped_objects": 0, "candidate_object_count": 0, "completed_objects": 0, "frontier_ledger_digest": "23837060c2c8186fb971d5c7515620cc0720508d025337e12154eb20b1935270", "id": "R8_3_941", "object_family": "one nontrivial dual-pair atom or sum of two distinct nonunit self-dual orbit atoms", "outcome": "inapplicable_noncommutative", "per_object_case_budget": 100000, "profile_derived_from_tensor": { "commutative": false, "dual": [ 0, 2, 1, 3, 4, 5, 6, 7 ], "fixed_point_involutions": [], "invertibles": [ 0, 3 ] }, "scope": "general_complex_fusion_category", "scope_note": "a modular survivor gives no exclusion; it is not a feasible categorification or an exact module witness", "source": "evidence/rank8_m3_20261021/noncommutative/split_module_object_results.json", "stage": "split_module_object_results", "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, { "attempt_kind": "exclusion", "id": "R8_3_941", "outcome": "no_conclusion", "report": { "lowest_numerical_eigenvalue": -5.644629592080283e-07, "scope": "numerical discovery only" }, "source": "evidence/rank8_m3_20261021/noncommutative/unitary_primary3_results.json", "stage": "unitary_primary3_results", "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, { "attempt_kind": "exclusion", "id": "R8_3_941", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/one_spectrum_m3.json", "stage": "one_spectrum", "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, { "attempt_kind": "construction", "id": "R8_3_941", "outcome": "implementation_ineligible", "reason": "Nonintegral noncommutative FP-field quotient not implemented in this diagnostic.", "scope": "weakly_group_theoretical_models_only", "source": "evidence/rank8_m3_20261021/pipeline/frobenius_results.json", "stage": "strong_frobenius_weak_group_theoretical", "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, { "attempt_kind": "exclusion", "id": "R8_3_941", "outcome": "scope_inapplicable", "reason": "No closed based subring of rank6or7 exists.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/unified_scope_attempts.json", "stage": "expanded_exact_negative_subring_transfer", "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, { "attempt_kind": "exclusion", "id": "R8_3_941", "outcome": "scope_inapplicable", "reason": "Regular Casimir polynomial has no irreducible quadratic factor.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/unified_scope_attempts.json", "stage": "multiquadratic_casimir_compression", "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, { "attempt_kind": "construction", "family": "72 explicitly constructed rank-eight tensors", "id": "R8_3_941", "matching_test_numbers": [], "outcome": "no_match_in_tested_family", "scope": "negative means no match in this finite list; crossed products are only abstract ring tests unless separately certified", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "72 explicitly constructed rank-eight tensors", "tested_tensors": 72 }, { "attempt_kind": "construction", "family": "ordinary finite-group representation categories", "id": "R8_3_941", "outcome": "family_excluded", "reason": "noncommutative ring", "rejected_by_elementary_filter": true, "scope": "This construction family only; no arbitrary categorification exclusion.", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "ordinary finite-group representation categories" }, { "attempt_kind": "construction", "family": "pointed and ordinary Tambara-Yamagami categories", "id": "R8_3_941", "outcome": "family_excluded", "reason": "their fusion multiplicities are at most1, but this source has maximum3", "rejected": true, "scope": "This construction family only; no arbitrary categorification exclusion.", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "pointed and ordinary Tambara-Yamagami categories" }, { "attempt_kind": "construction", "family": "Morita sources obtained by cyclic-prime extension/equivariantization", "id": "R8_3_941", "outcome": "no_conclusion", "scope": "if applicable, excludes Morita equivalence to a nontrivial cyclic-prime extension or equivariantization; not arbitrary categorification", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "Morita sources obtained by cyclic-prime extension/equivariantization", "structural_filter": { "adjoint_basis": [ 0, 1, 2, 3, 4, 5, 6, 7 ], "center_has_no_nontrivial_invertible_if_categorified": false, "central_invertible_indices": [ 0, 3 ], "invertible_indices": [ 0, 3 ], "scope": "if applicable, excludes Morita equivalence to a nontrivial cyclic-prime extension or equivariantization; not arbitrary categorification" } }, { "attempt_kind": "construction", "family": "bounded classical WZW dimension-ratio search", "id": "R8_3_941", "leads": [], "outcome": "heuristic_search_completed", "scope": "heuristic numerical leads only; no tensor identification or general exclusion", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "bounded classical WZW dimension-ratio search", "tested_models": 1505 }, { "attempt_kind": "construction", "casimir_characteristic_polynomial": [ 1, -608, 30336, -665600, 8114176, -58982400, 254803968, -603979776, 603979776 ], "casimir_matrix": [ [ 8, 8, 8, 0, 8, 8, 12, 12 ], [ 8, 88, 80, 8, 68, 68, 104, 104 ], [ 8, 80, 88, 8, 68, 68, 104, 104 ], [ 0, 8, 8, 8, 8, 8, 12, 12 ], [ 8, 68, 68, 8, 64, 56, 88, 88 ], [ 8, 68, 68, 8, 56, 64, 88, 88 ], [ 12, 104, 104, 12, 88, 88, 144, 136 ], [ 12, 104, 104, 12, 88, 88, 136, 144 ] ], "families": [ { "all_larger_odd_orders": { "first_order": 9, "source_upper_bound": 688, "strict_target_lower_bound": 729 }, "family": "odd_Haagerup_Izumi_and_all_Morita_duals", "outcome": "specific_family_Morita_model_excluded", "tested_odd_orders": [ { "casimir_remainder": [ -226103671089, 740498581755 ], "dimension_polynomial": [ 1, -39, 117 ], "odd_order": 3, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ -2191077832954791, 11361702442419851 ], "dimension_polynomial": [ 1, -145, 725 ], "odd_order": 5, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ -392067593047965885, 2798307055732470115 ], "dimension_polynomial": [ 1, -371, 2597 ], "odd_order": 7, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" } ] }, { "all_larger_odd_orders": { "first_order": 7, "source_upper_bound": 688, "strict_target_lower_bound": 2401 }, "family": "specified_near_group_and_all_Morita_duals", "outcome": "specific_family_Morita_model_excluded", "tested_odd_orders": [ { "casimir_remainder": [ -449749166719707, 4418811550063323 ], "dimension_polynomial": [ 1, -117, 1053 ], "odd_order": 3, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ 16136908813725850149, -418384855372263643349 ], "dimension_polynomial": [ 1, -725, 18125 ], "odd_order": 5, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" } ] } ], "global_dimension_diagonal_lower_bound": 144, "global_dimension_upper_bound": 688, "id": "R8_3_941", "outcome": "no_model_from_either_tested_family", "scope": "specific_model_families_only_not_categorification_exclusion", "source": "evidence/rank8_m3_20261021/models/fresh_202609/odd_hi_dimension_filter.json", "source_dual": [ 0, 2, 1, 3, 4, 5, 6, 7 ], "stage": "odd_hi_and_specified_near_group_Morita_dimension_filter", "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, { "AH_dimension_polynomial_remainder": [ -88555211961925632, 359589063046463488 ], "attempt_kind": "construction", "casimir_characteristic_polynomial": [ 1, -608, 30336, -665600, 8114176, -58982400, 254803968, -603979776, 603979776 ], "family_outcome": "different_global_dimension_so_no_AH_Morita_model", "general_categorification_excluded": false, "id": "R8_3_941", "invertibles": [ 0, 3 ], "maximum_multiplicity": 3, "outcome": "family_excluded", "rank": 8, "scope": "known_Asaeda_Haagerup_Morita_family_only", "source": "evidence/rank8_m3_20261021/models/asaeda_haagerup_attempt.json", "stage": "known_Asaeda_Haagerup_Morita_family_filter", "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" } ], "prior_stage_completed": [ { "domain": "integral_proper_subcategories_and_braided_Morita_representatives", "evidence": [ "evidence/rank8_research_20261022/braided_morita/PROOF.md", "evidence/rank8_research_20261022/braided_morita/verification.json", "site/BRAIDED_CATEGORIFICATION.md" ], "limitation": "This restricts braided Morita constructions; arbitrary unbraided categorification remains unresolved for the six surviving rings.", "outcome": "Nonintegral total FP dimension and integral proper fusion subcategories force any braided Morita representative to yield a braiding on the original category. The general braiding exclusions therefore exclude every braided Morita representative.", "scope": "general_complex_fusion_category", "verifiers": [ "evidence/rank8_research_20261022/braided_morita/verify.py", "site/verify_braided_scope.py" ] }, { "domain": "pointed_center_and_rank_square_Morita_obstructions", "evidence": [ "evidence/rank8_research_20261022/r928_r941/PROOF.md", "evidence/rank8_research_20261022/r928_r941/verification.json" ], "limitation": "This excludes braided representatives only and supplies no unbraided categorification.", "outcome": "An independent center argument excludes braided Morita representatives: the R928 center has a semion and no nontrivial boson, while all fourteen R941 charge allocations have centralizer rank 22, incompatible with the rank-square formula for a forced fermionic center.", "scope": "general_complex_Morita_model_obstruction", "verifiers": [ "evidence/rank8_research_20261022/r928_r941/verify.py" ] }, { "domain": "dimension_block_conductor_bound_and_saturation", "evidence": [ "evidence/rank8_research_20261022/pointed_followup/dimension_block/PROOF.md", "evidence/rank8_research_20261022/pointed_followup/dimension_block/verification.json" ], "limitation": "All seven incoming rings pass the scalar screen. No refined cone run, model, or exclusion follows from this criterion; R1795 is separately excluded by C2 Fourier packing.", "outcome": "Every additional twist-one central simple outside the dimension orbit has dimension ratio in Q(sqrt(2)); its coordinates satisfy a+b=3c and a+2b=6d. All forced I(1) summands satisfy the applicable mask.", "scope": "necessary_center_conditions_in_a_pseudo_unitary_Galois_form", "verifiers": [ "evidence/rank8_research_20261022/pointed_followup/dimension_block/verify.py" ] } ], "prior_unitary_obstruction_source": null, "prior_unitary_status": "not_separately_decided", "screening_case_specific_focus": "General Galois-PU reduction; complete joint integer Gram/cubic domain forces a central fermion after the ordinary rank-four quotient excludes the bosonic branch; 93 complete charge allocations give 77 exact correlated-cone exclusions; the final permutation and mixed-fusion-trace filter retains fourteen moment witnesses, eleven with exact integer counts and three unresolved bounded integer attempts; superproduct and specified unbounded model routes tested.", "screening_definition": "Extensively investigated exotic candidate means that the ring survived the documented, case-specific campaign of known and newly developed exclusion tests and explicit model searches, including complete finite domains and specified unbounded construction families. It proves neither existence of a fusion category nor novelty of a future realization. Each case lists exactly what was exhausted, what survived and the remaining hypotheses.", "screening_stage": "extensively_investigated_exotic_candidate", "tensor_sha256": "8ae4007492dee167ab8b3f4f16310ee368482da03696e20ed7cceea34bec64e6" }, { "categorification_or_novelty_proved": false, "general_status": "unresolved", "id": "R8_3_1154", "latest_completed": [ { "domain": "saturated_dimension_character_conductor_and_signed_spectral_capacity", "evidence": [ "evidence/rank8_creative_20261022/unified/verification.json", "evidence/rank8_creative_20261022/unified/composite_verification.json", "evidence/rank8_creative_20261022/unified/prime_gauss_audit.json" ], "limitation": "These restrictions alone establish neither existence nor nonexistence.", "outcome": "Equality in a dimension-character bound forces the cyclic modular representation through its character conductor. Signed prime and composite refinements give exact twist support and spectral capacities, with no twist-order cutoff.", "proof": "evidence/rank8_creative_20261022/unified/SATURATED_CONDUCTOR.md", "scope": "Necessary conditions for a pseudo-unitary Galois form of any complex categorification.", "verifiers": [ "evidence/rank8_creative_20261022/unified/verify_saturated_conductor.py", "evidence/rank8_creative_20261022/unified/verify_composite_support.py", "evidence/rank8_creative_20261022/unified/prime_gauss_audit.py" ] }, { "domain": "four_object_semionic_packet_and_exact_fusion_closure", "evidence": [ "evidence/rank8_creative_20261022/r1154/semionic_packet.json", "evidence/rank8_creative_20261022/r1154/packet_closure.json", "evidence/rank8_creative_20261022/r1154/quartic_witness_replay.json", "evidence/rank8_creative_20261022/unified/audits/r1154_packet_verification.json" ], "limitation": "All three patterns survive. The forced packet and single quartic witness do not construct the remaining center or a source category.", "outcome": "All 25 histograms have uniquely forced four semionic S columns and four nonnegative integral fusion matrices. Each packet object and its dual generate the entire center within two steps. One quartic witness passes 4900 Gram and 560 restriction entries.", "proof": "evidence/rank8_creative_20261022/r1154/PROOF.md", "scope": "Every remaining three-pattern center domain and all 25 histograms; the quartic witness is one explicitly identified necessary solution.", "verifiers": [ "evidence/rank8_creative_20261022/r1154/verify.py", "evidence/rank8_creative_20261022/unified/audits/verify_r1154_packet.py" ] }, { "domain": "positive_field_coordinate_algebra_obstruction", "evidence": [ "evidence/rank8_creative_20261022/r1154/quantum_subgroup_route.json" ], "limitation": "Other parents, other constructions, and general categorification of R1154 remain open.", "outcome": "Although FPdim(SU(3)_7 times Fibonacci)=5 FPdim(R1154), every noninvertible parent dimension has positive sqrt(5) coefficient. Its three invertibles cannot support a connected separable algebra of dimension five, excluding that direct construction route.", "proof": "evidence/rank8_creative_20261022/r1154/QUANTUM_SUBGROUP_ROUTE.md", "scope": "Connected separable algebras in the specified SU(3)_7 Deligne Fibonacci parent.", "verifiers": [ "evidence/rank8_creative_20261022/r1154/verify.py" ] } ], "limitations": [ "No completed search here exhausts all complex associators or all constructions.", "A feasible necessary-condition system is not a categorification.", "Extensively investigated exotic candidate is a research-screening description, not a proof of categorifiability or novelty." ], "new_attempts": [ { "details": { "FP_global_dimension_minimal_polynomial": [ "1", "-1080", "3600" ], "NG_minimal_polynomial": [ "1", "-103680", "33177600" ], "deequivariantization_constraints": { "obstruction": "cyclic_Tannakian_order_does_not_divide_n", "order_divides_n": false, "required_cyclic_group": true, "required_group_order": 96 }, "general_complex_exclusion_claimed_by_this_audit": false, "invertible_count": 2, "invertibles_abelian": true, "largest_real_root_isolating_interval": [ "34453/32", "347760/323" ], "only_possible_group_order_n": "320", "quadratic_norm": "3600", "quadratic_trace": "1080", "rational_scaling_outcome": "unique_n_and_scale", "reciprocal_scale": "96", "routes": { "deequivariantization_of_NG": "excluded", "direct_or_Morita_equivalent_to_NG": "excluded", "equivariantization_of_NG": "excluded", "group_extension_of_NG": "excluded", "neutral_component_of_nontrivially_graded_NG": "excluded" }, "scale_target_over_NG": "1/96" }, "domain": "all_NG_Cn_n_and_five_single_operation_routes_all_positive_n", "evidence": [ "evidence/rank8_frontier_20261021/structural_models/near_group_scaling.json" ], "limitation": "No arbitrary chains, inserted Morita changes, or noncentral condensations are covered.", "outcome": "family_excluded", "proof": "evidence/rank8_frontier_20261021/structural_models/NEAR_GROUP_SCALING.md", "scope": "general_complex_construction_family", "verifiers": [ "evidence/rank8_frontier_20261021/structural_models/near_group_scaling.py" ] }, { "details": { "PSD_Gram_survivors": 81, "bounded_selfdual_vectors_tested": 927472, "entrywise_Gram_survivors": 52056, "independent_complete_Python_domain": null, "invertible_stabilizer_survivors": 463918, "one_plus_simple_surviving_indices": [ 3, 4, 5 ] }, "domain": "all_connected_separable_algebra_underlying_objects_entrywise_and_PSD_Gram", "evidence": [ "evidence/rank8_frontier_20261021/structural_models/algebra_candidates.json" ], "limitation": "Surviving objects are not algebra structures; no multiplication, associator, model or general exclusion is claimed.", "outcome": "complete_necessary_object_filter_survives", "proof": "evidence/rank8_frontier_20261021/structural_models/ALGEBRA_CANDIDATES.md", "scope": "general_complex_algebra_construction_constraints", "verifiers": [ "evidence/rank8_frontier_20261021/structural_models/algebra_candidates.py" ] }, { "details": { "conclusion": "Every categorification has a spherical structure and a pseudo-unitary Galois conjugate. No unitarity or existence claim." }, "domain": "complete_signed_sphericalization_and_general_Galois_PU_reduction", "evidence": [ "evidence/rank8_frontier_20261021/central_integrality/general_galois_pu_gate.json" ], "limitation": "This proves a scope reduction, not categorifiability or unitarity. Necessary PU constraints apply after Galois conjugation.", "outcome": "Galois_pseudo_unitary_reduction", "proof": "evidence/rank8_frontier_20261021/central_integrality/GENERAL_GALOIS_PU.md", "scope": "general_complex_fusion_category", "verifiers": [ "evidence/rank8_frontier_20261021/central_integrality/general_galois_pu_gate.py" ] }, { "details": { "allowed_order_sequence": [ 5, 10 ], "complete_order_candidates": [ 3, 4, 5, 6, 8, 10, 12 ], "dimension_bounded_product_count": 10, "exponent_box_size": 18, "positive_integer_scaling_matches": [] }, "domain": "all_levels_SU2_adjoint_finite_products_integer_multiplier_chains_and_Morita_duals", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative_followup/su2_all_frontier.json" ], "limitation": "Other Lie types, arbitrary algebra condensation, dimension-decreasing de-equivariantization, or general categorifications.", "outcome": "family_excluded_by_exact_global_dimension", "proof": "evidence/rank8_frontier_20261021/noncommutative_followup/SU2_FAMILY.md", "scope": "general_complex_construction_family_dimension_obstruction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative_followup/su2_all_frontier.py", "evidence/rank8_frontier_20261021/noncommutative_followup/su2_product_family.py" ] }, { "details": { "families": [], "forced_group_order_before_integer_multiplier": "10/3", "global_dimension_minimal_polynomial": [ "1", "-1080", "3600" ], "id": "R8_3_1154", "outcome": "family_dimension_excluded", "positive_root_interval": [ "34453/32", "347760/323" ], "scope": "specific_model_family_only", "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" }, "domain": "all_generalized_near_group_dimension_parameters_with_integer_scalings", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative_followup/generalized_near_group_all_frontier.json" ], "limitation": "This is a necessary dimension-family test; retained parameter triples are not categories, and integral targets lie outside its irrational quadratic parameter theorem.", "outcome": "family_dimension_excluded", "proof": "evidence/rank8_frontier_20261021/noncommutative_followup/GENERALIZED_NEAR_GROUP_TABLE.md", "scope": "specific_model_family_only", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative_followup/generalized_near_group_all_frontier.py" ] }, { "details": { "id": "R8_3_1154", "joint_spectrum": { "input_patterns": 1238, "status": "exact_complete", "surviving_patterns": 1238 } }, "domain": "latest_complete_joint_Galois_spectra_and_correlated_semion_integer_attempts", "evidence": [ "evidence/rank8_frontier_20261021/r928_semion_rotation/final_campaign.json" ], "limitation": "Six bounded R928 integer searches remain unresolved integer-feasibility problems; numerical negative solver outcomes are not exclusions. Exact positive integer witnesses are necessary data only.", "outcome": "documented_finite_domains_finished", "proof": "evidence/rank8_frontier_20261021/review/QUARTIC_CENTRAL_ORBITS.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/r928_semion_rotation/verify_all.py", "evidence/rank8_frontier_20261021/r928_semion_rotation/charged_joint_spectrum.py", "evidence/rank8_frontier_20261021/noncommutative_joint_spectrum/verify.py", "evidence/rank8_frontier_20261021/r928_semion_rotation/final_campaign.py" ] }, { "details": { "low_count": 59, "passing_low_count": 4 }, "domain": "complete_central_image_low_induction_and_Gram_moment_constraints", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/central_support_results.json" ], "limitation": "This PU necessary-condition test applies after the separately proved Galois reduction; surviving data need not lift to a center.", "outcome": "necessary_data_survives", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py", "evidence/rank8_frontier_20261021/noncommutative/verify_independent.py" ] }, { "details": { "central": false, "dual_fixed_basis_count": 6, "id": "R8_3_1154", "invertible": 3, "left_action": [ 3, 7, 6, 0, 5, 4, 2, 1 ], "matrix_block_formal_codegree": [ "5" ], "matrix_block_size": 2, "matrix_involution_trace": 2, "matrix_involution_type": "orthogonal", "pointed_bimodule_orbits": [ { "basis": [ 0, 3 ], "cohomology_attempts": [ { "checked_triples": 8, "omega_exponent": 0, "restricted_cocycle_trivial": true }, { "checked_triples": 8, "omega_exponent": 1, "restricted_cocycle_trivial": true } ], "stabilizer": [ [ 0, 0 ], [ 1, 1 ] ] }, { "basis": [ 1, 2, 6, 7 ], "cohomology_attempts": [ { "checked_triples": 1, "omega_exponent": 0, "restricted_cocycle_trivial": true }, { "checked_triples": 1, "omega_exponent": 1, "restricted_cocycle_trivial": true } ], "stabilizer": [ [ 0, 0 ] ] }, { "basis": [ 4, 5 ], "cohomology_attempts": [ { "checked_triples": 8, "omega_exponent": 0, "restricted_cocycle_trivial": true }, { "checked_triples": 8, "omega_exponent": 1, "restricted_cocycle_trivial": true } ], "stabilizer": [ [ 0, 0 ], [ 1, 1 ] ] } ], "right_action": [ 3, 6, 7, 0, 5, 4, 1, 2 ], "scalar_dimension": 4, "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" }, "domain": "exact_matrix_block_involution_and_pointed_bimodule_cocycle_constraints", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/matricial_structure.json" ], "limitation": "This restricted domain does not exhaust general categorifications.", "outcome": "necessary_structure_compatible", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "based_ring_and_pointed_coherence_constraints", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "exact_or_numeric_branch_statuses": [ "undecided_numeric" ], "height": 12, "low_index": 30 }, "domain": "finite_order_higher_rotation_LP", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/certified_1154_h12_low30.json" ], "limitation": "Individual infeasible branches do not exclude the ring; numeric undecided is not an exact result.", "outcome": "undecided_numeric", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "exact_or_numeric_branch_statuses": [ "numerically_feasible_no_exact_reconstruction" ], "height": 12, "low_index": 31 }, "domain": "finite_order_higher_rotation_LP", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/certified_1154_h12_low31.json" ], "limitation": "Individual infeasible branches do not exclude the ring; numeric undecided is not an exact result.", "outcome": "numerically_feasible_no_exact_reconstruction", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "exact_or_numeric_branch_statuses": [ "exact_feasible_relaxation" ], "height": 8, "low_index": 30 }, "domain": "finite_order_higher_rotation_LP", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/certified_1154_h8_low30.json" ], "limitation": "Individual infeasible branches do not exclude the ring; numeric undecided is not an exact result.", "outcome": "exact_feasible_relaxation", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "exact_or_numeric_branch_statuses": [ "exact_feasible_relaxation" ], "height": 8, "low_index": 31 }, "domain": "finite_order_higher_rotation_LP", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/certified_1154_h8_low31.json" ], "limitation": "Individual infeasible branches do not exclude the ring; numeric undecided is not an exact result.", "outcome": "exact_feasible_relaxation", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "exact_or_numeric_branch_statuses": [ "exact_infeasible_relaxation", "exact_feasible_relaxation" ], "height": 6, "low_index": null }, "domain": "joint_rotation_and_partial_STS_LP", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/modular_1154_h6.json" ], "limitation": "Individual infeasible branches do not exclude the ring; numeric undecided is not an exact result.", "outcome": "exact_feasible_relaxation; exact_infeasible_relaxation", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "combined_survivor_count": 1238, "cubic_survivor_count": 1238, "gram_pattern_count": 14211, "spectral_survivor_count": 7274 }, "domain": "all_integer_Gram_patterns_cubic_Verlinde_and_Galois_balance", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/central_integral_filters_1154.json" ], "limitation": "Necessary central fusion constraints; survivors are not constructed centers or categories.", "outcome": "exact_necessary_data_survives", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "case_count": 1238, "input_gram_patterns": 1238 }, "domain": "all_surviving_Gram_patterns_all_ten_low_pair_forgetful_tensor_cones", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative/cubic_tensor_1154.json" ], "limitation": "Complete rational cones for all distinguished pairs; integrality and full center associativity are not imposed.", "outcome": "exact_necessary_data_survives", "proof": "evidence/rank8_frontier_20261021/noncommutative/README.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative/verify_all.py" ] }, { "details": { "certificate_family_count": 434, "complete": true, "height": 6, "pattern_count": 1238, "unresolved_indices": [] }, "domain": "all_1238_cubic_surviving_integer_Gram_patterns_joint_height6_rotation_and_STS", "evidence": [ "evidence/rank8_frontier_20261021/central_integrality/fixed_gram_rotation_1154_h6.json" ], "limitation": "This finite-height exact system leaves feasible patterns; neither full center fusion nor associators are constructed.", "outcome": "complete_exact_necessary_data_survives", "proof": "evidence/rank8_frontier_20261021/central_integrality/RESEARCH_RECORD.md", "scope": "general_complex_necessary_after_proved_Galois_PU_reduction", "verifiers": [ "evidence/rank8_frontier_20261021/central_integrality/fixed_gram_rotation_1154.py" ] } ], "prior_published_attempts": [ { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "scope_inapplicable", "reason": "Not a rank-eight near-group ring.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "near_group", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "rank_two", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "zero_spectrum", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "lower_rank_subring", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "scope_inapplicable", "reason": "No rank-seven closed based subring.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "rank7_certified_subring", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "dnumber", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "noncomm_dnumber", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "scope_inapplicable", "reason": "No integral positive FP character; integral Drinfeld branch only.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "integral_drinfeld", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "scope_detail": "Bounded exact obstruction search; no conclusion is not a proof that all character fields are cyclotomic.", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "cyclotomic", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "scope_inapplicable", "reason": "The inherited module implementation requires an invertible involution with one or two fixed simples.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "involution_module", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "scope_detail": "Search restricted to fusion-matrix irreducible factors of degrees3through6; no conclusion is not a proof of cyclotomicity.", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "field_automorphisms", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "subring_dimension", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "scope_inapplicable", "reason": "Implementation requires commutativity.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "primitive_codegree", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "column_counts": [ 8181, 4690, 30, 1582 ], "completed": true, "id": "R8_3_1154", "multiplicities": [ 1, 1, 1, 2 ], "objects": [ { "cases": [ { "passes": true, "principal_minors": [ [ "1100/123", "-18460/123" ], [ "132000", "-2229600" ], [ "132000", "-2229600" ], [ "79875840000/41", "-1349176128000/41" ] ], "second_indicator": 0, "signs": [ 1, 1, 1, 1 ] } ], "lower_trivial_twist_mass": [], "object": 1, "total_mass": [ "1100/123", "-18460/123" ] }, { "cases": [ { "passes": true, "principal_minors": [ [ "1100/123", "-18460/123" ], [ "132000", "-2229600" ], [ "132000", "-2229600" ], [ "79875840000/41", "-1349176128000/41" ] ], "second_indicator": 0, "signs": [ 1, 1, 1, 1 ] } ], "lower_trivial_twist_mass": [], "object": 2, "total_mass": [ "1100/123", "-18460/123" ] }, { "cases": [ { "passes": true, "principal_minors": [ [ "80/123", "-940/123" ], [ "28800/41", "-486000/41" ], [], [] ], "second_indicator": -1, "signs": [ 1, 1, 0, 0 ] }, { "passes": true, "principal_minors": [ [ "80/123", "-940/123" ], [ "28800/41", "-486000/41" ], [], [] ], "second_indicator": 1, "signs": [ 1, 1, 0, 0 ] } ], "lower_trivial_twist_mass": [], "object": 3, "total_mass": [ "80/123", "-940/123" ] }, { "cases": [ { "passes": true, "principal_minors": [ [ "340/41", "-5840/41" ], [ "4651200/41", "-78562800/41" ], [ "4622400/41", "-78076800/41" ], [ "63245376000/41", "-1068272352000/41" ] ], "second_indicator": -1, "signs": [ 1, 1, 1, 1 ] }, { "passes": true, "principal_minors": [ [ "340/41", "-5840/41" ], [ "4651200/41", "-78562800/41" ], [ "4622400/41", "-78076800/41" ], [ "63245376000/41", "-1068272352000/41" ] ], "second_indicator": 1, "signs": [ 1, 1, 1, 1 ] } ], "lower_trivial_twist_mass": [], "object": 4, "total_mass": [ "340/41", "-5840/41" ] }, { "cases": [ { "passes": true, "principal_minors": [ [ "340/41", "-5840/41" ], [ "4651200/41", "-78562800/41" ], [ "4622400/41", "-78076800/41" ], [ "63245376000/41", "-1068272352000/41" ] ], "second_indicator": -1, "signs": [ 1, 1, 1, 1 ] }, { "passes": true, "principal_minors": [ [ "340/41", "-5840/41" ], [ "4651200/41", "-78562800/41" ], [ "4622400/41", "-78076800/41" ], [ "63245376000/41", "-1068272352000/41" ] ], "second_indicator": 1, "signs": [ 1, 1, 1, 1 ] } ], "lower_trivial_twist_mass": [], "object": 5, "total_mass": [ "340/41", "-5840/41" ] }, { "cases": [ { "passes": true, "principal_minors": [ [ "1100/123", "-18460/123" ], [ "132000", "-2229600" ], [ "5383200/41", "-90927600/41" ], [ "79450776000/41", "-1341996408000/41" ] ], "second_indicator": -1, "signs": [ 1, 1, 1, 1 ] }, { "passes": true, "principal_minors": [ [ "1100/123", "-18460/123" ], [ "132000", "-2229600" ], [ "5383200/41", "-90927600/41" ], [ "79450776000/41", "-1341996408000/41" ] ], "second_indicator": 1, "signs": [ 1, 1, 1, 1 ] } ], "lower_trivial_twist_mass": [], "object": 6, "total_mass": [ "1100/123", "-18460/123" ] }, { "cases": [ { "passes": true, "principal_minors": [ [ "1100/123", "-18460/123" ], [ "132000", "-2229600" ], [ "5383200/41", "-90927600/41" ], [ "79450776000/41", "-1341996408000/41" ] ], "second_indicator": -1, "signs": [ 1, 1, 1, 1 ] }, { "passes": true, "principal_minors": [ [ "1100/123", "-18460/123" ], [ "132000", "-2229600" ], [ "5383200/41", "-90927600/41" ], [ "79450776000/41", "-1341996408000/41" ] ], "second_indicator": 1, "signs": [ 1, 1, 1, 1 ] } ], "lower_trivial_twist_mass": [], "object": 7, "total_mass": [ "1100/123", "-18460/123" ] } ], "obstructed_objects": [], "outcome": "no_conclusion", "scope": "pseudo_unitary_only", "source": "evidence/rank8_m3_20261021/noncommutative/central_moments_results.json", "stage": "central_twist_moments", "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "no_conclusion", "report": { "original_components": 3, "scalar_rank": 4 }, "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/noncommutative/results.json", "stage": "noncommutative_scalar_sphericalization", "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" }, { "attempt_kind": "exclusion", "attempts": [], "budget_skipped_objects": 0, "candidate_object_count": 0, "completed_objects": 0, "frontier_ledger_digest": "23837060c2c8186fb971d5c7515620cc0720508d025337e12154eb20b1935270", "id": "R8_3_1154", "object_family": "one nontrivial dual-pair atom or sum of two distinct nonunit self-dual orbit atoms", "outcome": "inapplicable_noncommutative", "per_object_case_budget": 100000, "profile_derived_from_tensor": { "commutative": false, "dual": [ 0, 2, 1, 3, 4, 5, 6, 7 ], "fixed_point_involutions": [], "invertibles": [ 0, 3 ] }, "scope": "general_complex_fusion_category", "scope_note": "a modular survivor gives no exclusion; it is not a feasible categorification or an exact module witness", "source": "evidence/rank8_m3_20261021/noncommutative/split_module_object_results.json", "stage": "split_module_object_results", "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "no_conclusion", "report": { "lowest_numerical_eigenvalue": -2.745463290143382e-13, "scope": "numerical discovery only" }, "source": "evidence/rank8_m3_20261021/noncommutative/unitary_primary3_results.json", "stage": "unitary_primary3_results", "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/one_spectrum_m3.json", "stage": "one_spectrum", "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" }, { "attempt_kind": "construction", "id": "R8_3_1154", "outcome": "implementation_ineligible", "reason": "Nonintegral noncommutative FP-field quotient not implemented in this diagnostic.", "scope": "weakly_group_theoretical_models_only", "source": "evidence/rank8_m3_20261021/pipeline/frobenius_results.json", "stage": "strong_frobenius_weak_group_theoretical", "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "scope_inapplicable", "reason": "No closed based subring of rank6or7 exists.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/unified_scope_attempts.json", "stage": "expanded_exact_negative_subring_transfer", "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" }, { "attempt_kind": "exclusion", "id": "R8_3_1154", "outcome": "no_conclusion", "quadratic_factors": [ [ 1, -1080, 3600 ] ], "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/unified_scope_attempts.json", "stage": "multiquadratic_casimir_compression", "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" }, { "attempt_kind": "construction", "family": "72 explicitly constructed rank-eight tensors", "id": "R8_3_1154", "matching_test_numbers": [], "outcome": "no_match_in_tested_family", "scope": "negative means no match in this finite list; crossed products are only abstract ring tests unless separately certified", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "72 explicitly constructed rank-eight tensors", "tested_tensors": 72 }, { "attempt_kind": "construction", "family": "ordinary finite-group representation categories", "id": "R8_3_1154", "outcome": "family_excluded", "reason": "noncommutative ring", "rejected_by_elementary_filter": true, "scope": "This construction family only; no arbitrary categorification exclusion.", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "ordinary finite-group representation categories" }, { "attempt_kind": "construction", "family": "pointed and ordinary Tambara-Yamagami categories", "id": "R8_3_1154", "outcome": "family_excluded", "reason": "their fusion multiplicities are at most1, but this source has maximum3", "rejected": true, "scope": "This construction family only; no arbitrary categorification exclusion.", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "pointed and ordinary Tambara-Yamagami categories" }, { "attempt_kind": "construction", "family": "Morita sources obtained by cyclic-prime extension/equivariantization", "id": "R8_3_1154", "outcome": "family_excluded", "scope": "if applicable, excludes Morita equivalence to a nontrivial cyclic-prime extension or equivariantization; not arbitrary categorification", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "Morita sources obtained by cyclic-prime extension/equivariantization", "structural_filter": { "adjoint_basis": [ 0, 1, 2, 3, 4, 5, 6, 7 ], "center_has_no_nontrivial_invertible_if_categorified": true, "central_invertible_indices": [ 0 ], "invertible_indices": [ 0, 3 ], "scope": "if applicable, excludes Morita equivalence to a nontrivial cyclic-prime extension or equivariantization; not arbitrary categorification" } }, { "attempt_kind": "construction", "family": "bounded classical WZW dimension-ratio search", "id": "R8_3_1154", "leads": [], "outcome": "heuristic_search_completed", "scope": "heuristic numerical leads only; no tensor identification or general exclusion", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "bounded classical WZW dimension-ratio search", "tested_models": 1505 }, { "attempt_kind": "construction", "casimir_characteristic_polynomial": [ 1, -1144, 74360, -2023200, 29386000, -244640000, 1160000000, -2880000000, 2880000000 ], "casimir_matrix": [ [ 8, 12, 12, 0, 12, 16, 14, 14 ], [ 12, 198, 188, 14, 174, 172, 184, 184 ], [ 12, 188, 198, 14, 174, 172, 184, 184 ], [ 0, 14, 14, 8, 16, 12, 12, 12 ], [ 12, 174, 174, 16, 168, 160, 172, 172 ], [ 16, 172, 172, 12, 160, 168, 174, 174 ], [ 14, 184, 184, 12, 172, 174, 198, 188 ], [ 14, 184, 184, 12, 172, 174, 188, 198 ] ], "families": [ { "all_larger_odd_orders": { "first_order": 11, "source_upper_bound": 1126, "strict_target_lower_bound": 1331 }, "family": "odd_Haagerup_Izumi_and_all_Morita_duals", "outcome": "specific_family_Morita_model_excluded", "tested_odd_orders": [ { "casimir_remainder": [ -226941512025, 743239650771 ], "dimension_polynomial": [ 1, -39, 117 ], "odd_order": 3, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ -4388506630859375, 22756337451171875 ], "dimension_polynomial": [ 1, -145, 725 ], "odd_order": 5, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ -1397716043178309221, 9975929495050457851 ], "dimension_polynomial": [ 1, -371, 2597 ], "odd_order": 7, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ -55360253757587002875, 504246074530924726875 ], "dimension_polynomial": [ 1, -765, 6885 ], "odd_order": 9, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" } ] }, { "all_larger_odd_orders": { "first_order": 7, "source_upper_bound": 1126, "strict_target_lower_bound": 2401 }, "family": "specified_near_group_and_all_Morita_duals", "outcome": "specific_family_Morita_model_excluded", "tested_odd_orders": [ { "casimir_remainder": [ -829306761665571, 8147986851734451 ], "dimension_polynomial": [ 1, -117, 1053 ], "odd_order": 3, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ -41528978611131171875, 1076730115530018671875 ], "dimension_polynomial": [ 1, -725, 18125 ], "odd_order": 5, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" } ] } ], "global_dimension_diagonal_lower_bound": 198, "global_dimension_upper_bound": 1126, "id": "R8_3_1154", "outcome": "no_model_from_either_tested_family", "scope": "specific_model_families_only_not_categorification_exclusion", "source": "evidence/rank8_m3_20261021/models/fresh_202609/odd_hi_dimension_filter.json", "source_dual": [ 0, 2, 1, 3, 4, 5, 6, 7 ], "stage": "odd_hi_and_specified_near_group_Morita_dimension_filter", "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" }, { "AH_dimension_polynomial_remainder": [ -234906581945839616, 953866360138878976 ], "attempt_kind": "construction", "casimir_characteristic_polynomial": [ 1, -1144, 74360, -2023200, 29386000, -244640000, 1160000000, -2880000000, 2880000000 ], "family_outcome": "different_global_dimension_so_no_AH_Morita_model", "general_categorification_excluded": false, "id": "R8_3_1154", "invertibles": [ 0, 3 ], "maximum_multiplicity": 3, "outcome": "family_excluded", "rank": 8, "scope": "known_Asaeda_Haagerup_Morita_family_only", "source": "evidence/rank8_m3_20261021/models/asaeda_haagerup_attempt.json", "stage": "known_Asaeda_Haagerup_Morita_family_filter", "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" } ], "prior_stage_completed": [ { "domain": "integral_proper_subcategories_and_braided_Morita_representatives", "evidence": [ "evidence/rank8_research_20261022/braided_morita/PROOF.md", "evidence/rank8_research_20261022/braided_morita/verification.json", "site/BRAIDED_CATEGORIFICATION.md" ], "limitation": "This restricts braided Morita constructions; arbitrary unbraided categorification remains unresolved for the six surviving rings.", "outcome": "Nonintegral total FP dimension and integral proper fusion subcategories force any braided Morita representative to yield a braiding on the original category. The general braiding exclusions therefore exclude every braided Morita representative.", "scope": "general_complex_fusion_category", "verifiers": [ "evidence/rank8_research_20261022/braided_morita/verify.py", "site/verify_braided_scope.py" ] }, { "domain": "incompatible_center_simple_dimension_inventories", "evidence": [ "evidence/rank8_research_20261022/r1154_r1795/PROOF.md", "evidence/rank8_research_20261022/r1154_r1795/morita_obstruction.json" ], "limitation": "Neither ring is individually excluded by the pairwise comparison.", "outcome": "No complex categorifications of these two rings can be categorically Morita equivalent: R1795 forces four distinct center simples of dimension 54 + 24 sqrt(5), absent from all three final R1154 center patterns.", "scope": "general_complex_pairwise_Morita_obstruction", "verifiers": [ "evidence/rank8_research_20261022/r1154_r1795/morita_obstruction.py" ] }, { "domain": "full_Galois_covariance_of_forced_center_operators", "evidence": [ "evidence/rank8_research_20261022/r1154_r1795/PROOF.md", "evidence/rank8_research_20261022/r1154_r1795/galois_fixed_review.json", "evidence/rank8_research_20261022/r1154_r1795/forced_galois_probe.json" ], "limitation": "No additional pattern or histogram is excluded; the witnesses are not fusion or modular data.", "outcome": "Diagonal covariance forces the Galois permutation to fix A2 and A4. All 25 current pair histograms retain exact witnesses satisfying every entry of covariance for N2, N3-N1, N4+N1.", "scope": "necessary_pseudo_unitary_center_constraints_transferred_by_the_existing_Galois_gate", "verifiers": [ "evidence/rank8_research_20261022/r1154_r1795/galois_fixed_review.py", "evidence/rank8_research_20261022/r1154_r1795/forced_galois_probe.py" ] }, { "domain": "dimension_block_conductor_bound_and_saturation", "evidence": [ "evidence/rank8_research_20261022/pointed_followup/dimension_block/PROOF.md", "evidence/rank8_research_20261022/pointed_followup/dimension_block/verification.json" ], "limitation": "All seven incoming rings pass the scalar screen. No refined cone run, model, or exclusion follows from this criterion; R1795 is separately excluded by C2 Fourier packing.", "outcome": "The scalar dimension-block inequalities pass strictly for the nontrivial character; no saturation mask follows. All forced I(1) summands satisfy the applicable mask.", "scope": "necessary_center_conditions_in_a_pseudo_unitary_Galois_form", "verifiers": [ "evidence/rank8_research_20261022/pointed_followup/dimension_block/verify.py" ] } ], "prior_unitary_obstruction_source": null, "prior_unitary_status": "not_separately_decided", "screening_case_specific_focus": "General Galois-PU reduction; all 14,211 integer central Gram patterns filtered to 1,238 cubic/spectral survivors and 1,230 exact joint rotation/STS survivors; complete connected-algebra underlying-object necessary filters and specified unbounded model families tested.", "screening_definition": "Extensively investigated exotic candidate means that the ring survived the documented, case-specific campaign of known and newly developed exclusion tests and explicit model searches, including complete finite domains and specified unbounded construction families. It proves neither existence of a fusion category nor novelty of a future realization. Each case lists exactly what was exhausted, what survived and the remaining hypotheses.", "screening_stage": "extensively_investigated_exotic_candidate", "tensor_sha256": "c399505be76a64c8aba65feaf2cd55125dc7132f5894b74e58b64473d1387e25" }, { "categorification_or_novelty_proved": false, "general_status": "unresolved", "id": "R8_3_1768", "latest_completed": [ { "domain": "saturated_dimension_character_conductor_and_signed_spectral_capacity", "evidence": [ "evidence/rank8_creative_20261022/unified/verification.json", "evidence/rank8_creative_20261022/unified/composite_verification.json", "evidence/rank8_creative_20261022/unified/prime_gauss_audit.json" ], "limitation": "These restrictions alone establish neither existence nor nonexistence.", "outcome": "Equality in a dimension-character bound forces the cyclic modular representation through its character conductor. Signed prime and composite refinements give exact twist support and spectral capacities, with no twist-order cutoff.", "proof": "evidence/rank8_creative_20261022/unified/SATURATED_CONDUCTOR.md", "scope": "Necessary conditions for a pseudo-unitary Galois form of any complex categorification.", "verifiers": [ "evidence/rank8_creative_20261022/unified/verify_saturated_conductor.py", "evidence/rank8_creative_20261022/unified/verify_composite_support.py", "evidence/rank8_creative_20261022/unified/prime_gauss_audit.py" ] }, { "domain": "quadratic_Galois_restriction_complement_and_finite_central_Gram_branches", "evidence": [ "evidence/rank8_creative_20261022/r1768/verification.json", "evidence/rank8_creative_20261022/r1768/unit_orbit_cone.json" ], "limitation": "Rational dimension-ratio objects may also have primitive-37 twists. The final constrained cone remains exactly feasible, so no whole-ring exclusion or construction follows.", "outcome": "Exactly 39 of 40 Gram/indicator branches are rejected. All six noninvertible second indicators are +1 in a pseudo-unitary Galois form. Outside 1 and A_sigma, the irrational d_Z/D sector has either 18 objects of restriction 2z or 72 objects of restriction z, with nonsquare primitive-37 twists; A_sigma restricts uniquely to (1,0,2,2,2,2,2,2).", "proof": "evidence/rank8_creative_20261022/r1768/GALOIS_FIBRES.md", "scope": "All possible irrational dimension-ratio central columns and all eight admissible second-indicator assignments.", "verifiers": [ "evidence/rank8_creative_20261022/r1768/verify.py" ] }, { "domain": "forced_nineteen_dimensional_Weil_block_and_primitive_products", "evidence": [ "evidence/rank8_creative_20261022/unified/r1768_weil_verification.json" ], "limitation": "The necessary block and forced products survive; they do not complete modular data or prove categorifiability.", "outcome": "In the 18-object irrational-ratio option, the entire 19-dimensional saturated modular block and all products of its 18 distinguished simples are forced; exact cyclotomic checks give nonnegative integral coefficients.", "proof": "evidence/rank8_creative_20261022/unified/R1768_WEIL_BLOCK.md", "scope": "The 18-object option only; rational-ratio simples are retained and may have primitive-37 twists.", "verifiers": [ "evidence/rank8_creative_20261022/unified/verify_r1768_weil.py" ] }, { "domain": "cyclic_near_group_parent_divisibility_obstruction", "evidence": [ "evidence/rank8_creative_20261022/r3472/construction_routes.json" ], "limitation": "Noncyclic parents and arbitrary Morita constructions are not excluded; this is not a ring noncategorification theorem.", "outcome": "An ordinary de-equivariantization of NG(C_n,n) by a cyclic subgroup of order h requires h squared to divide n, because each descendant of the stable noninvertible has dimension d_rho/h. The tempting cyclic parents for both R1768 and R3472 fail with norm -1/4.", "proof": "evidence/rank8_creative_20261022/r3472/CONSTRUCTION_ROUTES.md", "scope": "Direct ordinary de-equivariantization of the specified cyclic near-group parent.", "verifiers": [ "evidence/rank8_creative_20261022/r3472/verify_construction_routes.py" ] } ], "limitations": [ "No completed search here exhausts all complex associators or all constructions.", "A feasible necessary-condition system is not a categorification.", "Extensively investigated exotic candidate is a research-screening description, not a proof of categorifiability or novelty." ], "new_attempts": [ { "details": { "FP_global_dimension_minimal_polynomial": [ "1", "-888", "5328" ], "NG_minimal_polynomial": [ "1", "-21312", "3068928" ], "deequivariantization_constraints": { "obstruction": "rank_mismatch", "order_divides_n": true, "required_cyclic_group": true, "required_group_order": 24, "required_invertible_count": 6, "required_invertible_group": "cyclic", "required_rank": 30 }, "general_complex_exclusion_claimed_by_this_audit": false, "invertible_count": 2, "invertibles_abelian": true, "largest_real_root_isolating_interval": [ "1180061/1338", "64383/73" ], "only_possible_group_order_n": "144", "quadratic_norm": "5328", "quadratic_trace": "888", "rational_scaling_outcome": "unique_n_and_scale", "reciprocal_scale": "24", "routes": { "deequivariantization_of_NG": "excluded", "direct_or_Morita_equivalent_to_NG": "excluded", "equivariantization_of_NG": "excluded", "group_extension_of_NG": "excluded", "neutral_component_of_nontrivially_graded_NG": "excluded" }, "scale_target_over_NG": "1/24" }, "domain": "all_NG_Cn_n_and_five_single_operation_routes_all_positive_n", "evidence": [ "evidence/rank8_frontier_20261021/structural_models/near_group_scaling.json" ], "limitation": "No arbitrary chains, inserted Morita changes, or noncentral condensations are covered.", "outcome": "family_excluded", "proof": "evidence/rank8_frontier_20261021/structural_models/NEAR_GROUP_SCALING.md", "scope": "general_complex_construction_family", "verifiers": [ "evidence/rank8_frontier_20261021/structural_models/near_group_scaling.py" ] }, { "details": { "affine_dimension_coordinates": [ { "constant": [ 1, 1 ], "slope": [ 0, 1 ] }, { "constant": [ 1, 1 ], "slope": [ 0, 1 ] }, { "constant": [ 0, 1 ], "slope": [ 1, 1 ] }, { "constant": [ 1, 1 ], "slope": [ 1, 1 ] }, { "constant": [ 2, 1 ], "slope": [ 1, 1 ] }, { "constant": [ 2, 1 ], "slope": [ 1, 1 ] }, { "constant": [ 1, 1 ], "slope": [ 1, 1 ] }, { "constant": [ 0, 1 ], "slope": [ 1, 1 ] } ], "class_count": 1, "complete_diagonal_domain": [ 0, 10 ], "dimension_minpoly": [ 1, -10, -12 ], "outcome": "exact_feasible_based_modules_not_categorifications", "reference_basis_index": 2, "scope": "all connected rank-two based modules, up to vertex permutation", "tested_diagonal_values": 11 }, "domain": "all_connected_rank_two_based_modules_up_to_permutation", "evidence": [ "evidence/rank8_frontier_20261021/quadratic/rank_two_and_models_results.json" ], "limitation": "No pentagon or algebra multiplication is supplied for the feasible based modules.", "outcome": "exact_feasible_based_modules_not_categorifications", "proof": "evidence/rank8_frontier_20261021/quadratic/RESIDUAL_QUADRATIC_SEARCH.md", "scope": "general_complex_necessary_module_constraints", "verifiers": [ "evidence/rank8_frontier_20261021/quadratic/rank_two_and_models.py" ] }, { "details": { "families": [ { "base_rank": 12, "base_rank_two_classification_excludes": false, "integer_dimension_multiplier": 1, "invertible_group_order": 6, "noninvertible_orbit_size": 6, "stabilizer_order": 1, "status": "dimension_match_only_no_model_or_Morita_equivalence", "total_noninvertible_multiplicity": 12 }, { "base_rank": 6, "base_rank_two_classification_excludes": false, "integer_dimension_multiplier": 2, "invertible_group_order": 3, "noninvertible_orbit_size": 3, "stabilizer_order": 1, "status": "dimension_match_only_no_model_or_Morita_equivalence", "total_noninvertible_multiplicity": 12 }, { "base_rank": 4, "base_rank_two_classification_excludes": false, "integer_dimension_multiplier": 3, "invertible_group_order": 2, "noninvertible_orbit_size": 2, "stabilizer_order": 1, "status": "dimension_match_only_no_model_or_Morita_equivalence", "total_noninvertible_multiplicity": 12 }, { "base_rank": 2, "base_rank_two_classification_excludes": true, "integer_dimension_multiplier": 6, "invertible_group_order": 1, "noninvertible_orbit_size": 1, "stabilizer_order": 1, "status": "impossible_rank_two_base", "total_noninvertible_multiplicity": 12 } ], "finite_domain_justification": "k divides G=Norm(D)/Trace(D); stabilizer divides G/k; K squared then uniquely determined", "forced_group_order_before_integer_multiplier": [ 6, 1 ], "global_dimension": "72*sqrt(37) + 444", "global_dimension_minpoly": [ [ 1, 1 ], [ -888, 1 ], [ 5328, 1 ] ], "outcome": "finite_family_dimension_matches_only", "scope": "all generalized-near-group base dimensions, arbitrary finite-group extensions/equivariantizations/pointed products and Morita equivalence; no de-equivariantization division" }, "domain": "all_generalized_near_group_dimension_families_with_integer_multipliers", "evidence": [ "evidence/rank8_frontier_20261021/quadratic/rank_two_and_models_results.json" ], "limitation": "Dimension matches are not models; division by de-equivariantization is outside this separate family sieve.", "outcome": "finite_family_dimension_matches_only", "proof": "evidence/rank8_frontier_20261021/quadratic/RESIDUAL_QUADRATIC_SEARCH.md", "scope": "construction_family_dimension_necessary_condition", "verifiers": [ "evidence/rank8_frontier_20261021/quadratic/rank_two_and_models.py" ] }, { "details": { "additional_generalized_near_group_branches_excluded": [ { "base_global_dimension": "24*sqrt(37) + 148", "base_rank": 4, "integer_dimension_multiplier": 3, "invertible_group_order": 2, "outcome": "excluded_by_pinned_unrestricted_rank_four_dimension_bound" } ], "all_unit_containing_subsets_tested": 128, "classification_status": "unresolved; these exclude construction routes only", "faithful_nontrivial_finite_group_gradings": "excluded: neutral proper subring pointed, total FP dimension irrational", "fusion_subrings": [ [ 0 ], [ 0, 1 ], [ 0, 1, 2, 3, 4, 5, 6, 7 ] ], "global_dimension": "72*sqrt(37) + 444", "id": "R8_3_1768", "nontrivial_finite_group_equivariantizations": { "only_possible_group": "C2", "outcome": "excluded", "rank_four_quotient": { "base_fp_dimensions": [ "1", "5 + sqrt(37)", "6 + sqrt(37)", "sqrt(37) + 7" ], "base_global_dimension": "36*sqrt(37) + 222", "base_tensor": [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 1, 0, 0 ], [ 1, 3, 3, 4 ], [ 0, 3, 4, 4 ], [ 0, 4, 4, 4 ] ], [ [ 0, 0, 1, 0 ], [ 0, 3, 4, 4 ], [ 1, 4, 4, 4 ], [ 0, 4, 4, 5 ] ], [ [ 0, 0, 0, 1 ], [ 0, 4, 4, 4 ], [ 0, 4, 4, 5 ], [ 1, 4, 5, 5 ] ] ], "forced_equivariantization_base_rank": 4, "free_involution": 1, "orbits": [ [ 0, 1 ], [ 2, 7 ], [ 3, 6 ], [ 4, 5 ] ], "outcome": "excluded_by_pinned_unrestricted_rank_four_dimension_bound", "scope": "hypothetical C2-equivariantization or central Tannakian C2 condensation only" } }, "proper_subring_dimensions": [ 1, 2 ], "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, "domain": "complete_subrings_gradings_equivariantizations_and_rank4_GNG_branches", "evidence": [ "evidence/rank8_frontier_20261021/quadratic/structural_constructions_results.json" ], "limitation": "These construction negatives do not exclude all complex categorifications; rank-four dependence is explicitly pinned in the receipt.", "outcome": "specified_construction_routes_excluded", "proof": "evidence/rank8_frontier_20261021/quadratic/STRUCTURAL_CONSTRUCTIONS.md", "scope": "general_complex_construction_families", "verifiers": [ "evidence/rank8_frontier_20261021/quadratic/structural_constructions.py" ] }, { "details": { "PSD_Gram_survivors": 115, "bounded_selfdual_vectors_tested": 9539712, "entrywise_Gram_survivors": 249139, "invertible_stabilizer_survivors": 4772040, "one_plus_simple_surviving_indices": [ 1, 3, 5, 6 ] }, "domain": "all_connected_separable_algebra_underlying_objects_entrywise_and_PSD_Gram", "evidence": [ "evidence/rank8_frontier_20261021/quadratic/connected_algebra_candidates_results.json" ], "limitation": "Every actual connected separable algebra passes this filter; surviving objects have no associative multiplication or module structure certified.", "outcome": "complete_necessary_object_filter_survives", "proof": "evidence/rank8_frontier_20261021/quadratic/CONNECTED_ALGEBRA_CANDIDATES.md", "scope": "general_complex_algebra_construction_constraints", "verifiers": [ "evidence/rank8_frontier_20261021/quadratic/connected_algebra_candidates.py" ] }, { "details": { "allowed_order_sequence": [], "complete_order_candidates": [ 3, 4, 5, 6, 8, 10, 12 ], "dimension_bounded_product_count": 1, "exponent_box_size": 1, "positive_integer_scaling_matches": [] }, "domain": "all_levels_SU2_adjoint_finite_products_integer_multiplier_chains_and_Morita_duals", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative_followup/su2_all_frontier.json" ], "limitation": "Other Lie types, arbitrary algebra condensation, dimension-decreasing de-equivariantization, or general categorifications.", "outcome": "family_excluded_by_exact_global_dimension", "proof": "evidence/rank8_frontier_20261021/noncommutative_followup/SU2_FAMILY.md", "scope": "general_complex_construction_family_dimension_obstruction", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative_followup/su2_all_frontier.py", "evidence/rank8_frontier_20261021/noncommutative_followup/su2_product_family.py" ] }, { "details": { "families": [ { "base_rank": 12, "base_rank_two_classification_excludes": false, "integer_dimension_multiplier": 1, "invertible_group_order": 6, "noninvertible_orbit_size": 6, "stabilizer_order": 1, "status": "dimension_match_only_no_model_or_Morita_equivalence", "total_noninvertible_multiplicity": 12 }, { "base_rank": 6, "base_rank_two_classification_excludes": false, "integer_dimension_multiplier": 2, "invertible_group_order": 3, "noninvertible_orbit_size": 3, "stabilizer_order": 1, "status": "dimension_match_only_no_model_or_Morita_equivalence", "total_noninvertible_multiplicity": 12 }, { "base_rank": 4, "base_rank_two_classification_excludes": false, "integer_dimension_multiplier": 3, "invertible_group_order": 2, "noninvertible_orbit_size": 2, "stabilizer_order": 1, "status": "dimension_match_only_no_model_or_Morita_equivalence", "total_noninvertible_multiplicity": 12 }, { "base_rank": 2, "base_rank_two_classification_excludes": true, "integer_dimension_multiplier": 6, "invertible_group_order": 1, "noninvertible_orbit_size": 1, "stabilizer_order": 1, "status": "impossible_rank_two_base", "total_noninvertible_multiplicity": 12 } ], "forced_group_order_before_integer_multiplier": "6", "global_dimension_minimal_polynomial": [ "1", "-888", "5328" ], "id": "R8_3_1768", "outcome": "finite_family_dimension_matches_only", "positive_root_interval": [ "1180061/1338", "64383/73" ], "scope": "specific_model_family_only", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, "domain": "all_generalized_near_group_dimension_parameters_with_integer_scalings", "evidence": [ "evidence/rank8_frontier_20261021/noncommutative_followup/generalized_near_group_all_frontier.json" ], "limitation": "This is a necessary dimension-family test; retained parameter triples are not categories, and integral targets lie outside its irrational quadratic parameter theorem.", "outcome": "finite_family_dimension_matches_only", "proof": "evidence/rank8_frontier_20261021/noncommutative_followup/GENERALIZED_NEAR_GROUP_TABLE.md", "scope": "specific_model_family_only", "verifiers": [ "evidence/rank8_frontier_20261021/noncommutative_followup/generalized_near_group_all_frontier.py" ] }, { "details": { "complete_exact_certificate": false, "frame_index": null, "height": 4, "high_columns": 497, "low_frames": null, "rows": 241, "variables": 18130, "variant": null }, "domain": "atomic_real_sts_global_column_generation", "evidence": [ "evidence/rank8_frontier_20261021/global_induction/atomic_results/R8_3_1768_h4.json" ], "limitation": "A running/partial computation is incomplete; a feasible relaxation does not construct integer center data or an associator.", "outcome": "replaced_by_complete_sparse_phase_one", "proof": "evidence/rank8_frontier_20261021/global_induction/ATOMIC_MODULAR.md", "scope": "incomplete_no_mathematical_conclusion", "verifiers": [ "evidence/rank8_frontier_20261021/global_induction/atomic_generation.py" ] }, { "details": { "complete_exact_certificate": true, "frame_index": null, "height": 4, "high_columns": 497, "low_frames": null, "rows": 241, "variables": 18130, "variant": "atomic_real_sts" }, "domain": "packed_complete_sparse_phase_one", "evidence": [ "evidence/rank8_frontier_20261021/global_induction/complete_results/R8_3_1768_atomic_real_sts_h4.json" ], "limitation": "A running/partial computation is incomplete; a feasible relaxation does not construct integer center data or an associator.", "outcome": "exact_feasible_relaxation", "proof": "evidence/rank8_frontier_20261021/global_induction/PROOF.md", "scope": "pseudo_unitary_induction_relaxation", "verifiers": [ "evidence/rank8_frontier_20261021/global_induction/complete_phase_one.py" ] }, { "details": { "completed_scope": "Integral-center and complete spectral-support reductions, integer Gram/S/cubic, joint Galois balance, FP bounds and source-unit product constraints, with exact recorded box/lattice/witness checks.", "exact_replay": [ [ "quadratic_atomic_integer/final_verify.py", "--ring", "1768" ] ] }, "domain": "final_joint_integer_Gram_full_S_cubic_Galois_bounds_unit_product_and_scoped_phase_screen", "evidence": [ "evidence/rank8_frontier_20261021/commutative_integer/final_campaign.json", "evidence/rank8_frontier_20261021/quadratic_atomic_integer/R8_3_1768_final.json" ], "limitation": "A capped Hermite attempt asserts neither lattice membership nor nonmembership; its resource limits are retained in the final receipt. When proved, signed lattice membership alone is weaker than bounded nonnegative integer feasibility. Numerical proposal failures are not exclusions; categorical general scope requires an independent lift.", "outcome": "integer_lattice_member_no_bounded_witness", "proof": "evidence/rank8_frontier_20261021/commutative_integer/PROOF.md", "scope": "necessary_center_data_scoped_by_each_source_receipt", "verifiers": [ "evidence/rank8_frontier_20261021/commutative_integer/verify_research.py", "evidence/rank8_frontier_20261021/quadratic_atomic_integer/final_verify.py" ] } ], "prior_published_attempts": [ { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "scope_inapplicable", "reason": "Not a rank-eight near-group ring.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "near_group", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "rank_two", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "zero_spectrum", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "lower_rank_subring", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "scope_inapplicable", "reason": "No rank-seven closed based subring.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "rank7_certified_subring", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "dnumber", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "scope_inapplicable", "reason": "Noncommutative Wedderburn branch only.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "noncomm_dnumber", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "scope_inapplicable", "reason": "No integral positive FP character; integral Drinfeld branch only.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "integral_drinfeld", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "scope_detail": "Bounded exact obstruction search; no conclusion is not a proof that all character fields are cyclotomic.", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "cyclotomic", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "scope_inapplicable", "reason": "The inherited module implementation requires an invertible involution with one or two fixed simples.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "involution_module", "status": "scope_inapplicable" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "scope_detail": "Search restricted to fusion-matrix irreducible factors of degrees3through6; no conclusion is not a proof of cyclotomicity.", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "field_automorphisms", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "subring_dimension", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/attempts.jsonl", "stage": "primitive_codegree", "status": "no_conclusion" }, { "attempt_kind": "exclusion", "budget_seconds": 90, "id": "R8_3_1768", "mode": "character", "outcome": "timeout", "source": "evidence/rank8_m3_20261021/criteria/advanced/candidate_character/R8_3_1768.json", "source_references": [ "evidence/rank8_m3_20261021/criteria/advanced/candidate_character/R8_3_1768.json", "evidence/rank8_m3_20261021/criteria/candidate_attempts.json" ], "stage": "character", "status": "timeout", "stdout": "", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, { "attempt_kind": "exclusion", "budget_seconds": 90, "id": "R8_3_1768", "mode": "dimension_order", "outcome": "timeout", "source": "evidence/rank8_m3_20261021/criteria/advanced/candidate_dimension_order/R8_3_1768.json", "source_references": [ "evidence/rank8_m3_20261021/criteria/advanced/candidate_dimension_order/R8_3_1768.json", "evidence/rank8_m3_20261021/criteria/candidate_attempts.json" ], "stage": "dimension_order", "status": "timeout", "stdout": "", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, { "attempt_kind": "exclusion", "budget_seconds": 90, "id": "R8_3_1768", "mode": "modular", "outcome": "timeout", "source": "evidence/rank8_m3_20261021/criteria/advanced/candidate_modular/R8_3_1768.json", "source_references": [ "evidence/rank8_m3_20261021/criteria/advanced/candidate_modular/R8_3_1768.json", "evidence/rank8_m3_20261021/criteria/candidate_attempts.json" ], "stage": "modular", "status": "timeout", "stdout": "", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, { "attempt_kind": "exclusion", "budget_seconds": 90, "id": "R8_3_1768", "mode": "tensor_rotation", "outcome": "timeout", "source": "evidence/rank8_m3_20261021/criteria/advanced/candidate_tensor_rotation/R8_3_1768.json", "source_references": [ "evidence/rank8_m3_20261021/criteria/advanced/candidate_tensor_rotation/R8_3_1768.json", "evidence/rank8_m3_20261021/criteria/candidate_attempts.json" ], "stage": "tensor_rotation", "status": "timeout", "stdout": "", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, { "attempt_kind": "exclusion", "attempts": [], "budget_skipped_objects": 0, "candidate_object_count": 0, "completed_objects": 0, "frontier_ledger_digest": "23837060c2c8186fb971d5c7515620cc0720508d025337e12154eb20b1935270", "id": "R8_3_1768", "object_family": "one nontrivial dual-pair atom or sum of two distinct nonunit self-dual orbit atoms", "outcome": "inapplicable_no_fixed_involution", "per_object_case_budget": 100000, "profile_derived_from_tensor": { "commutative": true, "dual": [ 0, 1, 2, 3, 4, 5, 6, 7 ], "fixed_point_involutions": [], "invertibles": [ 0, 1 ] }, "scope": "general_complex_fusion_category", "scope_note": "a modular survivor gives no exclusion; it is not a feasible categorification or an exact module witness", "source": "evidence/rank8_m3_20261021/noncommutative/split_module_object_results.json", "stage": "split_module_object_results", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, { "attempt_kind": "exclusion", "attempts": [], "id": "R8_3_1768", "outcome": "inapplicable_no_fixed_involution", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/noncommutative/split_module_polynomial_results.json", "stage": "split_module_polynomial_results", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "no_conclusion", "report": { "lowest_numerical_eigenvalue": -1.29528883101698e-12, "scope": "numerical discovery only" }, "source": "evidence/rank8_m3_20261021/noncommutative/unitary_primary3_results.json", "stage": "unitary_primary3_results", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "no_conclusion", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/one_spectrum_m3.json", "stage": "one_spectrum", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, { "attempt_kind": "construction", "failed_objects": [], "id": "R8_3_1768", "outcome": "no_conclusion", "scope": "weakly_group_theoretical_models_only", "source": "evidence/rank8_m3_20261021/pipeline/frobenius_results.json", "stage": "strong_frobenius_weak_group_theoretical", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "scope_inapplicable", "reason": "No closed based subring of rank6or7 exists.", "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/unified_scope_attempts.json", "stage": "expanded_exact_negative_subring_transfer", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, { "attempt_kind": "exclusion", "id": "R8_3_1768", "outcome": "no_conclusion", "quadratic_factors": [ [ 1, -888, 5328 ] ], "scope": "general_complex_fusion_category", "source": "evidence/rank8_m3_20261021/pipeline/unified_scope_attempts.json", "stage": "multiquadratic_casimir_compression", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, { "attempt_kind": "construction", "family": "72 explicitly constructed rank-eight tensors", "id": "R8_3_1768", "matching_test_numbers": [], "outcome": "no_match_in_tested_family", "scope": "negative means no match in this finite list; crossed products are only abstract ring tests unless separately certified", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "72 explicitly constructed rank-eight tensors", "tested_tensors": 72 }, { "attempt_kind": "construction", "family": "ordinary finite-group representation categories", "id": "R8_3_1768", "outcome": "family_excluded", "reason": "verified nonintegral simple dimension", "rejected_by_elementary_filter": true, "scope": "This construction family only; no arbitrary categorification exclusion.", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "ordinary finite-group representation categories" }, { "attempt_kind": "construction", "family": "pointed and ordinary Tambara-Yamagami categories", "id": "R8_3_1768", "outcome": "family_excluded", "reason": "their fusion multiplicities are at most1, but this source has maximum3", "rejected": true, "scope": "This construction family only; no arbitrary categorification exclusion.", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "pointed and ordinary Tambara-Yamagami categories" }, { "attempt_kind": "construction", "family": "Morita sources obtained by cyclic-prime extension/equivariantization", "id": "R8_3_1768", "outcome": "no_conclusion", "scope": "if applicable, excludes Morita equivalence to a nontrivial cyclic-prime extension or equivariantization; not arbitrary categorification", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "Morita sources obtained by cyclic-prime extension/equivariantization", "structural_filter": { "adjoint_basis": [ 0, 1, 2, 3, 4, 5, 6, 7 ], "center_has_no_nontrivial_invertible_if_categorified": false, "central_invertible_indices": [ 0, 1 ], "invertible_indices": [ 0, 1 ], "scope": "if applicable, excludes Morita equivalence to a nontrivial cyclic-prime extension or equivariantization; not arbitrary categorification" } }, { "attempt_kind": "construction", "family": "bounded classical WZW dimension-ratio search", "id": "R8_3_1768", "leads": [], "outcome": "heuristic_search_completed", "scope": "heuristic numerical leads only; no tensor identification or general exclusion", "source": "evidence/rank8_m3_20261021/models/model_search_attempts.json", "stage": "bounded classical WZW dimension-ratio search", "tested_models": 1505 }, { "attempt_kind": "construction", "casimir_characteristic_polynomial": [ 1, -936, 48888, -1096416, 13479696, -97977600, 421396992, -994332672, 994332672 ], "casimir_matrix": [ [ 8, 0, 10, 12, 12, 14, 12, 12 ], [ 0, 8, 12, 12, 14, 12, 12, 10 ], [ 10, 12, 130, 134, 144, 144, 132, 120 ], [ 12, 12, 134, 152, 156, 158, 144, 132 ], [ 12, 14, 144, 156, 178, 168, 158, 144 ], [ 14, 12, 144, 158, 168, 178, 156, 144 ], [ 12, 12, 132, 144, 158, 156, 152, 134 ], [ 12, 10, 120, 132, 144, 144, 134, 130 ] ], "families": [ { "all_larger_odd_orders": { "first_order": 11, "source_upper_bound": 974, "strict_target_lower_bound": 1331 }, "family": "odd_Haagerup_Izumi_and_all_Morita_duals", "outcome": "specific_family_Morita_model_excluded", "tested_odd_orders": [ { "casimir_remainder": [ -340169538249, 1114071991443 ], "dimension_polynomial": [ 1, -39, 117 ], "odd_order": 3, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ -3862722263274207, 20029914188405347 ], "dimension_polynomial": [ 1, -145, 725 ], "odd_order": 5, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ -1055249340601386837, 7531639257419115451 ], "dimension_polynomial": [ 1, -371, 2597 ], "odd_order": 7, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ -22151130624916601067, 201762454213914307227 ], "dimension_polynomial": [ 1, -765, 6885 ], "odd_order": 9, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" } ] }, { "all_larger_odd_orders": { "first_order": 7, "source_upper_bound": 974, "strict_target_lower_bound": 2401 }, "family": "specified_near_group_and_all_Morita_duals", "outcome": "specific_family_Morita_model_excluded", "tested_odd_orders": [ { "casimir_remainder": [ -763974391053843, 7506092532146931 ], "dimension_polynomial": [ 1, -117, 1053 ], "odd_order": 3, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" }, { "casimir_remainder": [ -20509058627110435347, 531742455171037524547 ], "dimension_polynomial": [ 1, -725, 18125 ], "odd_order": 5, "witness": "target_dimension_is_not_a_Casimir_eigenvalue" } ] } ], "global_dimension_diagonal_lower_bound": 178, "global_dimension_upper_bound": 974, "id": "R8_3_1768", "outcome": "no_model_from_either_tested_family", "scope": "specific_model_families_only_not_categorification_exclusion", "source": "evidence/rank8_m3_20261021/models/fresh_202609/odd_hi_dimension_filter.json", "source_dual": [ 0, 1, 2, 3, 4, 5, 6, 7 ], "stage": "odd_hi_and_specified_near_group_Morita_dimension_filter", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" }, { "AH_dimension_polynomial_remainder": [ -187926403654053888, 763097709492883456 ], "attempt_kind": "construction", "casimir_characteristic_polynomial": [ 1, -936, 48888, -1096416, 13479696, -97977600, 421396992, -994332672, 994332672 ], "family_outcome": "different_global_dimension_so_no_AH_Morita_model", "general_categorification_excluded": false, "id": "R8_3_1768", "invertibles": [ 0, 1 ], "maximum_multiplicity": 3, "outcome": "family_excluded", "rank": 8, "scope": "known_Asaeda_Haagerup_Morita_family_only", "source": "evidence/rank8_m3_20261021/models/asaeda_haagerup_attempt.json", "stage": "known_Asaeda_Haagerup_Morita_family_filter", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" } ], "prior_stage_completed": [ { "domain": "integral_proper_subcategories_and_braided_Morita_representatives", "evidence": [ "evidence/rank8_research_20261022/braided_morita/PROOF.md", "evidence/rank8_research_20261022/braided_morita/verification.json", "site/BRAIDED_CATEGORIFICATION.md" ], "limitation": "This restricts braided Morita constructions; arbitrary unbraided categorification remains unresolved for the six surviving rings.", "outcome": "Nonintegral total FP dimension and integral proper fusion subcategories force any braided Morita representative to yield a braiding on the original category. The general braiding exclusions therefore exclude every braided Morita representative.", "scope": "general_complex_fusion_category", "verifiers": [ "evidence/rank8_research_20261022/braided_morita/verify.py", "site/verify_braided_scope.py" ] }, { "domain": "unrestricted_sphericalization_and_dimension_character_sieve", "evidence": [ "evidence/rank8_research_20261022/pivotality/PROOF.md", "evidence/rank8_research_20261022/pivotality/general_spherical_gates.json", "evidence/rank8_research_20261022/r928_r941/PIVOTALITY_REVIEW.md", "evidence/rank8_research_20261022/r928_r941/pivotality_independent_review.json", "site/CATEGORIFICATION_SCOPE.md" ], "limitation": "Sphericality applies to the original category; pseudo-unitarity may require Galois conjugation. No unbraided existence or exclusion follows from this gate alone.", "outcome": "Every complex categorification admits a spherical structure and has a pseudo-unitary Galois conjugate with the same fusion rules; the existing braided obstruction now has general scope.", "scope": "general_complex_categorical_implication", "verifiers": [ "evidence/rank8_research_20261022/pivotality/verify_general_spherical_gates.py", "evidence/rank8_research_20261022/r928_r941/review_pivotality.py" ] }, { "domain": "pointed_C2_Fourier_fibre_packing_trivial_class_obstruction", "evidence": [ "evidence/rank8_research_20261022/pointed_followup/PROOF.md", "evidence/rank8_research_20261022/pointed_followup/R8_3_1768.json", "evidence/rank8_research_20261022/pointed_followup/verification_R8_3_1768.json", "evidence/rank8_research_20261022/r1768_r3472/R1768_POINTED_INDEPENDENT_AUDIT.md" ], "limitation": "The nontrivial pointed class retains an exact feasible relaxation. It is not a category, and the ring remains unresolved.", "outcome": "All eight even pointed-indicator branches are excluded on 10,711 exact columns. Every general categorification would have nontrivial pointed C2 associator; no connected separable algebra 1+g can exist.", "scope": "general_complex_fusion_category", "verifiers": [ "evidence/rank8_research_20261022/pointed_followup/verify.py", "evidence/rank8_research_20261022/r1768_r3472/r1768_pointed_independent_audit.py" ] }, { "domain": "nontrivial_pointed_associator_module_action_filter", "evidence": [ "evidence/rank8_research_20261022/pointed_followup/r1768_modules/PROOF.md", "evidence/rank8_research_20261022/pointed_followup/r1768_modules/verification.json", "evidence/rank8_research_20261022/pointed_followup/r1768_modules/INDEPENDENT_REVIEW.md" ], "limitation": "Fifteen additional retained induction factors still lack complete actions. This is neither a classification of actual module categories nor a model or ring exclusion.", "outcome": "The forced nontrivial C2 class removes g-fixed module simples. The saved 61 induction factors on 49 algebra objects reduce to 21 factors on 21 objects, of possible ranks4,6,8, 12. Of 9 completely determined action classes, 3 remain with ranks 8, 6, 6.", "scope": "necessary_conditions_for_indecomposable_semisimple_module_categories", "verifiers": [ "evidence/rank8_research_20261022/pointed_followup/r1768_modules/verify.py" ] }, { "domain": "dimension_block_conductor_bound_and_saturation", "evidence": [ "evidence/rank8_research_20261022/pointed_followup/dimension_block/PROOF.md", "evidence/rank8_research_20261022/pointed_followup/dimension_block/verification.json" ], "limitation": "All seven incoming rings pass the scalar screen. No refined cone run, model, or exclusion follows from this criterion; R1795 is separately excluded by C2 Fourier packing.", "outcome": "Every additional twist-one central simple outside the dimension orbit has rational dimension ratio; its coordinates satisfy 6a=37b. All forced I(1) summands satisfy the applicable mask.", "scope": "necessary_center_conditions_in_a_pseudo_unitary_Galois_form", "verifiers": [ "evidence/rank8_research_20261022/pointed_followup/dimension_block/verify.py" ] } ], "prior_unitary_obstruction_source": null, "prior_unitary_status": "not_separately_decided", "screening_case_specific_focus": "Complete rank-two based-module and connected-algebra underlying-object necessary filters; every proper subring, grading and nontrivial group-equivariantization tested; joint integer Gram/S/cubic/Galois and source-unit constraints pass signed lattice membership without a bounded nonnegative integer witness; specified unbounded construction families tested.", "screening_definition": "Extensively investigated exotic candidate means that the ring survived the documented, case-specific campaign of known and newly developed exclusion tests and explicit model searches, including complete finite domains and specified unbounded construction families. It proves neither existence of a fusion category nor novelty of a future realization. Each case lists exactly what was exhausted, what survived and the remaining hypotheses.", "screening_stage": "extensively_investigated_exotic_candidate", "tensor_sha256": "240509807c67da437c86f2a51efeb831364734feb93ed18a1d2f32087ebc1af0" } ], "schema": "audited-rank8-per-ring/v1", "scope": "research coverage and source binding; linked proof verifiers provide mathematical replay", "display_scope": "Four current rank-eight open cases; complete historical research is preserved separately." }