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complex categorification is excluded. The symmetric pairing on a multiplicity-one Galois summand forces four positive large second indicators in the zero-shift Morita dual. Morita invariance of the dimension-weighted second-indicator total contradicts the source duality. The spherical and Galois reductions cover arbitrary complex categorifications."],"audit_sources":[{"label":"GALOIS_FS2_EXCLUSION.md","path":"evidence/unresolved10_20261027/r9_commutative/GALOIS_FS2_EXCLUSION.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/unresolved10_20261027/r9_commutative/GALOIS_FS2_EXCLUSION.md","sha256":"c2a03c23310c0f21ef743cf41ac0ba9ec236633cedf94812ba1896bb57d86a35"},{"label":"AUDIT.md","path":"evidence/unresolved7_20261027/audit_fs2/AUDIT.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/unresolved7_20261027/audit_fs2/AUDIT.md","sha256":"c5f5de2f614daf153b185394d126f190156d395b0f8c7894f7fef29c17507cd5"}],"braided":{"status":"excluded","explanation":"General complex categorification is excluded.","sources":[{"label":"GALOIS_FS2_EXCLUSION.md","path":"evidence/unresolved10_20261027/r9_commutative/GALOIS_FS2_EXCLUSION.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/unresolved10_20261027/r9_commutative/GALOIS_FS2_EXCLUSION.md","sha256":"c2a03c23310c0f21ef743cf41ac0ba9ec236633cedf94812ba1896bb57d86a35"},{"label":"AUDIT.md","path":"evidence/unresolved7_20261027/audit_fs2/AUDIT.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/unresolved7_20261027/audit_fs2/AUDIT.md","sha256":"c5f5de2f614daf153b185394d126f190156d395b0f8c7894f7fef29c17507cd5"}]}},"download_url":"./downloads/fusion-tensors/R9_2_1607.txt","investigation_deferred":false},{"id":"R9_2_1608","rank":9,"multiplicity":2,"status":"excluded","name":"R9_2_1608","summary":"General complex fusion categorification is excluded by the recorded theoretical argument and exact certificate.","existence":"excluded","model_reference":"","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank9_m2_20261024/ledger.json","source_file":"evidence/rank9_m2_20261024/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank9_m2_20261024/ledger.json","source_location":"R9_2_1608","report_url":"./papers/rank9-m2.pdf","report_label":"Rank-nine multiplicity-two manuscript","case_url":"./case.html?id=R9_2_1608","fusion_tensor":[[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0],[0,2,2,0,0,1,1,2,2],[1,2,2,1,2,0,0,0,2],[0,1,0,0,0,0,0,0,0],[0,2,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,2,0,0,0,0,0,0],[0,2,2,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0],[1,2,2,1,2,0,0,0,2],[0,2,2,0,0,1,1,2,2],[0,0,1,0,0,0,0,0,0],[0,0,2,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,2,0,0,0,0,0,0,0],[0,2,2,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]],[[0,0,0,0,1,0,0,0,0],[0,2,0,0,0,0,0,0,0],[0,0,2,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[1,0,0,1,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,2]],[[0,0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1]],[[0,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1]],[[0,0,0,0,0,0,0,1,0],[0,0,2,0,0,0,0,0,0],[0,2,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,1,1,1,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0,0],[1,0,0,1,1,0,0,0,0],[0,0,0,0,0,0,0,0,2]],[[0,0,0,0,0,0,0,0,1],[0,2,2,0,0,0,0,0,0],[0,2,2,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,2],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,2],[1,0,0,1,2,1,1,2,1]]],"tensor_sha256":"b01ffcda11f1803c2920c5331faae3f3e53240e2c178e1761e810f4d3005d925","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8"],"unit_index":0,"tensor_convention":"Source basis; zero-based indices; X_0=1. 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The proof covers arbitrary complex cochains and does not impose a fourth-root normalization.","existence":"excluded","model_reference":"","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank9_m2_20261024/ledger.json","source_file":"evidence/frontier_20261027/rank9_m2_ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/frontier_20261027/rank9_m2_ledger.json","source_location":"R9_2_1975","report_url":"./papers/rank9-m2.pdf","report_label":"Rank-nine multiplicity-two manuscript","case_url":"./case.html?id=R9_2_1975","fusion_tensor":[[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0],[0,2,2,2,2,0,0,0,1],[1,2,2,2,2,0,0,1,0],[0,2,2,2,2,0,0,0,1],[0,2,2,2,2,1,1,0,0],[0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,1,0,0,0,0]],[[0,0,1,0,0,0,0,0,0],[1,2,2,2,2,0,1,0,0],[0,2,2,2,2,0,0,0,1],[0,2,2,2,2,1,0,1,0],[0,2,2,2,2,0,0,0,1],[0,0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,1,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0],[0,2,2,2,2,0,0,0,1],[0,2,2,2,2,1,1,0,0],[0,2,2,2,2,0,0,0,1],[1,2,2,2,2,0,0,1,0],[0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,1,0,0,0,0]],[[0,0,0,0,1,0,0,0,0],[0,2,2,2,2,1,0,1,0],[0,2,2,2,2,0,0,0,1],[1,2,2,2,2,0,1,0,0],[0,2,2,2,2,0,0,0,1],[0,0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,1,0,1,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1]],[[0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1]],[[0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]],[[0,0,0,0,0,0,0,0,1],[0,0,1,0,1,0,0,0,0],[0,1,0,1,0,0,0,0,0],[0,0,1,0,1,0,0,0,0],[0,1,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,1,1,1,0]]],"tensor_sha256":"8b30652d59e868a120b6c423e46a99ab51f6032af752dc0082a3d80fd9e61323","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8"],"unit_index":0,"tensor_convention":"Source basis; zero-based indices; X_0=1. 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complex categorification is excluded by the same Galois-pairing and Morita-invariant second-indicator argument as R9_2_1607. The required sum of four positive large indicators contradicts the source, which has no selfdual large simples. The proof retains the general-complex spherical and Galois reductions."],"audit_sources":[{"label":"GALOIS_FS2_EXCLUSION.md","path":"evidence/unresolved10_20261027/r9_commutative/GALOIS_FS2_EXCLUSION.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/unresolved10_20261027/r9_commutative/GALOIS_FS2_EXCLUSION.md","sha256":"c2a03c23310c0f21ef743cf41ac0ba9ec236633cedf94812ba1896bb57d86a35"},{"label":"AUDIT.md","path":"evidence/unresolved7_20261027/audit_fs2/AUDIT.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/unresolved7_20261027/audit_fs2/AUDIT.md","sha256":"c5f5de2f614daf153b185394d126f190156d395b0f8c7894f7fef29c17507cd5"}],"braided":{"status":"excluded","explanation":"General complex categorification is excluded.","sources":[{"label":"GALOIS_FS2_EXCLUSION.md","path":"evidence/unresolved10_20261027/r9_commutative/GALOIS_FS2_EXCLUSION.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/unresolved10_20261027/r9_commutative/GALOIS_FS2_EXCLUSION.md","sha256":"c2a03c23310c0f21ef743cf41ac0ba9ec236633cedf94812ba1896bb57d86a35"},{"label":"AUDIT.md","path":"evidence/unresolved7_20261027/audit_fs2/AUDIT.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/unresolved7_20261027/audit_fs2/AUDIT.md","sha256":"c5f5de2f614daf153b185394d126f190156d395b0f8c7894f7fef29c17507cd5"}]}},"download_url":"./downloads/fusion-tensors/R9_2_1976.txt","investigation_deferred":false},{"id":"R9_2_1977","rank":9,"multiplicity":2,"status":"excluded","name":"R9_2_1977","summary":"General complex fusion categorification is excluded by the recorded theoretical argument and exact certificate.","existence":"excluded","model_reference":"","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank9_m2_20261024/ledger.json","source_file":"evidence/rank9_m2_20261024/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank9_m2_20261024/ledger.json","source_location":"R9_2_1977","report_url":"./papers/rank9-m2.pdf","report_label":"Rank-nine multiplicity-two manuscript","case_url":"./case.html?id=R9_2_1977","fusion_tensor":[[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0],[0,2,2,2,2,0,0,1,1],[1,2,2,2,2,0,1,0,1],[0,2,2,2,2,0,0,1,1],[0,2,2,2,2,1,1,1,0],[0,0,0,1,0,0,0,0,0],[0,1,0,1,0,0,0,0,0],[0,1,1,0,1,0,0,0,0],[0,0,1,1,1,0,0,0,0]],[[0,0,1,0,0,0,0,0,0],[1,2,2,2,2,0,1,1,0],[0,2,2,2,2,0,0,1,1],[0,2,2,2,2,1,1,0,1],[0,2,2,2,2,0,0,1,1],[0,0,0,0,1,0,0,0,0],[0,0,1,0,1,0,0,0,0],[0,1,0,1,1,0,0,0,0],[0,1,1,1,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0],[0,2,2,2,2,0,0,1,1],[0,2,2,2,2,1,1,1,0],[0,2,2,2,2,0,0,1,1],[1,2,2,2,2,0,1,0,1],[0,1,0,0,0,0,0,0,0],[0,1,0,1,0,0,0,0,0],[0,0,1,1,1,0,0,0,0],[0,1,1,0,1,0,0,0,0]],[[0,0,0,0,1,0,0,0,0],[0,2,2,2,2,1,1,0,1],[0,2,2,2,2,0,0,1,1],[1,2,2,2,2,0,1,1,0],[0,2,2,2,2,0,0,1,1],[0,0,1,0,0,0,0,0,0],[0,0,1,0,1,0,0,0,0],[0,1,1,1,0,0,0,0,0],[0,1,0,1,1,0,0,0,0]],[[0,0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,1,0,0],[0,1,0,1,0,0,0,0,0],[0,0,1,0,1,0,0,0,0],[0,1,0,1,0,0,0,0,0],[0,0,1,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0],[1,0,0,0,0,1,1,0,0],[0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,1,1]],[[0,0,0,0,0,0,0,1,0],[0,0,1,1,1,0,0,0,0],[0,1,1,1,0,0,0,0,0],[0,1,1,0,1,0,0,0,0],[0,1,0,1,1,0,0,0,0],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,1],[1,0,0,0,0,0,1,1,1],[0,0,0,0,0,1,1,1,1]],[[0,0,0,0,0,0,0,0,1],[0,1,1,0,1,0,0,0,0],[0,1,0,1,1,0,0,0,0],[0,0,1,1,1,0,0,0,0],[0,1,1,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,1],[0,0,0,0,0,1,1,1,1],[1,0,0,0,0,0,1,1,1]]],"tensor_sha256":"1f06c9fe130170d97c97a028fbe06d3fecc19b88af80ccf3cbb0fab2134dcccd","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8"],"unit_index":0,"tensor_convention":"Source basis; zero-based indices; X_0=1. N[i][j][k] is the coefficient of X_k in X_i*X_j.","hash_convention":"SHA-256 of compact UTF-8 JSON; no trailing newline.","invariants":{"dual_indices":[0,2,1,4,3,5,6,7,8],"invertible_indices":[0,5],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,8.690415759823418,8.690415759823427,8.690415759823425,8.690415759823425,0.9999999999999967,1.9999999999999964,2.9999999999999964,2.9999999999999964],"global_fp_dimension_approx":326.0933043143494,"central_casimir_polynomial":"(t - 16)**2*(t - 8)**4*(t - 3)*(t**2 - 352*t + 8448)","dimensions_method":"FP dimensions are approximate; exact algebraic invariants and character certificates are in the proof archive.","character_method":"Exact characters and formal codegrees: hash-bound replay archive.","proper_fusion_subrings":[[0],[0,5],[0,5,6],[0,5,6,7,8],[0,1,2,3,4,5,6,7,8]],"original_physical_line":2101},"research":[{"title":"Current accepted case 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