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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_1.txt","investigation_deferred":false},{"id":"R12_1_2","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Rep(D36). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Rep(D36)","existence":"unitary","model_reference":"A unitary realization exists: Rep(D36). 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guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: Vec(C2) x 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_14.txt","investigation_deferred":false},{"id":"R12_1_15","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Fib x Vec_C_2boxtimesmathrmIsing; also R(0,0). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Fib x Vec_C_2boxtimesmathrmIsing; also R(0,0)","existence":"unitary","model_reference":"A unitary realization exists: Fib x Vec_C_2boxtimesmathrmIsing; also R(0,0). 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guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: Fib x 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_16.txt","investigation_deferred":false},{"id":"R12_1_17","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-zest(Vec(C2) x Rep(C_3⋊ C_4); h=5). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"C2-zest(Vec(C2) x Rep(C_3⋊ C_4); h=5)","existence":"unitary","model_reference":"A unitary realization exists: C2-zest(Vec(C2) x Rep(C_3⋊ C_4); h=5). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_17","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_17","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,1,1,0,0,0,0,0],[0,1,1,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,1,1,0,0],[1,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,1,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,1,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,1,1,0,0,0,0,0],[0,1,1,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,1,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,1,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,1,1,0,0],[1,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0]]],"tensor_sha256":"b4a4575d7a853bf98e9012ff21d3ef95ede72b537052b7c45ff4c0ff32804af7","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_17.txt","investigation_deferred":false},{"id":"R12_1_18","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-zest(Fib x Rep(C_3⋊ C_4); h=5). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"C2-zest(Fib x Rep(C_3⋊ C_4); h=5)","existence":"unitary","model_reference":"A unitary realization exists: C2-zest(Fib x Rep(C_3⋊ C_4); h=5). 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Rep(C_3⋊ C_4); h=5). The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_18.txt","investigation_deferred":false},{"id":"R12_1_19","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Rep(S3) x PSU(2)_6. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Rep(S3) x PSU(2)_6","existence":"unitary","model_reference":"A unitary realization exists: Rep(S3) x PSU(2)_6. 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_24.txt","investigation_deferred":false},{"id":"R12_1_25","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C(A5 x C2, graph(D10 reflection sign)). 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The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_25","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_25","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,1,1,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,1,0],[1,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,1,1,1,0,0,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[1,0,0,0,0,1,1,1,1,1,1,0],[0,0,0,0,0,1,1,1,1,1,1,1],[0,0,1,1,1,1,1,1,1,0,0,0],[0,1,0,1,1,1,1,1,1,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,1,1,1,1,1,1,1],[1,0,0,0,0,1,1,1,1,1,1,0],[0,1,0,1,1,1,1,1,1,0,0,0],[0,0,1,1,1,1,1,1,1,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,1,1,1,1,1,1,1,0,0,0],[0,1,0,1,1,1,1,1,1,0,0,0],[1,0,0,0,0,1,1,1,1,1,1,0],[0,0,0,0,0,1,1,1,1,1,1,1],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,1,0,1,1,1,1,1,1,0,0,0],[0,0,1,1,1,1,1,1,1,0,0,0],[0,0,0,0,0,1,1,1,1,1,1,1],[1,0,0,0,0,1,1,1,1,1,1,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,1,1,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,1,0,1,0,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,1,0],[1,0,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0]]],"tensor_sha256":"e5b471a12cc1ec259b6cfa6e45215f1c330e7c9012ada6f3832cfec696ff9b93","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_28","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_28","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,1,0,0,1],[0,0,0,0,1,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,1,0,0],[0,1,1,0,0,1,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,1,0,0],[1,0,0,1,0,1,1,0,0,0,1,1],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,1,1,1,0,0,0,1,1,0,1,0],[0,0,0,0,1,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,1,0,0],[0,1,1,1,0,0,0,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[1,0,0,1,0,1,1,0,0,0,1,1],[0,0,0,0,1,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,1,0,0,1,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,1,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0]]],"tensor_sha256":"982de39d0c770f0122687c18450e255f1a08b87c8d5de5392ed7627721ca365a","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_30","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_30","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,1,0,0,1],[0,0,0,0,1,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,1,0,0],[0,1,1,0,0,1,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,1,0,0],[1,0,0,1,1,1,1,0,0,1,1,1],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,1,1,1,1,0,0,1,1,1,1,0],[0,0,0,0,1,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,1,0,0],[0,1,1,1,1,0,0,1,1,1,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[1,0,0,1,1,1,1,0,0,1,1,1],[0,0,0,0,1,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,1,0,0,1,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,1,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0]]],"tensor_sha256":"309a70eb5b4d9b3792321f342a598a0332aefb18525451b7b7b96e16eb4224b0","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,1,2,3,4,5,6,7,8,9,10,11],"invertible_indices":[0,1,2,5,6,7,8,11],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0000000000000007,2.0000000000000018,4.000000000000007,1.0000000000000007,1.0000000000000007,1.0000000000000007,1.0000000000000007,4.000000000000007,2.0000000000000018,1.0000000000000007],"global_fp_dimension_approx":48.000000000000135,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t - 48)*(t - 24)*(t - 16)**5*(t - 8)**5","formal_codegree_polynomial":"(t - 48)*(t - 24)*(t - 16)**5*(t - 8)**5"},"research":[{"title":"Current accepted case decision","source":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102","data":{"id":"R12_1_30","input_line":30,"rank":12,"multiplicity":1,"tensor_sha256":"309a70eb5b4d9b3792321f342a598a0332aefb18525451b7b7b96e16eb4224b0","status":"unitarily_realized","commutative":true,"simple":false,"exclusion_route":null,"exclusion_evidence":[],"model_name":"C(S4xC2,H4)","model_constructor":"group_theoretical","model_certificate_count":1,"first_round_status":"unresolved","first_round_audit":"retained; no erroneous classification found","continuation_models":[{"name":"C(S4xC2,H4)","constructor":"group_theoretical","parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14,17,16,19,18,21,20,23,22,25,24,27,26,29,28,31,30,33,32,35,34,37,36,39,38,41,40,43,42,45,44,47,46],[2,3,0,1,8,9,10,11,4,5,6,7,14,15,12,13,20,21,22,23,16,17,18,19,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35],[3,2,1,0,9,8,11,10,5,4,7,6,15,14,13,12,21,20,23,22,17,16,19,18,37,36,39,38,41,40,43,42,45,44,47,46,25,24,27,26,29,28,31,30,33,32,35,34],[4,5,6,7,0,1,2,3,10,11,8,9,24,25,26,27,28,29,30,31,32,33,34,35,12,13,14,15,16,17,18,19,20,21,22,23,38,39,36,37,44,45,46,47,40,41,42,43],[5,4,7,6,1,0,3,2,11,10,9,8,25,24,27,26,29,28,31,30,33,32,35,34,13,12,15,14,17,16,19,18,21,20,23,22,39,38,37,36,45,44,47,46,41,40,43,42],[6,7,4,5,10,11,8,9,0,1,2,3,26,27,24,25,32,33,34,35,28,29,30,31,38,39,36,37,44,45,46,47,40,41,42,43,12,13,14,15,16,17,18,19,20,21,22,23],[7,6,5,4,11,10,9,8,1,0,3,2,27,26,25,24,33,32,35,34,29,28,31,30,39,38,37,36,45,44,47,46,41,40,43,42,13,12,15,14,17,16,19,18,21,20,23,22],[8,9,10,11,2,3,0,1,6,7,4,5,36,37,38,39,40,41,42,43,44,45,46,47,14,15,12,13,20,21,22,23,16,17,18,19,26,27,24,25,32,33,34,35,28,29,30,31],[9,8,11,10,3,2,1,0,7,6,5,4,37,36,39,38,41,40,43,42,45,44,47,46,15,14,13,12,21,20,23,22,17,16,19,18,27,26,25,24,33,32,35,34,29,28,31,30],[10,11,8,9,6,7,4,5,2,3,0,1,38,39,36,37,44,45,46,47,40,41,42,43,26,27,24,25,32,33,34,35,28,29,30,31,14,15,12,13,20,21,22,23,16,17,18,19],[11,10,9,8,7,6,5,4,3,2,1,0,39,38,37,36,45,44,47,46,41,40,43,42,27,26,25,24,33,32,35,34,29,28,31,30,15,14,13,12,21,20,23,22,17,16,19,18],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,24,25,26,27,34,35,32,33,40,41,42,43,36,37,38,39,46,47,44,45],[13,12,15,14,17,16,19,18,21,20,23,22,1,0,3,2,5,4,7,6,9,8,11,10,29,28,31,30,25,24,27,26,35,34,33,32,41,40,43,42,37,36,39,38,47,46,45,44],[14,15,12,13,20,21,22,23,16,17,18,19,2,3,0,1,8,9,10,11,4,5,6,7,40,41,42,43,36,37,38,39,46,47,44,45,28,29,30,31,24,25,26,27,34,35,32,33],[15,14,13,12,21,20,23,22,17,16,19,18,3,2,1,0,9,8,11,10,5,4,7,6,41,40,43,42,37,36,39,38,47,46,45,44,29,28,31,30,25,24,27,26,35,34,33,32],[16,17,18,19,12,13,14,15,22,23,20,21,28,29,30,31,24,25,26,27,34,35,32,33,0,1,2,3,4,5,6,7,8,9,10,11,42,43,40,41,46,47,44,45,36,37,38,39],[17,16,19,18,13,12,15,14,23,22,21,20,29,28,31,30,25,24,27,26,35,34,33,32,1,0,3,2,5,4,7,6,9,8,11,10,43,42,41,40,47,46,45,44,37,36,39,38],[18,19,16,17,22,23,20,21,12,13,14,15,30,31,28,29,34,35,32,33,24,25,26,27,42,43,40,41,46,47,44,45,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11],[19,18,17,16,23,22,21,20,13,12,15,14,31,30,29,28,35,34,33,32,25,24,27,26,43,42,41,40,47,46,45,44,37,36,39,38,1,0,3,2,5,4,7,6,9,8,11,10],[20,21,22,23,14,15,12,13,18,19,16,17,40,41,42,43,36,37,38,39,46,47,44,45,2,3,0,1,8,9,10,11,4,5,6,7,30,31,28,29,34,35,32,33,24,25,26,27],[21,20,23,22,15,14,13,12,19,18,17,16,41,40,43,42,37,36,39,38,47,46,45,44,3,2,1,0,9,8,11,10,5,4,7,6,31,30,29,28,35,34,33,32,25,24,27,26],[22,23,20,21,18,19,16,17,14,15,12,13,42,43,40,41,46,47,44,45,36,37,38,39,30,31,28,29,34,35,32,33,24,25,26,27,2,3,0,1,8,9,10,11,4,5,6,7],[23,22,21,20,19,18,17,16,15,14,13,12,43,42,41,40,47,46,45,44,37,36,39,38,31,30,29,28,35,34,33,32,25,24,27,26,3,2,1,0,9,8,11,10,5,4,7,6],[24,25,26,27,28,29,30,31,32,33,34,35,4,5,6,7,0,1,2,3,10,11,8,9,16,17,18,19,12,13,14,15,22,23,20,21,44,45,46,47,38,39,36,37,42,43,40,41],[25,24,27,26,29,28,31,30,33,32,35,34,5,4,7,6,1,0,3,2,11,10,9,8,17,16,19,18,13,12,15,14,23,22,21,20,45,44,47,46,39,38,37,36,43,42,41,40],[26,27,24,25,32,33,34,35,28,29,30,31,6,7,4,5,10,11,8,9,0,1,2,3,44,45,46,47,38,39,36,37,42,43,40,41,16,17,18,19,12,13,14,15,22,23,20,21],[27,26,25,24,33,32,35,34,29,28,31,30,7,6,5,4,11,10,9,8,1,0,3,2,45,44,47,46,39,38,37,36,43,42,41,40,17,16,19,18,13,12,15,14,23,22,21,20],[28,29,30,31,24,25,26,27,34,35,32,33,16,17,18,19,12,13,14,15,22,23,20,21,4,5,6,7,0,1,2,3,10,11,8,9,46,47,44,45,42,43,40,41,38,39,36,37],[29,28,31,30,25,24,27,26,35,34,33,32,17,16,19,18,13,12,15,14,23,22,21,20,5,4,7,6,1,0,3,2,11,10,9,8,47,46,45,44,43,42,41,40,39,38,37,36],[30,31,28,29,34,35,32,33,24,25,26,27,18,19,16,17,22,23,20,21,12,13,14,15,46,47,44,45,42,43,40,41,38,39,36,37,4,5,6,7,0,1,2,3,10,11,8,9],[31,30,29,28,35,34,33,32,25,24,27,26,19,18,17,16,23,22,21,20,13,12,15,14,47,46,45,44,43,42,41,40,39,38,37,36,5,4,7,6,1,0,3,2,11,10,9,8],[32,33,34,35,26,27,24,25,30,31,28,29,44,45,46,47,38,39,36,37,42,43,40,41,6,7,4,5,10,11,8,9,0,1,2,3,18,19,16,17,22,23,20,21,12,13,14,15],[33,32,35,34,27,26,25,24,31,30,29,28,45,44,47,46,39,38,37,36,43,42,41,40,7,6,5,4,11,10,9,8,1,0,3,2,19,18,17,16,23,22,21,20,13,12,15,14],[34,35,32,33,30,31,28,29,26,27,24,25,46,47,44,45,42,43,40,41,38,39,36,37,18,19,16,17,22,23,20,21,12,13,14,15,6,7,4,5,10,11,8,9,0,1,2,3],[35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,19,18,17,16,23,22,21,20,13,12,15,14,7,6,5,4,11,10,9,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no uniqueness claim","round4_baseline_status":"unitarily_realized","round4_audit":"retained after fresh proof, implementation, and tensor audit"}},{"title":"Categorical construction and exact 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unitary realization exists: C(S4xC2,H4). The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_30.txt","investigation_deferred":false},{"id":"R12_1_31","rank":12,"multiplicity":1,"status":"excluded","summary":"Excluded from fusion categorification over ℂ by Arithmetic / near-group. The argument applies to general complex fusion categories.","name":"R12_1_31","existence":"excluded","model_reference":"Excluded from fusion categorification over ℂ by Arithmetic / near-group. 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Indices are zero-based; X_0 is the unit. The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,1,2,3,4,5,6,7,8,9,10,11],"invertible_indices":[0,1,2,5,6,7,8,11],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0000000000000007,1.000000000000002,3.561552812808836,3.561552812808836,1.000000000000002,1.000000000000002,1.000000000000002,1.000000000000002,3.561552812808836,3.561552812808836,1.000000000000002],"global_fp_dimension_approx":58.738633753706154,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t - 24)**3*(t - 12)**3*(t - 8)**4*(t**2 - 68*t + 544)","formal_codegree_polynomial":"(t - 24)**3*(t - 12)**3*(t - 8)**4*(t**2 - 68*t + 544)"},"research":[{"title":"Current accepted case decision","source":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102","data":{"id":"R12_1_31","input_line":31,"rank":12,"multiplicity":1,"tensor_sha256":"5c54837e13bf70f6f19f88f67201a74860513fd477953c150dc75f4073014ad4","status":"excluded","commutative":true,"simple":false,"exclusion_route":"Arithmetic / near-group","exclusion_evidence":[{"source":"arithmetic.json","certificate":{"kind":"casimir_dnumber","factor":[1,-68,544],"multiplicity":1,"index":1,"nonzero_remainder":272,"scope":"general_complex_fusion_category","reference":"Ostrik, On formal codegrees of fusion categories, Theorem 1.2 and Lemma 2.7; d-number property invariant under multiplication by rational integers"}},{"source":"arithmetic.json","certificate":{"kind":"generalized_near_group_divisibility","invertibles":[0,1,2,5,6,7,8,11],"noninvertibles":[3,4,9,10],"object":3,"stabilizer":[0,11],"r":3,"s":2,"discriminant":17,"scope":"general_complex_fusion_category","reference":"Schopieray, Categorification of integral group rings extended by one dimension, Proposition 3.3 (2023)"}},{"source":"cyclic.json","certificate":{"kind":"cyclic_isotypic_integrality","g":11,"order":2,"X":3,"Y":3,"Z":9,"multiplicity":1,"dimension_polynomial":["1","-3","-2"],"normalized_dimension_polynomial":["1","-3/2","-1/2"],"scope":"general_complex_fusion_category"}}],"model_name":null,"model_constructor":null,"model_certificate_count":0,"first_round_status":"excluded","first_round_audit":"retained; no erroneous classification found","round4_baseline_status":"excluded","round4_audit":"retained after fresh proof, implementation, and tensor audit"}},{"title":"General complex exclusion certificate","source":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/tree/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","data":[{"source":"arithmetic.json","certificate":{"kind":"casimir_dnumber","factor":[1,-68,544],"multiplicity":1,"index":1,"nonzero_remainder":272,"scope":"general_complex_fusion_category","reference":"Ostrik, On formal codegrees of fusion categories, Theorem 1.2 and Lemma 2.7; d-number property invariant under multiplication by rational integers"}},{"source":"arithmetic.json","certificate":{"kind":"generalized_near_group_divisibility","invertibles":[0,1,2,5,6,7,8,11],"noninvertibles":[3,4,9,10],"object":3,"stabilizer":[0,11],"r":3,"s":2,"discriminant":17,"scope":"general_complex_fusion_category","reference":"Schopieray, Categorification of integral group rings extended by one dimension, Proposition 3.3 (2023)"}},{"source":"cyclic.json","certificate":{"kind":"cyclic_isotypic_integrality","g":11,"order":2,"X":3,"Y":3,"Z":9,"multiplicity":1,"dimension_polynomial":["1","-3","-2"],"normalized_dimension_polynomial":["1","-3/2","-1/2"],"scope":"general_complex_fusion_category"}}]}],"evidence":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"},{"label":"arithmetic.json","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/arithmetic.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/arithmetic.json","sha256":"7cd51b860f4ef52d01141f1870289583ee1664c9c5daf7f6540db82d6195cc76"},{"label":"cyclic.json","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/cyclic.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/cyclic.json","sha256":"13367358ca941e02f7cee3f88e7678e86a2b222b899bde8b4eff18a1b7a00841"}],"categorification_scope":{"general_status":"excluded","level_status":{"general":"excluded","pivotal":"excluded","spherical":"excluded","pseudo_unitary":"excluded","unitary":"excluded"},"strongest_excluded":"general","same_category_implications":[],"existence_implications":[],"notes":["Excluded from fusion categorification over ℂ by Arithmetic / near-group. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_41.txt","investigation_deferred":false},{"id":"R12_1_42","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Vec(C2) x PSU(2)_10 (the other 3311 even-part ring). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Vec(C2) x PSU(2)_10 (the other 3311 even-part ring)","existence":"unitary","model_reference":"A unitary realization exists: Vec(C2) x PSU(2)_10 (the other 3311 even-part ring). 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guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: Vec(C2) x PSU(2)_10 (the other 3311 even-part ring). The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_42.txt","investigation_deferred":false},{"id":"R12_1_43","rank":12,"multiplicity":1,"status":"excluded","summary":"Excluded from fusion categorification over ℂ by Arithmetic / near-group. The argument applies to general complex fusion categories.","name":"R12_1_43","existence":"excluded","model_reference":"Excluded from fusion categorification over ℂ by Arithmetic / near-group. 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Indices are zero-based; X_0 is the unit. The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,1,2,3,4,5,6,7,8,9,10,11],"invertible_indices":[0,1,2,11],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0000000000000027,1.0000000000000033,3.1700864866260425,3.1700864866260425,2.709275359436931,2.709275359436931,2.709275359436931,2.709275359436931,3.1700864866260425,3.1700864866260425,1.0000000000000033],"global_fp_dimension_approx":73.55848522376465,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t - 16)**4*(t - 8)**2*(t**3 - 104*t**2 + 2368*t - 9472)**2","formal_codegree_polynomial":"(t - 16)**4*(t - 8)**2*(t**3 - 104*t**2 + 2368*t - 9472)**2"},"research":[{"title":"Current accepted case decision","source":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102","data":{"id":"R12_1_43","input_line":43,"rank":12,"multiplicity":1,"tensor_sha256":"fbf45978d24d4304ef1fce7d11cd8bd8b618905dec5be30c8cbc5df24336105a","status":"excluded","commutative":true,"simple":false,"exclusion_route":"Arithmetic / near-group","exclusion_evidence":[{"source":"arithmetic.json","certificate":{"kind":"casimir_dnumber","factor":[1,-104,2368,-9472],"multiplicity":2,"index":1,"nonzero_remainder":2304,"scope":"general_complex_fusion_category","reference":"Ostrik, On formal codegrees of fusion categories, Theorem 1.2 and Lemma 2.7; 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d-number property invariant under multiplication by rational integers"}},{"source":"subrings.json","certificate":{"kind":"excluded_subring","subset":[0,7,8,9,10,11],"class_index":32,"source_to_class_basis_map":[0,1,2,3,4,5],"scope":"general_complex_fusion_category"}}]}],"evidence":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"},{"label":"arithmetic.json","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/arithmetic.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/arithmetic.json","sha256":"7cd51b860f4ef52d01141f1870289583ee1664c9c5daf7f6540db82d6195cc76"},{"label":"subrings.json","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/subrings.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/subrings.json","sha256":"81698328373db09aa6b305a65bac0017ed1fb799da6ff778e1cfef6a3f4ea432"}],"categorification_scope":{"general_status":"excluded","level_status":{"general":"excluded","pivotal":"excluded","spherical":"excluded","pseudo_unitary":"excluded","unitary":"excluded"},"strongest_excluded":"general","same_category_implications":[],"existence_implications":[],"notes":["Excluded from fusion categorification over ℂ by Arithmetic / near-group. 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guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: Fib x Vec_C_2boxtimes PSU(2)_5. The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_50.txt","investigation_deferred":false},{"id":"R12_1_51","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Fib x PSU(2)_10 (the other 3311 even-part ring). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Fib x PSU(2)_10 (the other 3311 even-part ring)","existence":"unitary","model_reference":"A unitary realization exists: Fib x PSU(2)_10 (the other 3311 even-part ring). 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classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: Fib x PSU(2)_10 (the other 3311 even-part ring). The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_51.txt","investigation_deferred":false},{"id":"R12_1_52","rank":12,"multiplicity":1,"status":"excluded","summary":"Excluded from fusion categorification over ℂ by Arithmetic / near-group. The argument applies to general complex fusion categories.","name":"R12_1_52","existence":"excluded","model_reference":"Excluded from fusion categorification over ℂ by Arithmetic / near-group. 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_52.txt","investigation_deferred":false},{"id":"R12_1_53","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: PSU(2)_5 x PSU(2)_6. 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guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: SU(2)_11. The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"realized","explanation":"The recorded classical quantum-group, representation, or abelian pointed construction provides a braided realization. This is an existence statement, not a statement about every associator.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_54.txt","investigation_deferred":false},{"id":"R12_1_55","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: PSU(2)_22. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"PSU(2)_22","existence":"unitary","model_reference":"A unitary realization exists: PSU(2)_22. 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The argument applies to general complex fusion categories.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_64","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_64","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,1,1,1,1,1,1,1,1],[0,0,0,1,1,1,1,1,1,1,1,1],[0,0,1,1,1,0,0,1,1,1,1,1],[0,0,1,1,0,1,0,1,1,1,1,1],[0,0,1,1,0,0,1,1,1,1,1,1],[0,0,1,1,1,1,1,1,0,0,1,1],[0,0,1,1,1,1,1,0,1,0,1,1],[0,0,1,1,1,1,1,0,0,1,1,1],[0,0,1,1,1,1,1,1,1,1,1,0],[0,1,1,1,1,1,1,1,1,1,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,1,1,1,1,1,1,1,1,1],[1,0,1,1,1,1,1,1,1,1,1,1],[0,0,1,1,0,1,1,1,1,1,1,1],[0,0,1,1,1,0,1,1,1,1,1,1],[0,0,1,1,1,1,0,1,1,1,1,1],[0,0,1,1,1,1,1,0,1,1,1,1],[0,0,1,1,1,1,1,1,0,1,1,1],[0,0,1,1,1,1,1,1,1,0,1,1],[0,1,1,1,1,1,1,1,1,1,1,1],[0,0,1,1,1,1,1,1,1,1,1,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,1,1,1,0,0,1,1,1,1,1],[0,0,1,1,0,1,1,1,1,1,1,1],[1,0,1,0,1,0,1,1,1,1,1,1],[0,0,0,1,0,1,1,1,1,1,1,1],[0,0,0,1,1,1,0,1,1,1,1,1],[0,0,1,1,1,1,1,0,1,1,1,0],[0,0,1,1,1,1,1,1,1,0,1,0],[0,1,1,1,1,1,1,1,0,1,0,1],[0,0,1,1,1,1,1,1,1,0,1,1],[0,0,1,1,1,1,1,0,0,1,1,1]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,1,1,0,1,0,1,1,1,1,1],[0,0,1,1,1,0,1,1,1,1,1,1],[0,0,0,1,0,1,1,1,1,1,1,1],[1,0,1,0,1,1,0,1,1,1,1,1],[0,0,0,1,1,0,1,1,1,1,1,1],[0,0,1,1,1,1,1,1,0,1,1,0],[0,1,1,1,1,1,1,0,1,1,0,1],[0,0,1,1,1,1,1,1,1,0,1,0],[0,0,1,1,1,1,1,1,0,1,1,1],[0,0,1,1,1,1,1,0,1,0,1,1]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,1,1,0,0,1,1,1,1,1,1],[0,0,1,1,1,1,0,1,1,1,1,1],[0,0,0,1,1,1,0,1,1,1,1,1],[0,0,0,1,1,0,1,1,1,1,1,1],[1,0,1,0,0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1,1,0,0,1],[0,0,1,1,1,1,1,1,0,1,1,0],[0,0,1,1,1,1,1,0,1,1,1,0],[0,0,1,1,1,1,1,0,1,1,1,1],[0,0,1,1,1,1,1,1,0,0,1,1]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,1,1,1,1,1,1,0,0,1,1],[0,0,1,1,1,1,1,0,1,1,1,1],[0,0,1,1,1,1,1,0,1,1,1,0],[0,0,1,1,1,1,1,1,0,1,1,0],[0,1,1,1,1,1,1,1,1,0,0,1],[1,0,1,0,0,1,1,1,1,1,1,1],[0,0,0,1,1,0,1,1,1,1,1,1],[0,0,0,1,1,1,0,1,1,1,1,1],[0,0,1,1,1,1,0,1,1,1,1,1],[0,0,1,1,0,0,1,1,1,1,1,1]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,1,1,1,1,1,0,1,0,1,1],[0,0,1,1,1,1,1,1,0,1,1,1],[0,0,1,1,1,1,1,1,1,0,1,0],[0,1,1,1,1,1,1,0,1,1,0,1],[0,0,1,1,1,1,1,1,0,1,1,0],[0,0,0,1,1,0,1,1,1,1,1,1],[1,0,1,0,1,1,0,1,1,1,1,1],[0,0,0,1,0,1,1,1,1,1,1,1],[0,0,1,1,1,0,1,1,1,1,1,1],[0,0,1,1,0,1,0,1,1,1,1,1]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,1,1,1,1,1,0,0,1,1,1],[0,0,1,1,1,1,1,1,1,0,1,1],[0,1,1,1,1,1,1,1,0,1,0,1],[0,0,1,1,1,1,1,1,1,0,1,0],[0,0,1,1,1,1,1,0,1,1,1,0],[0,0,0,1,1,1,0,1,1,1,1,1],[0,0,0,1,0,1,1,1,1,1,1,1],[1,0,1,0,1,0,1,1,1,1,1,1],[0,0,1,1,0,1,1,1,1,1,1,1],[0,0,1,1,1,0,0,1,1,1,1,1]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,1,1,1,1,1,1,1,1,0],[0,1,1,1,1,1,1,1,1,1,1,1],[0,0,1,1,1,1,1,1,1,0,1,1],[0,0,1,1,1,1,1,1,0,1,1,1],[0,0,1,1,1,1,1,0,1,1,1,1],[0,0,1,1,1,1,0,1,1,1,1,1],[0,0,1,1,1,0,1,1,1,1,1,1],[0,0,1,1,0,1,1,1,1,1,1,1],[1,0,1,1,1,1,1,1,1,1,1,1],[0,0,0,1,1,1,1,1,1,1,1,1]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,1,1,1,1,1,1,1,1,0,0],[0,0,1,1,1,1,1,1,1,1,1,0],[0,0,1,1,1,1,1,0,0,1,1,1],[0,0,1,1,1,1,1,0,1,0,1,1],[0,0,1,1,1,1,1,1,0,0,1,1],[0,0,1,1,0,0,1,1,1,1,1,1],[0,0,1,1,0,1,0,1,1,1,1,1],[0,0,1,1,1,0,0,1,1,1,1,1],[0,0,0,1,1,1,1,1,1,1,1,1],[1,0,0,0,1,1,1,1,1,1,1,1]]],"tensor_sha256":"18151ac759e385575928f429812449787dd39c154e372c27dbc6e85a9b27c1b1","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_67.txt","investigation_deferred":false},{"id":"R12_1_68","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Rep(S3) x Rep(D10). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Rep(S3) x Rep(D10)","existence":"unitary","model_reference":"A unitary realization exists: Rep(S3) x Rep(D10). 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The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_71","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_71","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,1,1,0,0],[0,0,0,1,0,0,0,0,1,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,1,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,1,0,0,0,0],[0,0,1,0,1,0,0,0,0,0,0,0],[0,1,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,1],[0,0,0,0,0,0,0,0,0,1,1,0],[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],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,1,1,0,0],[0,0,0,0,0,1,0,1,0,0,0,0],[1,0,0,0,0,0,1,0,1,1,1,1],[0,1,0,1,0,0,0,0,1,1,1,1],[0,0,1,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,1,1,0,0],[0,0,1,0,1,1,0,1,0,0,0,0],[0,0,1,0,1,1,0,1,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,1,1,0,0],[0,0,1,0,1,0,0,0,0,0,0,0],[0,1,0,1,0,0,0,0,1,1,1,1],[1,0,0,0,0,0,1,0,1,1,1,1],[0,0,0,0,0,1,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,1,1,0,0],[0,0,1,0,1,1,0,1,0,0,0,0],[0,0,1,0,1,1,0,1,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,1,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,1,0,0,0,0],[1,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,1],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,1,1,0,0],[0,0,0,0,0,0,1,0,1,1,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,1],[0,0,1,0,1,1,0,1,0,0,0,0],[0,0,1,0,1,1,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,1],[0,0,0,0,1,1,0,0,0,0,0,0],[1,1,0,1,0,0,1,0,0,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,1],[0,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,1,0,0,1,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,1,0,1,1,0,1,0,0,0,0],[0,0,1,0,1,1,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,1,1,1],[1,1,0,1,0,0,1,0,1,0,0,1],[0,0,0,1,0,0,1,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,1,0,0,1,0,1,0,0,0],[1,1,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,1]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,1,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,1],[1,1,0,0,0,0,0,0,0,0,1,0]]],"tensor_sha256":"b8ffd714c2db4a2c2d9118d55c133fd7c99c3e458211d66e09ee6839521a071d","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_72.txt","investigation_deferred":false},{"id":"R12_1_73","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Fib x mathrmFibboxtimesRep(S_3). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Fib x mathrmFibboxtimesRep(S_3)","existence":"unitary","model_reference":"A unitary realization exists: Fib x mathrmFibboxtimesRep(S_3). 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_100.txt","investigation_deferred":false},{"id":"R12_1_101","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Fib x Rep((C_3× C_3)⋊ C_2), inversion action. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_101","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_101","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[1,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,1,0,0,0,0],[0,0,0,0,0,1,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0,0,0,1]],[[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,0,0],[1,0,1,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,1,0,1,0,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,1,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[1,0,1,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,1,0,0,0,0,0],[0,0,0,1,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,1,0],[0,0,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,1,0,0,0,0,0],[0,0,0,1,0,0,1,0,0,0,0,0],[1,0,1,0,0,1,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,1,0],[0,1,0,0,0,0,0,0,1,0,0,1],[0,0,0,0,0,0,0,1,0,0,1,0],[0,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,1,0,1,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[1,0,1,0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,1,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,0,0,0,0,0,1,0,1,0],[0,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,0,0,0,0,1,0,1,0],[0,0,0,0,0,0,0,0,0,1,1,0],[0,1,0,0,0,0,0,1,0,0,0,1],[1,1,1,0,0,0,1,1,0,0,0,1],[0,0,0,1,1,0,0,0,0,1,1,0],[0,0,0,1,0,1,0,0,1,0,1,0],[0,0,0,0,1,1,0,0,1,1,0,0],[0,0,0,0,0,0,1,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,1],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,1,1,0,0,0,0,1,1,0],[1,1,1,0,0,1,0,0,1,0,0,1],[0,0,0,1,0,0,1,1,0,0,1,0],[0,0,0,0,1,0,1,1,0,1,0,0],[0,0,0,0,0,1,0,0,1,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,1,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,1,0,0,1,0],[0,0,0,0,0,0,0,0,1,0,1,0],[0,0,0,1,0,1,0,0,1,0,1,0],[0,0,0,1,0,0,1,1,0,0,1,0],[1,1,1,0,1,0,0,0,0,1,0,1],[0,0,0,0,0,1,1,1,1,0,0,0],[0,0,0,0,1,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,1,1,0,0,1,1,0,0],[0,0,0,0,1,0,1,1,0,1,0,0],[0,0,0,0,0,1,1,1,1,0,0,0],[1,1,1,1,0,0,0,0,0,0,1,1],[0,0,0,1,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,1,0,0,0,0],[0,0,0,0,0,1,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,1,0],[1,1,0,0,0,0,0,0,0,0,0,0]]],"tensor_sha256":"27e7d9bd4e9252612c8a3f9cd852434fb2c0c780df0595c023a778e03cea8a1d","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_103.txt","investigation_deferred":false},{"id":"R12_1_104","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Fib x PSU(2)_11. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Fib x PSU(2)_11","existence":"unitary","model_reference":"A unitary realization exists: Fib x PSU(2)_11. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"realized","explanation":"The recorded classical quantum-group, representation, or abelian pointed construction provides a braided realization. This is an existence statement, not a statement about every associator.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_108.txt","investigation_deferred":false},{"id":"R12_1_109","rank":12,"multiplicity":1,"status":"excluded","summary":"Excluded from fusion categorification over ℂ by Arithmetic / near-group. The argument applies to general complex fusion categories.","name":"R12_1_109","existence":"excluded","model_reference":"Excluded from fusion categorification over ℂ by Arithmetic / near-group. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"realized","explanation":"The recorded classical quantum-group, representation, or abelian pointed construction provides a braided realization. This is an existence statement, not a statement about every associator.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_116.txt","investigation_deferred":false},{"id":"R12_1_117","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: PSU(2)_5 x PSU(2)_7. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"PSU(2)_5 x PSU(2)_7","existence":"unitary","model_reference":"A unitary realization exists: PSU(2)_5 x PSU(2)_7. 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The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,1,2,3,4,5,6,7,8,9,10,11],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.879385241571809,1.8019377358048325,4.562666514524199,2.2469796037174574,5.6895520821763474,4.222940305339482,3.3865351869029348,2.8793852415718075,6.469919909056942,5.188472922707768,2.5320888862379487],"global_fp_dimension_approx":178.801208239102,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t**3 - 42*t**2 + 441*t - 1323)*(t**9 - 378*t**8 + 51597*t**7 - 3524472*t**6 + 135025380*t**5 - 3024568512*t**4 + 39697461720*t**3 - 294753653271*t**2 + 1125423039762*t - 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The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,1,2,3,4,5,6,7,8,9,10,11],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,9.10977222864644,9.10977222864644,10.109772228646444,9.10977222864644,9.10977222864644,9.10977222864644,9.10977222864644,9.10977222864644,9.109772228646444,9.10977222864644,9.109772228646444],"global_fp_dimension_approx":933.0869950932903,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t - 21)**10*(t**2 - 935*t + 1785)","formal_codegree_polynomial":"(t - 21)**10*(t**2 - 935*t + 1785)"},"research":[{"title":"Current accepted case decision","source":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102","data":{"id":"R12_1_118","input_line":118,"rank":12,"multiplicity":1,"tensor_sha256":"e8ae7f1700ce10d17f38a4fb5d9f8d08b5e0f5687b6d836a389c51544fd34f1e","status":"excluded","commutative":true,"simple":true,"exclusion_route":"Arithmetic / near-group","exclusion_evidence":[{"source":"arithmetic.json","certificate":{"kind":"casimir_dnumber","factor":[1,-935,1785],"multiplicity":1,"index":1,"nonzero_remainder":1360,"scope":"general_complex_fusion_category","reference":"Ostrik, On formal codegrees of fusion categories, Theorem 1.2 and Lemma 2.7; d-number property invariant under multiplication by rational integers"}}],"model_name":null,"model_constructor":null,"model_certificate_count":0,"first_round_status":"excluded","first_round_audit":"retained; no erroneous classification found","round4_baseline_status":"excluded","round4_audit":"retained after fresh proof, implementation, and tensor audit"}},{"title":"General complex exclusion certificate","source":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/tree/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","data":[{"source":"arithmetic.json","certificate":{"kind":"casimir_dnumber","factor":[1,-935,1785],"multiplicity":1,"index":1,"nonzero_remainder":1360,"scope":"general_complex_fusion_category","reference":"Ostrik, On formal codegrees of fusion categories, Theorem 1.2 and Lemma 2.7; d-number property invariant under multiplication by rational integers"}}]}],"evidence":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"},{"label":"arithmetic.json","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/arithmetic.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/arithmetic.json","sha256":"7cd51b860f4ef52d01141f1870289583ee1664c9c5daf7f6540db82d6195cc76"}],"categorification_scope":{"general_status":"excluded","level_status":{"general":"excluded","pivotal":"excluded","spherical":"excluded","pseudo_unitary":"excluded","unitary":"excluded"},"strongest_excluded":"general","same_category_implications":[],"existence_implications":[],"notes":["Excluded from fusion categorification over ℂ by Arithmetic / near-group. The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_118.txt","investigation_deferred":false},{"id":"R12_1_119","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-zest(Vec(C2) x Rep((C_3× C_3)⋊ C_2), inversion action; h=1). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"C2-zest(Vec(C2) x Rep((C_3× C_3)⋊ C_2), inversion action; h=1)","existence":"unitary","model_reference":"A unitary realization exists: C2-zest(Vec(C2) x Rep((C_3× C_3)⋊ C_2), inversion action; h=1). 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guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: C2-zest(Vec(C2) x 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_119.txt","investigation_deferred":false},{"id":"R12_1_120","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-zest(Rep(D36); h=1). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"C2-zest(Rep(D36); h=1)","existence":"unitary","model_reference":"A unitary realization exists: C2-zest(Rep(D36); h=1). 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crossed C2[parity=0];h=1). 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"Noncommutative fusion rules exclude braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_155.txt","investigation_deferred":false},{"id":"R12_1_156","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2 Morita dual of Liptrap feudal category 'group': 'F20', 'sigma': [0, 3, 2, 1], 'p': 0. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"C2 Morita dual of Liptrap feudal category 'group': 'F20', 'sigma': [0, 3, 2, 1], 'p': 0","existence":"unitary","model_reference":"A unitary realization exists: C2 Morita dual of Liptrap feudal category 'group': 'F20', 'sigma': [0, 3, 2, 1], 'p': 0. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_162","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_162","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,1],[1,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,0,1],[0,0,0,0,0,0,0,1,0,0,1,0],[0,1,0,1,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,1,0,0,0,1,0,0,0,0,0],[0,0,1,0,0,1,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,1,1,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0],[0,1,0,0,0,1,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,1,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,1,0,1,0],[0,0,0,0,0,0,0,1,0,1,0,0],[0,0,1,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0],[0,0,0,1,1,1,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,0,0,0,0,1,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,1],[0,0,1,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,1,0,0,0,1,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0],[0,0,0,1,1,0,1,0,0,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,1,1,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0],[0,1,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,0,0,1,1,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,1,0,0,0,0,0],[0,0,1,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,0,0,0,1,0,0,0,1],[0,0,0,0,0,0,0,1,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,1,1,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,1,0],[0,0,0,0,0,0,0,1,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,1,0,0,0,0,0,0],[0,1,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0],[0,0,0,1,1,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,0,0,1,0],[0,0,0,0,0,0,0,1,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,1]]],"tensor_sha256":"cc86a4386e3d6fef15feb2d598536f46231fecf88348410b0112350502899c11","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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S3,omega,diagonal C2). 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R8_1_19. The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"Noncommutative fusion rules exclude braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_163.txt","investigation_deferred":false},{"id":"R12_1_164","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: TY(C5)^C2 crossed C2[parity=0]. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"TY(C5)^C2 crossed C2[parity=0]","existence":"unitary","model_reference":"A unitary realization exists: TY(C5)^C2 crossed C2[parity=0]. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_183","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_183","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,1,1,1,1,1],[1,0,0,1,1,1,0,0,0,0,1,1],[0,1,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,1,1,1,0,0,0,0,1,1],[0,0,0,0,0,0,1,1,1,1,1,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,1,1,0,0,0,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,1,0,0,1,1,0,0],[0,0,0,1,1,0,1,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,1,1,0,0,0,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,1,1,0,1,1,0,0,0,0],[1,0,0,0,0,1,0,0,1,1,0,0]]],"tensor_sha256":"fcae559b4748d88e8e99cbfe2e644b477307b2e68d05665ff377052018da93a7","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_186.txt","investigation_deferred":false},{"id":"R12_1_187","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-zest(Half SO(28)_2; h=1). 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_187.txt","investigation_deferred":false},{"id":"R12_1_188","rank":12,"multiplicity":1,"status":"excluded","summary":"Excluded from fusion categorification over ℂ by Arithmetic / near-group. The argument applies to general complex fusion categories.","name":"R12_1_188","existence":"excluded","model_reference":"Excluded from fusion categorification over ℂ by Arithmetic / near-group. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_234","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_234","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,1,0,0,1,0],[0,0,0,0,0,0,0,0,1,0,1,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,1,0,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,1,1,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,1,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,1,0,0,0,0,0],[0,1,1,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,0,0,0,1,0,0,1],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,1,1,0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,0,0,0,0,1,0,1,0],[0,0,0,0,0,0,0,0,0,1,1,0],[1,0,0,0,0,0,0,1,0,0,0,1],[0,1,1,0,0,0,1,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,0,0,1,0,1,0,0,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,1,0,1,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,1,1,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,0,0,0,0,0,1,0,1,0],[0,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,1,0,0,0,0,0],[0,0,0,1,0,0,1,0,0,0,0,0],[0,1,1,0,0,1,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,1,0],[1,0,0,0,0,0,0,0,1,0,0,1],[0,0,0,0,0,0,0,1,0,0,1,0],[0,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,1,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,1,0,1,0,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,1,0],[0,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,1,1,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,1,0,0,0,0,0],[0,0,0,1,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0]]],"tensor_sha256":"6cd33c540e1fdfe62ea5154a924dc77b0b948883d432f279e625caf03ce2bff3","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_239.txt","investigation_deferred":false},{"id":"R12_1_240","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-algebra Morita dual of the matched-grading subcategory of Ising x Vec(D16,omega). 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The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_240","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_240","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,1,1,0,1,1,1],[0,0,0,0,0,1,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,1,0,0,0],[0,1,1,1,1,0,1,1,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,1,0,0,0],[1,0,0,1,1,0,0,0,0,0,0,1],[0,1,1,0,0,0,0,0,0,1,1,0],[0,0,0,0,0,1,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,1,0,0,0],[0,1,1,0,0,0,0,0,0,1,1,0],[1,0,0,1,1,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,1,1,1,1,0,1,1,0,0,0,0],[0,0,0,0,0,1,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,1,0,0,0],[1,0,0,0,0,0,1,1,0,1,1,1],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,0]]],"tensor_sha256":"1acaaa934006f8f389c9696d458105d5726560d41f3da24c129779e3be638078","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,2,1,4,3,5,6,7,8,9,10,11],"invertible_indices":[0,1,2,3,4,9,10,11],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0000000000000016,1.0000000000000013,1.0000000000000016,1.0000000000000016,2.828427124746196,2.000000000000005,2.000000000000005,2.828427124746196,1.0000000000000016,1.0000000000000016,1.0000000000000016],"global_fp_dimension_approx":32.00000000000013,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t - 32)**2*(t - 16)**5*(t - 8)**5","formal_codegree_polynomial":"(t - 32)**2*(t - 16)**5*(t - 8)**5"},"research":[{"title":"Current accepted case 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weak integrality","braided":"not settled here","unitarizable":"necessary by two-prime solvability and unitarizability"},"categorification":{"existence":"verified unitary model","spherical":"canonical spherical structure on the model","pseudo_unitary":"the supplied model is pseudo-unitary","braided":"not asserted for the supplied construction"},"round4_baseline_status":"unitarily_realized","round4_audit":"retained after fresh proof, implementation, and tensor audit"}},{"title":"Categorical construction and exact certificate","source":"evidence/rank12_m1_20261026/baseline/prior_round","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/tree/main/evidence/rank12_m1_20261026/baseline/prior_round","data":{"model_name":"C2-algebra Morita dual of the matched-grading subcategory of Ising x Vec(D16,omega)","constructor":"twisted_Ising_pointed_Morita_dual","parameters":{"group":"D16","group_order":16,"grading":"rotation exponent modulo 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Indices are zero-based; X_0 is the unit. The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,2,1,4,3,5,6,7,8,9,10,11],"invertible_indices":[0,1,2,3,4,9,10,11],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,0.999999999999996,0.9999999999999982,0.9999999999999981,0.9999999999999981,3.6038754716096726,4.493959207434925,4.493959207434925,3.6038754716096726,0.9999999999999981,0.9999999999999981,0.9999999999999981],"global_fp_dimension_approx":74.36717554591792,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t - 16)**4*(t - 8)**5*(t**3 - 112*t**2 + 3136*t - 25088)","formal_codegree_polynomial":"(t - 16)**4*(t - 8)**5*(t**3 - 112*t**2 + 3136*t - 25088)"},"research":[{"title":"Current accepted case decision","source":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102","data":{"id":"R12_1_241","input_line":241,"rank":12,"multiplicity":1,"tensor_sha256":"d6d040fdf7d676a55715ee36d8ee17a23af5c72a483805d615b917bd7efd28fc","status":"excluded","commutative":true,"simple":false,"exclusion_route":"Cyclic isotypic summands","exclusion_evidence":[{"source":"cyclic.json","certificate":{"kind":"cyclic_isotypic_integrality","g":3,"order":4,"X":6,"Y":6,"Z":6,"multiplicity":1,"dimension_polynomial":["1","-4","-4","8"],"normalized_dimension_polynomial":["1","-1","-1/4","1/8"],"scope":"general_complex_fusion_category"}}],"model_name":null,"model_constructor":null,"model_certificate_count":0,"first_round_status":"excluded","first_round_audit":"retained; no erroneous classification found","round4_baseline_status":"excluded","round4_audit":"retained after fresh proof, implementation, and tensor audit"}},{"title":"General complex exclusion certificate","source":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/tree/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","data":[{"source":"cyclic.json","certificate":{"kind":"cyclic_isotypic_integrality","g":3,"order":4,"X":6,"Y":6,"Z":6,"multiplicity":1,"dimension_polynomial":["1","-4","-4","8"],"normalized_dimension_polynomial":["1","-1","-1/4","1/8"],"scope":"general_complex_fusion_category"}}]}],"evidence":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"},{"label":"cyclic.json","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/cyclic.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/cyclic.json","sha256":"13367358ca941e02f7cee3f88e7678e86a2b222b899bde8b4eff18a1b7a00841"}],"categorification_scope":{"general_status":"excluded","level_status":{"general":"excluded","pivotal":"excluded","spherical":"excluded","pseudo_unitary":"excluded","unitary":"excluded"},"strongest_excluded":"general","same_category_implications":[],"existence_implications":[],"notes":["Excluded from fusion categorification over ℂ by Cyclic isotypic summands. 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classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: Vec(C2) x mathcal I(C_4,0): cyclic, self-dual Ising zest. The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_246.txt","investigation_deferred":false},{"id":"R12_1_247","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Vec(C2) x Rep(C_3⋊ C_4). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Vec(C2) x Rep(C_3⋊ C_4)","existence":"unitary","model_reference":"A unitary realization exists: Vec(C2) x Rep(C_3⋊ C_4). 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classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: Fib x mathcal I(C_4,0): cyclic, self-dual Ising zest. The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_252.txt","investigation_deferred":false},{"id":"R12_1_253","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Fib x Rep(C_3⋊ C_4). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Fib x Rep(C_3⋊ C_4)","existence":"unitary","model_reference":"A unitary realization exists: Fib x Rep(C_3⋊ C_4). 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R8_1_3. The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"Noncommutative fusion rules exclude braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_305.txt","investigation_deferred":false},{"id":"R12_1_306","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Vec(C2) x mathcal I(C_2^2,1): non-self-dual Ising zest. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Vec(C2) x mathcal I(C_2^2,1): non-self-dual Ising zest","existence":"unitary","model_reference":"A unitary realization exists: Vec(C2) x mathcal I(C_2^2,1): non-self-dual Ising zest. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_306.txt","investigation_deferred":false},{"id":"R12_1_307","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Fib x mathcal I(C_2^2,1): non-self-dual Ising zest. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Fib x mathcal I(C_2^2,1): non-self-dual Ising zest","existence":"unitary","model_reference":"A unitary realization exists: Fib x mathcal I(C_2^2,1): non-self-dual Ising zest. 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guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: C2-zest(Ising x 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_308.txt","investigation_deferred":false},{"id":"R12_1_309","rank":12,"multiplicity":1,"status":"excluded","summary":"Excluded from fusion categorification over ℂ by D40 pointed Morita extension exhaustion. The argument applies to general complex fusion categories.","name":"R12_1_309","existence":"excluded","model_reference":"Excluded from fusion categorification over ℂ by D40 pointed Morita extension exhaustion. 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C4,<square of C4>). The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"Noncommutative fusion rules exclude braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_313.txt","investigation_deferred":false},{"id":"R12_1_314","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Vec(C2) x Zested SO(5)_2 category. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Vec(C2) x Zested SO(5)_2 category","existence":"unitary","model_reference":"A unitary realization exists: Vec(C2) x Zested SO(5)_2 category. 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unitary realization exists: S3-algebra dual of a (C3^2 semidirect C4)-extension of Vec(V4). 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The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_322","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_322","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,1,1,1,1,1],[1,0,0,0,0,1,0,0,1,1,1,1],[0,0,0,0,0,1,0,1,0,1,1,1],[0,0,0,0,0,1,0,1,0,1,1,1],[0,1,0,1,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,1,1,1,0,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,1,0,0,1,1,1,1],[0,0,0,0,0,0,1,1,1,1,1,1],[0,0,0,0,0,1,0,1,0,1,1,1],[0,0,0,0,0,1,0,1,0,1,1,1],[0,0,1,1,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,1,0,1,1,0,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,1,0,1,1,1],[0,0,0,0,0,1,0,1,0,1,1,1],[0,0,0,0,0,1,1,0,1,1,1,1],[1,0,0,0,0,0,0,1,1,1,1,1],[0,1,1,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,1,1,1,0,0,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,1,0,1,1,1],[0,0,0,0,0,1,0,1,0,1,1,1],[1,0,0,0,0,0,0,1,1,1,1,1],[0,0,0,0,0,1,1,0,1,1,1,1],[0,1,1,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,1,0,1,0,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,1,0,1,1,0,0,0,0,0,0,0],[0,0,1,1,1,0,0,0,0,0,0,0],[0,1,1,0,1,0,0,0,0,0,0,0],[0,1,1,1,0,0,0,0,0,0,0,0],[1,0,0,0,0,1,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,1,1,1,0,0,0],[0,0,0,0,0,1,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,1,1,1,0,0,0,0,0,0,0],[0,1,0,1,1,0,0,0,0,0,0,0],[0,1,1,1,0,0,0,0,0,0,0,0],[0,1,1,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,1,1,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,1,0,1,1,0,0,0],[0,0,0,0,0,1,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,1,0,0,0,0],[1,0,0,0,0,0,1,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,0,0,1,1,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,1,1],[1,0,0,0,0,1,1,1,0,0,1,1],[0,0,0,0,0,1,0,1,1,1,1,0],[0,0,0,0,0,1,0,1,1,1,0,1]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,1,0,1,1,1,1,0],[1,0,0,0,0,1,1,1,0,1,0,1],[0,0,0,0,0,1,0,1,1,0,1,1]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,0,0,1,0,1,1,1,0,1],[0,0,0,0,0,1,0,1,1,0,1,1],[1,0,0,0,0,1,1,1,0,1,1,0]]],"tensor_sha256":"55b234963b6fe30cdc07fd102523eab3420e12ee2b637ec294a7cb7e80e99d70","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"Noncommutative fusion rules exclude braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_325.txt","investigation_deferred":false},{"id":"R12_1_326","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Generalized Haagerup(C6). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Generalized Haagerup(C6)","existence":"unitary","model_reference":"A unitary realization exists: Generalized Haagerup(C6). 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bimodules over the subgroup algebra in Vec(G); equivariant-bundle convolution"}]}}],"evidence":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: Vec(A4)^C2. The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"Noncommutative fusion rules exclude braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_351.txt","investigation_deferred":false},{"id":"R12_1_352","rank":12,"multiplicity":1,"status":"excluded","summary":"Excluded from fusion categorification over ℂ by Arithmetic / near-group. 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_360.txt","investigation_deferred":false},{"id":"R12_1_361","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Group-theoretical category on A5 x C2 with graph D10 and relative binary cocycle. 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classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: C2-zest(Ising graded pullback R8_1_13;h=4). 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classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: C2-zest(Vec(C2) x mathcal I(C_4,0): cyclic, self-dual Ising zest; h=6). The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_389.txt","investigation_deferred":false},{"id":"R12_1_390","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-zest(Vec(C2) x Rep(C_3⋊ C_4); h=6). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"C2-zest(Vec(C2) x Rep(C_3⋊ C_4); h=6)","existence":"unitary","model_reference":"A unitary realization exists: C2-zest(Vec(C2) x Rep(C_3⋊ C_4); h=6). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_390","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_390","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,1,1],[1,0,0,0,0,0,0,1,0,1,0,0],[0,1,0,1,0,1,0,0,0,0,0,0],[0,0,1,0,1,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,1,1],[0,0,1,0,1,0,1,0,0,0,0,0],[0,1,0,1,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,1,0,1,0,1,0,0,0,0,0,0],[0,0,1,0,1,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,1,1],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,1,0,1,0,1,0,0,0,0,0],[0,1,0,1,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,1,1],[1,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,0]]],"tensor_sha256":"a2f047a65a6e76efbae5ebb5f176d8a8bd60966262347bb7acb959156c0e74ff","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: C2-zest(Vec(C2) x Rep(C_3⋊ C_4); h=6). The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_390.txt","investigation_deferred":false},{"id":"R12_1_391","rank":12,"multiplicity":1,"status":"excluded","summary":"Excluded from fusion categorification over ℂ by Cyclic isotypic summands. The argument applies to general complex fusion categories.","name":"R12_1_391","existence":"excluded","model_reference":"Excluded from fusion categorification over ℂ by Cyclic isotypic summands. 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no erroneous classification found","round4_baseline_status":"excluded","round4_audit":"retained after fresh proof, implementation, and tensor audit"}},{"title":"General complex exclusion certificate","source":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/tree/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","data":[{"source":"cyclic.json","certificate":{"kind":"cyclic_isotypic_integrality","g":9,"order":2,"X":5,"Y":5,"Z":8,"multiplicity":1,"dimension_polynomial":["1","-2","-2"],"normalized_dimension_polynomial":["1","-1","-1/2"],"scope":"general_complex_fusion_category"}},{"source":"modules.json","certificate":{"kind":"module_excluded_subring","subset":[0,7,8,9,10,11],"class_index":26,"source_to_class_basis_map":[0,3,4,5,1,2]}}]}],"evidence":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"},{"label":"cyclic.json","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/cyclic.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/cyclic.json","sha256":"13367358ca941e02f7cee3f88e7678e86a2b222b899bde8b4eff18a1b7a00841"},{"label":"modules.json","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/modules.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/modules.json","sha256":"0575311ec758ece2987d36cb664454825ea26402e66995ed323083627a2c0439"}],"categorification_scope":{"general_status":"excluded","level_status":{"general":"excluded","pivotal":"excluded","spherical":"excluded","pseudo_unitary":"excluded","unitary":"excluded"},"strongest_excluded":"general","same_category_implications":[],"existence_implications":[],"notes":["Excluded from fusion categorification over ℂ by Cyclic isotypic summands. 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_394.txt","investigation_deferred":false},{"id":"R12_1_395","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-zest(Vec(C2) x Vec_C_2boxtimes PSU(2)_5; h=1). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"C2-zest(Vec(C2) x Vec_C_2boxtimes PSU(2)_5; h=1)","existence":"unitary","model_reference":"A unitary realization exists: C2-zest(Vec(C2) x Vec_C_2boxtimes PSU(2)_5; h=1). 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classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: C2-zest(Vec(C2) x Vec_C_2boxtimes PSU(2)_5; h=1). The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_395.txt","investigation_deferred":false},{"id":"R12_1_396","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-zest(Vec(C2) x PSU(2)_10 (the other 3311 even-part ring); h=1). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"C2-zest(Vec(C2) x PSU(2)_10 (the other 3311 even-part ring); h=1)","existence":"unitary","model_reference":"A unitary realization exists: C2-zest(Vec(C2) x PSU(2)_10 (the other 3311 even-part ring); h=1). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_396","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_396","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,1,1],[1,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,1,1,0,1,0],[0,0,0,0,0,0,0,1,1,1,0,0],[0,0,0,1,0,1,1,0,0,0,0,0],[0,0,0,0,1,1,1,0,0,0,0,0],[0,1,0,1,0,1,0,0,0,0,0,0],[0,0,1,0,1,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,1,1],[0,0,0,0,0,0,0,1,1,1,0,0],[0,0,0,0,0,0,0,1,1,0,1,0],[0,0,0,0,1,1,1,0,0,0,0,0],[0,0,0,1,0,1,1,0,0,0,0,0],[0,0,1,0,1,0,1,0,0,0,0,0],[0,1,0,1,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,1,0,1,0],[0,0,0,0,0,0,0,1,1,1,0,0],[0,0,0,0,0,0,0,1,1,1,1,1],[1,0,0,0,0,0,0,1,1,1,1,0],[0,1,0,1,1,1,1,0,0,0,0,0],[0,0,1,1,1,1,1,0,0,0,0,0],[0,0,0,1,0,1,1,0,0,0,0,0],[0,0,0,0,1,1,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,1,1,0,0],[0,0,0,0,0,0,0,1,1,0,1,0],[1,0,0,0,0,0,0,1,1,1,1,0],[0,0,0,0,0,0,0,1,1,1,1,1],[0,0,1,1,1,1,1,0,0,0,0,0],[0,1,0,1,1,1,1,0,0,0,0,0],[0,0,0,0,1,1,1,0,0,0,0,0],[0,0,0,1,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,1,1,0,0,0,0,0],[0,0,0,0,1,1,1,0,0,0,0,0],[0,1,0,1,1,1,1,0,0,0,0,0],[0,0,1,1,1,1,1,0,0,0,0,0],[1,0,0,0,0,0,0,1,1,1,1,0],[0,0,0,0,0,0,0,1,1,1,1,1],[0,0,0,0,0,0,0,1,1,1,0,0],[0,0,0,0,0,0,0,1,1,0,1,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,1,1,0,0,0,0,0],[0,0,0,1,0,1,1,0,0,0,0,0],[0,0,1,1,1,1,1,0,0,0,0,0],[0,1,0,1,1,1,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,1,1,1,1],[1,0,0,0,0,0,0,1,1,1,1,0],[0,0,0,0,0,0,0,1,1,0,1,0],[0,0,0,0,0,0,0,1,1,1,0,0],[0,0,0,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,1,0,1,0,1,0,0,0,0,0,0],[0,0,1,0,1,0,1,0,0,0,0,0],[0,0,0,1,0,1,1,0,0,0,0,0],[0,0,0,0,1,1,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,1,1,0,0],[0,0,0,0,0,0,0,1,1,0,1,0],[1,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,1,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,1,0,1,0,0,0,0,0],[0,1,0,1,0,1,0,0,0,0,0,0],[0,0,0,0,1,1,1,0,0,0,0,0],[0,0,0,1,0,1,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,1,0,1,0],[0,0,0,0,0,0,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,1,0,1,1],[1,0,0,0,0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,0]]],"tensor_sha256":"1365cef513b570af39f90cef469e21258a2498343ac73e7a1986df5d4d653dec","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_407.txt","investigation_deferred":false},{"id":"R12_1_408","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-algebra Morita dual of the matched-grading subcategory of Ising x Vec(SD16,omega). 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Indices are zero-based; X_0 is the unit. The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,2,1,4,3,6,5,7,8,9,10,11],"invertible_indices":[0,1,2,3,4,9,10,11],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,0.9999999999999986,1.0000000000000002,1.0000000000000002,1.0000000000000002,3.603875471609679,3.603875471609679,4.493959207434935,4.493959207434935,1.0000000000000002,1.0000000000000002,1.0000000000000002],"global_fp_dimension_approx":74.36717554591823,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t - 16)**4*(t - 8)**5*(t**3 - 112*t**2 + 3136*t - 25088)","formal_codegree_polynomial":"(t - 16)**4*(t - 8)**5*(t**3 - 112*t**2 + 3136*t - 25088)"},"research":[{"title":"Current accepted case decision","source":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102","data":{"id":"R12_1_409","input_line":409,"rank":12,"multiplicity":1,"tensor_sha256":"1ece9dc9cbe24c737a584c6db0e36ba0906853385db6f7a387f0a70a48934769","status":"excluded","commutative":true,"simple":false,"exclusion_route":"Cyclic isotypic summands","exclusion_evidence":[{"source":"cyclic.json","certificate":{"kind":"cyclic_isotypic_integrality","g":3,"order":4,"X":7,"Y":7,"Z":8,"multiplicity":1,"dimension_polynomial":["1","-4","-4","8"],"normalized_dimension_polynomial":["1","-1","-1/4","1/8"],"scope":"general_complex_fusion_category"}}],"model_name":null,"model_constructor":null,"model_certificate_count":0,"first_round_status":"excluded","first_round_audit":"retained; no erroneous classification found","round4_baseline_status":"excluded","round4_audit":"retained after fresh proof, implementation, and tensor audit"}},{"title":"General complex exclusion certificate","source":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/tree/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","data":[{"source":"cyclic.json","certificate":{"kind":"cyclic_isotypic_integrality","g":3,"order":4,"X":7,"Y":7,"Z":8,"multiplicity":1,"dimension_polynomial":["1","-4","-4","8"],"normalized_dimension_polynomial":["1","-1","-1/4","1/8"],"scope":"general_complex_fusion_category"}}]}],"evidence":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"},{"label":"cyclic.json","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/cyclic.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/cyclic.json","sha256":"13367358ca941e02f7cee3f88e7678e86a2b222b899bde8b4eff18a1b7a00841"}],"categorification_scope":{"general_status":"excluded","level_status":{"general":"excluded","pivotal":"excluded","spherical":"excluded","pseudo_unitary":"excluded","unitary":"excluded"},"strongest_excluded":"general","same_category_implications":[],"existence_implications":[],"notes":["Excluded from fusion categorification over ℂ by Cyclic isotypic summands. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_414","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_414","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[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],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0,0,0,0],[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],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,1,1,0],[1,0,0,1,1,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,1,1,0,0,0,0,0,0,1],[0,1,1,0,0,0,0,0,0,1,1,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[1,0,0,0,0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,0,0,1,1,0,0,0],[0,1,1,1,1,1,1,1,1,0,0,0],[1,0,0,0,0,1,1,1,1,1,1,1],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,0]]],"tensor_sha256":"7898e6f2bf52455463f7e0e195f25e397d71c330b51248699905ace86b7eebcd","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_417","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_417","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[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],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0,0,0,0],[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],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,1,1,0,0,1,1,1,1,1,1,0],[1,0,0,1,1,1,1,1,1,0,0,1],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,1,1,1,1,1,1,0,0,1],[0,1,1,0,0,1,1,1,1,1,1,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,1,1],[0,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,0,0,0],[0,1,1,1,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,1,1],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,0]]],"tensor_sha256":"8047b17d81ecfb89071461433aa1e827ae2da09c1c09decc1ef6d0cbd533d77a","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,2,1,4,3,6,5,7,8,9,10,11],"invertible_indices":[0,1,2,3,4,9,10,11],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,0.9999999999999998,1.0000000000000002,1.0000000000000002,1.0000000000000002,4.0000000000000036,4.0000000000000036,2.0000000000000013,2.0000000000000013,1.0000000000000002,1.0000000000000002,1.0000000000000002],"global_fp_dimension_approx":48.00000000000007,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t - 48)*(t - 24)*(t - 16)**5*(t - 8)**5","formal_codegree_polynomial":"(t - 48)*(t - 24)*(t - 16)**5*(t - 8)**5"},"research":[{"title":"Current accepted case decision","source":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102","data":{"id":"R12_1_417","input_line":417,"rank":12,"multiplicity":1,"tensor_sha256":"8047b17d81ecfb89071461433aa1e827ae2da09c1c09decc1ef6d0cbd533d77a","status":"unitarily_realized","commutative":true,"simple":false,"exclusion_route":null,"exclusion_evidence":[],"model_name":"C(S4_fiber_C4,H4)","model_constructor":"group_theoretical","model_certificate_count":1,"first_round_status":"unresolved","first_round_audit":"retained; no erroneous classification found","continuation_models":[{"name":"C(S4_fiber_C4,H4)","constructor":"group_theoretical","parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14,17,16,19,18,21,20,23,22,25,24,27,26,29,28,31,30,33,32,35,34,37,36,39,38,41,40,43,42,45,44,47,46],[2,3,1,0,9,8,10,11,4,5,7,6,15,14,12,13,20,21,23,22,17,16,18,19,36,37,39,38,41,40,42,43,44,45,47,46,25,24,26,27,28,29,31,30,33,32,34,35],[3,2,0,1,8,9,11,10,5,4,6,7,14,15,13,12,21,20,22,23,16,17,19,18,37,36,38,39,40,41,43,42,45,44,46,47,24,25,27,26,29,28,30,31,32,33,35,34],[4,5,7,6,1,0,2,3,10,11,9,8,25,24,26,27,28,29,31,30,33,32,34,35,12,13,15,14,17,16,18,19,20,21,23,22,39,38,36,37,44,45,47,46,41,40,42,43],[5,4,6,7,0,1,3,2,11,10,8,9,24,25,27,26,29,28,30,31,32,33,35,34,13,12,14,15,16,17,19,18,21,20,22,23,38,39,37,36,45,44,46,47,40,41,43,42],[6,7,4,5,10,11,8,9,0,1,2,3,26,27,24,25,32,33,34,35,28,29,30,31,38,39,36,37,44,45,46,47,40,41,42,43,12,13,14,15,16,17,18,19,20,21,22,23],[7,6,5,4,11,10,9,8,1,0,3,2,27,26,25,24,33,32,35,34,29,28,31,30,39,38,37,36,45,44,47,46,41,40,43,42,13,12,15,14,17,16,19,18,21,20,23,22],[8,9,10,11,2,3,0,1,6,7,4,5,36,37,38,39,40,41,42,43,44,45,46,47,14,15,12,13,20,21,22,23,16,17,18,19,26,27,24,25,32,33,34,35,28,29,30,31],[9,8,11,10,3,2,1,0,7,6,5,4,37,36,39,38,41,40,43,42,45,44,47,46,15,14,13,12,21,20,23,22,17,16,19,18,27,26,25,24,33,32,35,34,29,28,31,30],[10,11,9,8,7,6,4,5,2,3,1,0,39,38,36,37,44,45,47,46,41,40,42,43,26,27,25,24,33,32,34,35,28,29,31,30,15,14,12,13,20,21,23,22,17,16,18,19],[11,10,8,9,6,7,5,4,3,2,0,1,38,39,37,36,45,44,46,47,40,41,43,42,27,26,24,25,32,33,35,34,29,28,30,31,14,15,13,12,21,20,22,23,16,17,19,18],[12,13,15,14,17,16,18,19,20,21,23,22,1,0,2,3,4,5,7,6,9,8,10,11,28,29,31,30,25,24,26,27,34,35,33,32,41,40,42,43,36,37,39,38,47,46,44,45],[13,12,14,15,16,17,19,18,21,20,22,23,0,1,3,2,5,4,6,7,8,9,11,10,29,28,30,31,24,25,27,26,35,34,32,33,40,41,43,42,37,36,38,39,46,47,45,44],[14,15,12,13,20,21,22,23,16,17,18,19,2,3,0,1,8,9,10,11,4,5,6,7,40,41,42,43,36,37,38,39,46,47,44,45,28,29,30,31,24,25,26,27,34,35,32,33],[15,14,13,12,21,20,23,22,17,16,19,18,3,2,1,0,9,8,11,10,5,4,7,6,41,40,43,42,37,36,39,38,47,46,45,44,29,28,31,30,25,24,27,26,35,34,33,32],[16,17,18,19,12,13,14,15,22,23,20,21,28,29,30,31,24,25,26,27,34,35,32,33,0,1,2,3,4,5,6,7,8,9,10,11,42,43,40,41,46,47,44,45,36,37,38,39],[17,16,19,18,13,12,15,14,23,22,21,20,29,28,31,30,25,24,27,26,35,34,33,32,1,0,3,2,5,4,7,6,9,8,11,10,43,42,41,40,47,46,45,44,37,36,39,38],[18,19,17,16,23,22,20,21,12,13,15,14,31,30,28,29,34,35,33,32,25,24,26,27,42,43,41,40,47,46,44,45,36,37,39,38,1,0,2,3,4,5,7,6,9,8,10,11],[19,18,16,17,22,23,21,20,13,12,14,15,30,31,29,28,35,34,32,33,24,25,27,26,43,42,40,41,46,47,45,44,37,36,38,39,0,1,3,2,5,4,6,7,8,9,11,10],[20,21,23,22,15,14,12,13,18,19,17,16,41,40,42,43,36,37,39,38,47,46,44,45,2,3,1,0,9,8,10,11,4,5,7,6,31,30,28,29,34,35,33,32,25,24,26,27],[21,20,22,23,14,15,13,12,19,18,16,17,40,41,43,42,37,36,38,39,46,47,45,44,3,2,0,1,8,9,11,10,5,4,6,7,30,31,29,28,35,34,32,33,24,25,27,26],[22,23,20,21,18,19,16,17,14,15,12,13,42,43,40,41,46,47,44,45,36,37,38,39,30,31,28,29,34,35,32,33,24,25,26,27,2,3,0,1,8,9,10,11,4,5,6,7],[23,22,21,20,19,18,17,16,15,14,13,12,43,42,41,40,47,46,45,44,37,36,39,38,31,30,29,28,35,34,33,32,25,24,27,26,3,2,1,0,9,8,11,10,5,4,7,6],[24,25,26,27,28,29,30,31,32,33,34,35,4,5,6,7,0,1,2,3,10,11,8,9,16,17,18,19,12,13,14,15,22,23,20,21,44,45,46,47,38,39,36,37,42,43,40,41],[25,24,27,26,29,28,31,30,33,32,35,34,5,4,7,6,1,0,3,2,11,10,9,8,17,16,19,18,13,12,15,14,23,22,21,20,45,44,47,46,39,38,37,36,43,42,41,40],[26,27,25,24,33,32,34,35,28,29,31,30,7,6,4,5,10,11,9,8,1,0,2,3,44,45,47,46,39,38,36,37,42,43,41,40,17,16,18,19,12,13,15,14,23,22,20,21],[27,26,24,25,32,33,35,34,29,28,30,31,6,7,5,4,11,10,8,9,0,1,3,2,45,44,46,47,38,39,37,36,43,42,40,41,16,17,19,18,13,12,14,15,22,23,21,20],[28,29,31,30,25,24,26,27,34,35,33,32,17,16,18,19,12,13,15,14,23,22,20,21,4,5,7,6,1,0,2,3,10,11,9,8,47,46,44,45,42,43,41,40,39,38,36,37],[29,28,30,31,24,25,27,26,35,34,32,33,16,17,19,18,13,12,14,15,22,23,21,20,5,4,6,7,0,1,3,2,11,10,8,9,46,47,45,44,43,42,40,41,38,39,37,36],[30,31,28,29,34,35,32,33,24,25,26,27,18,19,16,17,22,23,20,21,12,13,14,15,46,47,44,45,42,43,40,41,38,39,36,37,4,5,6,7,0,1,2,3,10,11,8,9],[31,30,29,28,35,34,33,32,25,24,27,26,19,18,17,16,23,22,21,20,13,12,15,14,47,46,45,44,43,42,41,40,39,38,37,36,5,4,7,6,1,0,3,2,11,10,9,8],[32,33,34,35,26,27,24,25,30,31,28,29,44,45,46,47,38,39,36,37,42,43,40,41,6,7,4,5,10,11,8,9,0,1,2,3,18,19,16,17,22,23,20,21,12,13,14,15],[33,32,35,34,27,26,25,24,31,30,29,28,45,44,47,46,39,38,37,36,43,42,41,40,7,6,5,4,11,10,9,8,1,0,3,2,19,18,17,16,23,22,21,20,13,12,15,14],[34,35,33,32,31,30,28,29,26,27,25,24,47,46,44,45,42,43,41,40,39,38,36,37,18,19,17,16,23,22,20,21,12,13,15,14,7,6,4,5,10,11,9,8,1,0,2,3],[35,34,32,33,30,31,29,28,27,26,24,25,46,47,45,44,43,42,40,41,38,39,37,36,19,18,16,17,22,23,21,20,13,12,14,15,6,7,5,4,11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no uniqueness claim","round4_baseline_status":"unitarily_realized","round4_audit":"retained after fresh proof, implementation, and tensor audit"}},{"title":"Categorical construction and exact 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Indices are zero-based; X_0 is the unit. The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,2,1,4,3,6,5,7,8,9,10,11],"invertible_indices":[0,1,2,3,4,9,10,11],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,0.9999999999999996,1.0000000000000013,1.0000000000000007,1.0000000000000007,4.493959207434938,4.493959207434938,3.6038754716096824,3.6038754716096824,1.0000000000000007,1.0000000000000007,1.0000000000000007],"global_fp_dimension_approx":74.36717554591834,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t - 16)**4*(t - 8)**5*(t**3 - 112*t**2 + 3136*t - 25088)","formal_codegree_polynomial":"(t - 16)**4*(t - 8)**5*(t**3 - 112*t**2 + 3136*t - 25088)"},"research":[{"title":"Current accepted case decision","source":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102","data":{"id":"R12_1_418","input_line":418,"rank":12,"multiplicity":1,"tensor_sha256":"10bfca9944c4a9cbecf2ca917025374487dffd5c129a4999a7981b971885ffb8","status":"excluded","commutative":true,"simple":false,"exclusion_route":"Cyclic isotypic summands","exclusion_evidence":[{"source":"cyclic.json","certificate":{"kind":"cyclic_isotypic_integrality","g":3,"order":4,"X":5,"Y":5,"Z":5,"multiplicity":1,"dimension_polynomial":["1","-4","-4","8"],"normalized_dimension_polynomial":["1","-1","-1/4","1/8"],"scope":"general_complex_fusion_category"}}],"model_name":null,"model_constructor":null,"model_certificate_count":0,"first_round_status":"excluded","first_round_audit":"retained; no erroneous classification found","round4_baseline_status":"excluded","round4_audit":"retained after fresh proof, implementation, and tensor audit"}},{"title":"General complex exclusion certificate","source":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/tree/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","data":[{"source":"cyclic.json","certificate":{"kind":"cyclic_isotypic_integrality","g":3,"order":4,"X":5,"Y":5,"Z":5,"multiplicity":1,"dimension_polynomial":["1","-4","-4","8"],"normalized_dimension_polynomial":["1","-1","-1/4","1/8"],"scope":"general_complex_fusion_category"}}]}],"evidence":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"},{"label":"cyclic.json","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/cyclic.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/cyclic.json","sha256":"13367358ca941e02f7cee3f88e7678e86a2b222b899bde8b4eff18a1b7a00841"}],"categorification_scope":{"general_status":"excluded","level_status":{"general":"excluded","pivotal":"excluded","spherical":"excluded","pseudo_unitary":"excluded","unitary":"excluded"},"strongest_excluded":"general","same_category_implications":[],"existence_implications":[],"notes":["Excluded from fusion categorification over ℂ by Cyclic isotypic summands. The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_418.txt","investigation_deferred":false},{"id":"R12_1_419","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Nonsplit generalized-Ising extension, rotation-cocycle twist, and C2 algebra dual. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Nonsplit generalized-Ising extension, rotation-cocycle twist, and C2 algebra dual","existence":"unitary","model_reference":"A unitary realization exists: Nonsplit generalized-Ising extension, rotation-cocycle twist, and C2 algebra dual. 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unitary realization exists: Liptrap generalized Tambara-Yamagami reconstruction. 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_440.txt","investigation_deferred":false},{"id":"R12_1_441","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-zest(TY(C5)^C2 crossed C2[parity=0];h=1). 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crossed C2[parity=0];h=1). 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_448.txt","investigation_deferred":false},{"id":"R12_1_449","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: TY(C5, chi)^C4, inversion with odd parity coherence. 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pullback R8_1_36;h=4). 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_484.txt","investigation_deferred":false},{"id":"R12_1_485","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Rep(S3) x pseudo-PSU(2)_6. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Rep(S3) x pseudo-PSU(2)_6","existence":"unitary","model_reference":"A unitary realization exists: Rep(S3) x pseudo-PSU(2)_6. 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_486.txt","investigation_deferred":false},{"id":"R12_1_487","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: PSU(2)_5 x TY(C3). 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classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: C2-zest(Vec(C2) x mathcal I(C_4,0): cyclic, self-dual Ising zest; h=6). The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_494.txt","investigation_deferred":false},{"id":"R12_1_495","rank":12,"multiplicity":1,"status":"excluded","summary":"Excluded from fusion categorification over ℂ by Cyclic isotypic summands. The argument applies to general complex fusion categories.","name":"R12_1_495","existence":"excluded","model_reference":"Excluded from fusion categorification over ℂ by Cyclic isotypic summands. 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_496.txt","investigation_deferred":false},{"id":"R12_1_497","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Vec(C2) x mathcal I(C_4,1): cyclic, non-self-dual Ising zest. 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_521.txt","investigation_deferred":false},{"id":"R12_1_522","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: TY(C8)^C2[u=5; parity=1]. 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The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_530.txt","investigation_deferred":false},{"id":"R12_1_531","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Relative-cocycle subgroup-bimodule category: S4 x_C2 C4 with C4. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Relative-cocycle subgroup-bimodule category: S4 x_C2 C4 with C4","existence":"unitary","model_reference":"A unitary realization exists: Relative-cocycle subgroup-bimodule category: S4 x_C2 C4 with C4. 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unitary realization exists: Liptrap generalized Tambara-Yamagami reconstruction. The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_540.txt","investigation_deferred":false},{"id":"R12_1_541","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Rep(C3 semidirect C8), inversion action. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"Rep(C3 semidirect C8), inversion action","existence":"unitary","model_reference":"A unitary realization exists: Rep(C3 semidirect C8), inversion action. The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_541","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_541","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[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],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,1,1,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,1,0,1,0,0,0,0],[0,0,0,0,1,0,1,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,1],[0,1,1,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,1,0,1,0,0,0],[0,0,0,1,0,1,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,1,0,1,0,0,0,0],[0,0,0,0,1,0,1,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,1],[0,1,1,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,1,0,1,0,0,0],[0,0,0,1,0,1,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,1],[0,0,0,0,0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0]]],"tensor_sha256":"c5271c707858f1642cc1e827eaec6cd6e3dec1e51c8b84c9aa191b333e2bc0d4","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,2,1,4,3,6,5,8,7,9,10,11],"invertible_indices":[0,1,2,3,4,7,8,11],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,0.9999999999999992,1.0000000000000004,1.0000000000000004,1.0000000000000004,2.0000000000000018,2.0000000000000018,1.0000000000000004,1.0000000000000004,2.0000000000000018,2.0000000000000018,1.0000000000000004],"global_fp_dimension_approx":24.000000000000032,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t - 24)**4*(t - 12)**4*(t - 8)**4","formal_codegree_polynomial":"(t - 24)**4*(t - 12)**4*(t - 8)**4"},"research":[{"title":"Current accepted case 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explicit induced representations and their complete tensor 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unitary realization exists: Rep(C3 semidirect C8), inversion action. 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R8_1_11. The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"Noncommutative fusion rules exclude braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_551.txt","investigation_deferred":false},{"id":"R12_1_552","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-zest(Vec((12,)); h=6). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"C2-zest(Vec((12,)); h=6)","existence":"unitary","model_reference":"A unitary realization exists: C2-zest(Vec((12,)); h=6). 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classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"}],"categorification_scope":{"general_status":"realized","level_status":{"general":"realized","pivotal":"realized","spherical":"realized","pseudo_unitary":"realized","unitary":"realized"},"strongest_excluded":null,"same_category_implications":[],"existence_implications":[],"notes":["A unitary realization exists: C2-zest(Ising x TY(C3); h=4). The categorical construction and complete tensor identification are recorded in the accepted proof archive."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]},"download_url":"./downloads/fusion-tensors/R12_1_578.txt","investigation_deferred":false},{"id":"R12_1_579","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-zest(Rep(S3) x TY(C3); h=4). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","name":"C2-zest(Rep(S3) x TY(C3); h=4)","existence":"unitary","model_reference":"A unitary realization exists: C2-zest(Rep(S3) x TY(C3); h=4). The categorical construction and complete tensor identification are recorded in the accepted proof archive.","model_reference_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_file":"evidence/rank12_m1_20261026/ledger.json","source_url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_location":"R12_1_579","report_url":"./papers/rank12.pdf","report_label":"Rank-twelve classification manuscript","case_url":"./case.html?id=R12_1_579","fusion_tensor":[[[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0,0,0,0,0],[0,1,0,0,1,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,1,0,0,0,0,1,0,0,0]],[[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,1],[0,0,1,1,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,1,1,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,1,0,0,1,0,0,1,0,0,0,0]],[[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,1,1,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,1,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,1,0,0,0],[0,1,1,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,1,0,0,1,0,0],[0,0,0,0,0,1,1,0,0,1,0,0],[0,0,0,1,1,0,0,0,0,0,0,1],[0,0,0,1,1,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[1,1,1,1,1,0,0,1,1,0,1,1],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,1,0,0,1,0,0]],[[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1]],[[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,1,0,0,0,0,1,0,0,0],[0,1,0,0,1,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,1,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,1]]],"tensor_sha256":"8a2989401658bc3753a89567e071226e26ce82b4def8ab678f411606391b53c3","basis":["X0","X1","X2","X3","X4","X5","X6","X7","X8","X9","X10","X11"],"unit_index":0,"tensor_convention":"N[i][j][k] is the coefficient of X_k in X_i * X_j. 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The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,2,1,4,3,6,5,8,7,9,10,11],"invertible_indices":[0,1,2,7,8,10],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,0.9999999999999994,1.000000000000001,2.000000000000001,2.000000000000001,1.7320508075688783,1.7320508075688783,1.0000000000000004,1.0000000000000004,3.4641016151377553,1.0000000000000004,2.000000000000001],"global_fp_dimension_approx":36.00000000000003,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t - 36)**2*(t - 18)**4*(t - 12)**2*(t - 9)**2*(t - 6)**2","formal_codegree_polynomial":"(t - 36)**2*(t - 18)**4*(t - 12)**2*(t - 9)**2*(t - 6)**2"},"research":[{"title":"Current accepted case 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_607.txt","investigation_deferred":false},{"id":"R12_1_608","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: Vec(Dic3). 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The supplied source basis is preserved.","hash_convention":"SHA-256 of UTF-8 compact JSON without a trailing newline.","invariants":{"dual_indices":[0,2,1,4,3,6,5,8,7,10,9,11],"invertible_indices":[0,1,2,11],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,0.9999999999999996,1.000000000000002,6.162277660168382,6.162277660168382,5.162277660168379,5.162277660168379,7.162277660168384,7.162277660168384,6.162277660168382,6.162277660168382,1.000000000000001],"global_fp_dimension_approx":311.7893276880825,"dimensions_method":"Approximate Perron eigenvalues for display; exact integrality and structural flags are independently certified.","character_method":"Exact character and formal-codegree certificates are retained in the source-bound proof archive.","central_casimir_polynomial":"(t - 16)**4*(t**2 - 320*t + 2560)*(t**2 - 64*t + 512)*(t**2 - 24*t + 96)**2"},"research":[{"title":"Current accepted case decision","source":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","source_sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102","data":{"id":"R12_1_614","input_line":614,"rank":12,"multiplicity":1,"tensor_sha256":"ad97e1b35689113bf00565f0af441cfe32b65b461da846d2ffa7370b006af43c","status":"excluded","commutative":false,"simple":false,"exclusion_route":"Cyclic isotypic summands","exclusion_evidence":[{"source":"cyclic.json","certificate":{"kind":"cyclic_isotypic_integrality","g":11,"order":2,"X":5,"Y":5,"Z":8,"multiplicity":1,"dimension_polynomial":["1","-8","6"],"normalized_dimension_polynomial":["1","-4","3/2"],"scope":"general_complex_fusion_category"}}],"model_name":null,"model_constructor":null,"model_certificate_count":0,"first_round_status":"excluded","first_round_audit":"retained; no erroneous classification found","round4_baseline_status":"excluded","round4_audit":"retained after fresh proof, implementation, and tensor audit"}},{"title":"General complex exclusion certificate","source":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/tree/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025","data":[{"source":"cyclic.json","certificate":{"kind":"cyclic_isotypic_integrality","g":11,"order":2,"X":5,"Y":5,"Z":8,"multiplicity":1,"dimension_polynomial":["1","-8","6"],"normalized_dimension_polynomial":["1","-4","3/2"],"scope":"general_complex_fusion_category"}}]}],"evidence":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"},{"label":"Proof archive and replay guide","path":"evidence/rank12_m1_20261026/README.md","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/README.md","sha256":"dd99f4fed0657537da81d7ce1332da1940bf1dcc2490eeeb809aeb7dbb4b68b4"},{"label":"report.pdf","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/report.pdf","sha256":"6adf3138488ee97f40fa52b14b39377a8fc941f80d3f7784d7bed67a2d8680a8"},{"label":"cyclic.json","path":"evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/cyclic.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/baseline/prior_round/prior_round/evidence/rank12-m1-offline-20261025/cyclic.json","sha256":"13367358ca941e02f7cee3f88e7678e86a2b222b899bde8b4eff18a1b7a00841"}],"categorification_scope":{"general_status":"excluded","level_status":{"general":"excluded","pivotal":"excluded","spherical":"excluded","pseudo_unitary":"excluded","unitary":"excluded"},"strongest_excluded":"general","same_category_implications":[],"existence_implications":[],"notes":["Excluded from fusion categorification over ℂ by Cyclic isotypic summands. The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_614.txt","investigation_deferred":false},{"id":"R12_1_615","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: C2-zest(Vec(C2) x Tambara--Yamagami category for C_5; h=6). 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The argument applies to general complex fusion categories."],"audit_sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}],"braided":{"status":"excluded","explanation":"General nonexistence excludes braided categorification.","sources":[{"label":"Accepted classification ledger","path":"evidence/rank12_m1_20261026/ledger.json","url":"https://github.com/sebastienpalcoux/grothendieck-ring-classification/blob/main/evidence/rank12_m1_20261026/ledger.json","sha256":"d136adfd495b7c46f096143901dc917305e7c362be406e94673b17a015fa5102"}]}},"download_url":"./downloads/fusion-tensors/R12_1_620.txt","investigation_deferred":false},{"id":"R12_1_621","rank":12,"multiplicity":1,"status":"unitarily_realized","summary":"A unitary realization exists: TY(C11). 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complex and pseudo-unitary realization exists by central C6 condensation of the algebraic C36+36 near-group category. 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