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All matrix identities N_i=p_i(H) were checked over Q."}}},{"rank":4,"multiplicity":1,"id":"F_{1,4}","name":"PSU2 6","existence":"unitary","fusion_tensor":[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]],[[0,0,1,0],[0,0,0,1],[1,0,1,1],[0,1,1,1]],[[0,0,0,1],[0,0,1,0],[0,1,1,1],[1,0,1,1]]],"tensor_sha256":"1d1ab5790300c7689c92231c924c558b3a29e482b3b5542d825dfe55ba43ae50","basis":["X0","X1","X2","X3"],"model_reference":"Rank-four manuscript, model catalogue; LPR, multiplicity-one classification.","model_reference_url":"https://arxiv.org/abs/2010.10264","source_file":"supplements/rank4/data/models.json","source_location":"F_{1,4}","invariants":{"dual_indices":[0,1,2,3],"invertible_indices":[0,1],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,2.4142135624,2.4142135624],"global_fp_dimension_approx":13.6568542495,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 4)**2*(t**2 - 16*t + 32)","formal_codegree_polynomial":"(t - 4)**2*(t**2 - 16*t + 32)","irreducible_characters_approx":[{"values_approx":[1.0,1.0,2.4142135624,2.4142135624],"formal_codegree_approx":13.6568542495},{"values_approx":[1.0,-1.0,-1.0,1.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,-1.0,1.0,-1.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,1.0,-0.4142135624,-0.4142135624],"formal_codegree_approx":2.3431457505}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4],"primitive_polynomial":"theta*(theta + 2)*(theta**2 - 20*theta + 2)","basis_values":["1","-11*theta**3/23 + 197*theta**2/23 + 438*theta/23 - 1","-26*theta**3/161 + 474*theta**2/161 + 891*theta/161 - 1","58*theta**3/161 - 1045*theta**2/161 - 2161*theta/161 + 1"],"codegree_value":"2*(32*theta**3 - 571*theta**2 - 1270*theta + 322)/161","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}}},{"rank":4,"multiplicity":1,"id":"F_{1,5}","name":"Fib times Fib","existence":"unitary","fusion_tensor":[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[1,1,0,0],[0,0,0,1],[0,0,1,1]],[[0,0,1,0],[0,0,0,1],[1,0,1,0],[0,1,0,1]],[[0,0,0,1],[0,0,1,1],[0,1,0,1],[1,1,1,1]]],"tensor_sha256":"07d25bc103cec754d7398b1c2054d08a0f06875c0dcaa1239e7600a65cb93621","basis":["X0","X1","X2","X3"],"model_reference":"Rank-four manuscript, model catalogue; LPR, multiplicity-one classification.","model_reference_url":"https://arxiv.org/abs/2010.10264","source_file":"supplements/rank4/data/models.json","source_location":"F_{1,5}","invariants":{"dual_indices":[0,1,2,3],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.6180339887,1.6180339887,2.6180339887],"global_fp_dimension_approx":13.0901699437,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 5)**2*(t**2 - 15*t + 25)","formal_codegree_polynomial":"(t - 5)**2*(t**2 - 15*t + 25)","irreducible_characters_approx":[{"values_approx":[1.0,1.6180339887,1.6180339887,2.6180339887],"formal_codegree_approx":13.0901699437},{"values_approx":[1.0,-0.6180339887,1.6180339887,-1.0],"formal_codegree_approx":5.0},{"values_approx":[1.0,1.6180339887,-0.6180339887,-1.0],"formal_codegree_approx":5.0},{"values_approx":[1.0,-0.6180339887,-0.6180339887,0.3819660113],"formal_codegree_approx":1.9098300563}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4],"primitive_polynomial":"(theta**2 - 19*theta - 11)*(theta**2 + theta - 1)","basis_values":["1","-2*theta**3/45 + 79*theta**2/90 - theta/30 - 83/90","-2*theta**3/45 + 7*theta**2/9 + 28*theta/15 + 8/45","theta**3/18 - 46*theta**2/45 - 17*theta/15 + 7/90"],"codegree_value":"-(11*theta**3 - 206*theta**2 - 228*theta - 233)/90","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}}},{"rank":4,"multiplicity":1,"id":"F_{1,6}","name":"PSU2 7","existence":"unitary","fusion_tensor":[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[1,0,0,1],[0,0,1,1],[0,1,1,0]],[[0,0,1,0],[0,0,1,1],[1,1,1,1],[0,1,1,1]],[[0,0,0,1],[0,1,1,0],[0,1,1,1],[1,0,1,1]]],"tensor_sha256":"54ce3e9b044bb568c6f07a2030b27679dc98b0f598fd73e5ca3f6d586cbc0174","basis":["X0","X1","X2","X3"],"model_reference":"Rank-four manuscript, model catalogue; LPR, multiplicity-one classification.","model_reference_url":"https://arxiv.org/abs/2010.10264","source_file":"supplements/rank4/data/models.json","source_location":"F_{1,6}","invariants":{"dual_indices":[0,1,2,3],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.8793852416,2.8793852416,2.5320888862],"global_fp_dimension_approx":19.2344223834,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 3)*(t**3 - 27*t**2 + 162*t - 243)","formal_codegree_polynomial":"(t - 3)*(t**3 - 27*t**2 + 162*t - 243)","irreducible_characters_approx":[{"values_approx":[1.0,1.8793852416,2.8793852416,2.5320888862],"formal_codegree_approx":19.2344223834},{"values_approx":[1.0,-1.5320888862,-0.5320888862,1.3472963553],"formal_codegree_approx":5.4456224073},{"values_approx":[1.0,1.0,-1.0,0.0],"formal_codegree_approx":3.0},{"values_approx":[1.0,-0.3472963553,0.6527036447,-0.8793852416],"formal_codegree_approx":2.3199552093}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4],"primitive_polynomial":"theta*(theta**3 - 24*theta**2 + 9*theta + 51)","basis_values":["1","113*theta**3/3043 - 2644*theta**2/3043 - 224*theta/3043 + 1","-66*theta**3/3043 + 1652*theta**2/3043 - 1835*theta/3043 - 1","-7*theta**3/3043 + 83*theta**2/3043 + 2249*theta/3043"],"codegree_value":"-(40*theta**3 - 909*theta**2 - 2853*theta - 9129)/3043","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}}},{"rank":4,"multiplicity":1,"id":"F_{1,7}","name":"Vec C4","existence":"unitary","fusion_tensor":[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]],[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]],[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]],"tensor_sha256":"b09a86392f0540e24eeb9f5bc4b0a11068d161d889cac4f3223256fab033f1a5","basis":["X0","X1","X2","X3"],"model_reference":"Rank-four manuscript, model catalogue; LPR, multiplicity-one classification.","model_reference_url":"https://arxiv.org/abs/2010.10264","source_file":"supplements/rank4/data/models.json","source_location":"F_{1,7}","invariants":{"dual_indices":[0,1,3,2],"invertible_indices":[0,1,2,3],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,1.0],"global_fp_dimension_approx":4.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 4)**4","formal_codegree_polynomial":"(t - 4)**4","irreducible_characters_approx":[{"values_approx":[1.0,-1.0,{"re":0.0,"im":-1.0},{"re":0.0,"im":1.0}],"formal_codegree_approx":4.0},{"values_approx":[1.0,-1.0,{"re":0.0,"im":1.0},{"re":0.0,"im":-1.0}],"formal_codegree_approx":4.0},{"values_approx":[1.0,1.0,-1.0,-1.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,1.0,1.0,1.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4],"primitive_polynomial":"(theta - 10)*(theta + 4)*(theta**2 + 2*theta + 2)","basis_values":["1","-4*theta**3/305 + 37*theta**2/305 + 82*theta/305 - 43/61","76*theta**3/2135 - 398*theta**2/2135 - 3083*theta/2135 - 647/427","-43*theta**3/2135 + 169*theta**2/2135 + 2559*theta/2135 + 529/427"],"codegree_value":"4","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}}},{"rank":4,"multiplicity":1,"id":"F_{1,8}","name":"Tambara Yamagami C3","existence":"unitary","fusion_tensor":[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[1,0,1,1],[0,1,0,0],[0,1,0,0]],[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]],[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]],"tensor_sha256":"27c8d482dae88c99ab82f120cbfe69533068604d8b484704ee32fe2998a09aa1","basis":["X0","X1","X2","X3"],"model_reference":"Rank-four manuscript, model catalogue; LPR, multiplicity-one classification.","model_reference_url":"https://arxiv.org/abs/2010.10264","source_file":"supplements/rank4/data/models.json","source_location":"F_{1,8}","invariants":{"dual_indices":[0,1,3,2],"invertible_indices":[0,2,3],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.7320508076,1.0,1.0],"global_fp_dimension_approx":6.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 6)**2*(t - 3)**2","formal_codegree_polynomial":"(t - 6)**2*(t - 3)**2","irreducible_characters_approx":[{"values_approx":[1.0,-1.7320508076,1.0,1.0],"formal_codegree_approx":6.0},{"values_approx":[1.0,1.7320508076,1.0,1.0],"formal_codegree_approx":6.0},{"values_approx":[1.0,0.0,{"re":-0.5,"im":-0.8660254038},{"re":-0.5,"im":0.8660254038}],"formal_codegree_approx":3.0},{"values_approx":[1.0,0.0,{"re":-0.5,"im":0.8660254038},{"re":-0.5,"im":-0.8660254038}],"formal_codegree_approx":3.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4],"primitive_polynomial":"(theta**2 - 16*theta + 52)*(theta**2 + 5*theta + 7)","basis_values":["1","41*theta**3/6558 - 69*theta**2/2186 - 591*theta/2186 - 1442/3279","-43*theta**3/3279 + 179*theta**2/1093 + 60*theta/1093 - 4573/3279","22*theta**3/3279 - 117*theta**2/1093 + 376*theta/1093 + 3331/3279"],"codegree_value":"-(7*theta**3 - 62*theta**2 - 436*theta - 3958)/1093","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. 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All matrix identities N_i=p_i(H) were checked over Q."}}},{"rank":4,"multiplicity":3,"id":"F_{3,19}","name":"Ad E8","existence":"unitary","fusion_tensor":[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[1,1,0,0],[0,0,1,1],[0,0,1,0]],[[0,0,1,0],[0,0,1,1],[1,1,3,2],[0,1,2,1]],[[0,0,0,1],[0,0,1,0],[0,1,2,1],[1,0,1,1]]],"tensor_sha256":"20a78fc6939a1b4a0209f4386ec7f92d47bf4c558609e014b4b0d40ee8c472a5","basis":["X0","X1","X2","X3"],"model_reference":"Rank-four manuscript, model catalogue; Evans--Jones, displayed fusion rules.","model_reference_url":"https://arxiv.org/abs/2404.14466","source_file":"supplements/rank4/data/models.json","source_location":"F_{3,19}","invariants":{"dual_indices":[0,1,2,3],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.6180339887,4.7833861168,2.9562952015],"global_fp_dimension_approx":35.2384980489,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"t**4 - 45*t**3 + 375*t**2 - 1125*t + 1125","formal_codegree_polynomial":"t**4 - 45*t**3 + 375*t**2 - 1125*t + 1125","irreducible_characters_approx":[{"values_approx":[1.0,1.6180339887,4.7833861168,2.9562952015],"formal_codegree_approx":35.2384980489},{"values_approx":[1.0,1.6180339887,-0.5473181393,-0.3382612127],"formal_codegree_approx":4.0320117823},{"values_approx":[1.0,-0.6180339887,-0.7472382749,1.2090569265],"formal_codegree_approx":3.4021497024},{"values_approx":[1.0,-0.6180339887,0.5111702974,-0.8270909153],"formal_codegree_approx":2.3273404664}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4],"primitive_polynomial":"theta**4 - 32*theta**3 + 44*theta**2 + 137*theta - 179","basis_values":["1","14*theta**3/661 - 429*theta**2/661 + 81*theta/661 + 1603/661","-8*theta**3/12559 + 434*theta**2/12559 - 3729*theta/12559 - 2899/12559","-127*theta**3/12559 + 3750*theta**2/12559 + 5167*theta/12559 - 16194/12559"],"codegree_value":"(119*theta**3 - 3316*theta**2 + 3663*theta + 50972)/12559","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}}},{"rank":4,"multiplicity":3,"id":"F_{3,24}","name":"near group C3 parameter3","existence":"unitary","fusion_tensor":[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[1,3,1,1],[0,1,0,0],[0,1,0,0]],[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]],[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]],"tensor_sha256":"1af350f3403164254cf2e653909f1ccba561b400008109ef9a9652014d4a8f39","basis":["X0","X1","X2","X3"],"model_reference":"Rank-four manuscript, model catalogue; Izumi near-group construction.","model_reference_url":"https://arxiv.org/abs/1512.04288","source_file":"supplements/rank4/data/models.json","source_location":"F_{3,24}","invariants":{"dual_indices":[0,1,3,2],"invertible_indices":[0,2,3],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,3.7912878475,1.0,1.0],"global_fp_dimension_approx":17.3738635424,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 3)**2*(t**2 - 21*t + 63)","formal_codegree_polynomial":"(t - 3)**2*(t**2 - 21*t + 63)","irreducible_characters_approx":[{"values_approx":[1.0,3.7912878475,1.0,1.0],"formal_codegree_approx":17.3738635424},{"values_approx":[1.0,-0.7912878475,1.0,1.0],"formal_codegree_approx":3.6261364576},{"values_approx":[1.0,0.0,{"re":-0.5,"im":-0.8660254038},{"re":-0.5,"im":0.8660254038}],"formal_codegree_approx":3.0},{"values_approx":[1.0,0.0,{"re":-0.5,"im":0.8660254038},{"re":-0.5,"im":-0.8660254038}],"formal_codegree_approx":3.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4],"primitive_polynomial":"(theta**2 - 22*theta + 100)*(theta**2 + 5*theta + 7)","basis_values":["1","41*theta**3/17538 - 77*theta**2/5846 - 631*theta/5846 - 1526/8769","-67*theta**3/8769 + 411*theta**2/2923 - 466*theta/2923 - 15331/8769","40*theta**3/8769 - 289*theta**2/2923 + 1238*theta/2923 + 10069/8769"],"codegree_value":"(23*theta**3 + 13*theta**2 - 349*theta + 16824)/5846","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. 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All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,1,2,4,5,3],"alternative_models":[]},{"rank":6,"multiplicity":1,"id":"R6_1_32","name":"Rep((C3 x C3) semidirect C2)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],[[0,0,1,0,0,0],[0,0,1,0,0,0],[1,1,1,0,0,0],[0,0,0,0,1,1],[0,0,0,1,0,1],[0,0,0,1,1,0]],[[0,0,0,1,0,0],[0,0,0,1,0,0],[0,0,0,0,1,1],[1,1,0,1,0,0],[0,0,1,0,0,1],[0,0,1,0,1,0]],[[0,0,0,0,1,0],[0,0,0,0,1,0],[0,0,0,1,0,1],[0,0,1,0,0,1],[1,1,0,0,1,0],[0,0,1,1,0,0]],[[0,0,0,0,0,1],[0,0,0,0,0,1],[0,0,0,1,1,0],[0,0,1,0,1,0],[0,0,1,1,0,0],[1,1,0,0,0,1]]],"tensor_sha256":"361893f00cbe2716e34d41cc1360618e6d1e5e9b5754f6534ba4704dab3efd2a","basis":["X0","X1","X2","X3","X4","X5"],"model_reference":"Rank-six manuscript, positive model catalogue, R6_1_32; Generalized dihedral induced-character formula; LPR Section 3.6.","model_reference_url":"https://arxiv.org/abs/2010.10264","source_file":"supplements/rank6/data/complete_positive_models.json","source_location":"R6_1_32","invariants":{"dual_indices":[0,1,2,3,4,5],"invertible_indices":[0,1],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,2.0,2.0,2.0,2.0],"global_fp_dimension_approx":18.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 18)*(t - 9)**4*(t - 2)","formal_codegree_polynomial":"(t - 18)*(t - 9)**4*(t - 2)","irreducible_characters_approx":[{"values_approx":[1.0,1.0,2.0,2.0,2.0,2.0],"formal_codegree_approx":18.0},{"values_approx":[1.0,1.0,-1.0,-1.0,-1.0,2.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,1.0,-1.0,-1.0,2.0,-1.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,1.0,-1.0,2.0,-1.0,-1.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,1.0,2.0,-1.0,-1.0,-1.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,-1.0,0.0,0.0,0.0,0.0],"formal_codegree_approx":2.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6],"primitive_polynomial":"theta*(theta - 39)*(theta - 3)*(theta + 1)*(theta + 3)*(theta + 6)","basis_values":["1","theta**5/800 - 33*theta**4/800 - 243*theta**3/800 + 297*theta**2/800 + 1053*theta/400 + 1","-391*theta**5/552825 + 15007*theta**4/631800 + 114859*theta**3/737100 - 15007*theta**2/70200 - 110167*theta/81900 - 1","23*theta**5/631800 - 797*theta**4/315900 + 8167*theta**3/210600 + 6647*theta**2/35100 - 2492*theta/2925 - 1","34*theta**5/42525 - 1213*theta**4/48600 - 13681*theta**3/56700 - 587*theta**2/5400 + 13273*theta/6300 + 2","-13331*theta**5/17690400 + 61549*theta**4/2527200 + 1170851*theta**3/5896800 - 14749*theta**2/280800 - 401629*theta/327600 - 1"],"codegree_value":"(25799*theta**5 - 851347*theta**4 - 6269037*theta**3 + 7662123*theta**2 + 54331614*theta + 53071200)/5896800","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,1,2,3,4,5],"alternative_models":[]},{"rank":6,"multiplicity":1,"id":"R6_1_34","name":"PSU(2)_11","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],[[0,1,0,0,0,0],[1,1,1,0,0,0],[0,1,1,1,0,0],[0,0,1,1,1,0],[0,0,0,1,1,1],[0,0,0,0,1,1]],[[0,0,1,0,0,0],[0,1,1,1,0,0],[1,1,1,1,1,0],[0,1,1,1,1,1],[0,0,1,1,1,1],[0,0,0,1,1,0]],[[0,0,0,1,0,0],[0,0,1,1,1,0],[0,1,1,1,1,1],[1,1,1,1,1,1],[0,1,1,1,1,0],[0,0,1,1,0,0]],[[0,0,0,0,1,0],[0,0,0,1,1,1],[0,0,1,1,1,1],[0,1,1,1,1,0],[1,1,1,1,0,0],[0,1,1,0,0,0]],[[0,0,0,0,0,1],[0,0,0,0,1,1],[0,0,0,1,1,0],[0,0,1,1,0,0],[0,1,1,0,0,0],[1,1,0,0,0,0]]],"tensor_sha256":"bf3120c913cecea1166e95187ec71bbb7a8e38912d1d05426b2d31579d089f82","basis":["X0","X1","X2","X3","X4","X5"],"model_reference":"Rank-six manuscript, positive model catalogue, R6_1_34; Principal SU(2) even subcategory; LPR Theorem 1.1.","model_reference_url":"https://arxiv.org/abs/2010.10264","source_file":"supplements/rank6/data/complete_positive_models.json","source_location":"R6_1_34","invariants":{"dual_indices":[0,1,2,3,4,5],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,2.7709120513,3.9070415448,4.1481149053,3.4389051312,1.9418836349],"global_fp_dimension_approx":56.7467650487,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"t**6 - 91*t**5 + 2366*t**4 - 26364*t**3 + 142805*t**2 - 371293*t + 371293","formal_codegree_polynomial":"t**6 - 91*t**5 + 2366*t**4 - 26364*t**3 + 142805*t**2 - 371293*t + 371293","irreducible_characters_approx":[{"values_approx":[1.0,2.7709120513,3.9070415448,4.1481149053,3.4389051312,1.9418836349],"formal_codegree_approx":56.7467650487},{"values_approx":[1.0,2.1361294935,1.4269197194,-0.5149639155,-2.0119854118,-1.7709120513],"formal_codegree_approx":15.0485517235},{"values_approx":[1.0,1.2410733605,-0.7008102743,-1.4100200484,0.3608920029,1.4970214963],"formal_codegree_approx":7.390871062},{"values_approx":[1.0,0.2907902259,-1.2062312704,0.5646807809,0.8057541414,-1.1361294935],"formal_codegree_approx":4.7984471798},{"values_approx":[1.0,-0.4970214963,-0.2559481358,0.8801813576,-1.0617022772,0.7092097741],"formal_codegree_approx":3.7174492675},{"values_approx":[1.0,-0.9418836349,0.8290284165,-0.6679930799,0.4681364136,-0.2410733605],"formal_codegree_approx":3.2979157185}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6],"primitive_polynomial":"theta**6 - 56*theta**5 - 574*theta**4 + 5196*theta**3 + 7880*theta**2 - 22793*theta - 4187","basis_values":["1","10291372071*theta**5/434097916688701 - 601042533259*theta**4/434097916688701 - 4643673124951*theta**3/434097916688701 + 74882118540238*theta**2/434097916688701 + 2957306071452*theta/434097916688701 - 410655903603005/434097916688701","-24751842564*theta**5/434097916688701 + 1412553697596*theta**4/434097916688701 + 12674734452884*theta**3/434097916688701 - 139693406928329*theta**2/434097916688701 - 76283844067011*theta/434097916688701 + 350890024263210/434097916688701","27926792529*theta**5/434097916688701 - 1546090383524*theta**4/434097916688701 - 16873709769601*theta**3/434097916688701 + 127024078855413*theta**2/434097916688701 + 214306948013235*theta/434097916688701 - 256548731252193/434097916688701","-15845568276*theta**5/434097916688701 + 846369863360*theta**4/434097916688701 + 11421564838386*theta**3/434097916688701 - 61151273060233*theta**2/434097916688701 - 293474889927760*theta/434097916688701 + 153951387098715/434097916688701","3532242469*theta**5/434097916688701 - 180510634829*theta**4/434097916688701 - 3058307037046*theta**3/434097916688701 + 11162672264004*theta**2/434097916688701 + 211196915722115*theta/434097916688701 - 68169698792854/434097916688701"],"codegree_value":"(8070973603*theta**5 - 481039934592*theta**4 - 3355734482296*theta**3 + 85569327697651*theta**2 - 23180866004484*theta + 1424954960818018)/434097916688701","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. 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All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,4,1,5,2,3],"alternative_models":[]},{"rank":6,"multiplicity":2,"id":"R6_2_14","name":"Fermionic grading zest of E6","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],[[0,1,0,0,0,0],[0,0,1,0,1,0],[0,1,0,1,0,1],[1,0,1,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]],[[0,0,1,0,0,0],[0,1,0,1,0,1],[1,0,2,0,1,0],[0,1,0,1,0,1],[0,0,1,0,0,0],[0,1,0,1,0,0]],[[0,0,0,1,0,0],[1,0,1,0,0,0],[0,1,0,1,0,1],[0,0,1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]],[[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]],[[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,1,0]]],"tensor_sha256":"6d3067a28664b7003ddb7f4395812b28605c0e4c4d325a375fb59642b03120e8","basis":["X0","X1","X2","X3","X4","X5"],"model_reference":"Rank-six manuscript, positive model catalogue, R6_2_14; Central image of V10 and the unitary fermionic C2-zesting theorem.","model_reference_url":null,"source_file":"supplements/rank6/data/additional_positive_models.json","source_location":"R6_2_14","invariants":{"dual_indices":[0,3,2,1,4,5],"invertible_indices":[0,4],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.9318516526,2.7320508076,1.9318516526,1.0,1.4142135624],"global_fp_dimension_approx":18.9282032303,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 4)**2*(t**2 - 24*t + 96)**2","formal_codegree_polynomial":"(t - 4)**2*(t**2 - 24*t + 96)**2","irreducible_characters_approx":[{"values_approx":[1.0,-1.9318516526,2.7320508076,-1.9318516526,1.0,-1.4142135624],"formal_codegree_approx":18.9282032303},{"values_approx":[1.0,1.9318516526,2.7320508076,1.9318516526,1.0,1.4142135624],"formal_codegree_approx":18.9282032303},{"values_approx":[1.0,-0.5176380902,-0.7320508076,-0.5176380902,1.0,1.4142135624],"formal_codegree_approx":5.0717967697},{"values_approx":[1.0,0.5176380902,-0.7320508076,0.5176380902,1.0,-1.4142135624],"formal_codegree_approx":5.0717967697},{"values_approx":[1.0,{"re":0.0,"im":-1.0},0.0,{"re":0.0,"im":1.0},-1.0,0.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,{"re":0.0,"im":1.0},0.0,{"re":0.0,"im":-1.0},-1.0,0.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6],"primitive_polynomial":"(theta**2 + 8*theta + 20)*(theta**4 - 36*theta**3 + 1944*theta + 2916)","basis_values":["1","68563*theta**5/281152620 - 4452737*theta**4/562305240 - 418064*theta**3/14057631 + 40090049*theta**2/93717540 + 13278539*theta/5206530 + 591615/173551","-51403*theta**5/281152620 + 1692391*theta**4/281152620 + 534511*theta**3/28115262 - 4683769*theta**2/15619590 - 5101742*theta/2603265 - 527582/173551","2489*theta**5/140576310 - 363857*theta**4/562305240 + 22252*theta**3/14057631 - 1113031*theta**2/93717540 + 1038629*theta/5206530 + 151299/173551","3*theta**5/173551 - 5*theta**4/49586 - 3258*theta**3/173551 + 5832*theta**2/173551 + 184680*theta/173551 + 437449/173551","-9043*theta**5/562305240 + 2917*theta**4/20082330 + 843229*theta**3/56230524 - 597211*theta**2/46858770 - 1883051*theta/2603265 - 427746/173551"],"codegree_value":"-(48973*theta**5 - 1678216*theta**4 - 2706130*theta**3 + 79583922*theta**2 + 401397336*theta + 78620220)/70288155","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,1,3,5,4,2],"alternative_models":[{"name":"Diagonal fermion: Ising x half-E6 (equal rotation phases)","reference":"Aasen--Lake--Walker, arXiv:1709.01941v2, Sections 2.1 and 7.1--7.2, equations (9)--(13), (235), (240), (467); diagonal-fermion fixed-point theorem in this manuscript","source_url":"https://arxiv.org/abs/1709.01941","basis_permutation_model_to_census":[0,4,5,2,1,3]}]},{"rank":6,"multiplicity":2,"id":"R6_2_140","name":"Even D10 category from SU(2)_16","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],[[0,1,0,0,0,0],[1,1,1,0,0,0],[0,1,1,1,0,0],[0,0,1,1,1,1],[0,0,0,1,0,1],[0,0,0,1,1,0]],[[0,0,1,0,0,0],[0,1,1,1,0,0],[1,1,1,1,1,1],[0,1,1,2,1,1],[0,0,1,1,1,0],[0,0,1,1,0,1]],[[0,0,0,1,0,0],[0,0,1,1,1,1],[0,1,1,2,1,1],[1,1,2,2,1,1],[0,1,1,1,0,1],[0,1,1,1,1,0]],[[0,0,0,0,1,0],[0,0,0,1,0,1],[0,0,1,1,1,0],[0,1,1,1,0,1],[1,0,1,0,1,0],[0,1,0,1,0,0]],[[0,0,0,0,0,1],[0,0,0,1,1,0],[0,0,1,1,0,1],[0,1,1,1,1,0],[0,1,0,1,0,0],[1,0,1,0,0,1]]],"tensor_sha256":"72ab12e016874761e66ed55634cd56207a133062cbd5531fd737df20f31510cd","basis":["X0","X1","X2","X3","X4","X5"],"model_reference":"Rank-six manuscript, positive model catalogue, R6_2_140; Local modules over 1+V16; resolved even D10 fusion rules.","model_reference_url":null,"source_file":"supplements/rank6/data/additional_positive_models.json","source_location":"R6_2_140","invariants":{"dual_indices":[0,1,2,3,4,5],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,2.8793852416,4.4114741278,5.4114741278,2.8793852416,2.8793852416],"global_fp_dimension_approx":74.6177343244,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 9)**3*(t**3 - 81*t**2 + 486*t - 729)","formal_codegree_polynomial":"(t - 9)**3*(t**3 - 81*t**2 + 486*t - 729)","irreducible_characters_approx":[{"values_approx":[1.0,2.8793852416,4.4114741278,5.4114741278,2.8793852416,2.8793852416],"formal_codegree_approx":74.6177343244},{"values_approx":[1.0,-1.0,1.0,-1.0,-1.0,2.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,-1.0,1.0,-1.0,2.0,-1.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,2.0,1.0,-1.0,-1.0,-1.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,0.6527036447,-1.2266815969,-0.2266815969,0.6527036447,0.6527036447],"formal_codegree_approx":3.8341984298},{"values_approx":[1.0,-0.5320888862,-0.1847925309,0.8152074691,-0.5320888862,-0.5320888862],"formal_codegree_approx":2.5480672457}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6],"primitive_polynomial":"(theta - 5)*(theta - 2)*(theta + 7)*(theta**3 - 75*theta**2 + 108*theta + 1153)","basis_values":["1","3552557059*theta**5/4194159587892 - 124407419747*theta**4/2097079793946 - 992682712103*theta**3/4194159587892 + 10068962167393*theta**2/4194159587892 + 15078616689719*theta/4194159587892 - 31409212167031/2097079793946","-134799751*theta**5/116504432997 + 9460821175*theta**4/116504432997 + 36171247265*theta**3/116504432997 - 378408008395*theta**2/116504432997 - 543390739814*theta/116504432997 + 2280488421317/116504432997","48441548*theta**5/116504432997 - 3412852961*theta**4/116504432997 - 12041775775*theta**3/116504432997 + 136492173839*theta**2/116504432997 + 157049953447*theta/116504432997 - 827183311207/116504432997","788007968*theta**5/1048539896973 - 54966518189*theta**4/1048539896973 - 234669129916*theta**3/1048539896973 + 2082350334134*theta**2/1048539896973 + 3872870808172*theta/1048539896973 - 11246462083558/1048539896973","-849026849*theta**5/1398053195964 + 29628949507*theta**4/699026597982 + 250348279417*theta**3/1398053195964 - 2253985507979*theta**2/1398053195964 - 3741637963969*theta/1398053195964 + 6088710501749/699026597982"],"codegree_value":"-3*(35223870*theta**5 - 2476011137*theta**4 - 9171402608*theta**3 + 99030105243*theta**2 + 130788415852*theta - 662341450457)/38834810999","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,1,4,5,3,2],"alternative_models":[]},{"rank":6,"multiplicity":2,"id":"R6_2_143","name":"Commutative even part of the Izumi C5 subfactor","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],[[0,1,0,0,0,0],[1,1,1,1,1,1],[0,1,0,1,1,1],[0,1,1,0,1,1],[0,1,1,1,2,1],[0,1,1,1,1,2]],[[0,0,1,0,0,0],[0,1,0,1,1,1],[1,0,1,0,1,1],[0,1,0,0,1,1],[0,1,1,1,1,1],[0,1,1,1,1,1]],[[0,0,0,1,0,0],[0,1,1,0,1,1],[0,1,0,0,1,1],[1,0,0,1,1,1],[0,1,1,1,1,1],[0,1,1,1,1,1]],[[0,0,0,0,1,0],[0,1,1,1,2,1],[0,1,1,1,1,1],[0,1,1,1,1,1],[1,2,1,1,1,2],[0,1,1,1,2,2]],[[0,0,0,0,0,1],[0,1,1,1,1,2],[0,1,1,1,1,1],[0,1,1,1,1,1],[0,1,1,1,2,2],[1,2,1,1,2,1]]],"tensor_sha256":"1077145cc746d74c9883a7177df8e63961f45645b9d8011ef49e5b61b76d45dc","basis":["X0","X1","X2","X3","X4","X5"],"model_reference":"The actual HI(C5) Q-system dual is identified from its canonical module. Both possible dualities are included; exact module equations force every fusion coefficient without numerical center data. See the current manuscript and the dated exact proof.","model_reference_url":"https://arxiv.org/abs/1102.2631","source_file":"supplements/rank6/data/additional_positive_models.json","source_location":"R6_2_143","invariants":{"dual_indices":[0,1,2,3,4,5],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,5.1925824036,4.1925824036,4.1925824036,6.1925824036,6.1925824036],"global_fp_dimension_approx":139.8145600892,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 5)**4*(t**2 - 145*t + 725)","formal_codegree_polynomial":"(t - 5)**4*(t**2 - 145*t + 725)","irreducible_characters_approx":[{"values_approx":[1.0,5.1925824036,4.1925824036,4.1925824036,6.1925824036,6.1925824036],"formal_codegree_approx":139.8145600892},{"values_approx":[1.0,-0.1925824036,-1.1925824036,-1.1925824036,0.8074175964,0.8074175964],"formal_codegree_approx":5.1854399108},{"values_approx":[1.0,-1.0,-0.6180339887,1.6180339887,0.0,0.0],"formal_codegree_approx":5.0},{"values_approx":[1.0,-1.0,1.6180339887,-0.6180339887,0.0,0.0],"formal_codegree_approx":5.0},{"values_approx":[1.0,1.0,0.0,0.0,-1.6180339887,0.6180339887],"formal_codegree_approx":5.0},{"values_approx":[1.0,1.0,0.0,0.0,0.6180339887,-1.6180339887],"formal_codegree_approx":5.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6],"primitive_polynomial":"(theta**2 - 110*theta + 125)*(theta**2 - 5*theta + 5)*(theta**2 + 5*theta + 5)","basis_values":["1","-231931*theta**5/260173500 + 1676701*theta**4/17344900 + 925727*theta**3/17344900 - 5030103*theta**2/3468980 - 8557483*theta/10406940 + 1676701/693796","321423*theta**5/86724500 - 20963323*theta**4/52034700 - 2004963*theta**3/17344900 + 4123285*theta**2/693796 + 2749709*theta/3468980 - 18881935/2081388","186739*theta**5/260173500 - 1361647*theta**4/17344900 + 507057*theta**3/17344900 + 4431839*theta**2/3468980 - 2935343*theta/10406940 - 1708545/693796","-332851*theta**5/260173500 + 2424843*theta**4/17344900 - 707843*theta**3/17344900 - 6927631*theta**2/3468980 + 6952007*theta/10406940 + 1731047/693796","-251941*theta**5/260173500 + 1822049*theta**4/17344900 + 945737*theta**3/17344900 - 1162609*theta**2/693796 - 3374023*theta/10406940 + 2168947/693796"],"codegree_value":"-(2575*theta**5 - 280303*theta**4 - 38625*theta**3 + 4204545*theta**2 + 64375*theta - 59042275)/10406940","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,2,3,1,4,5],"alternative_models":[],"construction_proof_file":"evidence/correctness_20261025/rank6_r143_module_repair.md","construction_notes":["The actual HI(C5) Q-system dual is identified from its canonical module. Both possible dualities are included; exact module equations force every fusion coefficient without numerical center data.","Exact proof and tensor checks remain subject to independent human verification."]},{"rank":6,"multiplicity":2,"id":"R6_2_144","name":"G2 level 3","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],[[0,1,0,0,0,0],[1,1,1,0,1,0],[0,1,1,1,1,1],[0,0,1,1,0,1],[0,1,1,0,0,1],[0,0,1,1,1,1]],[[0,0,1,0,0,0],[0,1,1,1,1,1],[1,1,2,1,1,2],[0,1,1,1,1,1],[0,1,1,1,1,1],[0,1,2,1,1,1]],[[0,0,0,1,0,0],[0,0,1,1,0,1],[0,1,1,1,1,1],[1,1,1,1,0,0],[0,0,1,0,1,1],[0,1,1,0,1,1]],[[0,0,0,0,1,0],[0,1,1,0,0,1],[0,1,1,1,1,1],[0,0,1,0,1,1],[1,0,1,1,1,0],[0,1,1,1,0,1]],[[0,0,0,0,0,1],[0,0,1,1,1,1],[0,1,2,1,1,1],[0,1,1,0,1,1],[0,1,1,1,0,1],[1,1,1,1,1,1]]],"tensor_sha256":"4ea25b7f405370515c5e71829bdc2ced8adadbe2b77f0748a17e4732440e0255","basis":["X0","X1","X2","X3","X4","X5"],"model_reference":"Rank-six manuscript, positive model catalogue, R6_2_144; Principal unitary WZW category; exact Weyl-character reconstruction.","model_reference_url":null,"source_file":"supplements/rank6/data/complete_positive_models.json","source_location":"R6_2_144","invariants":{"dual_indices":[0,1,2,3,4,5],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,3.7912878475,5.7912878475,3.7912878475,3.7912878475,4.7912878475],"global_fp_dimension_approx":100.617044797,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 7)**3*(t - 3)*(t**2 - 105*t + 441)","formal_codegree_polynomial":"(t - 7)**3*(t - 3)*(t**2 - 105*t + 441)","irreducible_characters_approx":[{"values_approx":[1.0,3.7912878475,5.7912878475,3.7912878475,3.7912878475,4.7912878475],"formal_codegree_approx":100.617044797},{"values_approx":[1.0,-1.2469796037,0.0,0.4450418679,1.8019377358,-1.0],"formal_codegree_approx":7.0},{"values_approx":[1.0,0.4450418679,0.0,1.8019377358,-1.2469796037,-1.0],"formal_codegree_approx":7.0},{"values_approx":[1.0,1.8019377358,0.0,-1.2469796037,0.4450418679,-1.0],"formal_codegree_approx":7.0},{"values_approx":[1.0,-0.7912878475,1.2087121525,-0.7912878475,-0.7912878475,0.2087121525],"formal_codegree_approx":4.382955203},{"values_approx":[1.0,0.0,-1.0,0.0,0.0,1.0],"formal_codegree_approx":3.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6],"primitive_polynomial":"(theta - 4)*(theta**2 - 86*theta - 251)*(theta**3 + 4*theta**2 - 11*theta - 43)","basis_values":["1","22046023*theta**5/42007426580 - 47990839*theta**4/1024571380 + 136594079*theta**3/42007426580 + 2217011162*theta**2/1500265235 - 18228034023*theta/42007426580 - 111978541737/10501856645","-4311231*theta**5/6001060940 + 1340329*theta**4/20909620 - 29964103*theta**3/6001060940 - 2743997503*theta**2/1500265235 + 1253183431*theta/6001060940 + 2587778442/214323605","85133863*theta**5/42007426580 - 184932059*theta**4/1024571380 - 395838881*theta**3/42007426580 + 6943315787*theta**2/1500265235 - 19870780543*theta/42007426580 - 287979711617/10501856645","-16233597*theta**5/42007426580 + 36184181*theta**4/1024571380 - 3220876381*theta**3/42007426580 - 1525240503*theta**2/1500265235 + 56945574057*theta/42007426580 + 74623894263/10501856645","-5069611*theta**5/6001060940 + 1557009*theta**4/20909620 + 429614177*theta**3/6001060940 - 2724848408*theta**2/1500265235 - 3645192989*theta/6001060940 + 2081126432/214323605"],"codegree_value":"(10893089*theta**5 - 960675797*theta**4 - 879582763*theta**3 + 23607262168*theta**2 + 7353098471*theta - 98053628024)/6001060940","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,1,3,5,4,2],"alternative_models":[]},{"rank":6,"multiplicity":2,"id":"R6_2_2","name":"Fermionic C2-zesting of AdE6 x VecC2","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],[[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]],[[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]],[[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,1,0,2,0],[0,1,0,1,0,2]],[[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,1,0,2],[1,0,1,0,2,0]]],"tensor_sha256":"45fed6fac272e313032df057c4d9a58acd800cea581670a1f8a8debbe5749900","basis":["X0","X1","X2","X3","X4","X5"],"model_reference":"Rank-six manuscript, positive model catalogue, R6_2_2; Central-fermion lemma and associative zesting theorem in this manuscript.","model_reference_url":null,"source_file":"supplements/rank6/data/constructed_positive_models.json","source_location":"R6_2_2","invariants":{"dual_indices":[0,3,2,1,4,5],"invertible_indices":[0,1,2,3],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,1.0,2.7320508076,2.7320508076],"global_fp_dimension_approx":18.9282032303,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 4)**2*(t**2 - 24*t + 96)**2","formal_codegree_polynomial":"(t - 4)**2*(t**2 - 24*t + 96)**2","irreducible_characters_approx":[{"values_approx":[1.0,-1.0,1.0,-1.0,2.7320508076,-2.7320508076],"formal_codegree_approx":18.9282032303},{"values_approx":[1.0,1.0,1.0,1.0,2.7320508076,2.7320508076],"formal_codegree_approx":18.9282032303},{"values_approx":[1.0,-1.0,1.0,-1.0,-0.7320508076,0.7320508076],"formal_codegree_approx":5.0717967697},{"values_approx":[1.0,1.0,1.0,1.0,-0.7320508076,-0.7320508076],"formal_codegree_approx":5.0717967697},{"values_approx":[1.0,{"re":0.0,"im":-1.0},-1.0,{"re":0.0,"im":1.0},0.0,0.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,{"re":0.0,"im":1.0},-1.0,{"re":0.0,"im":-1.0},0.0,0.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6],"primitive_polynomial":"(theta**2 - 42*theta + 78)*(theta**2 + 4*theta + 8)*(theta**2 + 6*theta + 6)","basis_values":["1","-6769*theta**5/23349807 + 1562557*theta**4/186798456 + 2821909*theta**3/23349807 + 882967*theta**2/2394852 - 165119*theta/15566538 - 275889/199571","75*theta**5/112801 - 4949*theta**4/225602 - 20718*theta**3/112801 - 1668*theta**2/8677 + 83808*theta/112801 + 16795/8677","-58333*theta**5/93399228 + 3936433*theta**4/186798456 + 3579529*theta**3/23349807 + 444127*theta**2/2394852 - 382469*theta/15566538 - 214203/199571","-183197*theta**5/342463836 + 583457*theta**4/31133076 + 18011713*theta**3/171231918 + 42979*theta**2/399142 - 12700674*theta/28538653 - 2865308/2195281","1286993*theta**5/2054783016 - 1003385*theta**4/46699614 - 142096309*theta**3/1027391508 - 95853*theta**2/399142 + 15920980*theta/85615959 + 2479724/2195281"],"codegree_value":"-(69347*theta**5 - 2661736*theta**4 - 4573502*theta**3 - 3959670*theta**2 + 25187544*theta - 398140236)/85615959","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,1,3,5,4,2],"alternative_models":[]},{"rank":6,"multiplicity":2,"id":"R6_2_34","name":"TT3","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],[[0,1,0,0,0,0],[1,2,1,1,1,1],[0,1,0,0,1,0],[0,1,0,1,1,1],[0,1,1,1,1,0],[0,1,0,1,0,0]],[[0,0,1,0,0,0],[0,1,0,1,0,0],[1,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,1],[0,0,0,0,1,0]],[[0,0,0,1,0,0],[0,1,1,1,1,0],[0,0,0,1,0,1],[0,1,0,0,1,0],[1,1,0,0,0,1],[0,1,0,0,0,0]],[[0,0,0,0,1,0],[0,1,0,1,1,1],[0,1,0,0,0,0],[1,1,1,0,0,0],[0,1,0,1,0,0],[0,0,1,0,1,0]],[[0,0,0,0,0,1],[0,1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,1]]],"tensor_sha256":"8b70f11f014a113f7a8011ebe252b9208b235d177846fc5adb92a3bd896ee3a7","basis":["X0","X1","X2","X3","X4","X5"],"model_reference":"Rank-six manuscript, positive model catalogue, R6_2_34; Edie-Michell arXiv:1904.08909, Definition 2.5.","model_reference_url":"https://arxiv.org/abs/1904.08909","source_file":"supplements/rank6/data/constructed_positive_models.json","source_location":"R6_2_34","invariants":{"dual_indices":[0,1,2,4,3,5],"invertible_indices":[0],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,4.2360679775,1.6180339887,2.6180339887,2.6180339887,1.6180339887],"global_fp_dimension_approx":37.88854382,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 8)**4*(t**2 - 40*t + 80)","irreducible_characters_approx":[{"degree":2,"values_approx":[2.0,0.0,1.0,-1.0,-1.0,1.0],"formal_codegree_approx":4.0,"formal_codegree_minimal_polynomial":"u - 4","formal_codegree_exact":"4"},{"degree":1,"values_approx":[1.0,-0.2360679775,-0.6180339887,0.3819660113,0.3819660113,-0.6180339887],"formal_codegree_approx":2.11145618,"formal_codegree_minimal_polynomial":"u**2 - 40*u + 80"},{"degree":1,"values_approx":[1.0,4.2360679775,1.6180339887,2.6180339887,2.6180339887,1.6180339887],"formal_codegree_approx":37.88854382,"formal_codegree_minimal_polynomial":"u**2 - 40*u + 80"}],"character_method":"Numerical regular-algebra block traces after exact rational center-kernel computation; χ_E(X_i) and f_E=(Σ_i χ_E(X_i)χ_E(X_i*)) / dim(E) are approximate. The displayed codegree minimal polynomials are exact factors of the rational central-Casimir polynomial rescaled by each irreducible degree; factor association uses the validated numerical eigenvalues. Orthogonality and block dimensions validated."},"basis_permutation_model_to_census":[0,3,4,2,5,1],"alternative_models":[]},{"rank":6,"multiplicity":2,"id":"R6_2_36","name":"C(A4,C2)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]],[[0,0,1,0,0,0],[0,0,1,0,0,0],[0,0,0,2,0,0],[1,1,0,0,1,1],[0,0,1,0,0,0],[0,0,1,0,0,0]],[[0,0,0,1,0,0],[0,0,0,1,0,0],[1,1,0,0,1,1],[0,0,2,0,0,0],[0,0,0,1,0,0],[0,0,0,1,0,0]],[[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]],[[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]],"tensor_sha256":"a5097fb946d75a6a71893e2c4089ee1db000968903c454b5be47912c09ddcda7","basis":["X0","X1","X2","X3","X4","X5"],"model_reference":"Rank-six manuscript, positive model catalogue, R6_2_36; Construction in the manuscript.","model_reference_url":null,"source_file":"supplements/rank6/data/group_theoretical_models.json","source_location":"R6_2_36","invariants":{"dual_indices":[0,1,3,2,4,5],"invertible_indices":[0,1,4,5],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,2.0,2.0,1.0,1.0],"global_fp_dimension_approx":12.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 12)**3*(t - 4)**3","formal_codegree_polynomial":"(t - 12)**3*(t - 4)**3","irreducible_characters_approx":[{"values_approx":[1.0,1.0,2.0,2.0,1.0,1.0],"formal_codegree_approx":12.0},{"values_approx":[1.0,1.0,{"re":-1.0,"im":-1.7320508076},{"re":-1.0,"im":1.7320508076},1.0,1.0],"formal_codegree_approx":12.0},{"values_approx":[1.0,1.0,{"re":-1.0,"im":1.7320508076},{"re":-1.0,"im":-1.7320508076},1.0,1.0],"formal_codegree_approx":12.0},{"values_approx":[1.0,-1.0,0.0,0.0,-1.0,1.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,-1.0,0.0,0.0,1.0,-1.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,1.0,0.0,0.0,-1.0,-1.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6],"primitive_polynomial":"theta*(theta - 28)*(theta + 2)*(theta + 8)*(theta**2 - 14*theta + 52)","basis_values":["1","31*theta**5/1572480 - 7*theta**4/14040 - 27*theta**3/7280 + 1681*theta**2/28080 + 6401*theta/49140 - 1","50287*theta**5/1658180160 - 280057*theta**4/414545040 - 129097*theta**3/23030280 + 119407*theta**2/5601960 + 1532851*theta/25909065","-90653*theta**5/1658180160 + 570173*theta**4/414545040 + 171683*theta**3/23030280 - 437873*theta**2/5601960 - 4515149*theta/25909065","815*theta**5/8963136 - 6611*theta**4/2240784 + 655*theta**3/124488 + 219155*theta**2/1120392 - 91717*theta/140049 - 1","-421*theta**5/6894720 + 1763*theta**4/861840 - 509*theta**3/95760 - 121957*theta**2/861840 + 23257*theta/30780 + 1"],"codegree_value":"(1111*theta**5 - 31444*theta**4 - 84312*theta**3 + 2553656*theta**2 + 5175232*theta + 44815680)/11203920","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. 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Sci. Ec. Norm. Super. 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All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,1,5,2,3,6,4]},{"rank":7,"multiplicity":1,"id":"R7_1_12","name":"SO(7)_2","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[1,0,0,0,1,1,1],[0,1,0,0,1,1,1],[0,0,1,1,0,0,0],[0,0,1,1,0,0,0],[0,0,1,1,0,0,0]],[[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,1,1,1],[1,0,0,0,1,1,1],[0,0,1,1,0,0,0],[0,0,1,1,0,0,0],[0,0,1,1,0,0,0]],[[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,1,1,0,0,0],[0,0,1,1,0,0,0],[1,1,0,0,0,0,1],[0,0,0,0,0,1,1],[0,0,0,0,1,1,0]],[[0,0,0,0,0,1,0],[0,0,0,0,0,1,0],[0,0,1,1,0,0,0],[0,0,1,1,0,0,0],[0,0,0,0,0,1,1],[1,1,0,0,1,0,0],[0,0,0,0,1,0,1]],[[0,0,0,0,0,0,1],[0,0,0,0,0,0,1],[0,0,1,1,0,0,0],[0,0,1,1,0,0,0],[0,0,0,0,1,1,0],[0,0,0,0,1,0,1],[1,1,0,0,0,1,0]]],"tensor_sha256":"264a832c81bd0ae47fa4d0cb0cf2cda7f935ecd20f9b2ecc65730ba140b48188","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_1_12; unitary odd metaplectic SO(7) at level two; see Vercleyen, On Low-Rank Multiplicity-Free Fusion Categories (2024), Appendix; construction: unitary odd metaplectic SO(7) category at level two.","model_reference_url":"https://arxiv.org/abs/2405.20075","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_1_12","invariants":{"dual_indices":[0,1,2,3,4,5,6],"invertible_indices":[0,1],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,2.6457513111,2.6457513111,2.0,2.0,2.0],"global_fp_dimension_approx":28.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 28)**2*(t - 7)**3*(t - 4)**2","formal_codegree_polynomial":"(t - 28)**2*(t - 7)**3*(t - 4)**2","irreducible_characters_approx":[{"values_approx":[1.0,1.0,-2.6457513111,-2.6457513111,2.0,2.0,2.0],"formal_codegree_approx":28.0},{"values_approx":[1.0,1.0,2.6457513111,2.6457513111,2.0,2.0,2.0],"formal_codegree_approx":28.0},{"values_approx":[1.0,1.0,0.0,0.0,-0.4450418679,1.2469796037,-1.8019377358],"formal_codegree_approx":7.0},{"values_approx":[1.0,1.0,0.0,0.0,-1.8019377358,-0.4450418679,1.2469796037],"formal_codegree_approx":7.0},{"values_approx":[1.0,1.0,0.0,0.0,1.2469796037,-1.8019377358,-0.4450418679],"formal_codegree_approx":7.0},{"values_approx":[1.0,-1.0,-1.0,1.0,0.0,0.0,0.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,-1.0,1.0,-1.0,0.0,0.0,0.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7],"primitive_polynomial":"theta*(theta + 2)*(theta**2 - 78*theta + 1178)*(theta**3 + 9*theta**2 + 20*theta - 1)","basis_values":["1","4054*theta**6/5122533 - 90343*theta**5/1707511 + 1410691*theta**4/5122533 + 40965926*theta**3/5122533 + 782323*theta**2/22971 + 200804074*theta/5122533 - 1","2186800876024*theta**6/4825180738866943 - 20970262440163*theta**5/689311534123849 + 801959563402734*theta**4/4825180738866943 + 21801861593817982*theta**3/4825180738866943 + 397859725139808*theta**2/21637581788641 + 94247133826768393*theta/4825180738866943 - 1","-6560048233187*theta**6/14475542216600829 + 20969482182347*theta**5/689311534123849 - 2406088344539153*theta**4/14475542216600829 - 65406341742852529*theta**3/14475542216600829 - 1193582725981517*theta**2/64912745365923 - 282741360912789437*theta/14475542216600829 + 1","-1394337647684*theta**6/2067934602371547 + 4418524285380*theta**5/98473076303407 - 454309790220638*theta**4/2067934602371547 - 14397903180648328*theta**3/2067934602371547 - 281388053015525*theta**2/9273249337989 - 73071180208709510*theta/2067934602371547","-603645914141*theta**6/2067934602371547 + 1974480254388*theta**5/98473076303407 - 289151912813264*theta**4/2067934602371547 - 5353818117594292*theta**3/2067934602371547 - 81460931553437*theta**2/9273249337989 - 17912260339688726*theta/2067934602371547","1179630369910*theta**6/2067934602371547 - 3787792888629*theta**5/98473076303407 + 458757554435242*theta**4/2067934602371547 + 11483019258749354*theta**3/2067934602371547 + 204940195545823*theta**2/9273249337989 + 50451757196338042*theta/2067934602371547"],"codegree_value":"(116888758153*theta**6 - 7814753019156*theta**5 + 40683286282687*theta**4 + 1181283802623224*theta**3 + 5030565366083002*theta**2 + 5790238300501996*theta + 393892305213628)/98473076303407","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. 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approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7],"primitive_polynomial":"(theta - 1)*(theta**2 - 3*theta - 9)*(theta**4 - 114*theta**3 + 1056*theta**2 + 5616*theta - 4464)","basis_values":["1","-251222650345*theta**6/2501130475279854 + 58865202396725*theta**5/5002260950559708 - 6544267908905*theta**4/46317231023701 - 139817133600820*theta**3/416855079213309 + 1794373555549837*theta**2/555806772284412 + 494595487104262*theta/138951693071103 - 7952901674094/1494104226571","174103992827*theta**6/3334840633706472 - 10156391087413*theta**5/1667420316853236 + 39182630412907*theta**4/555806772284412 + 229352911666237*theta**3/1111613544568824 - 295235319898539*theta**2/185268924094804 - 211024714371657*theta/92634462047402 + 3885873209029/1494104226571","212837190827*theta**6/3334840633706472 - 12514434981769*theta**5/1667420316853236 + 51752862714887*theta**4/555806772284412 + 186137065328557*theta**3/1111613544568824 - 383240982185579*theta**2/185268924094804 - 132926192580297*theta/92634462047402 + 4856437398176/1494104226571","-16663174829*theta**6/1667420316853236 + 1968280618145*theta**5/1667420316853236 - 702764794989*theta**4/46317231023701 - 12004576152053*theta**3/555806772284412 + 73106698418733*theta**2/185268924094804 - 2285113410453*theta/92634462047402 - 1993626270252/1494104226571","-743425612337*theta**6/10004521901119416 + 43553441074075*theta**5/5002260950559708 - 58221091105553*theta**4/555806772284412 - 779043360960439*theta**3/3334840633706472 + 1255391913210233*theta**2/555806772284412 + 305494729126157*theta/138951693071103 - 4672850473044/1494104226571","407035413755*theta**6/10004521901119416 - 11928180067295*theta**5/2501130475279854 + 31999360292587*theta**4/555806772284412 + 424807228428361*theta**3/3334840633706472 - 177203192904581*theta**2/138951693071103 - 131271562165961*theta/138951693071103 + 3047652013050/1494104226571"],"codegree_value":"(307960953779*theta**6 - 36068830801826*theta**5 + 432231262185510*theta**4 + 1005346631314695*theta**3 - 9365494224597750*theta**2 - 7666447912597944*theta + 32304328282075896)/3334840633706472","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. 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All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,1,4,3,6,2,5]},{"rank":7,"multiplicity":1,"id":"R7_1_25","name":"Rep(SD16)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[0,0,0,0,1,1,1],[1,0,0,1,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,1,0,0,0,0],[0,1,1,0,0,0,0]],[[0,0,1,0,0,0,0],[1,0,0,1,0,0,1],[0,0,0,0,1,1,1],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,1,0,0,0,0,0],[0,1,1,0,0,0,0]],[[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]],[[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]],[[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1]],[[0,0,0,0,0,0,1],[0,1,1,0,0,0,0],[0,1,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,0,1],[0,0,0,0,0,0,1],[1,0,0,1,1,1,0]]],"tensor_sha256":"b773b993c4445b06ac827f2d9495befb1d5c8cfd7e1f6218889a51afec49324a","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_1_25; ordinary complex representation category of the explicitly presented semidihedral group of order sixteen; construction: representations of C8 semidirect C2 with action a to a cubed.","model_reference_url":"https://www.ams.org/books/surv/205/","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_1_25","invariants":{"dual_indices":[0,2,1,3,4,5,6],"invertible_indices":[0,3,4,5],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,2.0,2.0,1.0,1.0,1.0,2.0],"global_fp_dimension_approx":16.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 16)**2*(t - 8)**3*(t - 4)**2","formal_codegree_polynomial":"(t - 16)**2*(t - 8)**3*(t - 4)**2","irreducible_characters_approx":[{"values_approx":[1.0,-2.0,-2.0,1.0,1.0,1.0,2.0],"formal_codegree_approx":16.0},{"values_approx":[1.0,2.0,2.0,1.0,1.0,1.0,2.0],"formal_codegree_approx":16.0},{"values_approx":[1.0,0.0,0.0,1.0,1.0,1.0,-2.0],"formal_codegree_approx":8.0},{"values_approx":[1.0,{"re":0.0,"im":-1.4142135624},{"re":0.0,"im":1.4142135624},1.0,-1.0,-1.0,0.0],"formal_codegree_approx":8.0},{"values_approx":[1.0,{"re":0.0,"im":1.4142135624},{"re":0.0,"im":-1.4142135624},1.0,-1.0,-1.0,0.0],"formal_codegree_approx":8.0},{"values_approx":[1.0,0.0,0.0,-1.0,-1.0,1.0,0.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,0.0,0.0,-1.0,1.0,-1.0,0.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7],"primitive_polynomial":"(theta - 40)*(theta - 20)*(theta - 2)*(theta + 2)*(theta + 4)*(theta**2 + 12*theta + 38)","basis_values":["1","10454903*theta**6/3025258346880 - 649393*theta**5/3501456420 + 106731443*theta**4/72029960640 + 222910099*theta**3/14005825680 + 2272542577*theta**2/126052431120 - 10119991*theta/166736020 - 23894468/248787693","-10356937*theta**6/3025258346880 + 5820893*theta**5/31513107780 - 108001997*theta**4/72029960640 - 2018942389*theta**3/126052431120 - 254224087*theta**2/14005825680 + 275115443*theta/4501872540 + 24086332/248787693","-theta**6/88704 + 19*theta**5/33264 - 53*theta**4/19008 - 2603*theta**3/33264 - 4139*theta**2/33264 + 955*theta/1188 + 1129/2079","74665*theta**6/2653735392 - 592573*theta**5/442289232 + 171115*theta**4/63184176 + 4257233*theta**3/20104056 + 25604102*theta**2/27643077 - 1285810*theta/3949011 - 310991969/82929231","-20017*theta**6/1516420224 + 58219*theta**5/94776264 - 379849*theta**4/758210112 - 19938907*theta**3/189552528 - 99292717*theta**2/189552528 + 19473155*theta/47388132 + 36775183/11847033","-1125875*theta**6/605051669376 + 1963883*theta**5/25210486224 + 164091*theta**4/533555264 - 354507565*theta**3/25210486224 - 3501097217*theta**2/25210486224 - 400665535*theta/900374508 - 111781064/248787693"],"codegree_value":"-(6838985*theta**6 - 346045752*theta**5 + 1676779986*theta**4 + 47288284776*theta**3 + 75215007816*theta**2 - 486142241760*theta - 2143281214208)/302525834688","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,4,3,5,1,6,2]},{"rank":7,"multiplicity":1,"id":"R7_1_28","name":"AnyonWiki FC(7,1,2,12,1,0,1)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[0,0,0,1,1,1,1],[1,0,0,0,1,1,1],[0,0,1,0,0,0,0],[0,1,1,0,0,0,0],[0,1,1,0,0,0,0],[0,1,1,0,0,0,0]],[[0,0,1,0,0,0,0],[1,0,0,0,1,1,1],[0,0,0,1,1,1,1],[0,1,0,0,0,0,0],[0,1,1,0,0,0,0],[0,1,1,0,0,0,0],[0,1,1,0,0,0,0]],[[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,0,0,0,1,0,0],[0,1,1,0,0,0,0],[0,1,1,0,0,0,0],[0,0,0,0,1,0,0],[1,0,0,1,0,0,1],[0,0,0,0,0,1,1],[0,0,0,0,1,1,0]],[[0,0,0,0,0,1,0],[0,1,1,0,0,0,0],[0,1,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,1,1],[1,0,0,1,1,0,0],[0,0,0,0,1,0,1]],[[0,0,0,0,0,0,1],[0,1,1,0,0,0,0],[0,1,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,1,0],[0,0,0,0,1,0,1],[1,0,0,1,0,1,0]]],"tensor_sha256":"5a8aee35466610b40ae273b1179d10e23f9c727fed8c0580c9d1f962f5643767","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_1_28; Vercleyen--Slingerland, On Low-Rank Multiplicity-Free Complex Fusion Categories, CMP 407, 231 (2026), Section 5; AnyonWiki category [7, 1, 2, 12, 1, 0, 1]; construction: Published skeletal fusion category; independent exact pentagon, invertibility, unit and rigidity replay over Q(zeta_28). A unitary gauge is recorded in the cited category list..","model_reference_url":"https://anyonwiki.github.io/pages/Lists/losmffc.html","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_1_28","invariants":{"dual_indices":[0,2,1,3,4,5,6],"invertible_indices":[0,3],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,2.6457513111,2.6457513111,1.0,2.0,2.0,2.0],"global_fp_dimension_approx":28.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 28)**2*(t - 7)**3*(t - 4)**2","formal_codegree_polynomial":"(t - 28)**2*(t - 7)**3*(t - 4)**2","irreducible_characters_approx":[{"values_approx":[1.0,-2.6457513111,-2.6457513111,1.0,2.0,2.0,2.0],"formal_codegree_approx":28.0},{"values_approx":[1.0,2.6457513111,2.6457513111,1.0,2.0,2.0,2.0],"formal_codegree_approx":28.0},{"values_approx":[1.0,0.0,0.0,1.0,-0.4450418679,1.2469796037,-1.8019377358],"formal_codegree_approx":7.0},{"values_approx":[1.0,0.0,0.0,1.0,-1.8019377358,-0.4450418679,1.2469796037],"formal_codegree_approx":7.0},{"values_approx":[1.0,0.0,0.0,1.0,1.2469796037,-1.8019377358,-0.4450418679],"formal_codegree_approx":7.0},{"values_approx":[1.0,{"re":0.0,"im":-1.0},{"re":0.0,"im":1.0},-1.0,0.0,0.0,0.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,{"re":0.0,"im":1.0},{"re":0.0,"im":-1.0},-1.0,0.0,0.0,0.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7],"primitive_polynomial":"(theta**2 - 82*theta + 1506)*(theta**2 + 6*theta + 10)*(theta**3 + 3*theta**2 - 4*theta - 13)","basis_values":["1","-73681100599*theta**6/1025089450863170 + 11093552828541*theta**5/2050178901726340 - 28356039271709*theta**4/410035780345268 - 700520659388511*theta**3/1025089450863170 - 1919711396633277*theta**2/2050178901726340 + 2802117221057983*theta/1025089450863170 + 1073650213070279/205017890172634","73690634821*theta**6/1025089450863170 - 5547197783837*theta**5/1025089450863170 + 7088576240252*theta**4/102508945086317 + 700508420387409*theta**3/1025089450863170 + 479929061760697*theta**2/512544725431585 - 1401029306777116*theta/512544725431585 - 536818513226801/102508945086317","75*theta**6/723404 - 1451*theta**5/180851 + 84641*theta**4/723404 + 514451*theta**3/723404 + 212943*theta**2/723404 - 517067*theta/180851 - 74797/32882","219803212149*theta**6/410035780345268 - 4156229506992*theta**5/102508945086317 + 217880116927019*theta**4/410035780345268 + 1972197042325757*theta**3/410035780345268 + 2834523494265585*theta**2/410035780345268 - 1973570251618249*theta/102508945086317 - 758329897210205/18637990015694","-388899561079*theta**6/410035780345268 + 7346808387444*theta**5/102508945086317 - 383406093390829*theta**4/410035780345268 - 3518357922655747*theta**3/410035780345268 - 5224760995428479*theta**2/410035780345268 + 3417844815014951*theta/102508945086317 + 1351255195574995/18637990015694","147836159189*theta**6/410035780345268 - 2779301164477*theta**5/102508945086317 + 141540314288755*theta**4/410035780345268 + 1400367982240221*theta**3/410035780345268 + 2329886672094609*theta**2/410035780345268 - 1297741504264445*theta/102508945086317 - 581047807443725/18637990015694"],"codegree_value":"(2276880834*theta**6 - 176215684518*theta**5 + 2570364928092*theta**4 + 15621998552973*theta**3 + 6465891716121*theta**2 - 62806262215428*theta + 30572425908794)/14644135012331","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,3,4,5,6,1,2]},{"rank":7,"multiplicity":1,"id":"R7_1_36","name":"AnyonWiki FC(7,1,4,3,1,0,1)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0]],[[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,1,0,0,0,0,0]],[[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,1,1,0,0,1,0],[1,0,0,0,0,1,1],[0,0,0,1,1,0,0],[0,0,0,1,0,0,0]],[[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,1,0,0,0],[1,0,0,0,0,1,1],[0,1,1,0,0,1,0],[0,0,0,1,1,0,0],[0,0,0,0,1,0,0]],[[0,0,0,0,0,1,0],[0,0,0,0,0,1,0],[0,0,0,0,0,1,0],[0,0,0,1,1,0,0],[0,0,0,1,1,0,0],[1,1,1,0,0,0,1],[0,0,0,0,0,1,0]],[[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0]]],"tensor_sha256":"8e1e5c46e0bd27bfcc315417498f23bba9bdc28407cad57fbc09c626d861503d","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_1_36; Vercleyen--Slingerland, On Low-Rank Multiplicity-Free Complex Fusion Categories, CMP 407, 231 (2026), Section 5; AnyonWiki category [7, 1, 4, 3, 1, 0, 1]; construction: Published skeletal fusion category; independent exact pentagon, invertibility, unit and rigidity replay over Q(zeta_16). A unitary gauge is recorded in the cited category list..","model_reference_url":"https://anyonwiki.github.io/pages/Lists/losmffc.html","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_1_36","invariants":{"dual_indices":[0,2,1,4,3,5,6],"invertible_indices":[0,1,2,6],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,2.0,2.0,2.0,1.0],"global_fp_dimension_approx":16.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 16)**2*(t - 8)**3*(t - 4)**2","formal_codegree_polynomial":"(t - 16)**2*(t - 8)**3*(t - 4)**2","irreducible_characters_approx":[{"values_approx":[1.0,1.0,1.0,-2.0,-2.0,2.0,1.0],"formal_codegree_approx":16.0},{"values_approx":[1.0,1.0,1.0,2.0,2.0,2.0,1.0],"formal_codegree_approx":16.0},{"values_approx":[1.0,-1.0,-1.0,{"re":0.0,"im":-1.4142135624},{"re":0.0,"im":1.4142135624},0.0,1.0],"formal_codegree_approx":8.0},{"values_approx":[1.0,-1.0,-1.0,{"re":0.0,"im":1.4142135624},{"re":0.0,"im":-1.4142135624},0.0,1.0],"formal_codegree_approx":8.0},{"values_approx":[1.0,1.0,1.0,0.0,0.0,-2.0,1.0],"formal_codegree_approx":8.0},{"values_approx":[1.0,{"re":0.0,"im":-1.0},{"re":0.0,"im":1.0},0.0,0.0,0.0,-1.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,{"re":0.0,"im":1.0},{"re":0.0,"im":-1.0},0.0,0.0,0.0,-1.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7],"primitive_polynomial":"(theta - 43)*(theta - 7)*(theta - 1)*(theta**2 - 6*theta + 11)*(theta**2 + 12*theta + 37)","basis_values":["1","1916929399*theta**6/77802600475800 - 43344540463*theta**5/38901300237900 + 56638653463*theta**4/77802600475800 + 9196555577*theta**3/127128432150 - 5881688099323*theta**2/77802600475800 - 1823712746411*theta/1556052009516 + 169269767380613/77802600475800","1548491827*theta**6/77802600475800 - 16929609707*theta**5/19450650118950 - 73750572791*theta**4/77802600475800 + 8767278851*theta**3/127128432150 - 1943694886039*theta**2/77802600475800 - 2536786195504*theta/1945065011895 + 175992089044979/77802600475800","-2332236701*theta**6/108923640666120 + 48487120679*theta**5/54461820333060 + 58750054667*theta**4/21784728133224 - 10225265557*theta**3/177979805010 - 352778333093*theta**2/3112104019032 + 37886877294919*theta/54461820333060 - 8222434669813/15560520095160","2991496049*theta**6/108923640666120 - 61011700361*theta**5/54461820333060 - 91051066859*theta**4/21784728133224 + 11543652103*theta**3/177979805010 + 415591247345*theta**2/3112104019032 - 45064882414081*theta/54461820333060 + 9870520474717/15560520095160","-9112181*theta**6/309090921300 + 197664377*theta**5/154545460650 + 417207583*theta**4/309090921300 - 40831573*theta**3/505050525 + 2261010851*theta**2/44155845900 + 40828537097*theta/30909092130 - 145444113001/44155845900","4169*theta**6/1798737700 - 204101*theta**5/1798737700 + 218392*theta**4/449684425 + 147438*theta**3/26452025 - 75897253*theta**2/1798737700 + 35244457*theta/359747540 + 421928312/449684425"],"codegree_value":"-(1016633204*theta**6 - 31723503271*theta**5 - 535986900727*theta**4 + 499284084204*theta**3 + 16985780157892*theta**2 - 62566686084125*theta - 1043588091048377)/136154550832650","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,6,1,2,5,3,4]},{"rank":7,"multiplicity":1,"id":"R7_1_39","name":"TY(C6)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0]],[[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]],[[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,1,0,0,0,0,0]],[[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0]],[[0,0,0,0,0,1,0],[0,0,0,0,0,1,0],[0,0,0,0,0,1,0],[0,0,0,0,0,1,0],[0,0,0,0,0,1,0],[1,1,1,1,1,0,1],[0,0,0,0,0,1,0]],[[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0]]],"tensor_sha256":"fd67a74df977b0a87170fc121ca609bf4e7fd28d2d0c7584f6bdea6f1dee875e","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_1_39; 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All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,1,4,6,3,2,5]},{"rank":7,"multiplicity":1,"id":"R7_1_43","name":"Vec(C7)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0]],[[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0]],[[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0]],[[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0]],[[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0]],[[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0]]],"tensor_sha256":"c1db33c04c7d99b815f79b2ba8f8759045b5a61e38c410e50a43959bee824512","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_1_43; EGNO, Tensor Categories, pointed categories; construction: pointed fusion category on the cyclic group of order seven.","model_reference_url":"https://www.ams.org/books/surv/205/","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_1_43","invariants":{"dual_indices":[0,2,1,4,3,6,5],"invertible_indices":[0,1,2,3,4,5,6],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,1.0,1.0,1.0,1.0],"global_fp_dimension_approx":7.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 7)**7","formal_codegree_polynomial":"(t - 7)**7","irreducible_characters_approx":[{"values_approx":[1.0,1.0,1.0,1.0,1.0,1.0,1.0],"formal_codegree_approx":7.0},{"values_approx":[1.0,{"re":-0.222520934,"im":-0.9749279122},{"re":-0.222520934,"im":0.9749279122},{"re":0.6234898019,"im":-0.7818314825},{"re":0.6234898019,"im":0.7818314825},{"re":-0.9009688679,"im":-0.4338837391},{"re":-0.9009688679,"im":0.4338837391}],"formal_codegree_approx":7.0},{"values_approx":[1.0,{"re":-0.222520934,"im":0.9749279122},{"re":-0.222520934,"im":-0.9749279122},{"re":0.6234898019,"im":0.7818314825},{"re":0.6234898019,"im":-0.7818314825},{"re":-0.9009688679,"im":0.4338837391},{"re":-0.9009688679,"im":-0.4338837391}],"formal_codegree_approx":7.0},{"values_approx":[1.0,{"re":-0.9009688679,"im":-0.4338837391},{"re":-0.9009688679,"im":0.4338837391},{"re":-0.222520934,"im":0.9749279122},{"re":-0.222520934,"im":-0.9749279122},{"re":0.6234898019,"im":-0.7818314825},{"re":0.6234898019,"im":0.7818314825}],"formal_codegree_approx":7.0},{"values_approx":[1.0,{"re":-0.9009688679,"im":0.4338837391},{"re":-0.9009688679,"im":-0.4338837391},{"re":-0.222520934,"im":-0.9749279122},{"re":-0.222520934,"im":0.9749279122},{"re":0.6234898019,"im":0.7818314825},{"re":0.6234898019,"im":-0.7818314825}],"formal_codegree_approx":7.0},{"values_approx":[1.0,{"re":0.6234898019,"im":-0.7818314825},{"re":0.6234898019,"im":0.7818314825},{"re":-0.9009688679,"im":0.4338837391},{"re":-0.9009688679,"im":-0.4338837391},{"re":-0.222520934,"im":0.9749279122},{"re":-0.222520934,"im":-0.9749279122}],"formal_codegree_approx":7.0},{"values_approx":[1.0,{"re":0.6234898019,"im":0.7818314825},{"re":0.6234898019,"im":-0.7818314825},{"re":-0.9009688679,"im":-0.4338837391},{"re":-0.9009688679,"im":0.4338837391},{"re":-0.222520934,"im":-0.9749279122},{"re":-0.222520934,"im":0.9749279122}],"formal_codegree_approx":7.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; 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All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,1,6,4,3,5,2]},{"rank":7,"multiplicity":1,"id":"R7_1_6","name":"SU(2)_6","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]],[[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[1,0,0,0,1,0,0],[0,0,0,0,0,1,1],[0,0,1,0,0,0,1],[0,1,0,1,0,0,0],[0,0,0,1,1,0,0]],[[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,1],[1,0,0,1,1,0,0],[0,1,0,1,1,0,0],[0,0,1,0,0,0,1],[0,0,1,0,0,1,1]],[[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,1],[0,1,0,1,1,0,0],[1,0,0,1,1,0,0],[0,0,0,0,0,1,1],[0,0,1,0,0,1,1]],[[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,1,0,1,0,0,0],[0,0,1,0,0,0,1],[0,0,0,0,0,1,1],[1,0,0,0,1,0,0],[0,0,0,1,1,0,0]],[[0,0,0,0,0,0,1],[0,0,0,0,0,0,1],[0,0,0,1,1,0,0],[0,0,1,0,0,1,1],[0,0,1,0,0,1,1],[0,0,0,1,1,0,0],[1,1,0,1,1,0,0]]],"tensor_sha256":"01203661d2904c7965e0080fac30377858131b599ba6238cc286fe980b0d7077","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_1_6; 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All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,5,1,2,4,3,6]},{"rank":7,"multiplicity":2,"id":"R7_2_195","name":"TY(C3)-twisted-C4-equivariantization","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]],[[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0]],[[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,1,0],[1,0,0,0,0,1,1],[0,0,0,0,2,0,0],[0,1,1,1,0,0,0],[0,0,0,1,0,0,0]],[[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,0,0,2,0,0],[1,1,1,2,0,2,1],[0,0,0,0,2,0,0],[0,0,0,0,1,0,0]],[[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,1,0,0,0],[0,1,1,1,0,0,0],[0,0,0,0,2,0,0],[1,0,0,0,0,1,1],[0,0,0,0,0,1,0]],[[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0]]],"tensor_sha256":"d9ba378c7d570f70991119f50d2bd41d552df4bb81955714b71b0ed3963a3f5f","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_2_195; Tambara--Yamagami, J. Algebra 209 (1998), 692-707; Gelaki--Naidu--Nikshych, Centers of graded fusion categories, Section 4A and Proposition 4.8; explicit coherent actions in papers/ty_equivariant_models.md.; construction: The C2 equivariantization of the diagonal subcategory of TY(C3,chi,+1/sqrt(3)) Deligne Vec(C4,omega1); simultaneous inversion, explicit scalar tensorator U, and identity square coherence. All phases, cocycle equations and equivariant Hom invariants are replayed in software/ty_equivariant_models.py..","model_reference_url":"https://msp.org/ant/2009/3-8/ant-v3-n8-p05-s.pdf","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_2_195","invariants":{"dual_indices":[0,2,1,3,4,5,6],"invertible_indices":[0,1,2,6],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,2.0,3.4641016151,2.0,1.0],"global_fp_dimension_approx":24.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 24)**2*(t - 12)*(t - 6)**2*(t - 4)**2","formal_codegree_polynomial":"(t - 24)**2*(t - 12)*(t - 6)**2*(t - 4)**2","irreducible_characters_approx":[{"values_approx":[1.0,1.0,1.0,2.0,-3.4641016151,2.0,1.0],"formal_codegree_approx":24.0},{"values_approx":[1.0,1.0,1.0,2.0,3.4641016151,2.0,1.0],"formal_codegree_approx":24.0},{"values_approx":[1.0,-1.0,-1.0,-2.0,0.0,2.0,1.0],"formal_codegree_approx":12.0},{"values_approx":[1.0,-1.0,-1.0,1.0,0.0,-1.0,1.0],"formal_codegree_approx":6.0},{"values_approx":[1.0,1.0,1.0,-1.0,0.0,-1.0,1.0],"formal_codegree_approx":6.0},{"values_approx":[1.0,{"re":0.0,"im":-1.0},{"re":0.0,"im":1.0},0.0,0.0,0.0,-1.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,{"re":0.0,"im":1.0},{"re":0.0,"im":-1.0},0.0,0.0,0.0,-1.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7],"primitive_polynomial":"(theta - 7)*(theta - 3)*(theta - 1)*(theta**2 - 66*theta + 789)*(theta**2 + 12*theta + 37)","basis_values":["1","-16401817659*theta**6/3544544818174400 + 496048839443*theta**5/1772272409087200 - 244155982895*theta**4/141781792726976 - 15683037084651*theta**3/443068102271800 + 457525720323267*theta**2/3544544818174400 + 1737780109531353*theta/1772272409087200 - 7347038257172017/3544544818174400","-22817729003*theta**6/3544544818174400 + 725144255011*theta**5/1772272409087200 - 2652730829819*theta**4/708908963634880 - 13989850831347*theta**3/443068102271800 + 676782334552179*theta**2/3544544818174400 + 1327863438017001*theta/1772272409087200 - 6753299038741729/3544544818174400","24551403431*theta**6/7089089636348800 - 799277011853*theta**5/3544544818174400 + 15952249287607*theta**4/7089089636348800 + 17694996542071*theta**3/886136204543600 - 1153811533066799*theta**2/7089089636348800 - 2383862394970863*theta/3544544818174400 + 514187509614137/283563585453952","83741*theta**6/635678769400 - 8246569*theta**5/1271357538800 - 19132899*theta**4/1271357538800 + 33070017*theta**3/79459846175 + 310541623*theta**2/635678769400 - 7660709159*theta/1271357538800 + 6537717627/1271357538800","221215861*theta**6/150831694390400 - 1237820971*theta**5/15083169439040 + 25720070501*theta**4/150831694390400 + 45060128201*theta**3/3770792359760 + 3941830043699*theta**2/150831694390400 - 776202977157*theta/3016633887808 - 117779343780941/150831694390400","19551*theta**6/26041383700 - 335246*theta**5/6510345925 + 3512747*theta**4/5208276740 + 11033458*theta**3/6510345925 - 1261808883*theta**2/26041383700 + 981522204*theta/6510345925 + 23316727633/26041383700"],"codegree_value":"(14109524839*theta**6 - 867291590162*theta**5 + 6174287488191*theta**4 + 91624383467896*theta**3 - 164213446132335*theta**2 - 681428284712742*theta + 22015965151000713)/3544544818174400","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,6,1,2,5,3,4]},{"rank":7,"multiplicity":2,"id":"R7_2_218","name":"Rep(C13rtimesC4)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0]],[[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,1,0,0,0,0,0]],[[0,0,0,1,0,0,0],[0,0,0,1,0,0,0],[0,0,0,1,0,0,0],[1,1,1,0,1,2,1],[0,0,0,1,2,1,0],[0,0,0,2,1,1,0],[0,0,0,1,0,0,0]],[[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,0,1,2,1,0],[1,1,1,2,0,1,1],[0,0,0,1,1,2,0],[0,0,0,0,1,0,0]],[[0,0,0,0,0,1,0],[0,0,0,0,0,1,0],[0,0,0,0,0,1,0],[0,0,0,2,1,1,0],[0,0,0,1,1,2,0],[1,1,1,1,2,0,1],[0,0,0,0,0,1,0]],[[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0]]],"tensor_sha256":"9d1e3661df855b64662b61e9e449f4b53e1bfe6c8b7a447f24fdac12c44bc5da","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_2_218; Character induction for the explicit finite semidirect product C13 semidirect_5 C4; 5 has order four modulo 13.; construction: Ordinary complex representations of C13 semidirect C4, generator acting by exponent 5; exact induced characters..","model_reference_url":"https://www.ams.org/books/surv/205/","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_2_218","invariants":{"dual_indices":[0,2,1,3,4,5,6],"invertible_indices":[0,1,2,6],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,4.0,4.0,4.0,1.0],"global_fp_dimension_approx":52.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 52)*(t - 13)**3*(t - 4)**3","formal_codegree_polynomial":"(t - 52)*(t - 13)**3*(t - 4)**3","irreducible_characters_approx":[{"values_approx":[1.0,1.0,1.0,4.0,4.0,4.0,1.0],"formal_codegree_approx":52.0},{"values_approx":[1.0,1.0,1.0,-2.6510934089,0.273890555,1.377202854,1.0],"formal_codegree_approx":13.0},{"values_approx":[1.0,1.0,1.0,0.273890555,1.377202854,-2.6510934089,1.0],"formal_codegree_approx":13.0},{"values_approx":[1.0,1.0,1.0,1.377202854,-2.6510934089,0.273890555,1.0],"formal_codegree_approx":13.0},{"values_approx":[1.0,-1.0,-1.0,0.0,0.0,0.0,1.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,{"re":0.0,"im":-1.0},{"re":0.0,"im":1.0},0.0,0.0,0.0,-1.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,{"re":0.0,"im":1.0},{"re":0.0,"im":-1.0},0.0,0.0,0.0,-1.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7],"primitive_polynomial":"(theta - 73)*(theta - 3)*(theta**2 + 12*theta + 37)*(theta**3 - 24*theta**2 + 179*theta - 421)","basis_values":["1","-24677467*theta**6/6042184700755 + 2062081073*theta**5/6042184700755 - 3326323243*theta**4/1208436940151 - 39238519918*theta**3/1208436940151 + 20522702929*theta**2/82769653435 + 5290121721022*theta/6042184700755 - 80095069625/16553930687","-6159036*theta**6/1208436940151 + 2639352649*theta**5/6042184700755 - 26775202101*theta**4/6042184700755 - 30979072393*theta**3/1208436940151 + 345556735960*theta**2/1208436940151 + 3397891892281*theta/6042184700755 - 356902196713/82769653435","8042861719*theta**6/3103472380809745 - 638688471383*theta**5/3103472380809745 + 3029905123311*theta**4/3103472380809745 + 11190251449591*theta**3/620694476161949 - 156797382950884*theta**2/3103472380809745 - 1073028547267742*theta/3103472380809745 + 41417724383988/42513320285065","-39913990462*theta**6/3103472380809745 + 654350252012*theta**5/620694476161949 - 22577681906049*theta**4/3103472380809745 - 56061338763713*theta**3/620694476161949 + 1418973416939137*theta**2/3103472380809745 + 1267523874818363*theta/620694476161949 - 317157317320332/42513320285065","38711287073*theta**6/3103472380809745 - 3210923643736*theta**5/3103472380809745 + 24740549235664*theta**4/3103472380809745 + 53875709338007*theta**3/620694476161949 - 1648662676326838*theta**2/3103472380809745 - 6531056974632829*theta/3103472380809745 + 354876214784547/42513320285065","315222*theta**6/863169242965 - 28756909*theta**5/863169242965 + 423894184*theta**4/863169242965 + 13336244*theta**3/172633848593 - 30874715037*theta**2/863169242965 + 167817270149*theta/863169242965 + 8331673672/11824236205"],"codegree_value":"-(61557377631*theta**6 - 5200686985446*theta**5 + 46735097780538*theta**4 + 405203721132180*theta**3 - 3478375952264571*theta**2 - 11398113209770494*theta + 39578681894831582)/3103472380809745","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,1,6,2,3,4,5]},{"rank":7,"multiplicity":2,"id":"R7_2_233","name":"Rep(H24)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0]],[[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0]],[[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,1,0,0],[0,0,0,1,0,1,1],[0,1,0,0,1,0,0],[0,0,1,0,1,0,0]],[[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,1,1],[1,1,1,0,2,0,0],[0,0,0,1,0,1,1],[0,0,0,1,0,1,1]],[[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,1,0,1,0,0],[0,0,0,1,0,1,1],[1,0,0,0,1,0,0],[0,1,0,0,1,0,0]],[[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,1,0,0,1,0,0],[0,0,0,1,0,1,1],[0,0,1,0,1,0,0],[1,0,0,0,1,0,0]]],"tensor_sha256":"fd581e0897ae4a1e5b46cdf207f5c08efdcbf5cd19bcd4395f1cb8d6dd1ca7fb","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_2_233; Alekseyev--Bruns--Dong--Palcoux, Classifying integral Grothendieck rings up to rank 5 and beyond, Theorem 1.6; published full rank-7 noncommutative tensor appendix, FPdim 24; construction: representation category of the dimension-24 Kac algebra identified in Alekseyev--Bruns--Dong--Palcoux, Theorem 1.6.","model_reference_url":"https://github.com/sebastienpalcoux/Fusion-Categories/blob/main/IntegralPaper/Data/Noncommutative/InvestNCRank7DataOnly.txt","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_2_233","invariants":{"dual_indices":[0,2,1,3,4,5,6],"invertible_indices":[0,1,2],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,2.0,3.0,2.0,2.0],"global_fp_dimension_approx":24.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 24)**2*(t - 6)**4*(t - 4)","irreducible_characters_approx":[{"degree":2,"values_approx":[2.0,-1.0,-1.0,0.0,0.0,0.0,0.0],"formal_codegree_approx":3.0,"formal_codegree_minimal_polynomial":"u - 3","formal_codegree_exact":"3"},{"degree":1,"values_approx":[1.0,1.0,1.0,-2.0,3.0,-2.0,-2.0],"formal_codegree_approx":24.0,"formal_codegree_minimal_polynomial":"u - 24","formal_codegree_exact":"24"},{"degree":1,"values_approx":[1.0,1.0,1.0,0.0,-1.0,0.0,0.0],"formal_codegree_approx":4.0,"formal_codegree_minimal_polynomial":"u - 4","formal_codegree_exact":"4"},{"degree":1,"values_approx":[1.0,1.0,1.0,2.0,3.0,2.0,2.0],"formal_codegree_approx":24.0,"formal_codegree_minimal_polynomial":"u - 24","formal_codegree_exact":"24"}],"character_method":"Numerical regular-algebra block traces after exact rational center-kernel computation; χ_E(X_i) and f_E=(Σ_i χ_E(X_i)χ_E(X_i*)) / dim(E) are approximate. The displayed codegree minimal polynomials are exact factors of the rational central-Casimir polynomial rescaled by each irreducible degree; factor association uses the validated numerical eigenvalues. Orthogonality and block dimensions validated."},"basis_permutation_model_to_census":[0,1,2,3,5,6,4]},{"rank":7,"multiplicity":2,"id":"R7_2_234","name":"C(A5,1,A4,1)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0]],[[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0]],[[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,1,1,1,1],[0,0,0,1,0,1,1],[0,1,0,1,1,1,1],[0,0,1,1,1,1,1]],[[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,1,1],[1,1,1,0,2,0,0],[0,0,0,1,0,1,1],[0,0,0,1,0,1,1]],[[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,1,1,1,1,1],[0,0,0,1,0,1,1],[1,0,0,1,1,1,1],[0,1,0,1,1,1,1]],[[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,1,0,1,1,1,1],[0,0,0,1,0,1,1],[0,0,1,1,1,1,1],[1,0,0,1,1,1,1]]],"tensor_sha256":"23c5e89f25ed62b649e372939e3504d8262f4074c98d63887ef24d800f489a15","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_2_234; Alekseyev--Bruns--Dong--Palcoux, Classifying integral Grothendieck rings up to rank 5 and beyond, Theorem 1.6 and Section 6.2; published full rank-7 noncommutative tensor appendix, FPdim 60; construction: group-theoretical fusion category of A4-bimodules in Vec(A5) with trivial cocycles, constructed in Alekseyev--Bruns--Dong--Palcoux, Section 6.2.","model_reference_url":"https://github.com/sebastienpalcoux/Fusion-Categories/blob/main/IntegralPaper/Data/Noncommutative/InvestNCRank7DataOnly.txt","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_2_234","invariants":{"dual_indices":[0,2,1,3,4,5,6],"invertible_indices":[0,1,2],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,4.0,3.0,4.0,4.0],"global_fp_dimension_approx":60.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 60)*(t - 15)*(t - 6)**4*(t - 4)","irreducible_characters_approx":[{"degree":2,"values_approx":[2.0,-1.0,-1.0,0.0,0.0,0.0,0.0],"formal_codegree_approx":3.0,"formal_codegree_minimal_polynomial":"u - 3","formal_codegree_exact":"3"},{"degree":1,"values_approx":[1.0,1.0,1.0,-1.0,3.0,-1.0,-1.0],"formal_codegree_approx":15.0,"formal_codegree_minimal_polynomial":"u - 15","formal_codegree_exact":"15"},{"degree":1,"values_approx":[1.0,1.0,1.0,0.0,-1.0,0.0,0.0],"formal_codegree_approx":4.0,"formal_codegree_minimal_polynomial":"u - 4","formal_codegree_exact":"4"},{"degree":1,"values_approx":[1.0,1.0,1.0,4.0,3.0,4.0,4.0],"formal_codegree_approx":60.0,"formal_codegree_minimal_polynomial":"u - 60","formal_codegree_exact":"60"}],"character_method":"Numerical regular-algebra block traces after exact rational center-kernel computation; χ_E(X_i) and f_E=(Σ_i χ_E(X_i)χ_E(X_i*)) / dim(E) are approximate. The displayed codegree minimal polynomials are exact factors of the rational central-Casimir polynomial rescaled by each irreducible degree; factor association uses the validated numerical eigenvalues. Orthogonality and block dimensions validated."},"basis_permutation_model_to_census":[0,1,2,4,3,5,6]},{"rank":7,"multiplicity":2,"id":"R7_2_24","name":"Rep(S5)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]],[[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[1,0,0,0,1,1,1],[0,0,0,1,1,1,1],[0,0,1,1,1,0,1],[0,1,1,1,0,0,1],[0,0,1,1,1,1,1]],[[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,1,1,1],[1,0,1,1,1,1,1],[0,1,1,1,1,1,1],[0,0,1,1,1,0,1],[0,0,1,1,1,1,2]],[[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,1,1,0,1],[0,1,1,1,1,1,1],[1,0,1,1,1,1,1],[0,0,0,1,1,1,1],[0,0,1,1,1,1,2]],[[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,1,1,1,0,0,1],[0,0,1,1,1,0,1],[0,0,0,1,1,1,1],[1,0,0,0,1,1,1],[0,0,1,1,1,1,1]],[[0,0,0,0,0,0,1],[0,0,0,0,0,0,1],[0,0,1,1,1,1,1],[0,0,1,1,1,1,2],[0,0,1,1,1,1,2],[0,0,1,1,1,1,1],[1,1,1,2,2,1,1]]],"tensor_sha256":"d7f9bb803744a1491fcac9644528e9ebb90f69fa2cbf2bfe0e8e028b99312e8a","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_2_24; 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All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,5,4,6,3,2,1]},{"rank":7,"multiplicity":2,"id":"R7_2_240","name":"D12 even part; local bosonic modularization PSU(2)_20","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[1,1,1,0,0,0,0],[0,1,1,1,0,0,0],[0,0,1,1,1,0,0],[0,0,0,1,1,1,1],[0,0,0,0,1,0,1],[0,0,0,0,1,1,0]],[[0,0,1,0,0,0,0],[0,1,1,1,0,0,0],[1,1,1,1,1,0,0],[0,1,1,1,1,1,1],[0,0,1,1,2,1,1],[0,0,0,1,1,1,0],[0,0,0,1,1,0,1]],[[0,0,0,1,0,0,0],[0,0,1,1,1,0,0],[0,1,1,1,1,1,1],[1,1,1,1,2,1,1],[0,1,1,2,2,1,1],[0,0,1,1,1,0,1],[0,0,1,1,1,1,0]],[[0,0,0,0,1,0,0],[0,0,0,1,1,1,1],[0,0,1,1,2,1,1],[0,1,1,2,2,1,1],[1,1,2,2,2,1,1],[0,1,1,1,1,1,0],[0,1,1,1,1,0,1]],[[0,0,0,0,0,1,0],[0,0,0,0,1,0,1],[0,0,0,1,1,1,0],[0,0,1,1,1,0,1],[0,1,1,1,1,1,0],[0,1,0,1,0,0,1],[1,0,1,0,1,0,0]],[[0,0,0,0,0,0,1],[0,0,0,0,1,1,0],[0,0,0,1,1,0,1],[0,0,1,1,1,1,0],[0,1,1,1,1,0,1],[1,0,1,0,1,0,0],[0,1,0,1,0,1,0]]],"tensor_sha256":"40dfbace27e1ef951e644367f8e85f82461e47c3c1b2cf729b6386e37cc63534","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_2_240; 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approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7],"primitive_polynomial":"(theta**2 + 7*theta + 15)*(theta**5 - 120*theta**4 - 543*theta**3 + 10942*theta**2 + 9496*theta - 156727)","basis_values":["1","-6027529870459556*theta**6/1276150917342057233851 + 700535551132361370*theta**5/1276150917342057233851 + 5965473851813336854*theta**4/1276150917342057233851 - 49251116167569874026*theta**3/1276150917342057233851 - 234726877567597759053*theta**2/1276150917342057233851 + 524194153551963571013*theta/1276150917342057233851 + 1981072505522601919334/1276150917342057233851","11213042510529840*theta**6/1276150917342057233851 - 1315428018842271578*theta**5/1276150917342057233851 - 9654544117709143687*theta**4/1276150917342057233851 + 100165985084823050788*theta**3/1276150917342057233851 + 422618793465425896016*theta**2/1276150917342057233851 - 1166896293878697823033*theta/1276150917342057233851 - 5130723853570310201309/1276150917342057233851","-8307239558155641*theta**6/1276150917342057233851 + 980307947171840533*theta**5/1276150917342057233851 + 6533198408089561881*theta**4/1276150917342057233851 - 85796373379408651991*theta**3/1276150917342057233851 - 315025414921914678721*theta**2/1276150917342057233851 + 1160196040903694112195*theta/1276150917342057233851 + 4330843679375232867449/1276150917342057233851","1508380424601636*theta**6/1276150917342057233851 - 181623148325018898*theta**5/1276150917342057233851 - 796605945193279008*theta**4/1276150917342057233851 + 23024066484729778065*theta**3/1276150917342057233851 + 35986984785111886399*theta**2/1276150917342057233851 - 558958634086681643951*theta/1276150917342057233851 - 635976720536691428219/1276150917342057233851","5727331820943355*theta**6/1276150917342057233851 - 652266985383510042*theta**5/1276150917342057233851 - 7307980842253220996*theta**4/1276150917342057233851 + 43229871738467241203*theta**3/1276150917342057233851 + 435590239425823977195*theta**2/1276150917342057233851 - 388739511748858185214*theta/1276150917342057233851 - 5499516103779712491139/1276150917342057233851","-4323000372388022*theta**6/1276150917342057233851 + 492242688500126372*theta**5/1276150917342057233851 + 5532972256661175835*theta**4/1276150917342057233851 - 33329970322306733413*theta**3/1276150917342057233851 - 333111046579132392889*theta**2/1276150917342057233851 + 602128224169429206169*theta/1276150917342057233851 + 4143925877169210021363/1276150917342057233851"],"codegree_value":"(3138140520779110*theta**6 - 370548433701908460*theta**5 - 2428488961080859477*theta**4 + 31085473239206973953*theta**3 + 110562696680827081311*theta**2 - 446967844826716681286*theta - 808851210089479687469)/116013719758368839441","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. 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All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,3,4,5,1,2,6]},{"rank":7,"multiplicity":2,"id":"R7_2_313","name":"Adj(SU(3)_5)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[0,0,1,0,1,0,0],[1,0,0,0,0,0,1],[0,0,1,0,0,1,0],[0,0,0,1,0,0,1],[0,0,0,0,1,1,1],[0,1,0,1,0,1,0]],[[0,0,1,0,0,0,0],[1,0,0,0,0,0,1],[0,1,0,1,0,0,0],[0,0,0,0,1,0,1],[0,1,0,0,0,1,0],[0,0,0,1,0,1,1],[0,0,1,0,1,1,0]],[[0,0,0,1,0,0,0],[0,0,1,0,0,1,0],[0,0,0,0,1,0,1],[0,1,0,0,1,1,0],[1,0,0,0,0,1,1],[0,0,1,1,1,1,1],[0,1,0,1,0,1,1]],[[0,0,0,0,1,0,0],[0,0,0,1,0,0,1],[0,1,0,0,0,1,0],[1,0,0,0,0,1,1],[0,0,1,1,0,1,0],[0,1,0,1,1,1,1],[0,0,1,0,1,1,1]],[[0,0,0,0,0,1,0],[0,0,0,0,1,1,1],[0,0,0,1,0,1,1],[0,0,1,1,1,1,1],[0,1,0,1,1,1,1],[1,1,1,1,1,2,2],[0,1,1,1,1,2,1]],[[0,0,0,0,0,0,1],[0,1,0,1,0,1,0],[0,0,1,0,1,1,0],[0,1,0,1,0,1,1],[0,0,1,0,1,1,1],[0,1,1,1,1,2,1],[1,0,0,1,1,1,2]]],"tensor_sha256":"86ad414bcaeef419a4d0e19ee1c5be723a959ddc62f30999bdab8dbdb95e388b","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_2_313; Wenzl, C*-tensor categories from quantum groups, J. 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Soc. 11 (1998), 261-282; adjoint SU(3)_5 fusion reconstructed by exact Weyl alternants.; construction: The triality-zero fusion subcategory of the standard unitary SU(3) level-five category; exact Weyl alternants over Q(zeta24)..","model_reference_url":"https://doi.org/10.1090/S0894-0347-98-00253-7","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_2_313","invariants":{"dual_indices":[0,2,1,4,3,5,6],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,2.4142135624,2.4142135624,3.4142135624,3.4142135624,5.8284271247,4.8284271247],"global_fp_dimension_approx":93.2548339959,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 16)**2*(t - 8)**2*(t - 4)*(t**2 - 96*t + 256)","formal_codegree_polynomial":"(t - 16)**2*(t - 8)**2*(t - 4)*(t**2 - 96*t + 256)","irreducible_characters_approx":[{"values_approx":[1.0,2.4142135624,2.4142135624,3.4142135624,3.4142135624,5.8284271247,4.8284271247],"formal_codegree_approx":93.2548339959},{"values_approx":[1.0,{"re":-1.0,"im":-1.4142135624},{"re":-1.0,"im":1.4142135624},{"re":0.0,"im":-1.4142135624},{"re":0.0,"im":1.4142135624},-1.0,2.0],"formal_codegree_approx":16.0},{"values_approx":[1.0,{"re":-1.0,"im":1.4142135624},{"re":-1.0,"im":-1.4142135624},{"re":0.0,"im":1.4142135624},{"re":0.0,"im":-1.4142135624},-1.0,2.0],"formal_codegree_approx":16.0},{"values_approx":[1.0,{"re":0.0,"im":-1.0},{"re":0.0,"im":1.0},{"re":-1.0,"im":1.0},{"re":-1.0,"im":-1.0},1.0,0.0],"formal_codegree_approx":8.0},{"values_approx":[1.0,{"re":0.0,"im":1.0},{"re":0.0,"im":-1.0},{"re":-1.0,"im":-1.0},{"re":-1.0,"im":1.0},1.0,0.0],"formal_codegree_approx":8.0},{"values_approx":[1.0,1.0,1.0,0.0,0.0,-1.0,0.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,-0.4142135624,-0.4142135624,0.5857864376,0.5857864376,0.1715728753,-0.8284271247],"formal_codegree_approx":2.7451660041}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[14,11,27,19,43,34,26],"primitive_polynomial":"(theta - 18)*(theta**2 - 660*theta + 12100)*(theta**2 + 12*theta + 3236)*(theta**2 + 28*theta + 260)","basis_values":["1","395007528208433*theta**6/683530000037122362572800 - 5535187160448311*theta**5/15534772728116417331200 - 1250187596898748741*theta**4/170882500009280590643200 - 24049284471164611367*theta**3/21360312501160073830400 - 510744582837651884457*theta**2/42720625002320147660800 + 750535444367995325663*theta/2136031250116007383040 + 253866139922549856719/38836931820291043328","28365286948609*theta**6/42720625002320147660800 - 802113719190651*theta**5/1941846591014552166400 - 16122392507110537*theta**4/2670039062645009228800 - 3490396425780149877*theta**3/2670039062645009228800 - 18498058564172088431*theta**2/2670039062645009228800 + 18724194149944828235*theta/53400781252900184576 + 7219913727188714755/1213654119384095104","-116666274127219*theta**6/683530000037122362572800 + 1606660097087753*theta**5/15534772728116417331200 + 560573317953275383*theta**4/170882500009280590643200 + 7569891205195057921*theta**3/21360312501160073830400 + 305175180873227957931*theta**2/42720625002320147660800 - 200272071646290006641*theta/2136031250116007383040 - 125351283782624411925/38836931820291043328","-18332518667647*theta**6/68353000003712236257280 + 259810992678859*theta**5/1553477272811641733120 + 7471481023631027*theta**4/3417650000185611812864 + 226678015646807715*theta**3/427206250023201476608 + 8589156065283556047*theta**2/4272062500232014766080 - 98714713048078904839*theta/1068015625058003691520 - 50849023676858797833/19418465910145521664","-6347759005397*theta**6/11585254237917328179200 + 89598256634059*theta**5/263301232679939276800 + 15585395132035849*theta**4/2896313559479332044800 + 383557017584091563*theta**3/362039194934916505600 + 5314938133158112333*theta**2/724078389869833011200 - 12089998567460702643*theta/36203919493491650560 - 3116943347477720827/658253081699848192","239783078009227*theta**6/683530000037122362572800 - 3378298053630269*theta**5/15534772728116417331200 - 629512386869600359*theta**4/170882500009280590643200 - 14697434102373356333*theta**3/21360312501160073830400 - 251695374186227719603*theta**2/42720625002320147660800 + 392507057302619605813*theta/2136031250116007383040 + 132039566853725224629/38836931820291043328"],"codegree_value":"(14727245763025*theta**6 - 9102402294074276*theta**5 - 170426702018079412*theta**4 - 29752039254423620192*theta**3 - 376189024816365504912*theta**2 + 4441713534415078227904*theta + 386743708547518229652800)/34176500001856118128640","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. 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approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7],"primitive_polynomial":"(theta - 64)*(theta**2 - 29*theta + 213)*(theta**4 + 14*theta**3 + 36*theta**2 - 71*theta + 211)","basis_values":["1","2000764329539*theta**6/1649162737143323401 - 166469420901513*theta**5/1649162737143323401 + 2369175953612203*theta**4/1649162737143323401 + 8607052582371512*theta**3/1649162737143323401 - 178266859365377010*theta**2/1649162737143323401 - 276273217236646724*theta/1649162737143323401 - 17610736195630801/23227644185117231","-2126289882010*theta**6/1649162737143323401 + 180934350130858*theta**5/1649162737143323401 - 2813654850196144*theta**4/1649162737143323401 - 7659316394572973*theta**3/1649162737143323401 + 256070937337310729*theta**2/1649162737143323401 + 122276083998649825*theta/1649162737143323401 - 16269166603309969/23227644185117231","47613415011*theta**6/270215496240235943 - 3010055431208*theta**5/270215496240235943 - 9412685868927*theta**4/270215496240235943 + 429107602821617*theta**3/270215496240235943 + 1723552617214843*theta**2/270215496240235943 - 3640354520379277*theta/270215496240235943 + 119902196003045/3805852059721633","-59524047058*theta**6/270215496240235943 + 3834508149313*theta**5/270215496240235943 + 8341007656863*theta**4/270215496240235943 - 595854735045568*theta**3/270215496240235943 - 2319150897966877*theta**2/270215496240235943 + 4880397985873104*theta/270215496240235943 - 163052342058459/3805852059721633","404084681617*theta**6/1649162737143323401 - 24160557714846*theta**5/1649162737143323401 - 339569019093713*theta**4/1649162737143323401 + 15536996836947016*theta**3/1649162737143323401 - 34293032040007325*theta**2/1649162737143323401 - 961995155736339730*theta/1649162737143323401 + 20941200020638127/23227644185117231","86697914796*theta**6/1649162737143323401 - 15490056682629*theta**5/1649162737143323401 + 816468976158265*theta**4/1649162737143323401 - 11392975572952771*theta**3/1649162737143323401 - 25318103471893618*theta**2/1649162737143323401 + 1078113353725630810*theta/1649162737143323401 - 8971047948342097/23227644185117231"],"codegree_value":"(72052285624*theta**6 - 4955543708760*theta**5 + 6337675817690*theta**4 + 1001581510413220*theta**3 + 3579438131345000*theta**2 - 7452761568533726*theta + 1369485892077055585)/270215496240235943","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. 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All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,1,2,3,4,5,6]},{"rank":7,"multiplicity":4,"id":"R7_4_2","name":"Vec(Q8)^S3","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]],[[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]],[[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[1,0,0,0,1,1,0],[0,1,1,1,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,2]],[[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,1,0,0,0],[0,1,1,1,0,0,0],[1,0,0,0,1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,2]],[[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1]],[[0,0,0,0,0,0,1],[0,0,0,0,0,0,1],[0,0,0,0,0,0,1],[0,0,0,0,0,0,2],[0,0,0,0,0,0,2],[0,0,0,0,0,0,1],[1,1,1,2,2,1,4]]],"tensor_sha256":"90ad3405faf59d5ea8556f95c4cf881d8b7d04d4fb59038505e9ec117c211bbb","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_4_2; Explicit quaternion group and signed-axis S3 action; orbit/stabilizer equivariantization theory of Burciu--Natale, Fusion rules of equivariantizations of fusion categories. Full construction and proof in the manuscript.; construction: The unitary equivariantization Vec(Q8)^S3 for the strict action e_a -> sign(p)*e_p(a) on the three quaternion axes. Fixed central grades carry the three ordinary S3 irreducibles; the six noncentral grades form one free orbit..","model_reference_url":"https://arxiv.org/abs/1206.6625","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_4_2","invariants":{"dual_indices":[0,1,2,3,4,5,6],"invertible_indices":[0,1,2,5],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,2.0,2.0,1.0,6.0],"global_fp_dimension_approx":48.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 48)*(t - 16)*(t - 12)*(t - 6)**2*(t - 4)**2","formal_codegree_polynomial":"(t - 48)*(t - 16)*(t - 12)*(t - 6)**2*(t - 4)**2","irreducible_characters_approx":[{"values_approx":[1.0,1.0,1.0,2.0,2.0,1.0,6.0],"formal_codegree_approx":48.0},{"values_approx":[1.0,1.0,1.0,2.0,2.0,1.0,-2.0],"formal_codegree_approx":16.0},{"values_approx":[1.0,-1.0,-1.0,-2.0,2.0,1.0,0.0],"formal_codegree_approx":12.0},{"values_approx":[1.0,-1.0,-1.0,1.0,-1.0,1.0,0.0],"formal_codegree_approx":6.0},{"values_approx":[1.0,1.0,1.0,-1.0,-1.0,1.0,0.0],"formal_codegree_approx":6.0},{"values_approx":[1.0,-1.0,1.0,0.0,0.0,-1.0,0.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,1.0,-1.0,0.0,0.0,-1.0,0.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7],"primitive_polynomial":"(theta - 72)*(theta - 16)*(theta - 4)*(theta - 3)*(theta - 1)*(theta + 4)*(theta + 6)","basis_values":["1","-607*theta**6/36119475 + 32041*theta**5/23116464 - 3127819*theta**4/288955800 - 187661*theta**3/1359792 + 29189287*theta**2/96318600 + 297509*theta/160531 - 1725059/573325","-299447*theta**6/14912726400 + 2509267*theta**5/1491272640 - 25278557*theta**4/1656969600 - 2380501*theta**3/17544384 + 786481879*theta**2/1864090800 + 23148863*theta/15534090 - 11921687/4315025","10779959*theta**6/3509079235200 - 1252590139*theta**5/4561803005760 + 2439547837*theta**4/661130870400 + 768106645*theta**3/53668270656 - 1272812815069*theta**2/5702253757200 - 19420965761*theta/47518781310 + 3043808946/1885665925","3004249*theta**6/206416425600 - 319139069*theta**5/268341353280 + 331833427*theta**4/38890051200 + 6454912627*theta**3/53668270656 - 33235331159*theta**2/335426691600 - 3829250731*theta/2795222430 + 265063662/776450675","2341*theta**6/1643241600 - 4153*theta**5/32864832 + 916373*theta**4/547747200 + 130387*theta**3/32864832 - 16565687*theta**2/205405200 + 74929*theta/342342 + 407131/475475","132071*theta**6/3801502504800 - 65953*theta**5/27153589320 - 1038337*theta**4/165282717600 + 25703*theta**3/279522243 + 703737*theta**2/15085327400 - 2242512*theta/2639932295 + 9497562/13199661475"],"codegree_value":"(363676289*theta**6 - 29953775470*theta**5 + 232922400171*theta**4 + 2831735200450*theta**3 - 4100583944384*theta**2 - 26235089172480*theta + 95727650701824)/11404507514400","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,5,4,1,2,3,6]},{"rank":7,"multiplicity":4,"id":"R7_4_2019","name":"C(Q8rtimesS3,Chern,S3,1)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]],[[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0]],[[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,1,0],[1,0,0,0,0,1,1],[0,0,0,0,2,0,0],[0,1,1,1,0,0,0],[0,0,0,1,0,0,0]],[[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,0,0,2,0,0],[1,1,1,2,4,2,1],[0,0,0,0,2,0,0],[0,0,0,0,1,0,0]],[[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,1,0,0,0],[0,1,1,1,0,0,0],[0,0,0,0,2,0,0],[1,0,0,0,0,1,1],[0,0,0,0,0,1,0]],[[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0]]],"tensor_sha256":"e53b4a423763328d160bf33d459e1e205a57e9d0af120da0b654177d5bc9f73f","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_4_2019; Explicit characteristic-class construction in papers/r7_4_2019_chern_proof.md; Gelaki--Naidu, Some properties of group-theoretical categories, Section 3 and Theorem 5.1; Atiyah, Characters and cohomology of finite groups, appendix.; construction: The unitary group-theoretical category C(Q8 semidirect S3, Omega, S3, 1), where [Omega]=c2(V)-infl c2(R), V(q,p)=qR(p), and R implements the signed-axis quaternion action. The global characteristic class has a representative trivial on S3; its explicit restriction to C2 times S3 determines all fusion rules..","model_reference_url":"https://arxiv.org/abs/0709.4326","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_4_2019","invariants":{"dual_indices":[0,2,1,3,4,5,6],"invertible_indices":[0,1,2,6],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,2.0,6.0,2.0,1.0],"global_fp_dimension_approx":48.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 48)*(t - 16)*(t - 12)*(t - 6)**2*(t - 4)**2","formal_codegree_polynomial":"(t - 48)*(t - 16)*(t - 12)*(t - 6)**2*(t - 4)**2","irreducible_characters_approx":[{"values_approx":[1.0,1.0,1.0,2.0,6.0,2.0,1.0],"formal_codegree_approx":48.0},{"values_approx":[1.0,1.0,1.0,2.0,-2.0,2.0,1.0],"formal_codegree_approx":16.0},{"values_approx":[1.0,-1.0,-1.0,-2.0,0.0,2.0,1.0],"formal_codegree_approx":12.0},{"values_approx":[1.0,-1.0,-1.0,1.0,0.0,-1.0,1.0],"formal_codegree_approx":6.0},{"values_approx":[1.0,1.0,1.0,-1.0,0.0,-1.0,1.0],"formal_codegree_approx":6.0},{"values_approx":[1.0,{"re":0.0,"im":-1.0},{"re":0.0,"im":1.0},0.0,0.0,0.0,-1.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,{"re":0.0,"im":1.0},{"re":0.0,"im":-1.0},0.0,0.0,0.0,-1.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7],"primitive_polynomial":"(theta - 63)*(theta - 23)*(theta - 7)*(theta - 3)*(theta - 1)*(theta**2 + 12*theta + 37)","basis_values":["1","-94527534459*theta**6/42693981115878400 + 92467205813*theta**5/520658306291200 - 16917187678991*theta**4/8538796223175680 - 153858970496331*theta**3/5336747639484800 + 5853452261169507*theta**2/42693981115878400 + 18890945472541743*theta/21346990557939200 - 12145907771872571/6099140159411200","-141993774203*theta**6/42693981115878400 + 145373208501*theta**5/520658306291200 - 34788706328591*theta**4/8538796223175680 - 123989890600907*theta**3/5336747639484800 + 8588860914421859*theta**2/42693981115878400 + 13400260052733231*theta/21346990557939200 - 10989896869805627/6099140159411200","41343637691*theta**6/21346990557939200 - 42592656369*theta**5/260329153145600 + 51475316409779*theta**4/21346990557939200 + 42683066628683*theta**3/2668373819742400 - 715875644096071*theta**2/4269398111587840 - 6559987004709139*theta/10673495278969600 + 5379550598940451/3049570079705600","525939*theta**6/61253918387200 - 16118869*theta**5/30626959193600 - 66423773*theta**4/61253918387200 + 258427843*theta**3/7656739798400 + 2305481317*theta**2/61253918387200 - 14866072759*theta/30626959193600 + 3636768147/8750559769600","35810373*theta**6/60991401594112 - 812205173*theta**5/18594939510400 + 405710005721*theta**4/1524785039852800 + 1808002119261*theta**3/190598129981600 + 35295502563787*theta**2/1524785039852800 - 168854902777463*theta/762392519926400 - 1237174758256609/1524785039852800","30871*theta**6/69867349700 - 16656*theta**5/426020425 + 9878291*theta**4/13973469940 + 15699858*theta**3/17466837425 - 3454383803*theta**2/69867349700 + 2836192254*theta/17466837425 + 61867474313/69867349700"],"codegree_value":"(43590907785*theta**6 - 3512115965902*theta**5 + 41585666166057*theta**4 + 441963020170696*theta**3 - 1124153697537281*theta**2 - 2503232177271802*theta + 131229249061165647)/21346990557939200","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"basis_permutation_model_to_census":[0,6,5,1,2,3,4]},{"rank":7,"multiplicity":4,"id":"R7_4_2665","name":"Reflection Morita dual of the inversion extension of NG(C16,16) orbifold","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]],[[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]],[[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]],[[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[0,0,0,0,1,0,0],[1,1,1,1,0,0,0],[0,0,0,0,0,1,1],[0,0,0,0,0,1,1]],[[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,1],[0,1,0,1,1,4,4],[1,0,1,0,1,4,4]],[[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,0,1,1],[1,0,1,0,1,4,4],[0,1,0,1,1,4,4]]],"tensor_sha256":"974ba9644a7d0991c504b58a3c9868b54a3966f6d77c0936872e7fd9c596af82","basis":["X0","X1","X2","X3","X4","X5","X6"],"model_reference":"Rank-seven manuscript, positive model catalogue, R7_4_2665; R2665_INVERSION_CONSTRUCTION.md; Budinski, Theorem 3.1.1 and Section 3.3.2; Izumi, Theorems 11.3, 12.5, 13.4 and Corollary 13.5; Grossman-Jordan-Snyder, Lemma 4.7 and Remark 4.8; Liu-Ng, Proposition 5.3.; construction: Start with Budinski NG(C16,16), centrally de-equivariantize by 4C16, descend the regular-C16 self-Morita equivalence to a coherent inversion C2 action, take the crossed product, and dualize its reflection algebra 1+s..","model_reference_url":"https://arxiv.org/abs/1512.04288","source_file":"supplements/rank7/data/model_catalog.json","source_location":"R7_4_2665","invariants":{"dual_indices":[0,3,2,1,4,6,5],"invertible_indices":[0,1,2,3],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,1.0,2.0,8.472135955,8.472135955],"global_fp_dimension_approx":151.55417528,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 8)**3*(t - 4)**2*(t**2 - 160*t + 1280)","formal_codegree_polynomial":"(t - 8)**3*(t - 4)**2*(t**2 - 160*t + 1280)","irreducible_characters_approx":[{"values_approx":[1.0,1.0,1.0,1.0,2.0,8.472135955,8.472135955],"formal_codegree_approx":151.55417528},{"values_approx":[1.0,1.0,1.0,1.0,2.0,-0.472135955,-0.472135955],"formal_codegree_approx":8.44582472},{"values_approx":[1.0,-1.0,1.0,-1.0,0.0,{"re":0.0,"im":-1.4142135624},{"re":0.0,"im":1.4142135624}],"formal_codegree_approx":8.0},{"values_approx":[1.0,-1.0,1.0,-1.0,0.0,{"re":0.0,"im":1.4142135624},{"re":0.0,"im":-1.4142135624}],"formal_codegree_approx":8.0},{"values_approx":[1.0,1.0,1.0,1.0,-2.0,0.0,0.0],"formal_codegree_approx":8.0},{"values_approx":[1.0,{"re":0.0,"im":-1.0},-1.0,{"re":0.0,"im":1.0},0.0,0.0,0.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,{"re":0.0,"im":1.0},-1.0,{"re":0.0,"im":-1.0},0.0,0.0,0.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7],"primitive_polynomial":"theta*(theta**2 - 144*theta + 1804)*(theta**2 + 4*theta + 6)*(theta**2 + 4*theta + 8)","basis_values":["1","930152191*theta**6/29654731661280 - 61762936517*theta**5/14827365830640 + 107890830227*theta**4/14827365830640 + 3355417318729*theta**3/7413682915320 + 2476401519649*theta**2/1235613819220 + 14882213279*theta/4109580330 + 1","-597*theta**6/8928160 + 5093*theta**5/558010 - 221909*theta**4/4464080 - 796709*theta**3/1116020 - 2468789*theta**2/1116020 - 17259*theta/6805 + 1","1210749029*theta**6/14827365830640 - 166648726691*theta**5/14827365830640 + 548436178963*theta**4/7413682915320 + 5420370013897*theta**3/7413682915320 + 1444394549401*theta**2/617806909610 + 13407474827*theta/4109580330 + 1","-685497661*theta**6/29654731661280 + 2913119222*theta**5/926710364415 - 233274188957*theta**4/14827365830640 - 79115733307*theta**3/336985587060 - 1315579411099*theta**2/1235613819220 - 2014268489047*theta/926710364415 - 2","43040879*theta**6/587669987190 - 982762927669*theta**5/96377877899160 + 153093851677*theta**4/1853420728830 + 24766604427263*theta**3/48188938949580 + 5620731257769*theta**2/4015744912465 + 9827584284358*theta/12047234737395","-21517783*theta**6/293834993595 + 982656546271*theta**5/96377877899160 - 153110787763*theta**4/1853420728830 - 24768280310777*theta**3/48188938949580 - 5620994003231*theta**2/4015744912465 - 9828264396682*theta/12047234737395"],"codegree_value":"-(8592948669*theta**6 - 1172897834336*theta**5 + 6387636720426*theta**4 + 91735799613704*theta**3 + 284266094106504*theta**2 + 325912723259616*theta - 514015348795520)/64251918599440","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. 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The displayed codegree minimal polynomials are exact factors of the rational central-Casimir polynomial rescaled by each irreducible degree; factor association uses the validated numerical eigenvalues. 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approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[40,24,7,21,8,19,45,26],"primitive_polynomial":"(theta**2 - 252*theta + 3588)*(theta**2 - 72*theta + 1971)*(theta**2 - 30*theta + 237)*(theta**2 - 24*theta - 1884)","basis_values":["1","1755867121353757570277677*theta**7/17140923310581363679652273788388880 - 69605622415301733528876131*theta**6/1904547034509040408850252643154320 + 3828337061624459974351141331*theta**5/1142728220705424245310151585892592 - 38668101593797469799144270805*theta**4/380909406901808081770050528630864 - 1176297586854242532101189689043*theta**3/952273517254520204425126321577160 + 1150298720829196872386835571483*theta**2/7935612643787668370209386013143 - 207039707254882812193114917759527*theta/79356126437876683702093860131430 + 15620921948927502156620588876/1066356613156449833401782635","407269199147074368035681*theta**7/5713641103527121226550757929462960 - 50049360294431756784395849*theta**6/1904547034509040408850252643154320 + 2998619926316232369263851177*theta**5/1142728220705424245310151585892592 - 36222230931379348841663285951*theta**4/380909406901808081770050528630864 - 29752074395227583024364424303*theta**3/105808168583835578269458480175240 + 924399634485614894917783744136*theta**2/7935612643787668370209386013143 - 225535869903256247435223148860463*theta/79356126437876683702093860131430 + 22946216655152217166801465934/1066356613156449833401782635","12281632744999380071*theta**7/52717457982332187225015989040 - 498127320563237036243*theta**6/5857495331370243025001776560 + 29093350533796507883591*theta**5/3514497198822145815001065936 - 333367366162868561337757*theta**4/1171499066274048605000355312 - 2393335149749948860174517*theta**3/1464373832842560756250444140 + 18001348328564791200285677*theta**2/48812461094752025208348138 - 995825423930249795057706958*theta/122031152736880063020870345 + 28339178678120553900228386/514899378636624738484685","14084494261436518969*theta**7/52717457982332187225015989040 - 565359110999513886637*theta**6/5857495331370243025001776560 + 32119115647332200078617*theta**5/3514497198822145815001065936 - 345883863270335583874739*theta**4/1171499066274048605000355312 - 3901345844156147705767483*theta**3/1464373832842560756250444140 + 19660470501991115735266531*theta**2/48812461094752025208348138 - 978017414344378024780833122*theta/122031152736880063020870345 + 25032236540971844375239304/514899378636624738484685","-101617023466649084291*theta**7/351627669107471399009726900736 + 3074397969207679172719*theta**6/29302305758955949917477241728 - 11347214780338030793917*theta**5/1127011759959844227595278528 + 6555263345442225725716051*theta**4/19534870505970633278318161152 + 31944138047497496662697893*theta**3/13023247003980422185545440768 - 222846333730995416177563059*theta**2/500894115537708545597901568 + 30386381132212243913928591585*theta/3255811750995105546386360192 - 7563456649619889526798155603/125223528884427136399475392","-846461463660296009645*theta**7/2905554949993316297080374916608 + 76741727374902000701603*theta**6/726388737498329074270093729152 - 94171846575595086059891*theta**5/9312676121773449670129406784 + 18071634224896365053587359*theta**4/53806573148024375871858794752 + 814173934585556618264848385*theta**3/322839438888146255231152768512 - 5556014151165233545492672183*theta**2/12416901495697932893505875712 + 252015849494723210008804591791*theta/26903286574012187935929397376 - 399161856585474183310205697/6590712046548796652603968","244938713015*theta**7/739372672479640478184 - 411889368445*theta**6/3423021631850187399 + 474943684500685*theta**5/41076259582202248788 - 2646189332412451*theta**4/6846043263700374798 - 230156127669387077*theta**3/82152519164404497576 + 1169755009091023299*theta**2/2282014421233458266 - 24563919044578842069*theta/2282014421233458266 + 6319506635146135/91994453810911"],"codegree_value":"2*(463539504332558*theta**7 - 168273169551068184*theta**6 + 16142528001352507374*theta**5 - 537382301766775431339*theta**4 - 3983975312598795294900*theta**3 + 715097825896631144345928*theta**2 - 14991211429933882175988912*theta + 99775822920550015012931574)/926818881545924529272433","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":[],"basis_permutation_census_to_model":[0,5,7,4,6,2,3,1],"alternative_models":[]},{"rank":8,"multiplicity":1,"id":"R8_1_3","name":"C(C5 semidirect C4, inflated omega_2, C2, 1)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0]],[[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,1,0],[1,0,0,0,0,0,1,1],[0,0,0,1,1,0,0,0],[0,1,1,0,1,0,0,0],[0,0,0,1,0,0,0,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,1,0,1],[0,0,0,0,0,1,1,0],[0,1,1,1,0,0,0,0],[0,0,0,1,1,0,0,0],[0,0,0,0,1,0,0,0]],[[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0],[0,1,1,0,1,0,0,0],[0,0,0,1,1,0,0,0],[1,0,0,0,0,0,1,1],[0,0,0,0,0,1,1,0],[0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,1,0,0,0],[0,1,1,1,0,0,0,0],[0,0,0,0,0,1,1,0],[1,0,0,0,0,1,0,1],[0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0]]],"tensor_sha256":"8e5341d955006cc35e17908037e4552c31d1d97cbed4844002f68b353ee140b1","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_1_3; Explicit homogeneous bimodules over C2 in Vec_G^omega, G=C5 semidirect_2 C4, omega(x,y,z)=(-1)^(b(x)floor((b(y)+b(z))/4)); this manuscript construction.","model_reference_url":"https://math.mit.edu/~etingof/egnobookfinal.pdf","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_1_3","invariants":{"dual_indices":[0,2,1,4,3,5,6,7],"invertible_indices":[0,1,2,7],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,2.0,2.0,2.0,2.0,1.0],"global_fp_dimension_approx":20.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 20)**2*(t - 10)**4*(t - 4)**2","irreducible_characters_approx":[{"degree":2,"values_approx":[2.0,0.0,0.0,0.0,0.0,-1.0,-1.0,2.0],"formal_codegree_approx":5.0,"formal_codegree_minimal_polynomial":"u - 5","formal_codegree_exact":"5"},{"degree":1,"values_approx":[1.0,-1.0,-1.0,-2.0,-2.0,2.0,2.0,1.0],"formal_codegree_approx":20.0,"formal_codegree_minimal_polynomial":"u - 20","formal_codegree_exact":"20"},{"degree":1,"values_approx":[1.0,1.0,1.0,2.0,2.0,2.0,2.0,1.0],"formal_codegree_approx":20.0,"formal_codegree_minimal_polynomial":"u - 20","formal_codegree_exact":"20"},{"degree":1,"values_approx":[1.0,{"re":0.0,"im":-1.0},{"re":0.0,"im":1.0},0.0,0.0,0.0,0.0,-1.0],"formal_codegree_approx":4.0,"formal_codegree_minimal_polynomial":"u - 4","formal_codegree_exact":"4"},{"degree":1,"values_approx":[1.0,{"re":0.0,"im":1.0},{"re":0.0,"im":-1.0},0.0,0.0,0.0,0.0,-1.0],"formal_codegree_approx":4.0,"formal_codegree_minimal_polynomial":"u - 4","formal_codegree_exact":"4"}],"character_method":"Numerical regular-algebra block traces after exact rational center-kernel computation; χ_E(X_i) and f_E=(Σ_i χ_E(X_i)χ_E(X_i*)) / dim(E) are approximate. The displayed codegree minimal polynomials are exact factors of the rational central-Casimir polynomial rescaled by each irreducible degree; factor association uses the validated numerical eigenvalues. Orthogonality and block dimensions validated."},"construction_notes":[],"basis_permutation_census_to_model":[0,2,3,5,7,4,6,1],"alternative_models":[]},{"rank":8,"multiplicity":1,"id":"R8_1_30","name":"TY(C6), inversion equivariantization, odd coherence -1","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,0,0,1,1,1,1],[1,0,0,1,0,0,1,1],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,1,0,0,0,0,0],[0,1,1,0,0,0,0,0]],[[0,0,1,0,0,0,0,0],[1,0,0,1,0,0,1,1],[0,0,0,0,1,1,1,1],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,1,1,0,0,0,0,0],[0,1,1,0,0,0,0,0]],[[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]],[[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[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]],[[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,0,0,0,0,0,1,0],[0,1,1,0,0,0,0,0],[0,1,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[1,0,0,0,0,1,0,1],[0,0,0,1,1,0,1,0]],[[0,0,0,0,0,0,0,1],[0,1,1,0,0,0,0,0],[0,1,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[0,0,0,1,1,0,1,0],[1,0,0,0,0,1,0,1]]],"tensor_sha256":"b457a3eb2c23c080346c3bc0593a602b602b4f7d0bd231e203aaef7455b92a9a","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_1_30; 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Explicit unitary equivariantization or extension in this manuscript; Tambara-Yamagami standard associators, inversion symmetry and parity coherence.","model_reference_url":"https://arxiv.org/abs/0909.3140","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_1_35","invariants":{"dual_indices":[0,2,1,3,4,5,6,7],"invertible_indices":[0,3],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,3.0,3.0,1.0,2.0,2.0,2.0,2.0],"global_fp_dimension_approx":36.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 36)**2*(t - 9)**4*(t - 4)**2","formal_codegree_polynomial":"(t - 36)**2*(t - 9)**4*(t - 4)**2","irreducible_characters_approx":[{"values_approx":[1.0,-3.0,-3.0,1.0,2.0,2.0,2.0,2.0],"formal_codegree_approx":36.0},{"values_approx":[1.0,3.0,3.0,1.0,2.0,2.0,2.0,2.0],"formal_codegree_approx":36.0},{"values_approx":[1.0,0.0,0.0,1.0,-1.0,-1.0,-1.0,2.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,0.0,0.0,1.0,-1.0,-1.0,2.0,-1.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,0.0,0.0,1.0,-1.0,2.0,-1.0,-1.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,0.0,0.0,1.0,2.0,-1.0,-1.0,-1.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,{"re":0.0,"im":-1.0},{"re":0.0,"im":1.0},-1.0,0.0,0.0,0.0,0.0],"formal_codegree_approx":4.0},{"values_approx":[1.0,{"re":0.0,"im":1.0},{"re":0.0,"im":-1.0},-1.0,0.0,0.0,0.0,0.0],"formal_codegree_approx":4.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; 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All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":[],"basis_permutation_census_to_model":[0,6,7,1,2,3,4,5],"alternative_models":[]},{"rank":8,"multiplicity":1,"id":"R8_1_36","name":"TY(C3) crossed C2","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0]],[[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],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0]],[[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[1,1,1,0,0,0,0,0],[0,0,0,1,0,0,1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0]],[[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,1,1],[1,1,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0]]],"tensor_sha256":"561ea09c9b9f3828ce0301be037cbd506bf6abdb6397aef856b537be5e69955a","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"The TY(C3) crossed-C2 model is checked by tools/verify_anyonwiki_import.py. This corrects the obsolete verifier path; its tensor and categorical realization are unchanged. See the current manuscript and the dated exact proof.","model_reference_url":"https://www.ams.org/books/surv/205/","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_1_36","invariants":{"dual_indices":[0,2,1,3,4,5,6,7],"invertible_indices":[0,1,2,3,6,7],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,1.0,1.7320508076,1.7320508076,1.0,1.0],"global_fp_dimension_approx":12.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 12)**4*(t - 6)**4","irreducible_characters_approx":[{"degree":2,"values_approx":[2.0,-1.0,-1.0,0.0,0.0,0.0,0.0,0.0],"formal_codegree_approx":3.0,"formal_codegree_minimal_polynomial":"u - 3","formal_codegree_exact":"3"},{"degree":1,"values_approx":[1.0,1.0,1.0,-1.0,-1.7320508076,1.7320508076,-1.0,-1.0],"formal_codegree_approx":12.0,"formal_codegree_minimal_polynomial":"u - 12","formal_codegree_exact":"12"},{"degree":1,"values_approx":[1.0,1.0,1.0,-1.0,1.7320508076,-1.7320508076,-1.0,-1.0],"formal_codegree_approx":12.0,"formal_codegree_minimal_polynomial":"u - 12","formal_codegree_exact":"12"},{"degree":1,"values_approx":[1.0,1.0,1.0,1.0,-1.7320508076,-1.7320508076,1.0,1.0],"formal_codegree_approx":12.0,"formal_codegree_minimal_polynomial":"u - 12","formal_codegree_exact":"12"},{"degree":1,"values_approx":[1.0,1.0,1.0,1.0,1.7320508076,1.7320508076,1.0,1.0],"formal_codegree_approx":12.0,"formal_codegree_minimal_polynomial":"u - 12","formal_codegree_exact":"12"}],"character_method":"Numerical regular-algebra block traces after exact rational center-kernel computation; χ_E(X_i) and f_E=(Σ_i χ_E(X_i)χ_E(X_i*)) / dim(E) are approximate. The displayed codegree minimal polynomials are exact factors of the rational central-Casimir polynomial rescaled by each irreducible degree; factor association uses the validated numerical eigenvalues. Orthogonality and block dimensions validated."},"construction_notes":["The TY(C3) crossed-C2 model is checked by tools/verify_anyonwiki_import.py. This corrects the obsolete verifier path; its tensor and categorical realization are unchanged.","Exact proof and tensor checks remain subject to independent human verification."],"basis_permutation_census_to_model":[0,2,4,1,6,7,5,3],"alternative_models":[],"construction_proof_file":"evidence/correctness_20261025/TY_C3_C2.md"},{"rank":8,"multiplicity":1,"id":"R8_1_39","name":"TY(C3) x Fib","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0]],[[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0]],[[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,1,0,0,1,0,0,0],[1,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,1,0],[0,0,1,1,0,0,0,0]],[[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1],[0,0,1,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,1,0],[0,1,0,0,1,0,0,0]],[[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0],[1,1,1,0,0,0,0,0],[0,0,0,1,1,0,0,1],[0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,1,0],[0,0,0,0,0,1,1,0],[0,0,0,1,1,0,0,1],[1,1,1,1,1,0,0,1],[0,0,0,0,0,1,1,0]],[[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,1,0,0,0,0],[0,1,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,1,0],[1,0,0,0,0,0,0,1]]],"tensor_sha256":"2d9c65923de2459c23b9e869ae3917d5263762149966ba79d1a234d3d88a9089","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_1_39; 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All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":[],"basis_permutation_census_to_model":[0,4,6,7,5,2,3,1],"alternative_models":[]},{"rank":8,"multiplicity":1,"id":"R8_1_4","name":"Vec(C4×C2)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0]],[[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0]],[[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0]]],"tensor_sha256":"3ab01c8f197600c3114aec8763e8296989ae8fffc3b21857492d26d498753a44","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_1_4; 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All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":[],"basis_permutation_census_to_model":[0,2,3,4,5,7,6,1],"alternative_models":[{"name":"Fib x Fib x Vec(C2)","reference":"Rank-four Grothendieck rings through multiplicity 128 (2026), Table 1; Pointed category Vec(C2); Deligne tensor product","source_url":null,"basis_permutation_census_to_model":[0,1,3,5,7,6,4,2]}]},{"rank":8,"multiplicity":1,"id":"R8_1_71","name":"PSU(2)_6 x Fib","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[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],[1,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,1,0],[0,0,0,1,1,0,0,0],[0,0,0,0,1,0,0,0],[0,1,1,0,0,0,0,0]],[[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[1,0,0,1,0,0,1,0],[0,0,0,0,1,1,0,1],[0,0,1,0,1,1,0,0],[0,1,0,1,0,0,1,0],[0,0,0,0,1,0,0,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,1,0],[0,0,0,0,1,1,0,1],[1,0,0,1,1,1,1,1],[0,1,1,1,1,1,1,0],[0,0,1,0,1,1,0,0],[0,0,0,1,1,0,0,0]],[[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,1,0,0,0],[0,0,1,0,1,1,0,0],[0,1,1,1,1,1,1,0],[1,0,0,1,1,1,1,1],[0,0,0,0,1,1,0,1],[0,0,0,0,0,1,1,0]],[[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,1,0,0,1,0],[0,0,1,0,1,1,0,0],[0,0,0,0,1,1,0,1],[1,0,0,1,0,0,1,0],[0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,1,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,1,0,0,0],[0,0,0,0,0,1,1,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1]]],"tensor_sha256":"dedb8c72dec425404e5894ed43eb5f821d458a9db4a44d66c1a9e1c64ab5915a","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_1_71; 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This manuscript, Proposition 4.1 (explicit dihedral character multiplication); Serre, Linear Representations of Finite Groups, Chapter 5.","model_reference_url":null,"source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_1_91","invariants":{"dual_indices":[0,1,2,3,4,5,6,7],"invertible_indices":[0,1],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,2.0,2.0,2.0,2.0,2.0,2.0],"global_fp_dimension_approx":26.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 26)*(t - 13)**6*(t - 2)","formal_codegree_polynomial":"(t - 26)*(t - 13)**6*(t - 2)","irreducible_characters_approx":[{"values_approx":[1.0,1.0,2.0,2.0,2.0,2.0,2.0,2.0],"formal_codegree_approx":26.0},{"values_approx":[1.0,1.0,-0.7092097741,1.7709120513,-1.9418836349,1.1361294935,0.2410733605,-1.4970214963],"formal_codegree_approx":13.0},{"values_approx":[1.0,1.0,-1.4970214963,1.1361294935,1.7709120513,-0.7092097741,-1.9418836349,0.2410733605],"formal_codegree_approx":13.0},{"values_approx":[1.0,1.0,-1.9418836349,-1.4970214963,-0.7092097741,0.2410733605,1.1361294935,1.7709120513],"formal_codegree_approx":13.0},{"values_approx":[1.0,1.0,0.2410733605,-0.7092097741,1.1361294935,-1.4970214963,1.7709120513,-1.9418836349],"formal_codegree_approx":13.0},{"values_approx":[1.0,1.0,1.1361294935,-1.9418836349,0.2410733605,1.7709120513,-1.4970214963,-0.7092097741],"formal_codegree_approx":13.0},{"values_approx":[1.0,1.0,1.7709120513,0.2410733605,-1.4970214963,-1.9418836349,-0.7092097741,1.1361294935],"formal_codegree_approx":13.0},{"values_approx":[1.0,-1.0,0.0,0.0,0.0,0.0,0.0,0.0],"formal_codegree_approx":2.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7,8],"primitive_polynomial":"(theta - 69)*(theta + 1)*(theta**6 + 15*theta**5 - 20*theta**4 - 1644*theta**3 - 12267*theta**2 - 37487*theta - 42509)","basis_values":["1","-theta**7/548765 + 54*theta**6/548765 + 211*theta**5/109753 + 264*theta**4/548765 - 101169*theta**3/548765 - 808936*theta**2/548765 - 363442*theta/78395 - 2384356/548765","-14661204018537*theta**7/5501334924318905 + 865056149493538*theta**6/5501334924318905 + 2213300460525146*theta**5/1100266984863781 - 47108479107827177*theta**4/5501334924318905 - 1196749826769506543*theta**3/5501334924318905 - 6158172167042548812*theta**2/5501334924318905 - 1739133002651403689*theta/785904989188415 - 7155213414229842722/5501334924318905","-295008916708547*theta**7/5501334924318905 + 17329056132275743*theta**6/5501334924318905 + 45537532603134193*theta**5/1100266984863781 - 918324085269354972*theta**4/5501334924318905 - 24498158241427066513*theta**3/5501334924318905 - 127822093688498741712*theta**2/5501334924318905 - 36442477284558554574*theta/785904989188415 - 150645017861602165172/5501334924318905","87235592482998*theta**7/5501334924318905 - 5150815571483342*theta**6/5501334924318905 - 13125388836934242*theta**5/1100266984863781 + 282843761045738398*theta**4/5501334924318905 + 7100957455838495737*theta**3/5501334924318905 + 36368030369562579088*theta**2/5501334924318905 + 10233265779019895766*theta/785904989188415 + 42022554885348743743/5501334924318905","663387969667403*theta**7/5501334924318905 - 38917455682038617*theta**6/5501334924318905 - 103047164195539796*theta**5/1100266984863781 + 2043278972740793468*theta**4/5501334924318905 + 55362078046019443522*theta**3/5501334924318905 + 290174190469084560088*theta**2/5501334924318905 + 82989550942954724886*theta/785904989188415 + 343595810227551171208/5501334924318905","-335120979918857*theta**7/5501334924318905 + 19644770109690938*theta**6/5501334924318905 + 52249631662957406*theta**5/1100266984863781 - 1026255785273699237*theta**4/5501334924318905 - 28048617935031328938*theta**3/5501334924318905 - 147357017394800155052*theta**2/5501334924318905 - 42206824010385398724*theta/785904989188415 - 174871844560430088482/5501334924318905","-105827449027782*theta**7/5501334924318905 + 6229118188869893*theta**6/5501334924318905 + 16171030674924530*theta**5/1100266984863781 - 334435707440468772*theta**4/5501334924318905 - 8719002392279170343*theta**3/5501334924318905 - 45200882822203399247*theta**2/5501334924318905 - 12832560682769251354*theta/785904989188415 - 52937088434283336827/5501334924318905"],"codegree_value":"-(55137149384*theta**7 - 2977406615501*theta**6 - 58169700831595*theta**5 - 14556196462076*theta**4 + 5578171168199556*theta**3 + 44602431805795679*theta**2 + 140274111496491651*theta + 90206600049653084)/5501334924318905","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. 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All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":[],"basis_permutation_census_to_model":[0,1,4,3,2,5,6,7],"alternative_models":[]},{"rank":8,"multiplicity":2,"id":"R8_2_202","name":"C(S4,1,A4,1)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0]],[[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],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0]],[[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[1,1,1,0,0,2,0,0],[0,0,0,1,2,0,1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0]],[[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,2,0,1,1],[1,1,1,0,0,2,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0]],[[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0]]],"tensor_sha256":"a18f3462b805591e9e071ca8d52a3d2e79117fe869f3299cb53e348566dcb23b","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_2_202; https://bimsa.net/doc/publication/14753.pdf, Alekseyev–Bruns–Dong–Palcoux archived preprint Appendix C.3.2, PDF pages 46–49 (p45–48 zero based); Appendix C.3.2(6), Lemma-free group-theoretical construction.","model_reference_url":"https://bimsa.net/doc/publication/14753.pdf","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_2_202","invariants":{"dual_indices":[0,2,1,3,4,5,6,7],"invertible_indices":[0,1,2,3,6,7],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,1.0,3.0,3.0,1.0,1.0],"global_fp_dimension_approx":24.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 24)**2*(t - 8)**2*(t - 6)**4","irreducible_characters_approx":[{"degree":2,"values_approx":[2.0,-1.0,-1.0,0.0,0.0,0.0,0.0,0.0],"formal_codegree_approx":3.0,"formal_codegree_minimal_polynomial":"u - 3","formal_codegree_exact":"3"},{"degree":1,"values_approx":[1.0,1.0,1.0,-1.0,-3.0,3.0,-1.0,-1.0],"formal_codegree_approx":24.0,"formal_codegree_minimal_polynomial":"u - 24","formal_codegree_exact":"24"},{"degree":1,"values_approx":[1.0,1.0,1.0,-1.0,1.0,-1.0,-1.0,-1.0],"formal_codegree_approx":8.0,"formal_codegree_minimal_polynomial":"u - 8","formal_codegree_exact":"8"},{"degree":1,"values_approx":[1.0,1.0,1.0,1.0,-1.0,-1.0,1.0,1.0],"formal_codegree_approx":8.0,"formal_codegree_minimal_polynomial":"u - 8","formal_codegree_exact":"8"},{"degree":1,"values_approx":[1.0,1.0,1.0,1.0,3.0,3.0,1.0,1.0],"formal_codegree_approx":24.0,"formal_codegree_minimal_polynomial":"u - 24","formal_codegree_exact":"24"}],"character_method":"Numerical regular-algebra block traces after exact rational center-kernel computation; χ_E(X_i) and f_E=(Σ_i χ_E(X_i)χ_E(X_i*)) / dim(E) are approximate. The displayed codegree minimal polynomials are exact factors of the rational central-Casimir polynomial rescaled by each irreducible degree; factor association uses the validated numerical eigenvalues. 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All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":[],"basis_permutation_census_to_model":[0,2,3,4,5,7,6,1],"alternative_models":[]},{"rank":8,"multiplicity":2,"id":"R8_2_37","name":"Nonsplit C2 extension of the unitary 2D2 category","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0]],[[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,2,2,0],[0,0,0,0,0,2,2,1],[0,1,0,2,2,0,0,0],[0,0,1,2,2,0,0,0],[0,0,0,0,1,0,0,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,2,2,1],[1,0,0,0,0,2,2,0],[0,0,1,2,2,0,0,0],[0,1,0,2,2,0,0,0],[0,0,0,1,0,0,0,0]],[[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,2,2,0,0,0],[0,1,0,2,2,0,0,0],[1,0,0,0,0,2,2,0],[0,0,0,0,0,2,2,1],[0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,2,2,0,0,0],[0,0,1,2,2,0,0,0],[0,0,0,0,0,2,2,1],[1,0,0,0,0,2,2,0],[0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0]]],"tensor_sha256":"1cfa463b5ec59daa44cc51752a6a58ea5378684a46ab7ee72147ab69153d5f6a","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_2_37; Grossman-Izumi-Snyder, Theorem 4.1; Grossman, Section 4.1; the explicit crossed-product extension in the accompanying proof.","model_reference_url":"https://arxiv.org/abs/2201.11901","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_2_37","invariants":{"dual_indices":[0,2,1,3,4,5,6,7],"invertible_indices":[0,1,2,7],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,4.2360679775,4.2360679775,4.2360679775,4.2360679775,1.0],"global_fp_dimension_approx":75.77708764,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 8)**4*(t**2 - 80*t + 320)**2","irreducible_characters_approx":[{"degree":2,"values_approx":[2.0,0.0,0.0,0.0,0.0,0.0,0.0,-2.0],"formal_codegree_approx":4.0,"formal_codegree_minimal_polynomial":"u - 4","formal_codegree_exact":"4"},{"degree":1,"values_approx":[1.0,-1.0,-1.0,-4.2360679775,-4.2360679775,4.2360679775,4.2360679775,1.0],"formal_codegree_approx":75.77708764,"formal_codegree_minimal_polynomial":"u**2 - 80*u + 320"},{"degree":1,"values_approx":[1.0,-1.0,-1.0,0.2360679775,0.2360679775,-0.2360679775,-0.2360679775,1.0],"formal_codegree_approx":4.22291236,"formal_codegree_minimal_polynomial":"u**2 - 80*u + 320"},{"degree":1,"values_approx":[1.0,1.0,1.0,-0.2360679775,-0.2360679775,-0.2360679775,-0.2360679775,1.0],"formal_codegree_approx":4.22291236,"formal_codegree_minimal_polynomial":"u**2 - 80*u + 320"},{"degree":1,"values_approx":[1.0,1.0,1.0,4.2360679775,4.2360679775,4.2360679775,4.2360679775,1.0],"formal_codegree_approx":75.77708764,"formal_codegree_minimal_polynomial":"u**2 - 80*u + 320"}],"character_method":"Numerical regular-algebra block traces after exact rational center-kernel computation; χ_E(X_i) and f_E=(Σ_i χ_E(X_i)χ_E(X_i*)) / dim(E) are approximate. The displayed codegree minimal polynomials are exact factors of the rational central-Casimir polynomial rescaled by each irreducible degree; factor association uses the validated numerical eigenvalues. Orthogonality and block dimensions validated."},"construction_notes":[],"basis_permutation_census_to_model":[0,1,3,5,7,4,6,2],"alternative_models":[]},{"rank":8,"multiplicity":2,"id":"R8_2_39","name":"Asaeda-Haagerup AH4","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0]],[[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0]],[[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,2,2,2,2,0],[0,0,0,2,2,2,2,1],[0,1,0,2,2,2,2,0],[0,0,1,2,2,2,2,0],[0,0,0,0,1,0,0,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,2,2,2,2,1],[1,0,0,2,2,2,2,0],[0,0,1,2,2,2,2,0],[0,1,0,2,2,2,2,0],[0,0,0,1,0,0,0,0]],[[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,2,2,2,2,0],[0,1,0,2,2,2,2,0],[1,0,0,2,2,2,2,0],[0,0,0,2,2,2,2,1],[0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,2,2,2,2,0],[0,0,1,2,2,2,2,0],[0,0,0,2,2,2,2,1],[1,0,0,2,2,2,2,0],[0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0]]],"tensor_sha256":"0e9b85072acea8dbee8095f365279fbe6b4c2db116e6979d1583ae402244fdfc","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_2_39; Grossman-Izumi-Snyder, The Asaeda-Haagerup fusion categories, Theorem 4.9, arXiv:1501.07324.","model_reference_url":"https://arxiv.org/pdf/1501.07324","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_2_39","invariants":{"dual_indices":[0,2,1,3,4,5,6,7],"invertible_indices":[0,1,2,7],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,8.1231056256,8.1231056256,8.1231056256,8.1231056256,1.0],"global_fp_dimension_approx":267.9393800198,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 8)**6*(t**2 - 272*t + 1088)","irreducible_characters_approx":[{"degree":2,"values_approx":[2.0,0.0,0.0,0.0,0.0,0.0,0.0,-2.0],"formal_codegree_approx":4.0,"formal_codegree_minimal_polynomial":"u - 4","formal_codegree_exact":"4"},{"degree":1,"values_approx":[1.0,-1.0,-1.0,-1.0,-1.0,1.0,1.0,1.0],"formal_codegree_approx":8.0,"formal_codegree_minimal_polynomial":"u - 8","formal_codegree_exact":"8"},{"degree":1,"values_approx":[1.0,-1.0,-1.0,1.0,1.0,-1.0,-1.0,1.0],"formal_codegree_approx":8.0,"formal_codegree_minimal_polynomial":"u - 8","formal_codegree_exact":"8"},{"degree":1,"values_approx":[1.0,1.0,1.0,-0.1231056256,-0.1231056256,-0.1231056256,-0.1231056256,1.0],"formal_codegree_approx":4.0606199802,"formal_codegree_minimal_polynomial":"u**2 - 272*u + 1088"},{"degree":1,"values_approx":[1.0,1.0,1.0,8.1231056256,8.1231056256,8.1231056256,8.1231056256,1.0],"formal_codegree_approx":267.9393800198,"formal_codegree_minimal_polynomial":"u**2 - 272*u + 1088"}],"character_method":"Numerical regular-algebra block traces after exact rational center-kernel computation; χ_E(X_i) and f_E=(Σ_i χ_E(X_i)χ_E(X_i*)) / dim(E) are approximate. 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All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":[],"basis_permutation_census_to_model":[0,1,5,2,3,6,7,4],"alternative_models":[]},{"rank":8,"multiplicity":2,"id":"R8_2_820","name":"Nonpointed Mueger factor of Z(NG(C5,5)), Gprime=C3 x C3","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[1,0,0,1,1,1,1,1],[0,0,1,1,1,1,1,1],[0,1,1,1,0,0,1,1],[0,1,1,0,1,0,1,1],[0,1,1,0,0,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1]],[[0,0,1,0,0,0,0,0],[0,0,1,1,1,1,1,1],[1,1,1,1,1,1,1,1],[0,1,1,0,1,1,1,1],[0,1,1,1,0,1,1,1],[0,1,1,1,1,0,1,1],[0,1,1,1,1,1,2,1],[0,1,1,1,1,1,1,2]],[[0,0,0,1,0,0,0,0],[0,1,1,1,0,0,1,1],[0,1,1,0,1,1,1,1],[1,1,0,0,1,1,1,1],[0,0,1,1,1,0,1,1],[0,0,1,1,0,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1]],[[0,0,0,0,1,0,0,0],[0,1,1,0,1,0,1,1],[0,1,1,1,0,1,1,1],[0,0,1,1,1,0,1,1],[1,1,0,1,0,1,1,1],[0,0,1,0,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1]],[[0,0,0,0,0,1,0,0],[0,1,1,0,0,1,1,1],[0,1,1,1,1,0,1,1],[0,0,1,1,0,1,1,1],[0,0,1,0,1,1,1,1],[1,1,0,1,1,0,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1]],[[0,0,0,0,0,0,1,0],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,2,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[1,1,2,1,1,1,1,2],[0,1,1,1,1,1,2,2]],[[0,0,0,0,0,0,0,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,2],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,2,2],[1,1,2,1,1,1,2,1]]],"tensor_sha256":"88d4e0d03d1f45b727bb5d973ee4c47ef114ff4870f397d9f08342d5db1d0373","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"The actual NG(C5,5) center factor is identified by its known tube blocks, exhaustive associative completions, and exact third indicators with balancing. No numerical recognition of the unknown tube block is used. See the current manuscript and the dated exact proof.","model_reference_url":"https://arxiv.org/abs/1208.1500","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_2_820","invariants":{"dual_indices":[0,1,2,3,4,5,6,7],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,5.8541019662,6.8541019662,5.8541019662,5.8541019662,5.8541019662,7.8541019662,7.8541019662],"global_fp_dimension_approx":308.4345884812,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 9)**4*(t - 5)**2*(t**2 - 315*t + 2025)","formal_codegree_polynomial":"(t - 9)**4*(t - 5)**2*(t**2 - 315*t + 2025)","irreducible_characters_approx":[{"values_approx":[1.0,5.8541019662,6.8541019662,5.8541019662,5.8541019662,5.8541019662,7.8541019662,7.8541019662],"formal_codegree_approx":308.4345884812},{"values_approx":[1.0,-2.0,-1.0,1.0,1.0,1.0,0.0,0.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,1.0,-1.0,-2.0,1.0,1.0,0.0,0.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,1.0,-1.0,1.0,-2.0,1.0,0.0,0.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,1.0,-1.0,1.0,1.0,-2.0,0.0,0.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,-0.8541019662,0.1458980338,-0.8541019662,-0.8541019662,-0.8541019662,1.1458980338,1.1458980338],"formal_codegree_approx":6.5654115188},{"values_approx":[1.0,0.0,1.0,0.0,0.0,0.0,-1.6180339887,0.6180339887],"formal_codegree_approx":5.0},{"values_approx":[1.0,0.0,1.0,0.0,0.0,0.0,0.6180339887,-1.6180339887],"formal_codegree_approx":5.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7,8],"primitive_polynomial":"theta*(theta - 9)*(theta - 3)*(theta + 3)*(theta**2 - 243*theta + 981)*(theta**2 + 7*theta + 11)","basis_values":["1","6167414005569277*theta**7/2387699144818748145000 - 385028414148048893*theta**6/596924786204687036250 + 1130466635380746001*theta**5/170549938915624867500 + 1732916637861572377*theta**4/132649952489930452500 - 4760579309409089057*theta**3/22108325414988408750 - 320953269855158197*theta**2/4912961203330757500 + 41244912091528501177*theta/29477767219984545000 + 1","-1260116644323029*theta**7/198974928734895678750 + 156338123186415497*theta**6/99487464367447839375 - 392361288963192229*theta**5/28424989819270811250 - 961138070805404083*theta**4/22108325414988408750 + 4503353111408932034*theta**3/11054162707494204375 + 648461824432573089*theta**2/2456480601665378750 - 3124418075138305602*theta/1228240300832689375 - 1","75938503794047*theta**7/14738883609992272500 - 28009934521045913*theta**6/22108325414988408750 + 26569709563887383*theta**5/3158332202141201250 + 126179618374628341*theta**4/2456480601665378750 - 4496913068329293349*theta**3/22108325414988408750 - 3878821764879414931*theta**2/7369441804996136250 + 3171109073521444269*theta/4912961203330757500 + 1","-522769240675159*theta**7/596924786204687036250 + 60330761650483612*theta**6/298462393102343518125 + 155347908122789341*theta**5/85274969457812433750 - 2053926922727641043*theta**4/66324976244965226250 - 879480159114326362*theta**3/11054162707494204375 + 4389746001092052569*theta**2/7369441804996136250 + 2097075260496254608*theta/3684720902498068125 - 2","-4395903838675837*theta**7/795899714939582715000 + 135799864841266204*theta**6/99487464367447839375 - 609647040834655231*theta**5/56849979638541622500 - 187910873155608943*theta**4/4912961203330757500 + 951863751127371421*theta**3/3684720902498068125 + 987514595440868263*theta**2/14738883609992272500 - 9357147117492900837*theta/9825922406661515000 + 1","547524532892423*theta**7/99487464367447839375 - 135058123006514878*theta**6/99487464367447839375 + 142478877288203323*theta**5/14212494909635405625 + 540449344775295646*theta**4/11054162707494204375 - 3130757035004448841*theta**3/11054162707494204375 - 405391221768780768*theta**2/1228240300832689375 + 2128477173190993773*theta/1228240300832689375","-96777327553702*theta**7/99487464367447839375 + 48390914779032619*theta**6/198974928734895678750 - 36307764308717552*theta**5/14212494909635405625 - 163272080266361833*theta**4/22108325414988408750 + 2355874658037741943*theta**3/22108325414988408750 + 57440582743664607*theta**2/1228240300832689375 - 1845823965819729579*theta/2456480601665378750"],"codegree_value":"(278522626060307*theta**7 - 69121476193696702*theta**6 + 609713228777402449*theta**5 + 1900491129681594501*theta**4 - 18043205316896765196*theta**3 - 11505580595444917647*theta**2 + 112799033326703324592*theta + 198974928734895678750)/22108325414988408750","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":["The actual NG(C5,5) center factor is identified by its known tube blocks, exhaustive associative completions, and exact third indicators with balancing. No numerical recognition of the unknown tube block is used.","Exact proof and tensor checks remain subject to independent human verification."],"basis_permutation_census_to_model":[0,4,1,5,6,7,2,3],"alternative_models":[],"construction_proof_file":"evidence/correctness_20261025/rank8-neargroup-exact-repair.md"},{"rank":8,"multiplicity":2,"id":"R8_2_839","name":"Haagerup H1 x Fib","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[1,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,1,0,0],[0,0,0,1,0,0,1,0],[0,0,1,0,0,0,0,1]],[[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,1,0,0,0],[0,0,0,1,1,0,0,0],[0,0,1,1,1,0,0,0],[0,0,0,0,0,1,1,1],[0,0,0,0,0,1,1,0],[0,1,0,0,0,1,0,0]],[[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,1,1,0,0,0],[1,0,1,1,1,0,0,0],[0,0,1,1,2,0,0,0],[0,0,0,0,0,2,1,1],[0,1,0,0,0,1,1,1],[0,0,0,0,0,1,1,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,1,1,0,0,0],[0,0,1,1,2,0,0,0],[1,0,1,2,2,0,0,0],[0,1,0,0,0,2,2,1],[0,0,0,0,0,2,1,1],[0,0,0,0,0,1,1,1]],[[0,0,0,0,0,1,0,0],[0,0,0,0,1,1,0,0],[0,0,0,0,0,1,1,1],[0,0,0,0,0,2,1,1],[0,1,0,0,0,2,2,1],[1,1,1,2,2,2,2,1],[0,0,1,1,2,2,1,1],[0,0,1,1,1,1,1,1]],[[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,1,0],[0,0,0,0,0,1,1,0],[0,1,0,0,0,1,1,1],[0,0,0,0,0,2,1,1],[0,0,1,1,2,2,1,1],[1,1,1,1,1,1,1,1],[0,0,0,1,1,1,1,0]],[[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,1],[0,1,0,0,0,1,0,0],[0,0,0,0,0,1,1,0],[0,0,0,0,0,1,1,1],[0,0,1,1,1,1,1,1],[0,0,0,1,1,1,1,0],[1,1,0,0,1,1,0,0]]],"tensor_sha256":"af65229ca5082858e20ec88896351f6cae7ffb8460e9d626a9177b9ef89e729e","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_2_839; Rank-four Grothendieck rings through multiplicity 128 (2026), Table 1; Ostrik, Fusion categories of rank 2; Fibonacci category; Deligne tensor product.","model_reference_url":null,"source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_2_839","invariants":{"dual_indices":[0,1,2,3,4,5,6,7],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.6180339887,2.3027756377,3.3027756377,4.3027756377,6.9620372278,5.3440032391,3.7259692503],"global_fp_dimension_approx":129.2541945633,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t**2 - 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All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":[],"basis_permutation_census_to_model":[0,1,2,4,6,7,5,3],"alternative_models":[]},{"rank":8,"multiplicity":2,"id":"R8_2_855","name":"D14 even part; local modules SU(2)_24","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[1,1,0,0,0,0,0,1],[0,0,0,1,0,0,1,0],[0,0,1,0,0,0,1,0],[0,0,0,0,1,1,0,1],[0,0,0,0,1,1,1,0],[0,0,1,1,0,1,1,0],[0,1,0,0,1,0,0,1]],[[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,1,0],[1,0,1,0,0,1,0,1],[0,1,0,0,1,0,1,0],[0,0,0,1,1,1,1,0],[0,0,1,0,1,1,1,1],[0,1,0,1,1,1,1,1],[0,0,1,0,0,1,1,0]],[[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,1,0],[0,1,0,0,1,0,1,0],[1,0,0,1,0,1,0,1],[0,0,1,0,1,1,1,0],[0,0,0,1,1,1,1,1],[0,1,1,0,1,1,1,1],[0,0,0,1,0,1,1,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,1,1,0,1],[0,0,0,1,1,1,1,0],[0,0,1,0,1,1,1,0],[1,1,1,1,1,1,1,1],[0,1,1,1,1,1,2,1],[0,0,1,1,1,2,2,1],[0,1,0,0,1,1,1,1]],[[0,0,0,0,0,1,0,0],[0,0,0,0,1,1,1,0],[0,0,1,0,1,1,1,1],[0,0,0,1,1,1,1,1],[0,1,1,1,1,1,2,1],[1,1,1,1,1,2,2,1],[0,1,1,1,2,2,2,1],[0,0,1,1,1,1,1,1]],[[0,0,0,0,0,0,1,0],[0,0,1,1,0,1,1,0],[0,1,0,1,1,1,1,1],[0,1,1,0,1,1,1,1],[0,0,1,1,1,2,2,1],[0,1,1,1,2,2,2,1],[1,1,1,1,2,2,2,2],[0,0,1,1,1,1,2,0]],[[0,0,0,0,0,0,0,1],[0,1,0,0,1,0,0,1],[0,0,1,0,0,1,1,0],[0,0,0,1,0,1,1,0],[0,1,0,0,1,1,1,1],[0,0,1,1,1,1,1,1],[0,0,1,1,1,1,2,0],[1,1,0,0,1,1,0,1]]],"tensor_sha256":"c02ff92d7a4c765ad745ceee85ac93742f85bd89fdf2c04b8360e75d2ac7500b","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_2_855; D14 local bosonic modularization of SU(2)_24; NONINTEGRAL_PROOFS.md.","model_reference_url":"https://arxiv.org/abs/math/0101219","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_2_855","invariants":{"dual_indices":[0,1,2,3,4,5,6,7],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,2.9418836349,4.1481149053,4.1481149053,6.2098171825,7.345946676,8.05515645,4.7127956862],"global_fp_dimension_approx":223.6891444762,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 13)**2*(t**6 - 273*t**5 + 11830*t**4 - 184548*t**3 + 1285245*t**2 - 4084223*t + 4826809)","formal_codegree_polynomial":"(t - 13)**2*(t**6 - 273*t**5 + 11830*t**4 - 184548*t**3 + 1285245*t**2 - 4084223*t + 4826809)","irreducible_characters_approx":[{"values_approx":[1.0,2.9418836349,4.1481149053,4.1481149053,6.2098171825,7.345946676,8.05515645,4.7127956862],"formal_codegree_approx":223.6891444762},{"values_approx":[1.0,2.4970214963,-1.4100200484,-1.4100200484,1.6019653634,-0.3399182715,-2.1108303228,2.7380948569],"formal_codegree_approx":25.8460349805},{"values_approx":[1.0,-1.0,-1.3027756377,2.3027756377,-1.0,1.0,-1.0,1.0],"formal_codegree_approx":13.0},{"values_approx":[1.0,-1.0,2.3027756377,-1.3027756377,-1.0,1.0,-1.0,1.0],"formal_codegree_approx":13.0},{"values_approx":[1.0,1.7092097741,0.8801813576,0.8801813576,-1.5587237736,-1.3176504131,0.6242332218,0.2121882777],"formal_codegree_approx":10.0713498902},{"values_approx":[1.0,0.7589266395,-0.6679930799,-0.6679930799,-0.4737472213,1.29716483,0.1610353366,-1.1829569954],"formal_codegree_approx":5.8007918121},{"values_approx":[1.0,-0.1361294935,0.5646807809,0.5646807809,1.0965443673,-0.400477129,-0.6415504895,-0.8453392675],"formal_codegree_approx":4.1452369958},{"values_approx":[1.0,-0.7709120513,-0.5149639155,-0.5149639155,0.1241440816,-0.5850656924,0.9119558039,0.3652174422],"formal_codegree_approx":3.4474418451}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7,8],"primitive_polynomial":"(theta**2 - 9*theta + 17)*(theta**6 - 209*theta**5 + 736*theta**4 + 11743*theta**3 - 6208*theta**2 - 81305*theta + 72175)","basis_values":["1","-93341745845685557*theta**7/29111076264204221521000 + 20302572694609424311*theta**6/29111076264204221521000 - 29496195801892797246*theta**5/3638884533025527690125 - 242451927592221181997*theta**4/29111076264204221521000 + 14609968878817016247*theta**3/46355216981216913250 - 11603381225367626356341*theta**2/29111076264204221521000 - 79241505061978151883283*theta/29111076264204221521000 + 3194914743711941605509/727776906605105538025","-68605217216637438*theta**7/3638884533025527690125 + 29562579207089229173*theta**6/7277769066051055380250 - 286165399323903153549*theta**5/7277769066051055380250 - 470694187597048335173*theta**4/3638884533025527690125 + 35903722103597796267*theta**3/23177608490608456625 + 4074864387555731441287*theta**2/7277769066051055380250 - 42513212162703311032322*theta/3638884533025527690125 + 14933275988109979528971/1455553813210211076050","83005209434280871*theta**7/29111076264204221521000 - 2214130217435867226*theta**6/3638884533025527690125 + 137013197280602771929*theta**5/29111076264204221521000 + 778418094952202954241*theta**4/29111076264204221521000 - 30008930735657135239*theta**3/185420867924867653000 - 10816050735201636800227*theta**2/29111076264204221521000 + 11592336112478375025087*theta/14555538132102110760500 + 3330555639091242451509/5822215252840844304200","55123472511658287*theta**7/5822215252840844304200 - 5955882384470310593*theta**6/2911107626420422152100 + 122510013889952708853*theta**5/5822215252840844304200 + 311619270514253183597*theta**4/5822215252840844304200 - 30143720978029676723*theta**3/37084173584973530600 + 333495102653478997741*theta**2/5822215252840844304200 + 17386719150618793500019*theta/2911107626420422152100 - 7725655109708553496337/1164443050568168860840","-223697313249706813*theta**7/29111076264204221521000 + 48145524692648085199*theta**6/29111076264204221521000 - 113897410986572421353*theta**5/7277769066051055380250 - 1630413305183207228273*theta**4/29111076264204221521000 + 27932065142807978273*theta**3/46355216981216913250 + 13592489822313304608031*theta**2/29111076264204221521000 - 124070122080805398724247*theta/29111076264204221521000 + 1342260247400804021206/727776906605105538025","324429210788780619*theta**7/29111076264204221521000 - 34934001503078778931*theta**6/14555538132102110760500 + 669666817942952258181*theta**5/29111076264204221521000 + 2321261301722139260849*theta**4/29111076264204221521000 - 169653289191592921271*theta**3/185420867924867653000 - 14297719993836685908003*theta**2/29111076264204221521000 + 101249036810562496655143*theta/14555538132102110760500 - 28102393944306568368449/5822215252840844304200","-100714942707641811*theta**7/29111076264204221521000 + 21686632899298509503*theta**6/29111076264204221521000 - 51844033549282309541*theta**5/7277769066051055380250 - 698617383263408806881*theta**4/29111076264204221521000 + 6441611356097596703*theta**3/23177608490608456625 + 3470539524689127672807*theta**2/29111076264204221521000 - 53403656375096155745609*theta/29111076264204221521000 + 2373976809740768400739/1455553813210211076050"],"codegree_value":"-(148086155642087003*theta**7 - 31937314639215113519*theta**6 + 315621786178424895022*theta**5 + 953558345951057169663*theta**4 - 12382620227333622208514*theta**3 - 2866902577416925209861*theta**2 + 92450425202093654342807*theta - 98018275545553241482190)/2239313558784940117000","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":[],"basis_permutation_census_to_model":[0,1,6,7,3,4,5,2],"alternative_models":[]},{"rank":8,"multiplicity":2,"id":"R8_2_868","name":"unitary HI(C7) Q-dual","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[1,1,0,0,1,1,1,1],[0,0,1,1,0,1,1,1],[0,0,1,1,1,1,1,1],[0,1,0,1,0,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1]],[[0,0,1,0,0,0,0,0],[0,0,1,1,0,1,1,1],[1,1,1,0,0,1,1,1],[0,1,0,1,1,1,1,1],[0,0,0,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1]],[[0,0,0,1,0,0,0,0],[0,0,1,1,1,1,1,1],[0,1,0,1,1,1,1,1],[1,1,1,1,1,1,1,1],[0,1,1,1,0,1,1,1],[0,1,1,1,1,2,1,1],[0,1,1,1,1,1,2,1],[0,1,1,1,1,1,1,2]],[[0,0,0,0,1,0,0,0],[0,1,0,1,0,1,1,1],[0,0,0,1,1,1,1,1],[0,1,1,1,0,1,1,1],[1,0,1,0,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1]],[[0,0,0,0,0,1,0,0],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,2,1,1],[0,1,1,1,1,1,1,1],[1,1,1,2,1,1,1,2],[0,1,1,1,1,1,2,2],[0,1,1,1,1,2,2,1]],[[0,0,0,0,0,0,1,0],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,2,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,2,2],[1,1,1,2,1,2,1,1],[0,1,1,1,1,2,1,2]],[[0,0,0,0,0,0,0,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,2],[0,1,1,1,1,1,1,1],[0,1,1,1,1,2,2,1],[0,1,1,1,1,2,1,2],[1,1,1,2,1,1,2,1]]],"tensor_sha256":"1cfda199ea64692ba0d9bd267cb99212d408bc60fdc740757192771fcf0229bd","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"The actual HI(C7) center identification uses Gannon–Schopieray–Yadav, Theorem A(2)/5.1 (arXiv:2609.25185v1), followed by the canonical-algebra argument. Its generator matrix alone is not a full fusion-tensor proof. See the current manuscript and the dated exact proof.","model_reference_url":"https://arxiv.org/abs/1006.1326","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_2_868","invariants":{"dual_indices":[0,1,2,3,4,5,6,7],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,6.1400549446,6.1400549446,7.1400549446,6.1400549446,8.1400549446,8.1400549446,8.1400549446],"global_fp_dimension_approx":363.8626922874,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 7)**6*(t**2 - 371*t + 2597)","formal_codegree_polynomial":"(t - 7)**6*(t**2 - 371*t + 2597)","irreducible_characters_approx":[{"values_approx":[1.0,6.1400549446,6.1400549446,7.1400549446,6.1400549446,8.1400549446,8.1400549446,8.1400549446],"formal_codegree_approx":363.8626922874},{"values_approx":[1.0,-1.1400549446,-1.1400549446,-0.1400549446,-1.1400549446,0.8599450554,0.8599450554,0.8599450554],"formal_codegree_approx":7.1373077126},{"values_approx":[1.0,-1.2469796037,0.4450418679,-1.0,1.8019377358,0.0,0.0,0.0],"formal_codegree_approx":7.0},{"values_approx":[1.0,0.0,0.0,1.0,0.0,-0.4450418679,1.2469796037,-1.8019377358],"formal_codegree_approx":7.0},{"values_approx":[1.0,0.0,0.0,1.0,0.0,-1.8019377358,-0.4450418679,1.2469796037],"formal_codegree_approx":7.0},{"values_approx":[1.0,0.0,0.0,1.0,0.0,1.2469796037,-1.8019377358,-0.4450418679],"formal_codegree_approx":7.0},{"values_approx":[1.0,0.4450418679,1.8019377358,-1.0,-1.2469796037,0.0,0.0,0.0],"formal_codegree_approx":7.0},{"values_approx":[1.0,1.8019377358,-1.2469796037,-1.0,0.4450418679,0.0,0.0,0.0],"formal_codegree_approx":7.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7,8],"primitive_polynomial":"(theta**2 - 269*theta + 1859)*(theta**3 - theta**2 - 16*theta - 13)*(theta**3 + 6*theta**2 + 5*theta - 13)","basis_values":["1","-89616817187661*theta**7/28955683693971726965 + 3459253271621981*theta**6/4136526241995960995 - 27622365068390074*theta**5/4136526241995960995 - 344790560440519*theta**4/24476486639029355 + 690809965541520768*theta**3/4136526241995960995 - 194509883409075594*theta**2/4136526241995960995 - 339047735539770863*theta/318194326307381615 + 171521776821468302/171335406473205485","-1097725978104036*theta**7/28955683693971726965 + 41832519439815896*theta**6/4136526241995960995 - 193358037898929809*theta**5/4136526241995960995 - 124130240092366532*theta**4/318194326307381615 + 3974433672579043358*theta**3/4136526241995960995 + 19662819755498957321*theta**2/4136526241995960995 - 257060606618983283*theta/318194326307381615 - 852078606065148233/171335406473205485","915344037475419*theta**7/28955683693971726965 - 34964841953057064*theta**6/4136526241995960995 + 183334384646869726*theta**5/4136526241995960995 + 94253761660913908*theta**4/318194326307381615 - 4041181467308151202*theta**3/4136526241995960995 - 13817342169943418774*theta**2/4136526241995960995 + 735813059686870112*theta/318194326307381615 + 509440471951133462/171335406473205485","733348645872709*theta**7/28955683693971726965 - 27944330549032099*theta**6/4136526241995960995 + 128616249357358921*theta**5/4136526241995960995 + 81660449826418978*theta**4/318194326307381615 - 2627253144220588262*theta**3/4136526241995960995 - 12539610803274851544*theta**2/4136526241995960995 + 226694978144056467*theta/318194326307381615 + 510541055459263057/171335406473205485","-766202196764096*theta**7/28955683693971726965 + 29317765036151691*theta**6/4136526241995960995 - 167008287946038584*theta**5/4136526241995960995 - 69454752569279052*theta**4/318194326307381615 + 3405110194716147833*theta**3/4136526241995960995 + 10063572345850681466*theta**2/4136526241995960995 - 623993709395458903*theta/318194326307381615 - 525239232933998333/171335406473205485","1704542278143599*theta**7/28955683693971726965 - 65126600273331659*theta**6/4136526241995960995 + 345984729908869286*theta**5/4136526241995960995 + 164801584196859488*theta**4/318194326307381615 - 6746562471582267062*theta**3/4136526241995960995 - 24346616300437310029*theta**2/4136526241995960995 + 623083549419537447*theta/318194326307381615 + 881797851842574917/171335406473205485","-1398786322124801*theta**7/28955683693971726965 + 53393070923888571*theta**6/4136526241995960995 - 270117915403319679*theta**5/4136526241995960995 - 142605574361192442*theta**4/318194326307381615 + 5348918346786794618*theta**3/4136526241995960995 + 21176607903053286681*theta**2/4136526241995960995 - 365859762723487288*theta/318194326307381615 - 696219984039447618/171335406473205485"],"codegree_value":"-(2582860171*theta**7 - 663542386292*theta**6 - 3426192058642*theta**5 + 11171740439782*theta**4 + 85535484542874*theta**3 + 98455568640768*theta**2 - 96296802893122*theta - 4136640563176088838)/590932320285137285","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":["The actual HI(C7) center identification uses Gannon–Schopieray–Yadav, Theorem A(2)/5.1 (arXiv:2609.25185v1), followed by the canonical-algebra argument. Its generator matrix alone is not a full fusion-tensor proof.","Exact proof and tensor checks remain subject to independent human verification."],"basis_permutation_census_to_model":[0,2,3,1,4,5,7,6],"alternative_models":[],"construction_proof_file":"evidence/correctness_20261025/HI7_EXACT_CENTER.md"},{"rank":8,"multiplicity":2,"id":"R8_2_869","name":"Nonpointed Mueger factor of Z(NG(C5,5)), Gprime=C9","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[1,1,0,0,1,1,1,1],[0,0,1,1,1,0,1,1],[0,0,1,1,1,1,1,1],[0,1,1,1,0,0,1,1],[0,1,0,1,0,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1]],[[0,0,1,0,0,0,0,0],[0,0,1,1,1,0,1,1],[1,1,1,0,0,1,1,1],[0,1,0,1,1,1,1,1],[0,1,0,1,1,0,1,1],[0,0,1,1,0,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1]],[[0,0,0,1,0,0,0,0],[0,0,1,1,1,1,1,1],[0,1,0,1,1,1,1,1],[1,1,1,1,1,1,1,1],[0,1,1,1,0,1,1,1],[0,1,1,1,1,0,1,1],[0,1,1,1,1,1,2,1],[0,1,1,1,1,1,1,2]],[[0,0,0,0,1,0,0,0],[0,1,1,1,0,0,1,1],[0,1,0,1,1,0,1,1],[0,1,1,1,0,1,1,1],[1,0,1,0,1,1,1,1],[0,0,0,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1]],[[0,0,0,0,0,1,0,0],[0,1,0,1,0,1,1,1],[0,0,1,1,0,1,1,1],[0,1,1,1,1,0,1,1],[0,0,0,1,1,1,1,1],[1,1,1,0,1,0,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1]],[[0,0,0,0,0,0,1,0],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,2,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[1,1,1,2,1,1,1,2],[0,1,1,1,1,1,2,2]],[[0,0,0,0,0,0,0,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,2],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,1],[0,1,1,1,1,1,2,2],[1,1,1,2,1,1,2,1]]],"tensor_sha256":"6fd67687c6a4590e43cf539442042092ebc28973b626127f40d6f27fb66de112","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"The second actual NG(C5,5) center factor is distinguished by exact third indicators and balancing after all associative completions and dualities are exhausted. No numerical recognition of the unknown tube block is used. See the current manuscript and the dated exact proof.","model_reference_url":"https://arxiv.org/abs/1208.1500","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_2_869","invariants":{"dual_indices":[0,1,2,3,4,5,6,7],"invertible_indices":[0],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,5.8541019662,5.8541019662,6.8541019662,5.8541019662,5.8541019662,7.8541019662,7.8541019662],"global_fp_dimension_approx":308.4345884812,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 9)**4*(t - 5)**2*(t**2 - 315*t + 2025)","formal_codegree_polynomial":"(t - 9)**4*(t - 5)**2*(t**2 - 315*t + 2025)","irreducible_characters_approx":[{"values_approx":[1.0,5.8541019662,5.8541019662,6.8541019662,5.8541019662,5.8541019662,7.8541019662,7.8541019662],"formal_codegree_approx":308.4345884812},{"values_approx":[1.0,-0.3472963553,1.8793852416,-1.0,-1.5320888862,1.0,0.0,0.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,-1.5320888862,-0.3472963553,-1.0,1.8793852416,1.0,0.0,0.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,1.0,1.0,-1.0,1.0,-2.0,0.0,0.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,1.8793852416,-1.5320888862,-1.0,-0.3472963553,1.0,0.0,0.0],"formal_codegree_approx":9.0},{"values_approx":[1.0,-0.8541019662,-0.8541019662,0.1458980338,-0.8541019662,-0.8541019662,1.1458980338,1.1458980338],"formal_codegree_approx":6.5654115188},{"values_approx":[1.0,0.0,0.0,1.0,0.0,0.0,-1.6180339887,0.6180339887],"formal_codegree_approx":5.0},{"values_approx":[1.0,0.0,0.0,1.0,0.0,0.0,0.6180339887,-1.6180339887],"formal_codegree_approx":5.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7,8],"primitive_polynomial":"(theta + 5)*(theta**2 - 245*theta + 1225)*(theta**2 + 5*theta + 5)*(theta**3 - 9*theta**2 + 6*theta - 1)","basis_values":["1","-1935231745940099077*theta**7/42497863760620047312225 + 314304720928712029349*theta**6/28331909173746698208150 - 3224343557094719624923*theta**5/84995727521240094624450 - 375424948881693529207*theta**4/566638183474933964163 + 28596132395910750514597*theta**3/42497863760620047312225 + 169445545766764011596651*theta**2/16999145504248018924890 + 71240174455886776896272*theta/8499572752124009462445 - 1714594763719898301359/485689871549943397854","3050067241271717273*theta**7/42497863760620047312225 - 495423695691904332751*theta**6/28331909173746698208150 + 5121400080073913520527*theta**5/84995727521240094624450 + 593419703363623260428*theta**4/566638183474933964163 - 47675061458989043071778*theta**3/42497863760620047312225 - 268892315541477597175099*theta**2/16999145504248018924890 - 105610286393187569331853*theta/8499572752124009462445 + 3246241643984691514891/485689871549943397854","-6161899389665071*theta**7/1673144242544096350875 + 1004827286047784141*theta**6/1115429495029397567250 - 13189003705143027067*theta**5/3346288485088192701750 - 30080522969458467733*theta**4/557714747514698783625 + 44347458969699416942*theta**3/334628848508819270175 + 545951546349219379337*theta**2/669257697017638540350 - 41416136476659055073*theta/66925769701763854035 - 3414506790867271141/3824329697243648802","-1153153092032049502*theta**7/42497863760620047312225 + 187403716465712046749*theta**6/28331909173746698208150 - 2007838901638114929973*theta**5/84995727521240094624450 - 221145462166364222722*theta**4/566638183474933964163 + 19595942692767498699472*theta**3/42497863760620047312225 + 99665015992852692522401*theta**2/16999145504248018924890 + 34262845000386996433772*theta/8499572752124009462445 - 1530526959181972435109/485689871549943397854","498291982225271483*theta**7/212489318803100236561125 - 81760368621349479343*theta**6/141659545868733491040750 + 1441126673614305857741*theta**5/424978637606200473122250 + 1943986481073175969784*theta**4/70829772934366745520375 - 4086912070097980278241*theta**3/42497863760620047312225 - 36209580441071765002951*theta**2/84995727521240094624450 + 2834665229932306996354*theta/8499572752124009462445 + 457386802458197157239/485689871549943397854","8377427035061879*theta**7/1673144242544096350875 - 685418614169837192*theta**6/557714747514698783625 + 10755597647607192304*theta**5/1673144242544096350875 + 33626568924625911392*theta**4/557714747514698783625 - 62044695135417549298*theta**3/334628848508819270175 - 243604470849653691794*theta**2/334628848508819270175 + 38511125567742303052*theta/66925769701763854035 - 192657104042570507/1912164848621824401","-9600203733160067*theta**7/3346288485088192701750 + 789369646514209241*theta**6/1115429495029397567250 - 7604228280574507196*theta**5/1673144242544096350875 - 16473691613094774208*theta**4/557714747514698783625 + 88590360946210460389*theta**3/669257697017638540350 + 108438347394152528506*theta**2/334628848508819270175 - 36750249333062535271*theta/133851539403527708070 + 93603585553504918/1912164848621824401"],"codegree_value":"(2733123798913254*theta**7 - 668571383587491421*theta**6 + 2932505584394399009*theta**5 + 40037162131352732606*theta**4 - 98794935202124508020*theta**3 - 606686032511292386795*theta**2 + 460337708270471477650*theta + 3266567723438986703625)/371809831676465855750","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":["The second actual NG(C5,5) center factor is distinguished by exact third indicators and balancing after all associative completions and dualities are exhausted. No numerical recognition of the unknown tube block is used.","Exact proof and tensor checks remain subject to independent human verification."],"basis_permutation_census_to_model":[0,4,5,1,7,6,2,3],"alternative_models":[],"construction_proof_file":"evidence/correctness_20261025/rank8-neargroup-exact-repair.md"},{"rank":8,"multiplicity":3,"id":"R8_3_1456","name":"Ad(E8) x Vec(C2)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[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],[1,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,1,0],[0,0,0,1,1,0,0,0],[0,0,0,0,1,0,0,0],[0,1,1,0,0,0,0,0]],[[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,1,1,0],[0,0,0,0,0,2,1,1],[0,0,1,1,2,0,0,0],[0,1,0,1,1,0,0,0],[0,0,0,0,1,0,0,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,1,0],[0,0,0,0,0,2,1,1],[1,0,0,0,0,3,2,1],[0,1,1,2,3,0,0,0],[0,0,1,1,2,0,0,0],[0,0,0,1,1,0,0,0]],[[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,1,0,0,0],[0,0,1,1,2,0,0,0],[0,1,1,2,3,0,0,0],[1,0,0,0,0,3,2,1],[0,0,0,0,0,2,1,1],[0,0,0,0,0,1,1,0]],[[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,1,1,0,0,0],[0,0,1,1,2,0,0,0],[0,0,0,0,0,2,1,1],[1,0,0,0,0,1,1,0],[0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,1,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,1,0,0,0],[0,0,0,0,0,1,1,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1]]],"tensor_sha256":"5dab1eaad3760ca8fb51cf3c95f1dfede4633475254ea8c7a0441e632a8942b7","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_3_1456; Existing unitary near-group or E8 model and Deligne product.; Kirillov–Ostrik, arXiv:math/0101219, E8 construction and its adjoint category, followed by the stated Deligne product.","model_reference_url":"https://arxiv.org/abs/math/0101219","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_3_1456","invariants":{"dual_indices":[0,1,2,3,4,5,6,7],"invertible_indices":[0,1],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,1.6180339887,2.9562952015,4.7833861168,4.7833861168,2.9562952015,1.6180339887],"global_fp_dimension_approx":70.4769960978,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t**4 - 90*t**3 + 1500*t**2 - 9000*t + 18000)**2","formal_codegree_polynomial":"(t**4 - 90*t**3 + 1500*t**2 - 9000*t + 18000)**2","irreducible_characters_approx":[{"values_approx":[1.0,-1.0,-1.6180339887,-2.9562952015,-4.7833861168,4.7833861168,2.9562952015,1.6180339887],"formal_codegree_approx":70.4769960978},{"values_approx":[1.0,1.0,1.6180339887,2.9562952015,4.7833861168,4.7833861168,2.9562952015,1.6180339887],"formal_codegree_approx":70.4769960978},{"values_approx":[1.0,-1.0,-1.6180339887,0.3382612127,0.5473181393,-0.5473181393,-0.3382612127,1.6180339887],"formal_codegree_approx":8.0640235647},{"values_approx":[1.0,1.0,1.6180339887,-0.3382612127,-0.5473181393,-0.5473181393,-0.3382612127,1.6180339887],"formal_codegree_approx":8.0640235647},{"values_approx":[1.0,-1.0,0.6180339887,-1.2090569265,0.7472382749,-0.7472382749,1.2090569265,-0.6180339887],"formal_codegree_approx":6.8042994048},{"values_approx":[1.0,1.0,-0.6180339887,1.2090569265,-0.7472382749,-0.7472382749,1.2090569265,-0.6180339887],"formal_codegree_approx":6.8042994048},{"values_approx":[1.0,-1.0,0.6180339887,0.8270909153,-0.5111702974,0.5111702974,-0.8270909153,-0.6180339887],"formal_codegree_approx":4.6546809327},{"values_approx":[1.0,1.0,-0.6180339887,-0.8270909153,0.5111702974,0.5111702974,-0.8270909153,-0.6180339887],"formal_codegree_approx":4.6546809327}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; 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All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":["twisted finite-group bimodules","Explicit SylowD8 cocycle and360 stable-element witnesses; global class existence, not global cochain table","New derivation independently reviewed during this investigation; further specialist review appropriate."],"construction_proof_file":"evidence/rank8_m3_20261021/models/psl2_11/r360/proof.md","basis_permutation_census_to_model":[0,1,2,5,6,7,3,4],"alternative_models":[]},{"rank":8,"multiplicity":3,"id":"R8_3_342","name":"Rep(C13 semidirect C6)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0]],[[0,0,1,0,0,0,0,0],[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],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0]],[[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0]],[[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[1,1,1,1,1,2,3,1],[0,0,0,0,0,3,3,0],[0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,3,3,0],[1,1,1,1,1,3,2,1],[0,0,0,0,0,0,1,0]],[[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0]]],"tensor_sha256":"68dccbc1bf5cec5ddd9f76ef75e1d785f4834bac464cf77383c553b93400be12","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_3_342; Faithful finite Frobenius group; exact orbit convolution..","model_reference_url":null,"source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_3_342","invariants":{"dual_indices":[0,2,1,4,3,5,6,7],"invertible_indices":[0,1,2,3,4,7],"commutative":true,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,1.0,1.0,6.0,6.0,1.0],"global_fp_dimension_approx":78.0,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 78)*(t - 13)**2*(t - 6)**5","formal_codegree_polynomial":"(t - 78)*(t - 13)**2*(t - 6)**5","irreducible_characters_approx":[{"values_approx":[1.0,1.0,1.0,1.0,1.0,6.0,6.0,1.0],"formal_codegree_approx":78.0},{"values_approx":[1.0,1.0,1.0,1.0,1.0,-2.3027756377,1.3027756377,1.0],"formal_codegree_approx":13.0},{"values_approx":[1.0,1.0,1.0,1.0,1.0,1.3027756377,-2.3027756377,1.0],"formal_codegree_approx":13.0},{"values_approx":[1.0,-1.0,-1.0,1.0,1.0,0.0,0.0,-1.0],"formal_codegree_approx":6.0},{"values_approx":[1.0,{"re":-0.5,"im":-0.8660254038},{"re":-0.5,"im":0.8660254038},{"re":-0.5,"im":-0.8660254038},{"re":-0.5,"im":0.8660254038},0.0,0.0,1.0],"formal_codegree_approx":6.0},{"values_approx":[1.0,{"re":-0.5,"im":0.8660254038},{"re":-0.5,"im":-0.8660254038},{"re":-0.5,"im":0.8660254038},{"re":-0.5,"im":-0.8660254038},0.0,0.0,1.0],"formal_codegree_approx":6.0},{"values_approx":[1.0,{"re":0.5,"im":-0.8660254038},{"re":0.5,"im":0.8660254038},{"re":-0.5,"im":0.8660254038},{"re":-0.5,"im":-0.8660254038},0.0,0.0,-1.0],"formal_codegree_approx":6.0},{"values_approx":[1.0,{"re":0.5,"im":0.8660254038},{"re":0.5,"im":-0.8660254038},{"re":-0.5,"im":-0.8660254038},{"re":-0.5,"im":0.8660254038},0.0,0.0,-1.0],"formal_codegree_approx":6.0}],"character_method":"Numerical simultaneous eigencharacters of the integral fusion matrices; approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[40,24,7,21,8,19,45,26],"primitive_polynomial":"(theta - 510)*(theta - 12)*(theta**2 - 188*theta + 6639)*(theta**2 - 72*theta + 1971)*(theta**2 - 30*theta + 237)","basis_values":["1","608478613031978268273089*theta**7/133127905140251533863952441513807680 - 502825256964397685446586749*theta**6/133127905140251533863952441513807680 + 34298849161805969389728211*theta**5/37734666989867214814045476619560 - 434284635118190593990338700711*theta**4/4437596838008384462131748050460256 + 27853111770600801609030619940891*theta**3/4930663153342649402368608944955840 - 127553577179747399944090384196621*theta**2/704380450477521343195515563565120 + 39083988298506006158862573869609*theta/13044082416250395244361399325280 - 14903807891038791859699283193/791647103465864112281340344","3038418314043479229197383*theta**7/66563952570125766931976220756903840 - 1210923215197368728418446779*theta**6/33281976285062883465988110378451920 + 201890735415235306221592711*theta**5/25156444659911476542696984413040 - 104465767306041177091999019813*theta**4/138674901187762014441617126576883 + 30323165034511074104717306568559*theta**3/821777192223774900394768157492640 - 171393668564797122258322499203721*theta**2/176095112619380335798878890891280 + 15200691448389438645037769888849*theta/1222882726523474554158881186745 - 5941181129618661077145108258/98955887933233014035167543","-40038052665469147*theta**7/9751586288631391626216500160 + 33287086793188861187*theta**6/9751586288631391626216500160 - 6896679734123868679*theta**5/8292165211421251382837160 + 29467551037008538874741*theta**4/325052876287713054207216672 - 5711252065492102134434579*theta**3/1083509587625710180690722240 + 8861798097406134098709163*theta**2/51595694648843341937653440 - 24859542308470774076104463*theta/8599282441473890322942240 + 3198058730766567875237/173963866511523126584","-13466712942633319*theta**7/304737071519730988319265630 + 172147453803397607489*theta**6/4875793144315695813108250080 - 129876320581996568107*theta**5/16584330422842502765674320 + 120370971042311851127051*theta**4/162526438143856527103608336 - 4940072739172315437343247*theta**3/135438698453213772586340280 + 24911450398763955603434401*theta**2/25797847324421670968826720 - 53143006530293832365901431*theta/4299641220736945161471120 + 5199914679976162595053/86981933255761563292","1283709019136352751*theta**7/1249422592659067967889823951968 - 245386446460691335607*theta**6/312355648164766991972455987992 + 110233958700223723699*theta**5/708289451620786829869514712 - 2555085697002739964668645*theta**4/208237098776511327981637325328 + 23110900551260665548479143*theta**3/46274910839224739551474961184 - 34394086643104160912619683*theta**2/3305350774230338539391068656 + 4844943010191385893645761*theta/45907649642088035269320398 - 909103018245698933949399/2201051695168604430720841","-13589278841362012955*theta**7/7496535555954407807338943711808 + 10392956361981170142247*theta**6/7496535555954407807338943711808 - 129751575661758228835*theta**5/472192967747191219913009808 + 27073014022145641352877107*theta**4/1249422592659067967889823951968 - 81636886002788612643447799*theta**3/92549821678449479102949922368 + 729022118765984330746278875*theta**2/39664209290764062472692823872 - 410797845266772527373943229*theta/2203567182820225692927379104 + 12847427804675250417713773/17608413561348835445766728","11460404875627*theta**7/4105273614296157622990476 - 17766902953284001*theta**6/8210547228592315245980952 + 202311510782164853*theta**5/456141512699573069221164 - 49933359479542187855*theta**4/1368424538098719207663492 + 227032856678683914413*theta**3/152047170899857689740388 - 9382230421853952604247*theta**2/304094341799715379480776 + 5229870852679519233515*theta/16894130099984187748932 - 39197403322717409316/17820812341755472309"],"codegree_value":"(41209421986923486215*theta**7 - 31525846696609142357497*theta**6 + 6252546953018615258796444*theta**5 - 493185475345161089750458098*theta**4 + 20080368740923303500065157327*theta**3 - 418456022141861391265160639805*theta**2 + 4244623735730560891414823879118*theta + 28366037830142121658579440154344)/7496535555954407807338943711808","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":[],"basis_permutation_census_to_model":[0,5,1,2,4,6,7,3],"alternative_models":[]},{"rank":8,"multiplicity":3,"id":"R8_3_448","name":"Unitary Izumi–Xu Z2 extension (2221)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0]],[[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],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0]],[[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,1,0,1,1],[0,1,0,0,0,1,0,0],[0,0,1,0,0,1,0,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[1,1,1,0,0,0,0,0],[0,0,0,1,0,0,1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0]],[[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,1,0,1,1],[0,0,0,1,0,0,1,1],[1,1,1,0,0,3,0,0],[0,0,0,1,1,0,1,1],[0,0,0,1,1,0,1,1]],[[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,1,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,1,0,1,1],[1,0,0,0,0,1,0,0],[0,1,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,1,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,1,0,1,1],[0,0,1,0,0,1,0,0],[1,0,0,0,0,1,0,0]]],"tensor_sha256":"eba29e824021c7b6b76c7654568131585719036261a9ecfb915312aa0ce9d4dc","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_3_448; CMS Appendix A.6.1; Morrison–Penneys arXiv:1208.3637v2 §5.1 and Theorem5.5; GJS arXiv:1211.6414v2 Theorem4.9 and Remarks4.10–4.11..","model_reference_url":"https://arxiv.org/abs/1208.3637","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_3_448","invariants":{"dual_indices":[0,2,1,3,4,5,6,7],"invertible_indices":[0,1,2],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,1.0,1.0,2.1889010593,1.7320508076,3.7912878475,2.1889010593,2.1889010593],"global_fp_dimension_approx":34.7477270849,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 6)**4*(t**2 - 42*t + 252)**2","irreducible_characters_approx":[{"degree":2,"values_approx":[2.0,-1.0,-1.0,0.0,0.0,0.0,0.0,0.0],"formal_codegree_approx":3.0,"formal_codegree_minimal_polynomial":"u - 3","formal_codegree_exact":"3"},{"degree":1,"values_approx":[1.0,1.0,1.0,-0.4568502517,1.7320508076,-0.7912878475,-0.4568502517,-0.4568502517],"formal_codegree_approx":7.2522729151,"formal_codegree_minimal_polynomial":"u**2 - 42*u + 252"},{"degree":1,"values_approx":[1.0,1.0,1.0,-2.1889010593,-1.7320508076,3.7912878475,-2.1889010593,-2.1889010593],"formal_codegree_approx":34.7477270849,"formal_codegree_minimal_polynomial":"u**2 - 42*u + 252"},{"degree":1,"values_approx":[1.0,1.0,1.0,0.4568502517,-1.7320508076,-0.7912878475,0.4568502517,0.4568502517],"formal_codegree_approx":7.2522729151,"formal_codegree_minimal_polynomial":"u**2 - 42*u + 252"},{"degree":1,"values_approx":[1.0,1.0,1.0,2.1889010593,1.7320508076,3.7912878475,2.1889010593,2.1889010593],"formal_codegree_approx":34.7477270849,"formal_codegree_minimal_polynomial":"u**2 - 42*u + 252"}],"character_method":"Numerical regular-algebra block traces after exact rational center-kernel computation; χ_E(X_i) and f_E=(Σ_i χ_E(X_i)χ_E(X_i*)) / dim(E) are approximate. The displayed codegree minimal polynomials are exact factors of the rational central-Casimir polynomial rescaled by each irreducible degree; factor association uses the validated numerical eigenvalues. 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All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":[],"construction_proof_file":"evidence/rank8_m3_20261021/models/FIBONACCI_CYCLIC_PROOF.md","basis_permutation_census_to_model":[0,1,2,5,4,6,7,3],"alternative_models":[]},{"rank":8,"multiplicity":3,"id":"R8_3_715","name":"Morita dual of the nonsplit unitary 2D2 extension","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,1],[1,0,0,0,0,0,1,1],[0,0,0,1,0,1,0,0],[0,1,0,0,0,0,1,0],[0,0,0,1,0,2,0,0],[0,0,0,0,0,0,0,1],[0,1,1,0,1,0,0,1]],[[0,0,1,0,0,0,0,0],[1,0,0,0,1,0,0,1],[0,1,0,0,0,0,0,1],[0,0,0,1,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,2,0,0],[0,0,1,0,1,0,0,0],[0,1,1,0,0,0,1,1]],[[0,0,0,1,0,0,0,0],[0,0,0,1,0,1,0,0],[0,0,0,1,0,1,0,0],[1,1,1,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,1,1,0,1,0,1,2],[0,0,0,0,0,1,0,0],[0,0,0,1,0,2,0,0]],[[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,1,0],[0,0,0,0,0,1,0,0],[1,0,0,0,1,0,0,0],[0,0,0,1,0,1,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,1]],[[0,0,0,0,0,1,0,0],[0,0,0,1,0,2,0,0],[0,0,0,1,0,2,0,0],[0,1,1,0,1,0,1,2],[0,0,0,1,0,1,0,0],[1,2,2,0,1,0,1,3],[0,0,0,1,0,1,0,0],[0,0,0,2,0,3,0,0]],[[0,0,0,0,0,0,1,0],[0,1,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,1,0,0],[1,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,1]],[[0,0,0,0,0,0,0,1],[0,1,1,0,0,0,1,1],[0,1,1,0,1,0,0,1],[0,0,0,1,0,2,0,0],[0,0,1,0,0,0,0,1],[0,0,0,2,0,3,0,0],[0,1,0,0,0,0,0,1],[1,1,1,0,1,0,1,2]]],"tensor_sha256":"203af8504271bf5aae871b36d44f3271782cc60a190ae5ffcc4c7ecd41fa1ff8","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_3_715; Grossman arXiv:1612.07185 Lemma4.1 and §4.1; Edie-Michell arXiv:1904.08909 Definition2.5..","model_reference_url":"https://arxiv.org/abs/1612.07185","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_3_715","invariants":{"dual_indices":[0,2,1,3,4,5,6,7],"invertible_indices":[0],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,2.6180339887,2.6180339887,3.2360679775,1.6180339887,5.2360679775,1.6180339887,4.2360679775],"global_fp_dimension_approx":75.77708764,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 8)**4*(t**2 - 80*t + 320)**2","irreducible_characters_approx":[{"degree":2,"values_approx":[2.0,-1.0,-1.0,0.0,1.0,0.0,1.0,0.0],"formal_codegree_approx":4.0,"formal_codegree_minimal_polynomial":"u - 4","formal_codegree_exact":"4"},{"degree":1,"values_approx":[1.0,0.3819660113,0.3819660113,-1.2360679775,-0.6180339887,0.7639320225,-0.6180339887,-0.2360679775],"formal_codegree_approx":4.22291236,"formal_codegree_minimal_polynomial":"u**2 - 80*u + 320"},{"degree":1,"values_approx":[1.0,0.3819660113,0.3819660113,1.2360679775,-0.6180339887,-0.7639320225,-0.6180339887,-0.2360679775],"formal_codegree_approx":4.22291236,"formal_codegree_minimal_polynomial":"u**2 - 80*u + 320"},{"degree":1,"values_approx":[1.0,2.6180339887,2.6180339887,-3.2360679775,1.6180339887,-5.2360679775,1.6180339887,4.2360679775],"formal_codegree_approx":75.77708764,"formal_codegree_minimal_polynomial":"u**2 - 80*u + 320"},{"degree":1,"values_approx":[1.0,2.6180339887,2.6180339887,3.2360679775,1.6180339887,5.2360679775,1.6180339887,4.2360679775],"formal_codegree_approx":75.77708764,"formal_codegree_minimal_polynomial":"u**2 - 80*u + 320"}],"character_method":"Numerical regular-algebra block traces after exact rational center-kernel computation; χ_E(X_i) and f_E=(Σ_i χ_E(X_i)χ_E(X_i*)) / dim(E) are approximate. The displayed codegree minimal polynomials are exact factors of the rational central-Casimir polynomial rescaled by each irreducible degree; factor association uses the validated numerical eigenvalues. Orthogonality and block dimensions validated."},"construction_notes":[],"construction_proof_file":"evidence/rank8_m3_20261021/models/TT3_MORITA_PROOF.md","basis_permutation_census_to_model":[0,3,4,6,2,7,1,5],"alternative_models":[]},{"rank":8,"multiplicity":3,"id":"R8_3_721","name":"Bimodules over 1+rho in the C2 de-equivariantization of GH(C8)","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,1],[1,0,0,0,0,1,1,0],[0,0,0,1,0,1,1,1],[0,1,0,1,0,0,1,0],[0,1,0,1,1,0,0,1],[0,0,0,0,0,1,0,1],[0,0,1,1,1,1,0,1]],[[0,0,1,0,0,0,0,0],[1,0,0,0,1,1,0,0],[0,1,0,0,0,0,0,1],[0,0,0,1,1,1,0,1],[0,0,0,0,0,1,0,1],[0,0,1,1,0,0,1,1],[0,0,1,1,1,0,0,0],[0,1,0,1,0,1,1,1]],[[0,0,0,1,0,0,0,0],[0,0,0,1,1,1,0,1],[0,0,0,1,0,1,1,1],[1,1,1,2,1,2,1,2],[0,1,0,1,1,1,1,1],[0,1,1,2,1,1,1,2],[0,0,1,1,1,1,1,1],[0,1,1,2,1,2,1,3]],[[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,1],[0,0,1,1,0,0,1,0],[0,0,1,1,1,1,1,1],[1,0,0,1,1,1,0,0],[0,1,0,1,1,1,0,1],[0,0,1,1,0,0,0,1],[0,1,0,1,0,1,1,2]],[[0,0,0,0,0,1,0,0],[0,1,0,1,0,0,1,1],[0,0,1,1,1,0,0,1],[0,1,1,2,1,1,1,2],[0,0,1,1,1,1,0,1],[1,0,0,1,1,2,1,2],[0,1,0,1,0,1,1,1],[0,1,1,2,1,2,1,2]],[[0,0,0,0,0,0,1,0],[0,1,0,1,1,0,0,0],[0,0,0,0,0,1,0,1],[0,1,0,1,1,1,1,1],[0,1,0,1,0,0,0,1],[0,0,1,1,0,1,1,1],[1,0,0,1,0,1,1,0],[0,0,1,1,1,1,0,2]],[[0,0,0,0,0,0,0,1],[0,0,1,1,0,1,1,1],[0,1,0,1,1,1,0,1],[0,1,1,2,1,2,1,3],[0,0,1,1,0,1,1,2],[0,1,1,2,1,2,1,2],[0,1,0,1,1,1,0,2],[1,1,1,3,2,2,2,2]]],"tensor_sha256":"76e26a815651476dfda5c2aa654f0d1327bdc890e08086cb6dbfe39ef9a1cdc5","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"The actual inherited-Q-system dual is identified by a direct exact completion of its induction equations. All four dualities give one based-ring class in a 14-node search; census completeness is not assumed. See the current manuscript and the dated exact proof.","model_reference_url":"https://arxiv.org/abs/1501.07679","source_file":"supplements/rank8/output/creative_20261022/model_catalog.json","source_location":"R8_3_721","invariants":{"dual_indices":[0,2,1,3,4,5,6,7],"invertible_indices":[0],"commutative":false,"frobenius_perron_dimensions_approx":[1.0,3.5615528128,3.5615528128,8.1231056256,4.5615528128,7.1231056256,4.5615528128,9.1231056256],"global_fp_dimension_approx":267.9393800198,"dimensions_method":"Numerical Perron eigenvector, normalized at the tensor unit; approximate values.","central_casimir_polynomial":"(t - 16)**4*(t - 4)**2*(t**2 - 272*t + 1088)","irreducible_characters_approx":[{"degree":2,"values_approx":[2.0,-1.0,-1.0,0.0,1.0,2.0,1.0,-2.0],"formal_codegree_approx":8.0,"formal_codegree_minimal_polynomial":"u - 8","formal_codegree_exact":"8"},{"degree":1,"values_approx":[1.0,-0.5615528128,-0.5615528128,-0.1231056256,0.4384471872,-1.1231056256,0.4384471872,0.8768943744],"formal_codegree_approx":4.0606199802,"formal_codegree_minimal_polynomial":"u**2 - 272*u + 1088"},{"degree":1,"values_approx":[1.0,0.0,0.0,1.0,-1.0,0.0,-1.0,0.0],"formal_codegree_approx":4.0,"formal_codegree_minimal_polynomial":"u - 4","formal_codegree_exact":"4"},{"degree":1,"values_approx":[1.0,1.0,1.0,-1.0,0.0,0.0,0.0,0.0],"formal_codegree_approx":4.0,"formal_codegree_minimal_polynomial":"u - 4","formal_codegree_exact":"4"},{"degree":1,"values_approx":[1.0,3.5615528128,3.5615528128,8.1231056256,4.5615528128,7.1231056256,4.5615528128,9.1231056256],"formal_codegree_approx":267.9393800198,"formal_codegree_minimal_polynomial":"u**2 - 272*u + 1088"}],"character_method":"Numerical regular-algebra block traces after exact rational center-kernel computation; χ_E(X_i) and f_E=(Σ_i χ_E(X_i)χ_E(X_i*)) / dim(E) are approximate. The displayed codegree minimal polynomials are exact factors of the rational central-Casimir polynomial rescaled by each irreducible degree; factor association uses the validated numerical eigenvalues. Orthogonality and block dimensions validated."},"construction_notes":["The actual inherited-Q-system dual is identified by a direct exact completion of its induction equations. All four dualities give one based-ring class in a 14-node search; census completeness is not assumed.","Exact proof and tensor checks remain subject to independent human verification."],"construction_proof_file":"evidence/correctness_20261025/r721-exact-completion-proof.md","basis_permutation_census_to_model":[0,1,2,5,3,6,4,7],"alternative_models":[]},{"rank":8,"multiplicity":3,"id":"R8_3_724","name":"Rep(PSL(2,11))","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,1,1],[1,0,0,0,1,1,0,0],[0,0,0,1,1,1,1,1],[0,1,0,1,1,1,1,1],[0,1,0,1,1,1,1,1],[0,0,1,1,1,1,1,0],[0,0,1,1,1,1,0,1]],[[0,0,1,0,0,0,0,0],[1,0,0,0,1,1,0,0],[0,1,0,0,0,0,1,1],[0,0,0,1,1,1,1,1],[0,0,1,1,1,1,1,1],[0,0,1,1,1,1,1,1],[0,1,0,1,1,1,1,0],[0,1,0,1,1,1,0,1]],[[0,0,0,1,0,0,0,0],[0,0,0,1,1,1,1,1],[0,0,0,1,1,1,1,1],[1,1,1,2,2,2,2,2],[0,1,1,2,3,2,2,2],[0,1,1,2,2,3,2,2],[0,1,1,2,2,2,1,2],[0,1,1,2,2,2,2,1]],[[0,0,0,0,1,0,0,0],[0,1,0,1,1,1,1,1],[0,0,1,1,1,1,1,1],[0,1,1,2,3,2,2,2],[1,1,1,3,2,3,2,2],[0,1,1,2,3,3,2,2],[0,1,1,2,2,2,2,2],[0,1,1,2,2,2,2,2]],[[0,0,0,0,0,1,0,0],[0,1,0,1,1,1,1,1],[0,0,1,1,1,1,1,1],[0,1,1,2,2,3,2,2],[0,1,1,2,3,3,2,2],[1,1,1,3,3,2,2,2],[0,1,1,2,2,2,2,2],[0,1,1,2,2,2,2,2]],[[0,0,0,0,0,0,1,0],[0,0,1,1,1,1,1,0],[0,1,0,1,1,1,1,0],[0,1,1,2,2,2,1,2],[0,1,1,2,2,2,2,2],[0,1,1,2,2,2,2,2],[1,1,1,1,2,2,1,2],[0,0,0,2,2,2,2,1]],[[0,0,0,0,0,0,0,1],[0,0,1,1,1,1,0,1],[0,1,0,1,1,1,0,1],[0,1,1,2,2,2,2,1],[0,1,1,2,2,2,2,2],[0,1,1,2,2,2,2,2],[0,0,0,2,2,2,2,1],[1,1,1,1,2,2,1,2]]],"tensor_sha256":"d0424cf6cfd9e8e99dfdddeabc3a84aa014262259721e3ddc6b466b2772b0ee5","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_3_724; 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All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":[],"basis_permutation_census_to_model":[0,1,2,5,6,7,3,4],"alternative_models":[]},{"rank":8,"multiplicity":3,"id":"R8_3_982","name":"Triality-zero component of E21 from the conformal embedding SU(3)_21 into E7_1","existence":"unitary","fusion_tensor":[[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]],[[0,1,0,0,0,0,0,0],[0,1,1,2,2,0,1,2],[1,1,1,2,2,2,0,0],[0,2,2,2,2,1,0,2],[0,2,2,2,2,2,0,1],[0,2,0,2,1,1,0,1],[0,0,1,0,0,0,0,0],[0,0,2,1,2,1,0,1]],[[0,0,1,0,0,0,0,0],[1,1,1,2,2,2,0,0],[0,1,1,2,2,0,1,2],[0,2,2,2,2,2,0,1],[0,2,2,2,2,1,0,2],[0,0,2,1,2,1,0,1],[0,1,0,0,0,0,0,0],[0,2,0,2,1,1,0,1]],[[0,0,0,1,0,0,0,0],[0,2,2,2,2,1,0,2],[0,2,2,2,2,2,0,1],[0,2,2,3,3,2,1,2],[1,2,2,3,3,2,0,2],[0,2,1,2,2,1,0,1],[0,0,0,0,1,0,0,0],[0,1,2,2,2,1,0,1]],[[0,0,0,0,1,0,0,0],[0,2,2,2,2,2,0,1],[0,2,2,2,2,1,0,2],[1,2,2,3,3,2,0,2],[0,2,2,3,3,2,1,2],[0,1,2,2,2,1,0,1],[0,0,0,1,0,0,0,0],[0,2,1,2,2,1,0,1]],[[0,0,0,0,0,1,0,0],[0,2,0,2,1,1,0,1],[0,0,2,1,2,1,0,1],[0,2,1,2,2,1,0,1],[0,1,2,2,2,1,0,1],[1,1,1,1,1,2,0,0],[0,0,0,0,0,0,0,1],[0,1,1,1,1,0,1,2]],[[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0]],[[0,0,0,0,0,0,0,1],[0,0,2,1,2,1,0,1],[0,2,0,2,1,1,0,1],[0,1,2,2,2,1,0,1],[0,2,1,2,2,1,0,1],[0,1,1,1,1,0,1,2],[0,0,0,0,0,1,0,0],[1,1,1,1,1,2,0,0]]],"tensor_sha256":"f4a9fb77b3a78e9dcd8370ca9fa822ea331f1b71e3fad7990efdd386445fcd1c","basis":["X0","X1","X2","X3","X4","X5","X6","X7"],"model_reference":"Rank-eight manuscript, positive model catalogue, R8_3_982; Böckenhauer-Evans, Commun. Math. 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approximate values.","exact_character_presentation":{"exact_generator":"theta = Σ_i w_i X_i (character value at a root of P)","generator_weights":[1,2,3,4,5,6,7,8],"primitive_polynomial":"(theta**4 - 292*theta**3 + 7296*theta**2 - 41656*theta - 11996)*(theta**4 + 32*theta**3 + 380*theta**2 + 1920*theta + 3492)","basis_values":["1","1283310321688678783763*theta**7/40256731055626435216900178880 - 2028399334712112144329*theta**6/239623399140633542957739160 - 46256342468659205651191*theta**5/6709455175937739202816696480 + 3578737991525554901796541*theta**4/1258022845488326100528130590 + 16415542740936398149007453*theta**3/1437740394843801257746434960 - 509483826157444693446437839*theta**2/2516045690976652201056261180 - 4557404919945975263458751881*theta/5032091381953304402112522360 + 173465059197794417696209611/419340948496108700176043530","-1732430700396491916383*theta**7/40256731055626435216900178880 + 16322398151451340494119*theta**6/1437740394843801257746434960 + 192506738459090159030231*theta**5/6709455175937739202816696480 - 18678956996376921391700309*theta**4/5032091381953304402112522360 - 10374582536143813645964331*theta**3/479246798281267085915478320 + 138706859553363665036437841*theta**2/629011422744163050264065295 + 7158303578033321925020388241*theta/5032091381953304402112522360 + 213966254140169998420917342/209670474248054350088021765","-692854013531098487287*theta**7/8051346211125287043380035776 + 6568904186451320709269*theta**6/287548078968760251549286992 + 75456922750158097639009*theta**5/4025673105562643521690017888 - 2429860366520112561069903*theta**4/335472758796886960140834824 - 10405295445175336591691857*theta**3/287548078968760251549286992 + 60551976430227095593968257*theta**2/125802284548832610052813059 + 2654285678857935472622045509*theta/1006418276390660880422504472 + 25248784317646751375133959/41934094849610870017604353","238063422260266563489*theta**7/2683782070375095681126678592 - 1127202283426140586921*theta**6/47924679828126708591547832 - 106572335630689601152073*theta**5/4025673105562643521690017888 + 1242228918961170173368093*theta**4/167736379398443480070417412 + 11209039680101270454590533*theta**3/287548078968760251549286992 - 238523863271227885250684425*theta**2/503209138195330440211252236 - 2701043103860592097449649813*theta/1006418276390660880422504472 - 65894294755860508526420245/83868189699221740035208706","8211830994300652564*theta**7/629011422744163050264065295 - 23565642653022507368039*theta**6/6709455175937739202816696480 + 8446720304237438414691*theta**5/838681896992217400352087060 + 9667077653647860645096991*theta**4/10064182763906608804225044720 + 6565440450065228216700613*theta**3/2516045690976652201056261180 - 34177471663561776616019263*theta**2/718870197421900628873217480 - 361029233480860998273105143*theta/1258022845488326100528130590 - 100941896291699240101447793/119811699570316771478869580","-16984186040329*theta**7/1247811071630866992776 + 4431091251694765*theta**6/1247811071630866992776 + 12223334205316111*theta**5/623905535815433496388 - 362282781393473789*theta**4/311952767907716748194 - 3427880366162151531*theta**3/311952767907716748194 + 18603078269363716309*theta**2/311952767907716748194 + 135592801694939796789*theta/155976383953858374097 + 192477328610400336431/155976383953858374097","-21398613730686544039*theta**7/10064182763906608804225044720 + 13362485850539629794257*theta**6/20128365527813217608450089440 - 44548068423839535407287*theta**5/1677363793984434800704174120 - 307312272286833685515131*theta**4/10064182763906608804225044720 + 2789057313028988303838117*theta**3/419340948496108700176043530 + 5030454362407763867515163*theta**2/718870197421900628873217480 - 231942105771555690973343801*theta/629011422744163050264065295 - 104114156635243472473145347/119811699570316771478869580"],"codegree_value":"(114110017164108862324*theta**7 - 30819308994955008376001*theta**6 + 121089555888408577137147*theta**5 + 8075836496879220910842366*theta**4 + 12100468717885457226242792*theta**3 - 390618002683488181310886192*theta**2 - 1687451706807933556354803964*theta + 4634840170245443616776162448)/1258022845488326100528130590","instruction":"For each complex root θ of P(θ)=0, substitute θ into the displayed polynomials to obtain one exact irreducible character and its formal codegree. All matrix identities N_i=p_i(H) were checked over Q."}},"construction_notes":["Exact cyclic-unit reconstruction of all24 source fusion matrices from the fundamental graph and its transpose, followed by all512 neutral tensor coefficients.","The permutation identifies every source-neutral fusion coefficient with the canonical census tensor."],"construction_proof_file":"evidence/rank8_creative_20261022/r982/E21_MODEL.md","basis_permutation_census_to_model":[0,2,3,4,5,1,7,6],"alternative_models":[]}]}