{"id":"44.5.3.a","row":198,"stored":{"file":"master_catalog.json","row":198,"construction":null,"columns_sha256":null},"parameters":{"n":44,"k":5,"d":3,"N":12,"r":7,"level":3},"gate":{"string":"01+02+03+04+012+034","human":"CS01\u00b7CS02\u00b7CS03\u00b7CS04\u00b7CCZ012\u00b7CCZ034","monomials":[[0,1],[0,2],[0,3],[0,4],[0,1,2],[0,3,4]],"sk_key":[[0,1],[0,1,2],[0,2],[0,3],[0,3,4],[0,4]],"sk_fingerprint":"b8760968d227e25d","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[0,1,2,6],[3,4,7],[1,2,5,7],[0,1,2,6,7],[0,2,8],[0,1,3,4,7,8],[1,4,5,9],[2,6,9],[2,3,7,9],[1,3,4,6,7,9],[0,3,4,10],[0,1,2,6,10],[0,3,4,5,7,10],[0,1,2,6,7,10],[0,2,3,9,10],[0,2,3,5,9,10],[0,1,3,4,6,9,10],[0,2,6,7,9,10],[0,4,8,9,10],[0,1,2,3,7,8,9,10],[0,2,4,11],[0,2,3,4,6,11],[0,1,3,5,7,11],[0,1,6,7,11],[0,1,2,3,4,9,11],[0,1,2,3,4,5,9,11],[1,2,4,6,9,11],[3,6,7,9,11],[5,8,9,11],[1,3,6,8,9,11],[5,6,8,9,11],[5,7,8,9,11],[2,4,6,7,8,9,11],[5,6,7,8,9,11],[1,3,10,11],[2,4,5,7,10,11],[1,2,3,4,6,8,10,11],[6,7,8,10,11],[9,10,11],[5,9,10,11],[5,8,9,10,11],[5,6,8,9,10,11],[5,7,8,9,10,11],[5,6,7,8,9,10,11]]},"distance":{"d":3,"is_exact":true,"upper":3,"witness":[0,1,3],"A_d":112,"certified":null},"clifford":{"kind":"required","correction":{"S":[[0,2],[1,3],[2,2],[3,3],[6,2],[7,2],[9,3],[10,2]],"CZ":[[0,2],[0,4],[0,10],[0,11],[1,2],[1,3],[1,6],[1,10],[2,6],[2,9],[3,4],[3,6],[3,9],[4,6],[4,7],[4,10],[6,7],[7,9],[7,11],[10,11]]},"rotation_powers":null},"metrics":{"gamma":1.9795443250691258,"gamma_t":2.182658338644138,"gamma_rho":null,"gamma_claim":1.9795443250691258,"gamma_t_claim":2.182658338644138,"gamma_rho_claim":null,"v_ex":null,"rate":0.11363636363636363,"t_count":4,"poly_degree":2,"effective_width":5},"provenance":{"regimes":["exhaustive classification n<=54"],"discovery":"pre-existing","strongest_claim":"found within the exhaustive classification through length 54 but not one of its Pareto points: a Pareto point with the same output class (after removing spectator outputs) and the same exact distance strictly dominates it, which tests/test_master_catalog.py checks -- a verified witness, not an optimum","sources":[{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+03+04+012+034","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+012+013+014","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+012+013+014+023+024","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+03+012+013+014+023+024+034","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+03+04+12+13+14+012+013+014+023+024+034+123+124+134","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+03+04+12+13+24+34+012+013+014+023+024+034+123+124+134+234","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+03+12+13+012+013+014+023+024+034+123+124+134","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+12+012+013+014+023+024+123+124","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+13+23+012+013+014+023+024+123+134+234","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+03+04+012+013+023","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+03+04+12+13+14+012+013+023+123","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+03+04+12+13+24+34+012+013+023+123+234","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+03+14+24+34+012+013+023+124+134+234","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+04+012+013+023+034","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+04+12+14+012+013+023+034+123+134","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+04+13+23+34+012+013+023+034+123","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+04+14+24+012+013+023+034+124+134+234","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+14+24+012+013+023+124+134+234","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+04+012+013+024+034","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+04+14+012+013+024+034+124+134","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+03+04+12+13+14+012+034+134","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+03+04+12+13+24+34+012+034","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+02+03+14+24+34+012+124","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"01+03+14+34+012+023+124+234","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 1050, origin 3; census_shard_08.json"}]},"references":[{"key":"wills2026classification","short":"Wills et al. (2026)","full":"A. Wills, S. P. Jain, and S. Singh, \"Classification of Generalised Triorthogonal Codes through Length 54,\" arXiv:2609.30860 (2026).","url":"https://arxiv.org/abs/2609.30860","date":"2026-09-25"},{"key":"jain2026symmetry","short":"Jain et al. (2026)","full":"S. P. Jain, A. Wills, and S. Singh, \"Symmetry and AI-assisted discovery of magic-state factories,\" arXiv:2610.06535 (2026).","url":"https://arxiv.org/abs/2610.06535","date":"2026-10-05"}]}