{"id":"31.1.3.a","row":67,"stored":{"file":"master_catalog.json","row":67,"construction":null,"columns_sha256":null},"parameters":{"n":31,"k":1,"d":3,"N":6,"r":5,"level":3},"gate":{"string":"0","human":"T0","monomials":[[0]],"sk_key":[[0]],"sk_fingerprint":"bc179efd84e614ba","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[0,1],[2],[1,2],[3],[1,3],[2,3],[1,2,3],[4],[1,4],[2,4],[1,2,4],[3,4],[1,3,4],[0,2,3,4],[0,1,2,3,4],[0,5],[0,1,5],[0,2,5],[0,1,2,5],[0,3,5],[0,1,3,5],[2,3,5],[1,2,3,5],[0,4,5],[0,1,4,5],[2,4,5],[1,2,4,5],[3,4,5],[1,3,4,5],[2,3,4,5],[1,2,3,4,5]]},"distance":{"d":3,"is_exact":true,"upper":3,"witness":[0,1,2],"A_d":75,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,3]],"CZ":[[0,1]]},"rotation_powers":[[0,7],[1,7],[2,7],[27,3],[28,3],[29,3],[30,3]]},"metrics":{"gamma":3.1257498572570146,"gamma_t":3.1257498572570146,"gamma_rho":3.1257498572570146,"gamma_claim":3.1257498572570146,"gamma_t_claim":3.1257498572570146,"gamma_rho_claim":3.1257498572570146,"v_ex":1,"rate":0.03225806451612903,"t_count":1,"poly_degree":1,"effective_width":1},"provenance":{"regimes":["exhaustive classification n<=54","AI search"],"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/exhaustive_n38/catalog/classification_n38.json","label":"0","N":6,"d":3,"d_is_exact":true,"origin":null,"provenance":"quotient_catalog_k3"},{"regime":"AI search","file":"master_catalog/catalogue_new","label":"T0","N":6,"d":3,"d_is_exact":true,"origin":"frontier_campaign","provenance":"frontier_campaign \u00b7 catalogue.jsonl:47"},{"regime":"exhaustive classification n<=54","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0","N":7,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 3363, origin 7; 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"},{"key":"singh2026borrowed","short":"Singh et al. (2026)","full":"S. Singh, C. Gidney, and C. Jones, \"Borrowed Identities: Malleable Distillation Factories and a Unified Numerical Search,\" arXiv:2606.28518 (2026).","url":"https://arxiv.org/abs/2606.28518","date":"2026-06-26"}]}