{"id":"136.7.4.b","row":760,"stored":{"file":"master_catalog.json","row":760,"construction":null,"columns_sha256":null},"parameters":{"n":136,"k":7,"d":4,"N":17,"r":10,"level":3},"gate":{"string":"012+013+014+015+023+024+034+125+236+256+345+346+356+456","human":"CCZ012\u00b7CCZ013\u00b7CCZ014\u00b7CCZ015\u00b7CCZ023\u00b7CCZ024\u00b7CCZ034\u00b7CCZ125\u00b7CCZ236\u00b7CCZ256\u00b7CCZ345\u00b7CCZ346\u00b7CCZ356\u00b7CCZ456","monomials":[[0,1,2],[0,1,3],[0,1,4],[0,1,5],[0,2,3],[0,2,4],[0,3,4],[1,2,5],[2,3,6],[2,5,6],[3,4,5],[3,4,6],[3,5,6],[4,5,6]],"sk_key":[[0,1,2],[0,1,3],[0,1,4],[0,1,5],[0,2,3],[0,2,4],[0,3,4],[1,2,5],[2,3,6],[2,5,6],[3,4,5],[3,4,6],[3,5,6],[4,5,6]],"sk_fingerprint":"3d0714adaccaafd9","sk_canonical_frame":true,"relabelled_into_canonical_frame":true},"circuit":{"columns":[[0,2,3,4,5,6,7],[0,1,2,8],[4,5,9],[1,5,6,10],[2,3,5,6,11],[0,2,3,5,6,12],[3,5,6,13],[0,1,4,5,6,14],[0,1,2,3,4,5,7,9,14],[0,1,2,3,4,5,6,15],[4,5,16],[0,1,2,4,8,11,15],[0,2,4,5,11,13,16],[0,1,2,3,5,7,9,11,14,15],[0,1,6,7,8,9,10,13],[3,4,5,6,7,11,13],[0,3,5,6,7,9,10,12,13,14,16],[0,1,3,4,5,7,9,10,13,15],[0,2,4,5,6,7,10,11,12,13],[1,3,4,5,11,13,14,15,16],[0,1,3,5,6,7,9,10,14,16],[2,3,4,5,6,9,12,13,15,16],[1,2,3,6,8,11,12,15,16],[1,4,7,8,9,11,16],[3,4,5,6,7,9,11,12,16],[1,6,8,11,12,14,16],[1,2,3,4,5,9,10,11],[2,5,6,9,13,14,15,16],[1,2,4,5,6,7,9,14,15,16],[2,3,4,6,8,11,16],[2,3,4,5,6,7,11,12,14,16],[0,1,2,3,4,7,8,13,15,16],[0,1,2,5,6,7,9,12],[2,3,6,7,8,9,11,13,14,16],[1,2,3,6,8,10,11,12,13],[1,6,8,10,12,14,15],[1,4,7,8,11,12,13,14,16],[0,1,2,3,7,8,11,12,16],[0,1,8,9,11,12,16],[3,4,7,8,12,15,16],[0,1,2,5,6,13,14,15],[0,1,3,4,5,7,10,12,15,16],[4,5,6,9,11,12,14,16],[3,4,5,7,9,15],[0,2,4,5,7,9,11,12,14],[1,2,3,5,9,10,14,15,16],[1,3,6,8,9,10,11,14],[0,1,3,4,5,9,10,13,15,16],[1,4,5,6,10,11,15],[0,1,4,5,6,7,10,13,14,16],[1,2,3,4,6,7,8,13],[2,4,8,11,12,13,15],[4,5,6,7,14,16],[0,2,4,7,9,12,13,15],[0,1,4,5,8,9,11,15,16],[1,2,5,6,11,12,13,14,16],[1,2,3,5,8,10,11],[2,6,8,10,11,12,13,14,15],[0,1,3,4,5,6,7,11,13,14,15],[0,2,5,6,7,10,11,12,16],[1,2,6,7,8,9,10,11,12,13,15,16],[0,1,2,5,6,9,10,11,13,15],[1,5,6,8,9,10,12,14,15,16],[0,1,2,3,6,9,10,12,13,16],[0,3,4,5,6,10,12,13,14,15],[2,6,9,12,13,14,15],[1,6,10,12,15],[0,1,4,6,7,8,9,10,11,14,15],[6,9,11,12],[1,2,3,8,12,13,14,16],[0,1,3,4,6,8,9,10,11,13,14,15],[1,2,4,6,7,9,10,12,13,14,15],[0,1,2,3,5,8,11,13],[0,4,9,10,12,14,15],[1,3,4,5,7,9,10,11,12,13,15],[1,3,4,5,7,9,10,11,16],[1,2,4,6,7,8,9],[0,6,7,9,10,11,12],[2,3,4,6,8,9,13,14,16],[0,2,6,7,8,14,15,16],[3,7,8,14],[0,2,3,4,5,6,8,13,15],[3,4,6,7,8,11,15,16],[0,1,2,4,5,7,11,13,15,16],[1,2,4,7,8,11],[3,4,5,7,12,13,14,16],[3,5,6,7,10,11,13,14],[0,3,4,8,10,15],[0,3,5,13,14,16],[0,1,2,6,8,9,10,14,16],[2,3,4,5,6,10,11,14],[0,5,6,9,12,14,15,16],[2,3,4,6,8,13,16],[0,1,11,13,15],[1,3,7,9,10,11,12,13,16],[0,3,6,10,13,15],[0,1,2,7,9,10,15,16],[0,1,2,4,6,8,9,10,11,12],[1,2,3,6,7,9,10,12,14],[0,3,4,8,10,13,14,16],[0,1,3,4,5,6,8,9,10,11,12,14,16],[0,3,5,7,9,11,12,13,14,15],[0,8,9,12,13,14],[0,2,6,7,12,13],[2,3,8,9,10,11,16],[1,3,4,5,8,9,14],[0,3,4,5,8,9,13,14,15],[1,10,11,13,14,15],[2,3,5,6,8,9,10,11,12,15,16],[1,3,6,7,15,16],[2,3,4,5,6,12,14,16],[0,1,2,5,6,7,10,11,12,15],[1,2,3,7,8,10,13,14,15,16],[1,2,4,5,6,8,9,10,12,14],[0,2,6,8,9,10,12,15],[2,4,10,11,13],[2,3,5,6,7,8,12,14,16],[1,2,4,8,10,11,12,13,15,16],[0,2,4,5,6,8,10,13],[0,1,3,4,5,7,8,9,10,11,13,14,15,16],[0,2,3,5,6,7,8,9,15,16],[5,10,11,12,14,15],[0,3,4,6,7,8,9,10,14],[7,9,12,13,14,15,16],[1,3,9,13,15],[2,4,5,6,8,10,11,13,15],[2,4,5,6,7,8,9,11,12,14,16],[7,8,10,14,15],[7,8,9,11,12,15,16],[7,12,15],[9,12,15],[11,12,16],[7,8,9,10,11,12,14,15,16],[9,13,16],[8,10,11,13,14,15,16],[10,11,12,13,16]]},"distance":{"d":4,"is_exact":true,"upper":4,"witness":[0,1,2,76],"A_d":26019,"certified":null},"clifford":{"kind":"required","correction":{"S":[[0,3],[1,3],[2,3],[3,3],[4,3],[5,2],[6,3],[7,2],[8,2],[10,2],[12,2],[13,2],[14,2]],"CZ":[[0,2],[0,4],[0,5],[0,9],[0,11],[0,12],[0,15],[0,16],[1,2],[1,6],[1,7],[1,9],[1,11],[1,15],[1,16],[2,5],[2,8],[2,10],[2,12],[2,13],[2,16],[3,8],[3,9],[3,12],[3,14],[3,16],[4,5],[4,6],[4,8],[4,9],[4,10],[4,12],[4,13],[4,15],[4,16],[5,6],[5,7],[5,12],[5,14],[5,15],[6,9],[6,13],[6,15]]},"rotation_powers":null},"metrics":{"gamma":2.1400539595963677,"gamma_t":null,"gamma_rho":null,"gamma_claim":2.1400539595963677,"gamma_t_claim":null,"gamma_rho_claim":null,"v_ex":null,"rate":0.051470588235294115,"t_count":null,"poly_degree":null,"effective_width":7},"provenance":{"regimes":["AI search"],"discovery":"AI search","strongest_claim":"found by an AI search campaign; a verified witness, not a maximum","sources":[{"regime":"AI search","file":"master_catalog/catalogue_new","label":"CCZ012.CCZ013.CCZ023.CCZ026.CCZ046.CCZ056.CCZ123.CCZ134.CCZ145.CCZ156.CCZ234.CCZ246.CCZ256.CCZ456","N":17,"d":4,"d_is_exact":true,"origin":"frontier_campaign","provenance":"frontier_campaign \u00b7 catalogue.jsonl:331"}]},"references":[{"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"}]}