{"id":"52.5.3.g","row":303,"stored":{"file":"master_catalog.json","row":303,"construction":null,"columns_sha256":null},"parameters":{"n":52,"k":5,"d":3,"N":12,"r":7,"level":3},"gate":{"string":"0+1+2+3+4","human":"T0\u00b7T1\u00b7T2\u00b7T3\u00b7T4","monomials":[[0],[1],[2],[3],[4]],"sk_key":[[0],[1],[2],[3],[4]],"sk_fingerprint":"65a2b287ec4a80f7","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[1,2,4,6,7],[0,1,2,4,5,6,7],[1,3,6,8],[0,5,6,8],[3,7,8],[0,2,3,4,5,7,8],[0,1,3,9],[2,3,5,9],[0,1,2,3,4,6,7,9],[1,2,3,4,5,6,7,9],[0,3,4,8,9],[0,3,4,5,8,9],[0,1,6,8,9],[0,5,6,8,9],[0,4,7,8,9],[0,2,4,5,7,8,9],[0,2,4,6,10],[3,5,6,10],[0,8,10],[3,5,8,10],[0,3,4,6,7,8,10],[0,2,3,5,6,7,8,10],[0,9,10],[0,5,9,10],[3,4,6,9,10],[0,2,3,4,5,6,9,10],[6,7,8,9,10],[0,2,5,6,7,8,9,10],[3,4,7,11],[0,1,5,7,11],[8,11],[0,4,5,8,11],[1,3,6,7,8,11],[5,6,7,8,11],[2,3,9,11],[2,3,5,9,11],[0,3,7,9,11],[1,3,5,7,9,11],[0,1,6,7,8,9,11],[5,6,7,8,9,11],[2,4,6,10,11],[2,4,5,6,10,11],[7,10,11],[5,7,10,11],[3,4,6,7,10,11],[5,6,7,10,11],[0,4,6,8,10,11],[5,6,8,10,11],[0,3,7,8,10,11],[5,7,8,10,11],[6,7,8,9,10,11],[5,6,7,8,9,10,11]]},"distance":{"d":3,"is_exact":true,"upper":3,"witness":[0,2,4],"A_d":188,"certified":null},"clifford":{"kind":"required","correction":{"S":[[1,3],[2,1],[3,1],[4,3],[5,3],[6,2],[7,2],[8,2]],"CZ":[[0,1],[0,3],[0,9],[0,10],[0,11],[1,3],[1,9],[2,4],[2,5],[2,6],[2,10],[3,4],[3,5],[3,6],[3,8],[4,7],[4,11],[5,6],[5,7],[5,8]]},"rotation_powers":null},"metrics":{"gamma":2.1316035058977802,"gamma_t":2.1316035058977802,"gamma_rho":2.1316035058977802,"gamma_claim":2.1316035058977802,"gamma_t_claim":2.1316035058977802,"gamma_rho_claim":2.1316035058977802,"v_ex":5,"rate":0.09615384615384616,"t_count":5,"poly_degree":1,"effective_width":5},"provenance":{"regimes":["AI search (gamma frontier): logical restriction of a length-54 Pareto point"],"discovery":"AI search","strongest_claim":"a restriction to fewer logical rows of a Pareto point of the length-54 classification, in the pure-T distillation-exponent release (gamma = log(n/k)/log d, n < 1000); a verified witness, not a maximum -- only the release's own frontier points are the lowest gamma it knows","sources":[{"regime":"AI search (gamma frontier): logical restriction of a length-54 Pareto point","file":"gamma_frontier_release/protocols/logical_restriction_71_52_5_3_296cdb24ab.json","label":"logical_restriction_71_52_5_3_296cdb24ab","N":12,"d":3,"d_is_exact":true,"origin":"New verified T-to-T construction","provenance":"gamma frontier release record logical_restriction_71_52_5_3_296cdb24ab (source catalogue id logical_restriction_71_52_5_3_296cdb24ab); construction: New verified T-to-T construction; matrix 296cdb24abda4e67...; the release states d=3 (exact), gamma=2.1316035058977802 (exact)","notes":"matrix gamma_frontier_release/matrices/296cdb24abda4e672ed8d1cf577afecbfcdf94ee7ca805b7210166c89c19ba9d.matrix"}]},"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"}]}