{"id":"161.29.3.a","row":789,"stored":{"file":"master_catalog.json","row":789,"construction":null,"columns_sha256":null},"parameters":{"n":161,"k":29,"d":3,"N":40,"r":11,"level":3},"gate":{"string":"0+1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28","human":"T0\u00b7T1\u00b7T2\u00b7T3\u00b7T4\u00b7T5\u00b7T6\u00b7T7\u00b7T8\u00b7T9\u00b7T10\u00b7T11\u00b7T12\u00b7T13\u00b7T14\u00b7T15\u00b7T16\u00b7T17\u00b7T18\u00b7T19\u00b7T20\u00b7T21\u00b7T22\u00b7T23\u00b7T24\u00b7T25\u00b7T26\u00b7T27\u00b7T28","monomials":[[0],[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[25],[26],[27],[28]],"sk_key":[[0],[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[25],[26],[27],[28]],"sk_fingerprint":"944bc573b650cccf","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[5,6,8,11,12,14,15,17,21,22,23,25,26,27,29,37,38],[17,34],[0,2,3,4,7,8,10,11,19,20,21,22,25,27,28,30,35,37,38],[4,7,8,10,13,15,16,17,18,19,20,21,24,25,27,28,29,32,37],[4,5,6,7,10,11,12,13,14,15,16,18,19,20,23,24,25,28,29,32,36,38],[0,2,3,4,7,8,10,11,14,15,17,19,20,21,27,28,29,30,33,35,36,37,38],[14,33,35,36],[26,34,35,36],[1,2,4,5,9,11,13,14,16,18,19,21,24,27,28,31,32,33,39],[22,33,34,37],[15,33,35,37],[0,1,3,4,6,8,9,11,12,15,17,19,22,23,25,26,28,30,31,33,37,39],[0,1,3,4,5,9,15,17,19,21,26,27,28,30,31,33,35,37,38,39],[29],[1,2,4,6,8,9,12,13,14,15,16,18,19,23,24,28,29,31,32,33,34,38,39],[1,2,3,4,5,8,9,10,11,14,19,21,25,28,29,30,31,32,33,35,36,37],[0,1,3,5,6,8,9,10,11,12,13,16,17,18,19,20,26,27,29,30,31,32,34,36,37],[0,3,4,6,8,9,10,16,17,18,19,20,21,24,26,27,29,30,32,34,35,36,38],[0,1,3,5,6,8,9,10,16,17,18,19,20,21,24,26,27,28,29,31,32,33,35,37],[0,1,3,4,5,6,8,9,10,11,12,15,17,18,19,22,23,25,26,30,31,37,38,39],[5,13,28,30,33,34,35,37,38],[2,3,4,5,8,9,10,11,14,15,16,19,20,21,22,23,24,27,28,30,33,34,36,37,38,39],[0,3,6,8,10,12,15,17,18,19,21,22,23,24,25,26,28,33,34,36,39],[0,2,5,6,9,13,14,15,17,18,22,23,24,26,29,30,32,34,38,39],[12,36,37,38],[2,3,8,10,11,13,14,19,25,28,29,30,32,33,37],[27,32],[0,1,4,5,9,11,12,15,16,17,18,19,20,22,25,26,27,30,34,35,37,38],[0,3,8,9,10,16,17,18,19,20,21,24,26,27,29,30,31,34,35,36,39],[0,1,3,4,6,8,9,11,12,15,16,17,18,19,22,23,25,26,28,31,32,33,34,35,39],[20,33,36,39],[0,1,3,4,5,6,9,11,12,15,17,18,19,22,23,25,26,30,32,34,36,37,38],[3,5,28,30,31,32,33,34,35,37,38],[1,4,5,10,11,12,15,16,20,21,22,24,25,27,29,31,33,34,36,37,38,39],[0,2,3,5,8,16,19,27,28,30,31,33,34,35,37,38,39],[2,4,6,8,10,11,14,15,17,21,29,33,35,36,38,39],[2,5,7,8,9,10,17,18,19,20,21,23,24,26,27,28,29,32,33,34,37,38,39],[2,3,4,5,7,8,10,11,14,15,17,19,20,21,27,28,29,30,31,32,34,35,37,39],[2,3,5,7,8,19,20,23,28,30,31,32,33,36,38,39],[0,2,3,4,7,8,10,11,14,15,17,20,21,27,29,32,34,35,37,38],[2,33,35,36,37,39],[2,4,6,9,11,14,15,16,18,24,26,30,31,33,35,36,37,38],[1,2,7,8,9,10,11,15,17,18,20,34,35,37,38,39],[2,4,5,6,7,9,11,14,15,16,17,18,24,26,28,30,32,36],[1,4,5,10,11,12,14,16,17,20,21,23,24,27,28,31,32,33,36,37,38,39],[0,2,3,5,11,13,14,16,18,24,25,31,32,33,37],[0,1,2,3,4,5,6,8,10,11,13,15,17,18,20,22,29,34,37,39],[28,33,34,36],[1,2,6,7,9,10,12,18,20,21,22,23,25,26,27,28,29,30,31,35,36,37,38,39],[2,4,6,7,9,11,14,15,16,18,24,31,32,33,34,37,38],[1,2,4,5,9,11,13,14,16,18,19,21,24,27,28,30,31,32,33,39],[1,4,10,11,12,15,16,20,21,23,24,25,27,28,29,32,33,34,35,36,37,39],[0,3,6,8,9,10,15,16,17,18,19,20,21,24,26,27,29,31,32,34,35,36,37],[18,34,35,37],[1,4,6,8,10,15,16,18,20,24,29,31,32,33,36,39],[0,1,3,4,5,6,8,9,11,12,15,17,18,19,22,23,25,26,29,37,38,39],[5,25,28,31,33,34,35,37,38],[2,5,6,8,9,10,11,13,14,18,19,20,21,22,25,27,28,30,36,37],[1,2,3,4,5,8,9,10,11,14,19,21,25,28,29,30,32,33,35,36,37],[0,3,4,5,6,8,9,13,15,17,18,19,21,22,23,24,25,26,28,34,35,37,39],[4,33,35,38],[0,1,3,4,6,8,9,11,12,15,17,18,19,22,23,24,25,26,28,33,34,35,39],[1,33,36,38],[1,2,3,8,9,12,14,19,25,28,29,30,32],[0,1,2,3,5,7,8,9,14,15,17,19,20,24,27,28,29,30,31,33,35,36],[2,4,5,6,7,9,11,14,15,16,18,21,24,26,28,31,32,34,36],[1,3,6,7,8,9,12,13,15,16,18,19,25,26,29,30,33,34,35,36,38],[0,2,3,4,7,8,10,11,14,15,17,19,20,21,27,28,29,30,33],[0,1,2,3,7,8,14,16,17,19,22,23,24,25,28,30,31,32,34,39],[0,4,7,13,15,17,20,21,25,27,28,30,31,32,33,34,35,38],[1,2,6,7,9,10,11,12,14,18,20,21,22,23,25,26,27,28,29,33,37,38,39],[2,4,6,7,11,14,15,16,18,24,26,30,32,33,35,36,37,38],[0,5,9,10,17,18,19,21,23,24,26,28,29,34,36,37,38,39],[6,33,35,39],[0,3,5,6,8,9,10,16,17,18,19,21,24,26,27,28,29,33,35,37,38,39],[8,34,36,39],[0,1,3,6,7,8,9,10,12,14,16,18,19,20,21,22,23,25,26,28,29,31,32,33,37,38],[5,7,8,9,10,16,17,18,21,24,26,28,29,33,34,35,38,39],[0,1,3,9,10,12,14,16,18,20,21,22,25,26,27,28,29,31,32,34,35,36,37,38],[0,33,34,35,37,39],[0,1,5,12,14,16,17,20,22,24,25,31,32,33,34,37],[1,6,7,9,10,12,14,18,20,21,22,23,25,26,27,28,29,31,32,35,36,38],[5,6,8,11,12,14,15,17,21,22,23,25,26,27,29,33,37,38],[0,2,3,4,7,8,10,11,19,20,21,22,25,27,28,30,37,38],[4,7,8,10,13,15,16,17,18,19,20,21,24,25,27,28,29,32,33,34,35,37],[4,5,6,7,10,11,12,13,14,15,16,18,19,20,23,24,25,28,29,32,38],[1,2,4,5,9,11,13,14,16,18,19,21,24,27,28,31,32,33,37,39],[0,1,3,4,6,8,9,11,12,15,17,19,22,23,25,26,28,30,31,33,34,35,39],[0,1,3,4,5,9,15,17,19,21,26,27,28,30,31,35,36,38,39],[29,34,36,37],[25,35,36,37],[1,2,4,6,8,9,12,13,14,15,16,18,19,23,24,28,29,31,32,35,36,37,38,39],[1,2,3,4,5,8,9,10,11,14,19,21,25,28,29,30,31,32,33,35,36,37,38],[0,1,3,5,6,8,9,10,11,12,13,16,17,18,19,20,26,27,29,30,31,32,33,36,37,38],[0,3,4,6,8,9,10,16,17,18,19,20,21,24,26,27,29,30,32,33,34,36],[24,34,35,38],[0,1,3,5,6,8,9,10,16,17,18,19,20,21,24,26,27,28,29,31,32,35,36,37,38],[0,1,3,4,5,6,8,9,10,11,12,15,17,18,19,22,23,25,26,30,31,34,36,37,39],[5,13,28,30,33,34,36,37],[2,3,4,5,8,9,10,11,14,15,16,19,20,21,22,23,24,27,28,30,35,37,39],[0,3,6,8,10,12,15,17,18,19,21,22,23,24,25,26,28,34,36,37,38,39],[21,34,37,38],[0,2,5,6,9,13,14,15,17,18,22,23,24,26,29,30,32,34,35,37,39],[5,33,34,35,37,38],[2,3,8,10,11,13,14,19,25,28,29,30,32,34,36,38],[11,33,35,36,37,38],[9,34,35,36,37,38],[27,32,39],[0,1,4,5,9,11,12,15,16,17,18,19,20,22,25,26,27,30,33,35,37,38,39],[0,3,8,9,10,16,17,18,19,20,21,24,26,27,29,30,31,33,34,36],[0,1,3,4,6,8,9,11,12,15,16,17,18,19,22,23,25,26,28,31,32,33],[0,1,3,4,5,6,9,11,12,15,17,18,19,22,23,25,26,30,32,37,38,39],[3,5,28,30,31,32,33,34,36,37,38,39],[1,4,5,10,11,12,15,16,20,21,22,24,25,27,29,31,35,37,38],[0,2,3,5,8,16,19,27,28,30,31,34,35,38],[2,4,6,8,10,11,14,15,17,21,29,33,34,35,36,37,38],[2,5,7,8,9,10,17,18,19,20,21,23,24,26,27,28,29,32,33,34,35,38],[2,3,4,5,7,8,10,11,14,15,17,19,20,21,27,28,29,30,31,32,33],[2,3,5,7,8,19,20,23,28,30,31,32,33,37,38],[0,2,3,4,7,8,10,11,14,15,17,20,21,27,29,32,33,35,36,38,39],[2,4,6,9,11,14,15,16,18,24,26,30,31,33,34,38,39],[1,2,7,8,9,10,11,15,17,18,20,33,34,35,37,38,39],[2,4,5,6,7,9,11,14,15,16,17,18,24,26,28,30,32,34,36],[1,4,5,10,11,12,14,16,17,20,21,23,24,27,28,31,32,33,35,36,37,38,39],[0,2,3,5,11,13,14,16,18,24,25,31,32,34,35,37],[0,1,2,3,4,5,6,8,10,11,13,15,17,18,20,22,29,34,36,37,39],[1,2,6,7,9,10,12,18,20,21,22,23,25,26,27,28,29,30,31,33,37,38,39],[2,4,6,7,9,11,14,15,16,18,24,31,32,33,35,36,37,38],[1,2,4,5,9,11,13,14,16,18,19,21,24,27,28,30,31,32,33,37,39],[1,4,10,11,12,15,16,20,21,23,24,25,27,28,29,32,35,36,39],[0,3,6,8,9,10,15,16,17,18,19,20,21,24,26,27,29,31,32,33,34,36],[1,4,6,8,10,15,16,18,20,24,29,31,32,37,39],[0,1,3,4,5,6,8,9,11,12,15,17,18,19,22,23,25,26,29,34,36,38,39],[5,25,28,31,33,34,36,38],[2,5,6,8,9,10,11,13,14,18,19,20,21,22,25,27,28,30,33,34,35],[1,2,3,4,5,8,9,10,11,14,19,21,25,28,29,30,32,33,35,36,37,38],[0,3,4,5,6,8,9,13,15,17,18,19,21,22,23,24,25,26,28,33,35,37,38,39],[0,1,3,4,6,8,9,11,12,15,17,18,19,22,23,24,25,26,28,33,38,39],[10,34,36,38],[13,35,36,38],[1,2,3,8,9,12,14,19,25,28,29,30,32,33,34,35,36,38],[0,1,2,3,5,7,8,9,14,15,17,19,20,24,27,28,29,30,31,35,36,37,38],[2,4,5,6,7,9,11,14,15,16,18,21,24,26,28,31,32,36,37,38],[1,3,6,7,8,9,12,13,15,16,18,19,25,26,29,30,33,34,36,37],[0,1,2,3,7,8,14,16,17,19,22,23,24,25,28,30,31,32,34,36,37,38,39],[0,4,7,13,15,17,20,21,25,27,28,30,31,32,35,36,37],[1,2,6,7,9,10,11,12,14,18,20,21,22,23,25,26,27,28,29,35,36,39],[2,4,6,7,11,14,15,16,18,24,26,30,32,33,34],[27,39],[0,5,9,10,17,18,19,21,23,24,26,28,29,33,36,37,38],[16,34,35,39],[0,3,5,6,8,9,10,16,17,18,19,21,24,26,27,28,29,35,36,37,38],[3,35,36,39],[23,33,34,35,36,39],[0,1,3,6,7,8,9,10,12,14,16,18,19,20,21,22,23,25,26,28,29,31,32,38,39],[5,7,8,9,10,16,17,18,21,24,26,28,29,33,35,37,38],[0,1,3,9,10,12,14,16,18,20,21,22,25,26,27,28,29,31,32,34,36,38,39],[0,1,5,12,14,16,17,20,22,24,25,31,32,33,34,36,39],[19,33,34,36,37,39],[1,6,7,9,10,12,14,18,20,21,22,23,25,26,27,28,29,31,32,33,37,38,39],[7,34,35,36,37,39]]},"distance":{"d":3,"is_exact":true,"upper":3,"witness":[0,8,159],"A_d":412,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,2],[1,1],[2,2],[3,2],[4,2],[5,1],[6,1],[7,2],[8,2],[9,1],[10,2],[11,2],[12,1],[13,1],[14,2],[15,2],[16,1],[17,2],[18,1],[19,2],[20,2],[21,2],[22,1],[23,1],[24,1],[25,1],[26,1],[27,2],[28,2],[29,1],[30,1],[34,1],[35,1],[37,1],[38,1]],"CZ":[[0,2],[0,3],[0,4],[0,7],[0,8],[0,10],[0,11],[0,14],[0,15],[0,17],[0,19],[0,20],[0,21],[0,27],[0,28],[0,29],[0,30],[0,34],[0,35],[0,37],[0,38],[2,3],[2,4],[2,7],[2,8],[2,10],[2,11],[2,14],[2,15],[2,17],[2,19],[2,20],[2,21],[2,27],[2,28],[2,29],[2,30],[2,34],[2,35],[2,37],[2,38],[3,4],[3,7],[3,8],[3,10],[3,11],[3,14],[3,15],[3,17],[3,19],[3,20],[3,21],[3,27],[3,28],[3,29],[3,30],[3,34],[3,35],[3,37],[3,38],[4,7],[4,8],[4,10],[4,11],[4,14],[4,15],[4,17],[4,19],[4,20],[4,21],[4,27],[4,28],[4,29],[4,30],[4,34],[4,35],[4,37],[4,38],[7,8],[7,10],[7,11],[7,14],[7,15],[7,17],[7,19],[7,20],[7,21],[7,27],[7,28],[7,29],[7,30],[7,34],[7,35],[7,37],[7,38],[8,10],[8,11],[8,14],[8,15],[8,17],[8,19],[8,20],[8,21],[8,27],[8,28],[8,29],[8,30],[8,34],[8,35],[8,37],[8,38],[10,11],[10,14],[10,15],[10,17],[10,19],[10,20],[10,21],[10,27],[10,28],[10,29],[10,30],[10,34],[10,35],[10,37],[10,38],[11,14],[11,15],[11,17],[11,19],[11,20],[11,21],[11,27],[11,28],[11,29],[11,30],[11,34],[11,35],[11,37],[11,38],[14,15],[14,17],[14,19],[14,20],[14,21],[14,27],[14,28],[14,29],[14,30],[14,34],[14,35],[14,37],[14,38],[15,17],[15,19],[15,20],[15,21],[15,27],[15,28],[15,29],[15,30],[15,34],[15,35],[15,37],[15,38],[17,19],[17,20],[17,21],[17,27],[17,28],[17,29],[17,30],[17,34],[17,35],[17,37],[17,38],[19,20],[19,21],[19,27],[19,28],[19,29],[19,30],[19,34],[19,35],[19,37],[19,38],[20,21],[20,27],[20,28],[20,29],[20,30],[20,34],[20,35],[20,37],[20,38],[21,27],[21,28],[21,29],[21,30],[21,34],[21,35],[21,37],[21,38],[27,28],[27,29],[27,30],[27,34],[27,35],[27,37],[27,38],[28,29],[28,30],[28,34],[28,35],[28,37],[28,38],[29,30],[29,34],[29,35],[29,37],[29,38],[30,34],[30,35],[30,37],[30,38],[34,35],[34,37],[34,38],[35,37],[35,38],[37,38]]},"rotation_powers":[[0,3],[1,7],[2,7],[6,7],[8,7],[9,7],[10,3],[15,7],[17,5],[19,5],[20,7],[21,3],[22,5],[23,5],[24,7],[25,7],[26,3],[27,7],[28,7],[30,7],[36,3],[39,3],[42,5],[43,5],[44,3],[46,7],[47,3],[48,3],[50,5],[52,3],[54,3],[55,3],[56,3],[58,5],[59,3],[61,3],[62,3],[63,3],[64,3],[67,3],[70,3],[71,3],[72,7],[74,3],[79,3],[81,3],[82,5],[83,7],[84,3],[85,3],[86,5],[87,3],[88,3],[90,3],[91,3],[92,3],[93,3],[94,3],[95,3],[97,3],[99,3],[102,3],[103,3],[106,3],[107,7],[111,3],[112,3],[115,3],[121,3],[123,3],[124,3],[126,3],[127,3],[128,3],[130,3],[132,3],[133,3],[134,3],[137,3],[140,3],[141,3],[142,3],[143,3],[145,3],[147,3],[148,3],[150,3],[151,3],[153,3],[155,3],[156,3],[160,3]]},"metrics":{"gamma":1.5602488272510306,"gamma_t":null,"gamma_rho":1.5602488272510306,"gamma_claim":1.5602488272510306,"gamma_t_claim":null,"gamma_rho_claim":1.5602488272510306,"v_ex":29,"rate":0.18012422360248448,"t_count":null,"poly_degree":null,"effective_width":29},"provenance":{"regimes":["AI search (gamma frontier): contraction of an existing code"],"discovery":"AI search","strongest_claim":"a paired-column contraction of an existing triorthogonal code, 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): contraction of an existing code","file":"gamma_frontier_release/protocols/paired_column_contraction_161_29_3_87496b18d4f0.json","label":"paired_column_contraction_161_29_3_87496b18d4f0","N":40,"d":3,"d_is_exact":true,"origin":"Existing triorthogonal code with physical paired-column contraction","provenance":"gamma frontier release record paired_column_contraction_161_29_3_87496b18d4f0 (source catalogue id paired_column_contraction_161_29_3_87496b18d4f0); construction: Existing triorthogonal code with physical paired-column contraction; matrix 87496b18d4f0d366...; the release states d=3 (exact), gamma=1.5602488272510306 (exact)","notes":"matrix gamma_frontier_release/matrices/87496b18d4f0d366129e572542dc34493f382f927e6cd3a7895980a68fadbd6e.matrix"}]},"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"}]}