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:[0,31],"A_d":455886,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,3],[1,3],[2,3],[3,3],[4,3],[5,3],[6,3],[7,3],[8,3]],"CZ":[[0,1],[0,2],[0,3],[0,4],[0,5],[0,6],[0,7],[0,8],[1,2],[1,3],[1,4],[1,5],[1,6],[1,7],[1,8],[2,3],[2,4],[2,5],[2,6],[2,7],[2,8],[3,4],[3,5],[3,6],[3,7],[3,8],[4,5],[4,6],[4,7],[4,8],[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]]},"rotation_powers":[[0,7],[1,7],[3,5],[4,5],[16,7],[31,3],[62,3],[3998,3],[4308,3],[4494,3],[4525,3],[4556,3],[4587,3],[4618,3],[4649,3],[4773,3],[4804,3],[4835,3],[4928,3],[4959,3],[4990,3],[5083,3],[5114,3],[5145,3],[5176,3],[5207,3],[5238,3],[5254,3],[5285,3],[5309,3],[5316,3],[5324,3],[5331,3]]},"metrics":{"gamma":9.21053606364666,"gamma_t":null,"gamma_rho":null,"gamma_claim":9.21053606364666,"gamma_t_claim":null,"gamma_rho_claim":null,"v_ex":null,"rate":0.0016879219804951237,"t_count":null,"poly_degree":null,"effective_width":9},"provenance":{"regimes":["borrowed-identity search: two-group"],"discovery":"pre-existing","strongest_claim":"found by the two-group borrowed-identity search of Singh, Gidney and Jones (arXiv:2606.28518; borrowed_identities/): an identity circuit symmetric within its output and check blocks, with its output-only gates removed; a verified witness, not a maximum","sources":[{"regime":"borrowed-identity search: two-group","file":"borrowed_identities/circuits/large_unified_l3.json","label":"lu-1033","N":14,"d":2,"d_is_exact":true,"origin":"https://github.com/shraggy/Magic_state_factory_search at ae9fbf8, outputs/large_unified_factories_l3.csv line 1033","provenance":"larger unified search l=3, two-group s_total=1,s_O=3"}]},"references":[{"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"}]}