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14,15],[7],[7,8,9,10,11,12,13,14,15],[8],[9],[10],[11],[12],[13],[14],[15]]},"distance":{"d":2,"is_exact":true,"upper":2,"witness":[120,165],"A_d":675,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,2],[1,2],[2,2],[3,2],[4,2],[5,2],[6,1],[7,1],[8,1],[9,1],[10,1],[11,1],[12,1],[13,1],[14,1],[15,1]],"CZ":[]},"rotation_powers":[[120,5],[165,5],[210,5],[255,5],[300,5],[345,5],[391,3],[392,3],[393,3],[394,3],[395,3],[396,3],[397,3],[398,3],[399,3],[401,3]]},"metrics":{"gamma":6.09451759878429,"gamma_t":6.09451759878429,"gamma_rho":6.09451759878429,"gamma_claim":6.09451759878429,"gamma_t_claim":6.09451759878429,"gamma_rho_claim":6.09451759878429,"v_ex":6,"rate":0.014634146341463415,"t_count":6,"poly_degree":1,"effective_width":6},"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-0604","N":16,"d":2,"d_is_exact":true,"origin":"https://github.com/shraggy/Magic_state_factory_search at ae9fbf8, outputs/large_unified_factories_l3.csv line 604","provenance":"larger unified search l=3, two-group s_total=8,s_O=5"}]},"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"}]}