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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/unified_l3.json","label":"af-0942","N":11,"d":2,"d_is_exact":true,"origin":"https://github.com/shraggy/Magic_state_factory_search at 4627014, outputs/all_factories.csv line 942","provenance":"unified search l=3, two-group s_total=4,s_O=2"}]},"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"}]}