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search: Wills downset framework"],"discovery":"AI search","strongest_claim":"verified witness from the 2026-09-14 wills_downset_theory campaign in magic-states-AI, which solved the arXiv:2608.24000 downset framework EXACTLY at every n <= 1000; the d=7 and d=8 rows are proved optima of that framework, not merely best-found, but the framework is one family and they are not maxima over all factories","sources":[{"regime":"AI search: Wills downset framework","file":"magic-states-AI/campaigns/2026-09-14_wills_downset_theory/found/n988_k36_d7_wills_downset_optimum.json","label":"n988_k36_d7_wills_downset_optimum","N":176,"d":6,"d_is_exact":false,"origin":null,"provenance":"Wills arXiv:2608.24000 Prop 3.7. Delta = {|A| <= 3} on 2^[10] (RM(10,3)); Gamma = 36 points. CP-SAT proved OPTIMAL over every pair of downsets at every support size: no code in the framework reaches k > 36 at d >= 7. gamma = 1.702115; does NOT beat the bar 1.086033.","notes":"6 <= d <= 7 from the columns alone; d = 7 exactly by Prop 3.7 | A_7 >= 100: the eq (149) minimisers, each giving a distinct minimum-weight undetectable damaging fault via Prop 3.7's own codeword prod_(i in A)(1+x_i). A FLOOR, not a count -- tools/ad_floor.py"}]},"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"}]}