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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/n998_k26_d8_wills_downset_optimum.json","label":"n998_k26_d8_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), the ONLY 8-divisible downset admitting d >= 8 at m=10 -- Lemma B of tools/downset_ceiling.py forces rank <= 3 once Lemma A puts every 3-set in Delta); Gamma = {empty set} + all 10 singletons + the 15 edges of the PETERSEN graph (3-regular, girth 5, verified). CP-SAT proved OPTIMAL: no pair of downsets on any 2^[m] with m <= 10 reaches k > 26 at d >= 8, so this is the exact best the entire framework of Wills Sections 3-5 can do at d = 8. gamma = 1.754152, which does NOT beat the external bar 1.086033.","notes":"6 <= d <= 8 from the columns alone; d = 8 exactly by Prop 3.7 | A_8 >= 70: 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"}]}