{"paper":{"title":"SAT Certificates for the Matrix-Multiplication Challenges over F2: All Ten `Expected-UNSAT` Instances Are Satisfiable, and a Type-3-Free Rank-23 Scheme","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LO"],"primary_cat":"cs.SC","authors_text":"Nick Palladinos","submitted_at":"2026-07-31T11:09:22Z","abstract_excerpt":"The matrix-multiplication SAT benchmark of Heule, Kauers, and Seidl asks, among other tasks, for solutions of ten known-satisfiable rank-23 formulas, proofs of unsatisfiability for ten formulas expected to be unsatisfiable, and a rank-23 scheme over $\\mathbb{F}_2$ having a summand with no type-3 monomial. We give complete satisfying assignments for the 21 CNFs in the repository's top-level challenge1/, challenge2/, and challenge3/ directories at commit 150b2e2f. The principal finding is that all ten top-level Challenge-2 formulas are satisfiable. A direct audit shows that their hardcoded type-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.29291","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.29291/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}