🤖 AI Summary
This work resolves the satisfiability of ten Challenge-2 instances from the matrix multiplication SAT benchmark, previously conjectured to be unsatisfiable (UNSAT), and constructs the first rank-23 matrix multiplication scheme free of type-3 monomials. By leveraging the isotropic group action of GL(3,2)³, cyclic trace symmetry, perfect matching constraints, semantic repair techniques, and binary identities over 𝔽₂, the authors generate complete satisfying assignments for all 21 CNF instances—encompassing 26,541 variables and 2,461,316 clauses—and verify that all Brent equation residuals vanish identically. These results definitively establish the satisfiability of all Challenge-2 instances, correcting prior assumptions, and yield a reproducible type-3-free rank-23 scheme executable in under nine seconds.
📝 Abstract
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-3 pairings are imposed by positive unit clauses on the 621 base variables: the formulas require selected incidences but do not forbid additional type-3 incidences. Starting from exact 23-summand schemes, we use the $\mathrm{GL}(3,2)^3$ isotropy action, cyclic trace symmetry, and perfect matching of transformed summands to constrained slots to construct witnesses for all ten files. For Challenge 3, we combine a locked semantic repair with a two-term identity over $\mathbb{F}_2$ to obtain a distinguished summand of type-3 count zero. Every semantic decomposition has zero residual in all 729 Brent equations. The accompanying DIMACS models assign all 26,541 variables of each formula and satisfy all 2,461,316 clauses across the 21 instances. A separate parser and clause evaluator rechecks the emitted models. A deterministic one-file Python reproducer regenerates the 21 certificates from the original CNFs in approximately nine seconds on the reported test host.