Orthologic for SAT Solving

SAT solvers settle combinatorial problems beyond the reach of interactive theorem provers and produce LRAT certificates for independent verification. We present LRAT-Catcher, a standalone, general-purpose tool that imports a DIMACS formula together with an LRAT certificate into Lean 4 as a theorem. LRAT-Catcher runs the formally verified LRAT checker from Lean core as compiled native code via reflection. This scales to instances where Mathlib's explicit proof-term import exhausts memory. LRAT-Catcher also composes cube-and-conquer solving runs entirely inside Lean. Per-cube refutations are combined with a cover-completeness certificate, itself an LRAT proof, into a single unsatisfiability theorem. Verified encodings connect CNF-level results to the original combinatorial problems. We evaluate the tool against Mathlib's proof-term import and the external checker cake_lpr on establishing the Schur number S(4) = 44 and the Ramsey number R(4,4) = 18 as Lean theorems.

Large language models (LLMs) are increasingly used for tasks that implicitly reduce to Boolean satisfiability (SAT), yet their reasoning ability on SAT remains unclear. We present a systematic study of LLMs on 2-SAT and 3-SAT, together with two canonical reductions, Vertex Cover and discrete 3D packing, to probe representation-invariant reasoning. We first evaluate models using conventional metrics, including accuracy, precision, recall, and F1, as well as the SAT phase-transition setting. We find that these metrics can be misleading: many models obtain high scores by over-predicting satisfiable formulas, fail to reproduce the classical easy-hard-easy signature around the 3-SAT threshold, and degrade sharply as the number of variables grows. To address this problem, we introduce a paired-formula protocol based on minimally different satisfiable and unsatisfiable instances, together with Accurate Differentiation Rate (ADR), which requires both members of each pair to be classified correctly. ADR separates reasoning-oriented models from heuristic ones and correlates with witness validity. Beyond CNF, we test cross-representation consistency by converting CNF to Vertex Cover and 3-SAT to discrete 3D packing. Model decisions on CNF and on the corresponding graph or packing instances agree for most models on more than 80 percent of instances, suggesting stable decision rules across representations. Overall, our results show that SAT is a conservative probe for LLM reasoning, and that paired evaluation with ADR provides a more faithful and representation-robust assessment than conventional metrics.

Foundational optimization embeddings have recently emerged as powerful pre-trained representations for mixed-integer programming (MIP) problems. These embeddings were shown to enable cross-domain transfer and reduce reliance on solver-generated labels. In this work, we investigate whether such representations generalize beyond optimization to decision problems, focusing on Boolean satisfiability (SAT). We adapt the foundational optimization architecture to SAT by mapping CNF formulas into the same bipartite constraint-variable graph representation used for MIPs. This allows direct reuse of the pre-trained embedding model without architectural changes or supervised fine-tuning. Our results show that these embeddings capture structural regularities in SAT instances and support unsupervised tasks such as instance clustering and distribution identification. We demonstrate, for the first time, that foundational optimization embeddings can transfer to constraint satisfaction domains. Our findings is a step toward a unified representational framework for both optimization and decision problems.