LEVER: Adaptive Cost-Aware Proof Search Over AND/OR Graphs

Nihal Jain, Shuangjie Yao, Begum Cicekdag, Zhuo Zhang, Suman Jana

Mathematicians value proofs for more than correctness: among correct proofs, simplicity, purity and the computational cost of finding them vary widely. Yet LLM-powered theorem provers largely search for any correct proof, and improve its quality only after it is found. We propose LEVER, a proof search algorithm that makes the objective over correct proofs programmable and optimizes it during search. LEVER scores partial proofs over an AND/OR proof graph, combining realized objective values with predictions for open subgoals, so the objective guides search before a proof is complete. The same mechanism optimizes computational cost, proof length, topical impurity, and even their weighted combinations, while the Lean kernel enforces correctness. On PutnamBench in Lean 4, under matched budgets, LEVER costs 34% less than a strong single-conversation agent while raising the solve rate from 80% to 96%. On reducing topical impurity, i.e., how far a proof strays from its theorem's subject, it improves over post-hoc refactoring (42% reduction against 33%) at two-thirds of the cost and more reliably; on proof length, the metric refactoring is built for, it approaches refactoring. Varying…