Automated Conjecture Resolution with Formal Verification

Abstract

Recent advances in large language models (LLMs) have significantly improved their ability to perform mathematical reasoning, extending from elementary problem solving to increasingly capable performance on research-level problems. However, reliably solving and verifying such problems remains challenging due to the inherent ambiguity of natural language reasoning. In this paper, we propose an automated framework for tackling research-level mathematical problems that integrates natural language reasoning with formal verification, enabling end-to-end problem solving with minimal human intervention. Our framework consists of two components: an informal reasoning agent, Rethlas, and a formal verification agent, Archon. Rethlas mimics the workflow of human mathematicians by combining reasoning primitives with our mathematical theorem search engine, Matlas, to explore solution strategies and construct candidate proofs. Archon, equipped with our formal theorem search engine LeanSearch, translates informal arguments into fully formalized Lean 4 projects through structured task decomposition, iterative refinement, and automated proof synthesis, ensuring machine-checkable correctness. Using this framework, we automatically resolve an open problem in commutative algebra proposed by D. D. Anderson (2014) and formally verify the resulting proof in Lean 4 with essentially no human involvement. Additional research-level case studies further illustrate the capabilities of Rethlas in informal mathematical reasoning and discovery, as well as the ability of Archon to formalize research-level proofs in Lean 4. Our experiments demonstrate that strong theorem retrieval tools enable the discovery and application of deep, cross-domain mathematical techniques, while the formal agent is capable of autonomously filling nontrivial gaps in informal arguments. More broadly, our work illustrates a promising paradigm for mathematical research in which informal and formal reasoning systems, equipped with theorem retrieval tools, operate in tandem to produce verifiable results, substantially reduce human effort, and offer a concrete instantiation of human–AI collaborative mathematical research with minimal human involvement.