ProofCouncil: An LLM Agent for Solving Open Mathematical Problems

2026-07-10

AI Summary

This work addresses the limited capability of large language models in autonomously proving open-ended mathematical conjectures by introducing ProofCouncil, a novel agent framework featuring an author-critic collaborative architecture that emulates the iterative generation and verification cycles characteristic of real mathematical research. The proposed approach establishes an agent-based workflow tailored for open mathematical problems and is accompanied by an open-source agent development library. Evaluated on the FirstProof benchmark, the system produced solutions deemed acceptable by human reviewers for 6 out of 10 problems. Furthermore, when applied to 30 open problems posed by researchers, it yielded 5 fully correct proofs, 2 promising partial solutions, and 8 instances of meaningful progress—substantially outperforming existing methods.

Abstract

Large language models (LLMs) have shown increasing promise in solving open problems in mathematics. However, their performance can be further improved through agentic workflows tailored to real-world mathematical practice. To this end, we introduce ProofCouncil, a mathematical agent that is designed to tackle open problems using an author-critic architecture. ProofCouncil served as a submission to the second batch of FirstProof, a challenge consisting of 10 real-world mathematical problems that agents must solve autonomously. Its submissions for 6 of the 10 problems were judged by the referees to be correct up to at most minor revisions, showing the best performance among participating teams. We also evaluate ProofCouncil on 30 open problems collected from mathematical researchers. Among the 21 solutions that received human feedback, 5 were judged completely correct, 2 more were judged promising pending final verification, and a further 8 contained useful partial progress. In this short paper, we describe the development of ProofCouncil and the agent-building library used to create it, which we release as open source to the community.

Problem

• open mathematical problems
• large language models
• mathematical reasoning
• automated theorem proving
• AI for mathematics

Innovation

• ProofCouncil
• author-critic architecture
• mathematical reasoning
• LLM agent
• open mathematical problems