Maxproof (arxiv.org)

MaxProof has been introduced as an innovative framework for enhancing mathematical proof generation using advanced machine learning techniques. By integrating proof generation, verification, and critique-conditioned proof repair through a robust generative verifier, MaxProof effectively scales competition-level mathematical proof capabilities. This unique approach allows the model to operate as a generator, verifier, refiner, and ranker, searching a population of candidate proofs to produce one final output through a tournament selection process. Impressively, the M3 model that utilizes MaxProof achieved scores of 35/42 on the International Mathematical Olympiad (IMO) 2025 and 36/42 on the USA Mathematical Olympiad (USAMO) 2026, surpassing the human gold-medal benchmark. The significance of MaxProof lies in its potential to revolutionize the intersection of AI and mathematics, showcasing how machine learning can tackle complex problems traditionally reserved for human mathematicians. Key technical details include the defense-in-depth strategy that minimizes false positives and the population-level test-time scaling framework, which collectively enhance the robustness and accuracy of the proof generation process. This advancement signals a new era in AI-driven mathematical exploration and poses intriguing implications for future research in formal verification, education, and computational problem-solving.