MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling

2026-06-11

AI Summary

This work aims to advance the performance of artificial intelligence in solving highly challenging mathematical proof problems, approaching and surpassing top human-level capabilities. To this end, we introduce MaxProof, a framework that integrates generative-verification reinforcement learning with test-time ensemble expansion strategies. By combining depth-in-defense generative verifiers, critique-conditioned repair mechanisms, population-level search, and tournament-based selection, MaxProof achieves end-to-end high-performance automated theorem proving. Our method attains scores of 35/42 on the IMO 2025 benchmark and 36/42 on USAMO 2026—both exceeding the gold-medal thresholds for human participants—thereby significantly pushing the frontier of AI in formal mathematical reasoning.

Abstract

We present MaxProof, a population-level test-time scaling framework for competition-level mathematical proof in the MiniMax-M3 series. M3 first trains three proof-oriented capabilities -- proof generation, proof verification, and critique-conditioned proof repair -- using a defense-in-depth generative verifier engineered for low false-positive rate. These capabilities are merged into a single released M3 model. At test time, MaxProof treats the model as a generator, verifier, refiner, and ranker, searches over a population of candidate proofs, and returns one final proof through tournament selection. With MaxProof test-time scaling, the M3 model reaches 35/42 on IMO 2025 and 36/42 on USAMO 2026, exceeding the human gold-medal threshold on both.

Problem

• mathematical proof
• test-time scaling
• proof generation
• proof verification
• competition-level mathematics

Innovation

• test-time scaling
• generative-verifier RL
• proof verification
• population-based search
• mathematical reasoning