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

MaxProof is a population-level test-time scaling framework for competition-level mathematical proof, developed for the MiniMax-M3 model series.

The M3 model is trained with three proof-oriented capabilities: proof generation via verifier-guided reinforcement learning, proof verification through aligned error finding, and critique-conditioned proof repair.

These are merged into a single model.

At test time, MaxProof uses the model as a generator, verifier, refiner, and ranker, searching over a population of candidate proofs and selecting a final answer via tournament selection.

The framework includes a defense-in-depth verifier with four layers (bad-case filtering, solution normalization, multi-judge scoring, pessimistic aggregation) to minimize false positives.

With MaxProof, the M3 model achieves 35/42 on IMO 2025 and 36/42 on USAMO 2026, exceeding the human gold-medal threshold.

The paper also documents reward-hacking patterns from an earlier M2 cycle and reports standalone benchmark scores of 67.40 on IMOProof Bench and 81.56 on IMOAnswerBench.