For max@k (best-of-K) finite-horizon MDPs, Markovian policies are suboptimal, a compact (previous-best, current-cumulative) state augmentation restores optimality, exact planning is NP-hard but an FPTAS exists, and the minimax generative-model sample complexity is Θ(KH³SA/ε²).
MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling
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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.
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Theoretical Foundations of $\max$@$k$ Reinforcement Learning
For max@k (best-of-K) finite-horizon MDPs, Markovian policies are suboptimal, a compact (previous-best, current-cumulative) state augmentation restores optimality, exact planning is NP-hard but an FPTAS exists, and the minimax generative-model sample complexity is Θ(KH³SA/ε²).