MaxProof: Robust Test-Time Scaling for Proofs

• MaxProof is a test-time scaling framework that generates, verifies, and refines automated proofs to meet Olympiad-level precision.
• It employs a defense-in-depth verifier with multi-layer filtering and group-based credit assignment to drastically reduce false positives.
• Empirical results show significant score improvements on IMO and USAMO contests, converting best@K proof generation into reliable pass@1 outcomes.

MaxProof is a population-level test-time scaling framework designed to enable robust, competition-level automated mathematical proof, focused on performance at International Mathematical Olympiad (IMO) and USA Mathematical Olympiad (USAMO) standards. It operates as a wrapper around the MiniMax-M3 model—an architecture unifying proof-generating, verifying, and repair capabilities—elevating strong best@K proof generation to pass@1 reliability, a critical requirement in domains where a single misstep invalidates an entire mathematical argument (Chen et al., 11 Jun 2026).

1. Objectives and Motivations

Mathematical proof at the Olympiad level necessitates extremely low error tolerance and tight logical coupling across long reasoning chains. MaxProof addresses the discrepancy between models that can generate correct proofs among a group (best@K) and those capable of producing a single, unassailably valid proof (pass@1). The framework's goal is to convert non-trivial proof generation performance into robust pass@1 success by:

• Generating a population of diverse candidate proofs.
• Verifying and critiquing each candidate using a defense-in-depth verifier engineered for low false-positive rates.
• Iteratively refining promising candidates under model-generated critiques.
• Selecting the final proof via a pairwise tournament—mitigating noise or systematic bias in the verifier.

MaxProof operates agnostically as a test-time augmentation, wrapping around the M3 model (or comparable architectures) and producing substantial improvements in gold-medal–level math contest performance—35/42 on IMO 2025 and 36/42 on USAMO 2026—without altering the base model itself (Chen et al., 11 Jun 2026).

2. M3 Model Capabilities and Architecture

The M3 model fuses three distinct specialist roles into unified weights:

• Proof Expert: Trained via long-horizon RL under a learned generative verifier. The policy π_θ samples groups of full-length proofs (y_i), each spanning thousands of tokens. The generative verifier provides a reward R_i ∈ [0,7], evaluating the argument’s correctness relative to the rubric.
• Verifier Expert: Distilled from an external multi-judge system. Given a (problem, proof) pair, it outputs step-by-step assessment, a localized error list, and a verdict ({no_errors, minor_gaps, has_errors, fundamentally_wrong}), trained using the pessimistic-min teacher signal derived from the Proof Expert’s RL trajectory.
• Fixer Expert: Conditioned on critique, takes a (problem, flawed_proof, verification_analysis) triplet and produces a corrected proof. Fine-tuned using rejection sampling so that only verifier-approved (“no_errors”) repairs are retained.

These specialists are merged into the single M3 release model, facilitating generator, verifier, and refiner roles in low-latency deployment (Chen et al., 11 Jun 2026).

3. Generative-Verifier Reinforcement Learning Paradigm

The M3 Proof Expert is trained under a generative-verifier RL protocol with group-based credit assignment and tight variance control:

• The policy π_θ_old samples G proofs y₁,...,y_G for each problem p, each scored by the generative verifier with R_i∈[0,7].
• For cases with reward variance above a threshold (std({R_i})>τ_std), group-normed advantages are computed:

$$μ_R=\frac{1}{G} \sum_i R_i,\quad σ_R = \sqrt{\frac{1}{G}\sum_i(R_i-μ_R)^2},\quad A_i = \frac{R_i-μ_R}{σ_R+\epsilon}$$

• The CISPO (clipped incentive-structured policy optimization) objective implements clipped importance sampling for stable policy updates:

$$ρ_{i,t}(θ)=\frac{π_θ(y_{i,t}|\ldots)}{π_{θ_{\text{old}}}(y_{i,t}|\ldots)}$$

$\bar{ρ}_{i,t}=\mathrm{clip}(ρ_{i,t},1-ϵ,1+ϵ)$

$$J(θ)=\mathbb{E}\left[\mathbf{1}\{σ_R>τ_{std}\} \cdot \frac{1}{\sum_i T_i}\sum_{i,t} \mathrm{sg}(\bar{ρ}_{i,t})A_i\log π_θ(y_{i,t}|\ldots)\right]$$

The std-threshold filter $\mathbf{1}\{σ_R>τ_{std}\}$ discards low-diversity samples. This approach mitigates the risk of reward hacking by employing a multi-layer, defense-in-depth verifier strategy, described below (Chen et al., 11 Jun 2026).

4. Defense-in-Depth Generative Verifier

To suppress false positives during RL and ensure training signal integrity, the generative verifier employs four sequential filters:

Layer	Description	Effect
Bad-case filtering	Discards empty proofs, unclosed blocks, boilerplate loops, length violations	Immediate fail (0)
Solution normalization	Strips headers/verification blocks to reduce format bias	Standardization
Multi-judge scoring	Runs 3 judges: 2 rubric-based, 1 open-ended error finder	Redundant assessment
Pessimistic aggregation	Final reward = min(judge_score), favoring false negatives	Low false-positive rate

This defense-in-depth pipeline prioritizes minimizing spurious reward (false-positive proofs) even at the expense of some missed positives (false negatives), a critical quality in proof verification RL (Chen et al., 11 Jun 2026).

5. Population-Level Test-Time Scaling: The MaxProof Algorithm

At test time, MaxProof applies a structured population search loop encompassing the following roles and workflow:

• Generator (G): Samples an initial population of N candidate proofs, c_i ∼ G(p).
• Verifier (V): Each candidate is scored K_ver times; fitness is the pessimistic-min of those scores. Verifier also produces a localized critique a_i.
• Refiner (F): Invoked in two modes on selected parent candidates:
  • PATCH: Targeted repair of verifier-identified errors.
  • REWRITE: Full re-generation along a different approach.
• Ranker (Q): Final selection via pairwise tournament among top-K archived candidates, each match decided by K_ranker votes.

The search progresses for R rounds or halts early once ≥2 proofs achieve perfect score (7/7), reducing the risk of picking a spurious solution. Parent candidate selection enforces diversity, especially by forbidding parents with long common prefixes. The use of both PATCH and REWRITE strikes a balance between local exploitation and global exploration. Final selection via tournament mitigates residual verifier noise (Chen et al., 11 Jun 2026).

6. Empirical Performance and Analysis

The empirical evaluation demonstrates significant improvements from MaxProof:

• On Standalone (non-wrapped) M3:
  • IMOProofBench: 67.4/100
  • IMOAnswerBench: 81.6/100
• On Mathematical Contest Problems:
  • M3 one-shot: 27/42 (IMO 2025), 26/42 (USAMO 2026)
  • M3+MaxProof: 35/42 (IMO), 36/42 (USAMO)
  • These reflect +8 (IMO) and +10 (USAMO) point increases, converting best@32 capability into stable pass@1 performance.

Analysis of selection losses shows that selection errors (oracle-best vs. self-pick) are largely limited, with only one significant loss (USAMO P2, 4 points) attributed to ranker ambiguity. Most problems reach 7/7-grade solutions by round 4, and later rounds occasionally yield novel proof ideas, suggesting possible benefits from sustained search (Chen et al., 11 Jun 2026).

7. Limitations, Key Takeaways, and Future Directions

Critical insights from MaxProof include:

• Defense-in-depth verifier loops are essential to guard against reward hacking in proof-focused RL.
• The unification of generator, verifier, and fixer into a distilled model enables efficient low-latency deployment.
• Population search and tournament selection mechanisms are effective in boosting best@K models to meet rigorous pass@1 standards required for Olympiad-level proof.

Prospective directions for future research involve further calibration of the ranker to reduce final-selection loss, exploration of adaptive population and round budgets, integration of symbolic reasoning engines or formula interpreters to augment model reasoning, and possible extensions to collaborative or interactive theorem-proving paradigms (Chen et al., 11 Jun 2026).