Pessimistic Verification for Open-Ended Math Questions

Automatic verification is a critical component in building math-solving agents and reinforcement learning, yet it often falls short in generalizability, performance, and cost-efficiency. Identifying that the primary bottleneck of verification lies in error detection capability, we propose pessimistic verification, a paradigm of agentic workflows that rejects a solution if any of multiple parallel verifiers identifies a flaw. We further introduce progressive pessimistic verification, which employs fine-grained proof decomposition to significantly enhance verification accuracy and efficiency. Our approach surpasses the performance and token efficiency of extended long chain-of-thought (long CoT) and mainstream verification workflows, crucially, our analysis reveals that existing benchmarks underestimate its effectiveness on stronger models due to inherent annotation errors. To further validate the effectiveness of our method, we applied a verification-based solving workflow on the IMO 2025 and MathArena Apex 2025 datasets, where the workflow with progressive pessimistic verification exhibits remarkable improvements in both efficiency and accuracy on highly challenging contest-level math problems with state-of-the-art models.

Mathematical reasoning systems are becoming increasingly powerful, but they still struggle to reliably check whether a proof or solution is actually correct. This is especially difficult for open-ended math problems, where answers cannot be verified with simple rules or calculators. Current AI verification methods are often unreliable, expensive, or require many repeated checks. In this work, we propose a new approach called pessimistic verification. Instead of trusting a proof unless many checks fail, our method rejects a proof as soon as any verifier discovers a convincing mistake. We further introduce a more efficient strategy that breaks long proofs into smaller parts and checks them progressively, allowing the system to focus more carefully on detailed reasoning steps. Across several challenging mathematics benchmarks, our method improves both verification accuracy and efficiency compared to existing approaches, including methods that rely on very long reasoning chains. We also show that some current evaluation datasets contain annotation mistakes that underestimate the abilities of modern AI systems. Finally, we apply our verification method to difficult olympiad-level math problems, where it significantly improves the performance of AI problem-solving systems while reducing computational cost. Our findings suggest that better error detection may be a key ingredient for building more reliable and capable reasoning agents.