Pessimistic Verification for Open Ended Math Questions

While large language models have made significant progress in mathematical reasoning, they remain unreliable at judging the correctness of their own solutions. Existing approaches that equip models with self-verification typically treat solution generation and verification as two separate tasks, leading to substantially increased training time. In this paper, we propose FABSVer, which fuses these two tasks into a single generation pass, dramatically reducing training overhead while jointly optimizing both capabilities. We further identify a convergence bottleneck both theoretically and empirically: as training progresses, the reward reaches a plateau because the policy is constrained by a fixed reference model. To overcome this, we introduce Dynamic Reference Model Update (DRMU), which raises the reward ceiling and enables sustained reward growth. Extensive experiments on math benchmarks demonstrate that FABSVer achieves superior self-verification and reasoning performance across three model scales, while requiring only 51%--71% of the training time of existing methods. Analysis further reveals distinct learning phases in how models acquire self-verification, and that the gap between verify and answer rewards shrinks noticeably as model size increases.

Can AI make progress on important, unsolved mathematical problems? Large language models are now capable of sophisticated mathematical and scientific reasoning, but whether they can perform novel research is still widely debated and underexplored. We introduce HorizonMath, a benchmark of 113 predominantly unsolved problems spanning eight domains in mathematics and the mathematical sciences, paired with an open-source evaluation framework for automated verification. Our benchmark targets the generator-verifier gap: problems where discovery is hard and requires meaningful mathematical insight, but verification is computationally straightforward. This contrasts with most existing research-level benchmarks, which instead rely on formal proof verification or manual review, both of which are expensive to scale. Because these solutions are unknown, HorizonMath is resistant to data contamination, and most state-of-the-art models score under 10%. Using this framework, we identify six novel solutions to research problems that either resolve previously open questions or improve on the best-known published results, with GPT-5.4 Pro and GPT-5.6 Sol each discovering three of these solutions. Across seven frontier model families, reasoning efficiency and behavior also vary substantially. We release HorizonMath as an open challenge and a growing community resource, where each verified solution is a candidate contribution to the mathematical literature.

Large language models have demonstrated strong mathematical problem-solving capabilities, yet reliably verifying their candidate answers remains challenging. Existing representative methods mainly revise outputs through natural-language reflection or assist verification by directly generating verification programs; the former may not reliably support exact computation, whereas the latter prematurely couples mathematical modeling with low-level implementation. We propose AMTFV (Agentic Mathematical Tool-Flow Verification). By introducing Mathematical Tool Flow (MTF) as an interrupt--execute--resume interface, AMTFV decouples verification modeling from concrete execution and supports exact computation through a mathematical toolbox. Specifically, the verification agent first constructs a verification workflow, encodes the mathematical objects and computational intent requiring reliable execution in an MTF request, and sends it to the mathematical toolbox agent. The latter parses the request, generates executable calls, and dispatches them to the backend for exact computation. Tool outputs then support candidate-answer adjudication, answer revision, and verification-workflow revision. We evaluate AMTFV on five challenging mathematical reasoning datasets with seven model configurations from DeepSeek, GPT, and Gemini. Experimental results show that AMTFV outperforms the representative baselines evaluated in this study overall; under an individual model configuration, it improves average accuracy over the strongest baseline by up to 8.3 percentage points, with larger gains on samples of medium and high verification complexity.