We introduce FormalMATH, a large-scale Lean4 benchmark for formal mathematical reasoning, comprising 5,560 semantically validated Lean4 statements spanning competition-level and undergraduate mathematics. To improve reliability, FormalMATH is constructed through a human-in-the-loop pipeline that combines type checking, LLM-based semantic verification against the informal–formal statements alignment, negation-based disproving, and expert review. We conduct a systematic evaluation of frontier theorem provers under test-time scaling on FormalMATH. Results show that current provers remain far from solving the benchmark: the strongest model, DeepSeek-Prover-V2 (671B), achieves only 28.31% Pass@32. We further observe substantial domain imbalance across provers, with stronger performance in applied mathematics and algebra but weaker results in discrete mathematics. We identify common failure modes in current provers, including incomplete proofs, weak handling of inequalities, misuse of automation tactics, and redundant hypothesis introduction. Extreme test-time scaling (e.g., Pass@3200) yields only modest gains on a subset of FormalMATH, suggesting that current training paradigms do not equip provers to effectively exploit additional sampling budget. We also observe a counterintuitive negative effect when provers are given human-written natural-language solution references: such references can measurably degrade formal proof generation rather than aid it. Overall, FormalMATH provides a challenging and reliable testbed for measuring progress in formal mathematical reasoning.

Translating natural language mathematical statements into formal, executable code is a fundamental challenge in automated theorem proving. While prior work has focused on generation and compilation success, little attention has been paid to the critic phase—the evaluation of whether generated formalizations truly capture the semantic intent of the original problem. In this paper, we introduce CriticLean, a novel critic-guided reinforcement learning framework that elevates the role of the critic from a passive validator to an active learning component. Specifically, first, we propose the CriticLeanGPT, trained via supervised fine-tuning and reinforcement learning, to rigorously assess the semantic fidelity of Lean 4 formalizations. Then, we introduce CriticLeanBench, a benchmark designed to measure models’ ability to distinguish semantically correct from incorrect formalizations, and demonstrate that our trained CriticLeanGPT models can significantly outperform strong open- and closed-source baselines. Building on the CriticLean framework, we construct FineLeanCorpus, a dataset comprising over 509K problems that exhibits rich domain diversity, broad difficulty coverage, and high correctness based on human evaluation.Overall, our findings highlight that optimizing the critic phase is essential for producing reliable formalizations and we hope our CriticLean will provide valuable insights for future advances in formal mathematical reasoning.

We present Autonomous Data Selection (AutoDS), a method that leverages base language models as zero-shot “generative classifiers” to automatically curate high-quality mathematical texts. Unlike prior approaches that require human annotations or training a dedicated data filter, AutoDS relies solely on a model’s logits to determine whether a given passage is mathematically informative and educational. By integrating AutoDS into a continual pretraining pipeline, we substantially boost downstream performance on challenging math benchmarks (MATH, GSM8K, and BBH) while using far fewer tokens than previous methods. Empirically, our approach achieves roughly a twofold improvement in pretraining token efficiency over strong baselines, underscoring the potential of self-directed data selection in enhancing mathematical reasoning. We will release our curated dataset to facilitate future research in automated domain-specific data curation.