Large language models can now generate complex, multi-step mathematical proofs, but reliably determining their correctness and localizing early logical errors remains a critical challenge. Existing evaluation approaches largely depend on model-based natural-language judgments, which often overlook local reasoning gaps. While formal theorem provers like Lean offer a path to rigorous verification, using them to evaluate informal text requires solving locality and semantic mismatches: a prover might bypass a local flaw by proving an overly broad target, or validate an auto-formalized statement that drifts from the original mathematical intent. To address this, we introduce FaithSieve, a Lean-assisted framework for fine-grained evaluation of natural-language mathematical proofs. FaithSieve decomposes coarse proof steps into local reasoning units, extracts typed proof obligations, and verifies them through a formal evaluation agent. Formal validation is gated by semantic alignment scoring, so Lean evidence is incorporated only when the formal statement faithfully preserves the context, objects, and logical form of the original claim. We construct two expert-verified datasets, ProofLoc-Olympiad and ProofLoc-University, to benchmark first-error localization. On the 350-problem Olympiad dataset, FaithSieve using a GPT-5.4 backbone achieves 81.43% exact first-error accuracy, outperforming the direct-judging baseline of 72.29%. Furthermore, on the 200-problem ProofLoc-University benchmark spanning six advanced domains, FaithSieve reaches 84.5% exact accuracy, compared to 75.0% for the direct judge. Our work demonstrates that decomposing proofs into fine-grained units and grounding them with faithful formal evidence significantly improves reliable evaluation of natural-language reasoning.
Vincent Cohen-Addad, Dimitris Paparas, Ernest van Wijland +13cs.CL cs.AI
We introduce TCS-Bench, a benchmark for evaluating Large Language Models (LLMs) on research-level Theoretical Computer Science (TCS) proof generation. TCS-Bench consists of theorem-proving tasks from papers published at top theoretical computer science venues (STOC, FOCS, and SODA). Each task provides the necessary context to derive a self-contained proof for a target result. We evaluate state-of-the-art models on this benchmark. We verify the correctness of generated proofs via a verification agent, and further benchmark the verifier against human-expert proof judgements on a set of target statements and generated proofs pairs. Our reference verifier achieves over 90% accuracy on the expert labeled set.
Mikołaj Sienicki, Krzysztof Sienickics.AI cs.GL quant-ph
We present an independent human assessment of the proof developed in Chapter 6 of OpenAI's Ten Advances in Mathematics and Theoretical Computer Science. An initial audit appeared to identify a polarity error in a greedy conditioning lemma. Subsequent examination of the original typeset manuscript showed that this diagnosis resulted from automatic PDF text extraction, which removed an overbar from a mathematical symbol. The alleged error is therefore withdrawn. With the correctly rendered expression restored, we find no confirmed mathematical error in the examined lemma. We retain the broader analysis because it provides an independent reconstruction and assessment of the OpenAI argument and illustrates an important methodological hazard in auditing AI-generated mathematics: errors may arise not only in mathematical reasoning but also in the document-processing pipeline used by human reviewers.
Large language models (LLMs) have achieved remarkable performance on high-school and olympiad-style mathematics, yet their capabilities on advanced mathematics remain poorly understood. Existing benchmarks, however, fall short in both scope and evaluation granularity: they provide limited disciplinary coverage and often rely on final-answer correctness or coarse judgments, leaving the validity of the reasoning process inadequately assessed. To bridge this gap, we introduce AdvancedMathBench, a benchmark suite designed to evaluate advanced mathematical reasoning capabilities. Its core proof-generation benchmark, ProverBench, contains 296 problems spanning undergraduate and doctoral qualifying-exam levels. To provide reliable evaluation of the proofs, we develop a dedicated automatic verification pipeline trained on large-scale expert annotations to produce both correctness verdicts and fine-grained assessments of proof errors, which exhibits strong agreement with human experts on held-out proof trajectories. We further introduce VerifierBench, consisting of 888 model-generated proof trajectories paired with expert ground truth, to evaluate whether models can correctly judge proof validity and provide sound verification rationales. Experiments show that AdvancedMathBench remains challenging for frontier models. On proof generation, the best-performing model, GPT-5.5-xhigh, achieves only 75.8 and 66.1 on the UGD and QE splits, respectively, indicating substantial room for improvement on advanced mathematical proof construction. On proof verification, the best model attains a Balanced F1 of only 65.1, and models generally exhibit low true negative rates, suggesting that critical error detection remains a major bottleneck.
Jimmy Xin, Alex Schneidman, Chris Cummins +3cs.LO cs.AI
We present AXLE (Axiom Lean Engine), a cloud service for Lean 4 proof manipulation, extraction, and verification. Recent progress in AI for mathematics -- reinforcement learning pipelines, agentic proving workflows, dataset curation -- demands Lean 4 tooling that scales to millions of requests while remaining correct and robust; existing infrastructure offers parallel compilation but not scalable proof verification, higher-level proof manipulation, multi-version support, or per-request isolation at the throughput modern AI workflows require. AXLE provides 14 Lean 4 metaprogramming tools spanning strict proof verification, declaration metadata extraction, semantic source manipulation, deterministic proof repair and simplification, and lemma extraction. The service runs as a multi-tenant cloud deployment with per-request isolation and concurrent support for multiple Lean 4 and Mathlib versions, accessible via a Python SDK, command-line interface, web UI, MCP server, and raw HTTP API. AXLE is publicly available and free to use at https://axle.axiommath.ai and via the axiom-axle PyPI package, with no local Lean 4 installation required. It has served over 500 million requests to date and is the underlying infrastructure for Axiom Math's proving efforts, including its 12/12 score on the 2025 Putnam competition.
Large language models (LLMs) are increasingly capable of mathematical problem solving and can even assist with research-level proofs, yet we still lack a scalable and reproducible way to measure step-level reasoning in long proofs across diverse sources. This evaluation gap limits trustworthy AI assistance in proof-certified scientific progress. Existing evaluations often emphasize final answers or rely on costly expert grading, while end-to-end proof generation remains open-ended and hard to verify automatically. We introduce Mask-Proof, a pipeline that turns real proofs into automatically checkable masked-step tasks. It masks key formula steps, provides the necessary surrounding context, and evaluates model reconstructions with an LLM-based equivalence judge using repeated votes for stability. The resulting Mask-ProofBench contains 292 curated problems across diverse research areas. Experiments with 17 models show that reasoning-enhanced models outperform standard models by 12% to 27%. Our evaluator achieves 96.8% agreement with expert annotators, enabling faithful, reproducible, and comparable measurement of step-level mathematical reasoning. Benchmark, annotations, and code are available at https://github.com/weating/Mask-Proof.
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.
Large Language Models (LLMs) struggle to rigorously verify complex mathematical proofs. Standard global evaluation approaches suffer from "context poisoning," in which superficially plausible statements mask subtle logical flaws, leading to hallucination or over-skepticism. To address this, we shift from global evaluation to strict step-level verification: our framework maintains detailed context for each deduction step and strictly constrains the sources of applied theorems. We evaluate on a carefully curated adversarial diagnostic suite of research-level proofs drawn from the FirstProof challenge. A systematic ablation study demonstrates that these deductive constraints are indispensable, as unconstrained global prompting consistently fails to localize subtle logical errors. Beyond outperforming global evaluation, our approach fundamentally alters the failure taxonomy. Error analysis reveals that, rather than exhibiting severe logical hallucinations, remaining rejections are primarily instances of "pedantic hyper-rigor" stemming from unstated domain conventions, effectively exposing implicit ambiguities within the expert benchmark itself. Our findings suggest that prompting agents to organize their verification notes in a cautious, human-mathematician-like manner can substantially improve their ability to distinguish rigorous proofs from flawed ones, with the potential to strengthen agentic reasoning on frontier mathematical concepts that the base model does not already know well, and to lay a theoretical foundation for future automated proof-review systems. Code and prompts are available at GitHub.
Combinatorics is central to Olympiad-level mathematical problem solving, requiring deep discrete reasoning, creative constructions, and rigorous structural insight. Recent evidence suggests that even today's strongest frontier models remain uneven on Olympiad combinatorics, revealing a gap in creative mathematical reasoning. We introduce ComBench, an Olympiad-level combinatorics benchmark for evaluating and diagnosing the combinatorial reasoning capabilities of large language models. ComBench contains 100 human-annotated competition-level problems organized around two complementary settings: analysis-centric problems, which primarily require rigorous mathematical arguments, and construction-centric problems, which require explicit constructions in addition to correctness justifications. The evaluation protocol combines rubric-guided proof grading with deterministic construction verification, exposing cases where proof quality and construction validity diverge. Experiments on frontier open- and closed-source models show that ComBench is far from saturated: the strongest model reaches 65.4% overall Avg. and 75.3% overall Best@4. We further find that Rigorous Proof Reasoning and Constructive Realization are distinct capabilities: Kimi-K2.6 trails GPT-5.5 on analysis-centric proof grading but surpasses it on construction-centric Best@4, while Existence and Construction problems remain consistently hardest across representative frontier models.