LLM Mathematical Reasoning

LLM: Large Language Model

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Overview

The field investigates whether language models can carry out arithmetic, multi-step problem solving, and proof construction reliably—not just produce correct-looking answers. Common approaches include chain-of-thought prompting, supervised and reinforcement learning, test-time search and voting, tool use with code or computer algebra, and formal verification with proof assistants such as Lean. Evaluation spans word-problem and contest benchmarks such as GSM8K, MATH, and AIME, as well as formal proof tasks like miniF2F; newer tests probe robustness to equivalent transformations, multilingual inputs, and research-level problems. Models can perform strongly on familiar contest tasks, but generalization, faithful reasoning, proof quality, and reliable verification remain unresolved tensions, especially on open-ended research mathematics.

Latest papers

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273 papers20 papers in the last four weeks, down 5% on the four weeks before. 0.1% of all new papers.

Mar 3, 2026

Preventing Curriculum Collapse in Self-Evolving Reasoning Systems

Self-evolving reasoning frameworks let LLMs improve their reasoning capabilities by iteratively generating and solving problems without external supervision, using verifiable rewards. Ideally, such systems are expected to explore a diverse problem space and propose new challenges of high learning value. While prior work has largely focused on solver-side optimisation and verification, recent evidence suggests that self-evolving systems can exhibit diversity collapse in posing new problems after just a few iterations, even when surface-level variation is preserved. We introduce Prism, a question-centric self-evolution method that directly tackles this collapse. Prism defines a persistent diversity signal over an embedding-induced semantic partition of mathematical problems and uses it to encourage balanced exploration of underrepresented regions across iterations. This coverage signal is combined with a Zone-of-Proximal-Development (ZPD) gate to preserve edge-of-solvability difficulty. Evaluated on seven widely used mathematical reasoning benchmarks against five self-evolving baselines, Prism achieves the highest accuracy on six out of seven tasks, achieving gains of +3.98 absolute points over R-Zero on AMC and +3.68 on Minerva Math. Prism also generates semantically diverse and challenging questions across iterations, resulting in the construction of the Prism-Math dataset comprising 100k mathematical questions. These results demonstrate that cross-iteration semantic coverage is a high-leverage and under-explored axis for building more capable self-evolving reasoners. We release the code, dataset, and models to facilitate further research.

Vaibhav Mishra

Mathematical Problem GenerationLanguage Model Self-ImprovementSelf-Evolving AgentsLLM Mathematical Reasoning
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Mar 3, 2026

When Shallow Wins: Silent Failures and the Depth-Accuracy Paradox in Latent Reasoning

Mathematical reasoning models are widely deployed in education, automated tutoring, and decision support systems despite exhibiting fundamental computational instabilities. We demonstrate that state-of-the-art models (Qwen2.5-Math-7B) achieve 61% accuracy through a mixture of reliable and unreliable reasoning pathways: 18.4% of correct predictions employ stable, faithful reasoning while 81.6% emerge through computationally inconsistent pathways. Additionally, 8.8% of all predictions are silent failures -- confident yet incorrect outputs. Through comprehensive analysis using novel faithfulness metrics, we reveal: (1) reasoning quality shows weak negative correlation with correctness (r=-0.21, p=0.002), reflecting a binary classification threshold artifact rather than a monotonic inverse relationship; (2) scaling from 1.5B to 7B parameters (4.7x increase) provides zero accuracy benefit on our evaluated subset (6% of GSM8K), requiring validation on the complete benchmark; and (3) latent reasoning employs diverse computational strategies, with ~20% sharing CoT-like patterns. These findings highlight that benchmark accuracy can mask computational unreliability, demanding evaluation reforms measuring stability beyond single-sample metrics.

Subramanyam Sahoo, Aman Chadha, Vinija Jain +1

Independent · AWS Generative AI Innovation Center, Amazon Web Services · Stanford University +2

LLM Mathematical ReasoningReasoning Consistency in Language ModelsLLM Reliability
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Mar 3, 2026

NeuroProlog: Multi-Task Fine-Tuning for Neurosymbolic Mathematical Reasoning via the Cocktail Effect

Large Language Models (LLMs) achieve strong performance on natural language tasks but remain unreliable in mathematical reasoning, frequently generating fluent yet logically inconsistent solutions. We present NeuroProlog, a neurosymbolic framework that ensures verifiable reasoning by compiling math word problems into executable Prolog programs with formal verification guarantees. We propose a multi-task Cocktail training strategy that jointly optimizes three synergistic objectives in a unified symbolic representation space: (i) mathematical formula-to-rule translation (KB), (ii) natural language-to-program synthesis (SOLVE), and (iii) program-answer alignment. This joint supervision enables positive transfer, where symbolic grounding in formula translation directly improves compositional reasoning capabilities. At inference, we introduce an execution-guided decoding pipeline with fine-grained error taxonomy that enables iterative program repair and quantifies model self-debugging capacity. Evaluation on GSM8K across multiple model scales demonstrates that cocktail training improves accuracy over single-task baselines, with statistically significant gains for most evaluated models. Error analysis reveals scale-associated differences in repair behavior: larger models exhibit more readily correctable errors, whereas smaller models show reduced syntactic errors but persistent semantic failures. These findings suggest that model capacity influences the acquisition of reliable symbolic reasoning and self-correction capabilities.

Pratibha Zunjare, Michael Hsiao

Electrical and Computer Engineering, Virginia Tech, Blacksburg, USA

LLM Mathematical ReasoningNeuro-Symbolic ReasoningProgram SynthesisMulti-Task Learning
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Mar 1, 2026

Learn Hard Problems During RL with Reference Guided Fine-tuning

Reinforcement learning (RL) for mathematical reasoning can suffer from reward sparsity: for challenging problems, LLM fails to sample any correct trajectories, preventing RL from receiving meaningful positive feedback. At the same time, there often exist human-written reference solutions along with the problem (e.g., problems from AoPS), but directly fine-tuning on these solutions offers no benefit because models often cannot imitate human proofs that lie outside their own reasoning distribution. We introduce Reference-Guided Fine-Tuning (ReGFT), a simple and effective method that utilizes human-written reference solutions to synthesize positive trajectories on hard problems and train on them before RL. For each problem, we provide the model with a partial reference solution and let it generate its own reasoning trace, ensuring the resulting trajectories remain in the model's reasoning space while still benefiting from reference guidance. Fine-tuning on these reference-guided trajectories increases the number of solvable problems and produces a checkpoint that receives more positive rewards during RL. Across three benchmarks (AIME24, AIME25, BeyondAIME), ReGFT consistently improves supervised accuracy, accelerates DAPO training, and raises the final performance plateau of RL. Our results show that ReGFT effectively overcomes reward sparsity and unlocks stronger RL-based mathematical reasoning.

Yangzhen Wu, Shanda Li, Zixin Wen +5

UC Berkeley · Carnegie Mellon University · ByteDance Seed

LLM Mathematical ReasoningRL for Language Model ReasoningFine-TuningRL Fine-TuningSparse-Reward RL
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Mar 1, 2026

Evaluating AI Grading on Real-World Handwritten College Mathematics: A Large-Scale Study Toward a Benchmark

Grading in large undergraduate STEM courses often yields minimal feedback due to heavy instructional workloads. We present a large-scale empirical study of AI grading on real, handwritten single-variable calculus work from UC Irvine. Using OCR-conditioned large language models with structured, rubric-guided prompting, our system produces scores and formative feedback for thousands of free-response quiz submissions from nearly 800 students. In a setting with no single ground-truth label, we evaluate performance against official teaching-assistant grades, student surveys, and independent human review, finding strong alignment with TA scoring and a large majority of AI-generated feedback rated as correct or acceptable across quizzes. Beyond calculus, this setting highlights core challenges in OCR-conditioned mathematical reasoning and partial-credit assessment. We analyze key failure modes, propose practical rubric- and prompt-design principles, and introduce a multi-perspective evaluation protocol for reliable, real-course deployment. Building on the dataset and evaluation framework developed here, we outline a standardized benchmark for AI grading of handwritten mathematics to support reproducible comparison and future research.

Zhiqi Yu, Xingping Liu, Haobin Mao +4

UC Irvine Irvine, USA · Independent Shanghai, China · UC Berkeley Berkeley, USA

Automated GradingGenerative AI in EducationLLM Mathematical ReasoningRubric-Based Evaluation
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Feb 27, 2026

LemmaBench: A Live, Research-Level Benchmark to Evaluate LLM Capabilities in Mathematics

We present a new approach for benchmarking Large Language Model (LLM) capabilities on research-level mathematics. Existing benchmarks largely rely on static, hand-curated sets of contest or textbook-style problems as proxies for mathematical research. Instead, we establish an updatable benchmark evaluating models directly on the latest research results in mathematics. This consists of an automatic pipeline that extracts lemmas from arXiv and rewrites them into self-contained statements by making all assumptions and required definitions explicit. It results in a benchmark that can be updated regularly with new problems taken directly from human mathematical research, while previous instances can be used for training without compromising future evaluations. We benchmark current state-of-the-art LLMs, which obtain around 10-15%\% accuracy in theorem proving (pass@1) depending on the model, showing that there is currently a large margin of progression for LLMs to reach human-level proving capabilities in a research context.

Antoine Peyronnet, Fabian Gloeckle, Amaury Hayat

1Ecole Normale Superieure de Rennes · 2Ecole des Ponts Paris · 3Korean Institute for Advanced Study

Mathematical Reasoning BenchmarksAutomated Theorem ProvingLLM Mathematical ReasoningLarge Language Model-Guided Theorem Proving
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Feb 26, 2026

Strategy Executability in Mathematical Reasoning: Leveraging Human-Model Differences for Effective Guidance

Example-based guidance is widely used to improve mathematical reasoning at inference time, yet its effectiveness is highly unstable across problems and models-even when the guidance is correct and problem-relevant. We show that this instability arises from a previously underexplored gap between strategy usage-whether a reasoning strategy appears in successful solutions-and strategy executability-whether the strategy remains effective when instantiated as guidance for a target model. Through a controlled analysis of paired human-written and model-generated solutions, we identify a systematic dissociation between usage and executability: human- and model-derived strategies differ in structured, domain-dependent ways, leading to complementary strengths and consistent source-dependent reversals under guidance. Building on this diagnosis, we propose Selective Strategy Retrieval (SSR), a test-time framework that explicitly models executability by selectively retrieving and combining strategies using empirical, multi-route, source-aware signals. Across multiple mathematical reasoning benchmarks, SSR yields reliable and consistent improvements over direct solving, in-context learning, and single-source guidance, improving accuracy by up to +13+13 points on AIME25 and +5+5 points on Apex for compact reasoning models. Code and benchmark are publicly available at: https://github.com/lwd17/strategy-execute-pipeline.

Weida Liang, Yiyou Sun, Shuyuan Nan +3

National University of Singapore · University of California, Berkeley

LLM Mathematical ReasoningInference-Time Guidance
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Feb 25, 2026

GeoChallenge: A Multi-Answer Multiple-Choice Benchmark for Geometric Reasoning with Diagrams

Evaluating the symbolic reasoning of large language models (LLMs) calls for geometry benchmarks that require multi-step proofs grounded in both text and diagrams. However, existing benchmarks are often limited in scale and rarely provide visually grounded multiple-choice questions, limiting reliable evaluation of complex reasoning. We introduce GeoChallenge, a dataset of 90K automatically generated multiple-choice geometry proof problems, each requiring multi-step reasoning over aligned textual descriptions and diagrams. GeoChallenge provides fine-grained complexity ratings and formal language annotations to enable controlled evaluation. Experiments on multiple advanced LLMs show a clear performance gap between models and humans (the best-performing model, GPT-5-nano, achieves 75.89 exact match vs. 94.74 for humans). Further analysis also reveals three common failure patterns of LLMs: (1) exact match failures under the multiple-choice setting; (2) weak visual reliance; and (3) overextended reasoning without convergence.

Yushun Zhang, Weiping Fu, Zesheng Yang +6

School of Computer Science and Technology, Xi’an Jiaotong University, Xi’an, China · Shaanxi Province Key Laboratory of Big Data Knowledge Engineering, Xi’an, China

Mathematical Reasoning BenchmarksLLM Mathematical ReasoningVisually Grounded Reasoning
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Feb 24, 2026

Aletheia tackles FirstProof autonomously

We report the performance of Aletheia (Feng et al., 2026b), a mathematics research agent powered by Gemini 3 Deep Think, on the inaugural FirstProof challenge. Within the allowed timeframe of the challenge, Aletheia autonomously solved 6 problems (2, 5, 7, 8, 9, 10) out of 10 according to majority expert assessments; we note that experts were not unanimous on Problem 8 (only). For full transparency, we explain our interpretation of FirstProof and disclose details about our experiments as well as our evaluation. Raw prompts and outputs are available at https://github.com/google-deepmind/superhuman/tree/main/aletheia.

Tony Feng, Junehyuk Jung, Sang-hyun Kim +14

Google DeepMind · UC Berkeley · Brown University +7

LLM Mathematical ReasoningAutonomous Research AgentsAI Agent Evaluation
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Feb 24, 2026

Linear Reasoning vs. Proof by Cases: Obstacles for Large Language Models in FOL Problem Solving

To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets. However, most existing datasets primarily focus on linear reasoning, neglecting other parts such as proof by contradiction and proof by cases, which are crucial for investigating LLMs' reasoning abilities. To address this limitation, we first introduce a novel first-order logic (FOL) dataset named PC-FOL, annotated by professional mathematicians, focusing on case-based reasoning problems. All instances in this dataset are equipped with a manually written natural language proof, clearly distinguishing it from conventional linear reasoning datasets. Our experimental results over leading LLMs demonstrate a substantial performance gap between linear reasoning and case-based reasoning problems. To further investigate this phenomenon, we provide a theoretical analysis grounded in graphical model, which provides an explanation for the observed disparity between the two types of reasoning problems. We hope this work can reveal the core challenges in the field of automated natural language mathematical proof generation, paving the way for future research.

Yuliang Ji, Fuchen Shen, Jian Wu +2

Nanjing University of Science and Technology · Westlake University · Zhejiang University

Logical ReasoningMathematical ReasoningLLM Mathematical Reasoning
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Feb 24, 2026

Pipeline for Verifying LLM-Generated Mathematical Solutions

With the growing popularity of Large Reasoning Models and their results in solving mathematical problems, it becomes crucial to measure their capabilities. We introduce a pipeline for both automatic and interactive verification as a more accurate alternative to only checking the answer which is currently the most popular approach for benchmarks. The pipeline can also be used as a generator of correct solutions both in formal and informal languages. 3 AI agents, which can be chosen for the benchmark accordingly, are included in the structure. The key idea is the use of prompts to obtain the solution in the specific form which allows for easier verification using proof assistants and possible use of small models (≤8B\le 8B). Experiments on several datasets suggest low probability of False Positives. The open-source implementation with instructions on setting up a server is available at https://github.com/LogicEnj/lean4_verification_pipeline.

Varvara Sazonova, Dmitri Shmelkin, Stanislav Kikot +1

Moscow State University · Huawei Technologies Co., Ltd

LLM Answer VerificationLLM EvaluationLLM Mathematical ReasoningFormal VerificationInteractive Theorem Proving
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Feb 21, 2026

Early Evidence of Vibe-Proving with Consumer LLMs: A Case Study on Spectral Region Characterization with ChatGPT-5.2 (Thinking)

Large Language Models (LLMs) are increasingly used as scientific copilots, but evidence on their role in research-level mathematics remains limited, especially for workflows accessible to individual researchers. We present early evidence for vibe-proving with a consumer subscription LLM through an auditable case study that resolves Conjecture 20 of Ran and Teng (2024) on the exact nonreal spectral region of a 4-cycle row-stochastic nonnegative matrix family. We analyze seven shareable ChatGPT-5.2 (Thinking) threads and four versioned proof drafts, documenting an iterative pipeline of generate, referee, and repair. The model is most useful for high-level proof search, while human experts remain essential for correctness-critical closure. The final theorem provides necessary and sufficient region conditions and explicit boundary attainment constructions. Beyond the mathematical result, we contribute a process-level characterization of where LLM assistance materially helps and where verification bottlenecks persist, with implications for evaluation of AI-assisted research workflows and for designing human-in-the-loop theorem proving systems.

Brecht Verbeken, Brando Vagenende, Marie-Anne Guerry +2

Data Analytics Lab, Vrije Universiteit Brussel, Pleinlaan 5, 1050 Brussel, Belgium · imec-SMIT, Vrije Universiteit Brussel, Pleinlaan 9, 1050 Brussels, Belgium · School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

Large Language Model-Guided Theorem ProvingLLM Mathematical ReasoningAutomated Theorem ProvingLLM Evaluation
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Feb 19, 2026

Ontology-Guided Neuro-Symbolic Inference: Grounding Language Models with Mathematical Domain Knowledge

Language models exhibit fundamental limitations -- hallucination, brittleness, and lack of formal grounding -- that are particularly problematic in high-stakes specialist fields requiring verifiable reasoning. I investigate whether formal domain ontologies can enhance language model reliability through retrieval-augmented generation. Using mathematics as proof of concept, I implement a neuro-symbolic pipeline leveraging the OpenMath ontology with hybrid retrieval and cross-encoder reranking to inject relevant definitions into model prompts. Evaluation on the MATH benchmark with three open-source models reveals that ontology-guided context improves performance when retrieval quality is high, but irrelevant context actively degrades it -- highlighting both the promise and challenges of neuro-symbolic approaches.

Marcelo Labre

Advanced Institute for Artificial Intelligence (AI2)

Retrieval-Augmented GenerationLLM GroundingKnowledge Augmentation for Language ModelsLLM ReliabilityNeuro-Symbolic ReasoningLLM Mathematical Reasoning
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Feb 8, 2026

Process Rewards for Outcome-Guided Reasoning Steps in LLM Reinforcement Learning

Mathematical reasoning in large language models has improved substantially with reinforcement learning using verifiable rewards, where final answers can be checked automatically and converted into reliable training signals. Most such pipelines optimize outcome correctness only, which yields sparse feedback for long, multi-step solutions and offers limited guidance on intermediate reasoning errors. Recent work therefore introduces process reward models (PRMs) to score intermediate steps and provide denser supervision. In practice, PRM scores are often imperfectly aligned with final correctness and can reward locally fluent reasoning that still ends in an incorrect answer. When optimized as absolute rewards, such signals can amplify fluent failure modes and induce reward hacking. We propose PROGRS, a framework that leverages PRMs while keeping outcome correctness dominant. PROGRS treats process rewards as relative preferences within outcome groups rather than absolute targets. We introduce outcome-conditioned centering, which shifts PRM scores of incorrect trajectories to have zero mean within each prompt group. It removes systematic bias while preserving informative rankings. PROGRS combines a frozen quantile-regression PRM with a multi-scale coherence evaluator. We integrate the resulting centered process bonus into Group Relative Policy Optimization (GRPO) without auxiliary objectives or additional trainable components. Across MATH-500, AMC, AIME, MinervaMath, and OlympiadBench, PROGRS consistently improves Pass@1 over outcome-only baselines and achieves stronger performance with fewer rollouts. These results show that outcome-conditioned centering enables safe and effective use of process rewards for mathematical reasoning.

Mohammad Rezaei, Jens Lehmann, Sahar Vahdati

Faculty of Computer Science, Technical University of Dresden, Dresden, Germany · Amazon Science · Technical University of Dresden, Dresden, Germany +1

LLM Mathematical ReasoningReinforcement LearningProcess Reward ModelsReinforcement Learning with Verifiable Rewards
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Jan 29, 2026

MGSM-Pro: A Simple Strategy for Robust Multilingual Mathematical Reasoning Evaluation

Large language models have made substantial progress in mathematical reasoning. However, benchmark development for multilingual evaluation has lagged behind English in both difficulty and recency. Recently, GSM-Symbolic showed a strong evidence of high variance when models are evaluated on different instantiations of the same question; however, the evaluation was conducted only in English. In this paper, we introduce MGSM-Pro, an extension of MGSM dataset with GSM-Symbolic approach. Our dataset provides five instantiations per MGSM question by varying names, digits and irrelevant context. Evaluations across nine languages reveal that many low-resource languages suffer large performance drops when tested on digit instantiations different from those in the original test set. We further find that models robustness in HRL setting do not necessarily translate to LRL. Moreover, proprietary models, such as Gemini 2.5 Flash and GPT-4.1 are less robust to digit, whereas Gemini 3.0 Pro is more robust. Among open models, GPT-OSS 120B and DeepSeek v3 show stronger robustness. Based on these findings, we recommend evaluating each problem using at least five digit-varying instantiations to obtain a more robust and realistic assessment of math reasoning.

Tianyi Xu, Kosei Uemura, Alfred Malengo Kondoro +6

McGill University · Mila-Quebec AI Institute · University of Toronto +7

Numerical Reasoning in Language ModelsMathematical Reasoning BenchmarksMultilingual Language Model EvaluationLLM ReliabilityLanguage Model RobustnessLLM Mathematical Reasoning
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Jan 21, 2026

The Effect of Scripts and Formats on LLM Numeracy

Large language models (LLMs) have achieved impressive proficiency in basic arithmetic, rivaling human-level performance on standard numerical tasks. However, little attention has been given to how these models perform when numerical expressions deviate from the prevailing conventions present in their training corpora. In this work, we investigate numerical reasoning across a wide range of numeral scripts and formats. We show that LLM accuracy drops substantially when numerical inputs are rendered in underrepresented scripts or formats, despite the underlying mathematical reasoning being identical. We further demonstrate that targeted prompting strategies, such as few-shot prompting and explicit numeral mapping, can greatly narrow this gap. Our findings highlight an overlooked challenge in multilingual numerical reasoning and provide actionable insights for working with LLMs to reliably interpret, manipulate, and generate numbers across diverse numeral scripts and formatting styles.

Varshini Reddy, Craig W. Schmidt, Seth Ebner +3

Kensho Technologies · Ben-Gurion University · MIT Cambridge, MA

Numerical Reasoning in Language ModelsMultilingual Language Model EvaluationLanguage Model RobustnessLLM Mathematical Reasoning
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Dec 11, 2025

MINIF2F-DAFNY: LLM-Guided Mathematical Theorem Proving via Auto-Active Verification

LLMs excel at reasoning, but validating their steps remains challenging. Formal verification offers a solution through mechanically checkable proofs. Interactive theorem provers (ITPs) dominate mathematical reasoning but require detailed low-level proof steps, while auto-active verifiers offer automation but focus on software verification. Recent work has begun bridging this divide by evaluating LLMs for software verification in ITPs, but the complementary direction, LLMs for mathematical theorem proving in auto-active verifiers, remains unexplored. We present MINIF2F-DAFNY, the first translation of the widely-used mathematical benchmark miniF2F to an auto-active verifier: Dafny. We find that Dafny's automation alone solves 39-44% of problems with empty proofs, whereas many require substantial proof guidance in ITPs. We evaluate 8 off-the-shelf LLMs on proof generation, with the best model (Claude Opus 4.6) achieving 62.7% cumulative pass@4 on the full test set, improving over the 38.9% empty-proof baseline by 23.8 percentage points. These results show that auto-active verification offers a complementary empirical setting for AI-assisted mathematical reasoning, where LLMs provide high-level guidance while SMT automation handles low-level details. Our benchmark and evaluation infrastructure are publicly available on https://github.com/dafny-lang/miniF2F.

Mantas Baksys, Stefan Zetzsche, Olivier Bouissou +1

University of Cambridge, Cambridge, UK · Amazon Web Services, London, UK · Amazon Web Services, Boston, USA

Automated Theorem ProvingFormal VerificationLLM Mathematical Reasoning
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Dec 2, 2025

ScalePRM: Training Process Reward Models by Scaling Verification Compute Without Ground Truth

Training process reward models (PRMs) requires step-level correctness labels, obtained either through expensive human annotation or by relying on ground-truth answers, limiting the ability to scale process-level supervision. We propose ScalePRM, which scales verification compute as an alternative: given a problem and a candidate solution, we generate multiple independent verifications of each reasoning step and aggregate their judgments to produce synthetic step-level labels without ground truth. We explore two representative inference-time scaling strategies, parallel scaling through self-consistency and sequential scaling through meta-critique, and train generative PRMs on the resulting synthetic data. On ProcessBench, a benchmark for identifying erroneous steps in mathematical reasoning, PRMs trained on step-level self-consistency data achieve 67.5 F1, surpassing reference-guided training with ground-truth access (66.4 F1) and GPT-4o as a critic (61.9 F1). When deployed as reward signals in RL training with Qwen2.5-Math-7B, our best PRM achieves 47.4% average accuracy across six mathematical reasoning benchmarks, outperforming ground-truth-based RLVR (43.9%). We also identify and address reward exploitation patterns unique to generative PRM-based RL. Our results demonstrate that scaling verification compute is a viable alternative to ground-truth supervision for training process reward models.

Salman Rahman, Sruthi Gorantla, Arpit Gupta +3

2UCLA · Work done while as an intern at Amazon AGI. · 1Amazon AGI

Process Reward ModelsReinforcement Learning with Verifiable RewardsLLM Mathematical Reasoning
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Nov 26, 2025

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. Code is available at https://github.com/THUNLP-MT/pverify.

Yanxing Huang, Zihan Tang, Zejin Lin +2

Qiuzhen College, Tsinghua University, Beijing, China · Institute for AI Industry Research (AIR), Tsinghua University, Beijing, China · Dept. of Comp. Sci & Tech., Institute for AI, Tsinghua University, Beijing, China.

LLM Answer VerificationLLM Mathematical Reasoning
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Nov 13, 2025

Learning to Pose Problems: Reasoning-Driven and Solver-Adaptive Data Synthesis

Data synthesis for training large reasoning models offers a scalable alternative to limited, human-curated datasets, enabling the creation of high-quality data. However, existing approaches face several challenges: (i) indiscriminate generation that ignores the solver's ability and yields low-value problems, or reliance on complex data pipelines to balance problem difficulty; and (ii) a lack of reasoning in problem generation, leading to shallow problem variants. In this paper, we develop a problem generator that reasons explicitly to plan problem directions before synthesis and adapts difficulty to the solver's ability. We recover problem-design CoT from human-authored multi-part questions, using later subquestions as targets to infer the design rationale. These traces initialize the generator, which is then optimized with a seed-conditioned reward based on solver answer consistency. The reward guides difficulty adjustment relative to each seed while favoring problems near the solver's competence boundary. Across 10 mathematical and general reasoning benchmarks, our method outperforms seed-only training by 2.69 and 3.39 percentage points after one and three rounds, respectively. The framework also extends to vision-language models. Our code is publicly available at https://github.com/WalkerWorldPeace/Reasoning-Synthesis.

Yongxian Wei, Yilin Zhao, Li Shen +7

Tsinghua University · Tencent · Sun Yat-sen University +1

Synthetic Data GenerationSynthetic Data Generation for Language ModelsLLM Mathematical Reasoning
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Nov 1, 2025

OpenSIR: Open-Ended Self-Improving Reasoner

Recent advances in large language model (LLM) reasoning through reinforcement learning rely on annotated datasets for verifiable rewards, which may limit models' ability to surpass human-level performance. While self-play offers a promising alternative, prior methods yield only marginal or even negative gains on post-trained models because they generate problems that cluster around familiar concepts rather than discovering novel ones. We introduce Open-Ended Self-Improving Reasoner (OpenSIR), a self-play framework in which a single LLM alternates teacher and student roles to generate and solve novel problems without external verifiers or annotated data. Starting from a single seed problem, OpenSIR sustains open-ended exploration through diversity rewards that push the model toward unfamiliar concepts and difficulty calibration that keeps problems learnable. Across seven math benchmarks, OpenSIR consistently improves all models, averaging +3.6 points on instruction models and +3.1 on reasoning models, while recent self-play baselines yield marginal or even negative gains; starting from a single trivial seed, it also surpasses GRPO baselines trained on over 7K annotated examples. Despite training only on self-generated math, OpenSIR is the only self-play method that transfers to general reasoning, improving by at least +4.4 points on reasoning models.

Wai-Chung Kwan, Joshua Ong Jun Leang, Pavlos Vougiouklis +3

University of Edinburgh · Imperial College London · Huawei Technologies Research & Development (UK) Limited +1

Self-Play RLRL for Language Model ReasoningLanguage Model Self-ImprovementLLM Mathematical ReasoningLarge Reasoning Models
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Oct 13, 2025

TopoAlign: A Framework for Aligning Code to Math via Topological Decomposition

Large Language Models (LLMs) excel at both informal and formal (e.g. Lean 4) mathematical reasoning but still struggle with autoformalisation, the task of transforming informal into formal mathematical statements. Yet, the performance of current Math LLMs is constrained by the scarcity of large-scale corpora, particularly those containing pairs of informal and formal statements. Interestingly, the formal languages used in autoformalisation share structural similarities with programming languages, and code data is available at scale. However, current models trained on code do not transfer effectively to formal math, due to structural and syntactic differences between them. To address this, we propose TopoAlign, a framework that unlocks widely available code repositories as training resources for Math LLMs. TopoAlign decomposes code into docstrings, main functions, and dependency functions, and reassembles these components into analogues that structurally mirror formal statements. We train three state-of-the-art models, DeepSeek-Math, Qwen-3 and Herald, and evaluate them on the MiniF2F, Putnam, and ProofNet benchmarks. TopoAlign provides substantial gains for DeepSeek-Math, improving performance by 17.77% on BEq@10 and 68.82% on typecheck@10, and also measurably improves Herald by 0.12% on BEq@10 and 1.09% on typecheck@10 despite introducing no new mathematical knowledge.

Yupei Li, Philipp Borchert, Gerasimos Lampouras

1Imperial College London, UK · *Work conducted during an internship at Huawei Noah’s Ark Lab, London. · 2Huawei Noah’s Ark Lab, London, UK

Automated Theorem ProvingAutoformalizationLLM Mathematical Reasoning
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Oct 6, 2025

Aria: An Agent For Retrieval and Iterative Auto-Formalization via Dependency Graph

Accurate auto-formalization of theorem statements is essential for advancing automated discovery and verification of research-level mathematics, yet remains a major bottleneck for LLMs due to hallucinations, semantic mismatches, and their inability to synthesize new definitions. To tackle these issues, we present Aria (Agent for Retrieval and Iterative Autoformalization), a system for conjecture-level formalization in Lean that emulates human expert reasoning via a two-phase Graph-of-Thought process: recursively decomposing statements into a dependency graph and then constructing formalizations from grounded concepts. To ensure semantic correctness, we introduce AriaScorer, a checker that retrieves definitions from Mathlib for term-level grounding, enabling rigorous and reliable verification. We evaluate Aria on diverse benchmarks. On ProofNet, it achieves 91.6% compilation success rate and 68.5% final accuracy, surpassing previous methods. On FATE-X, a suite of challenging algebra problems from research literature, it outperforms the best baseline with 44.0% vs. 24.0% final accuracy. On a dataset of homological conjectures, Aria reaches 42.9% final accuracy while all other models score 0%.

Hanyu Wang, Ruohan Xie, Yutong Wang +3

Peking University · Institute for Artificial Intelligence, Peking University · IQuest Research +4

Lean Theorem ProvingAutoformalizationLLM Mathematical Reasoning
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Sep 30, 2025

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

Johannes Schmitt, Gergely Bérczi, Jasper Dekoninck +57

Mathematical Reasoning BenchmarksLLM EvaluationBenchmark DesignMathematical ReasoningLLM Mathematical Reasoning
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Sep 27, 2025

Learning How to Use Tools, Not Just When: Pattern-Aware Tool-Integrated Reasoning

Tool-integrated reasoning (TIR) has become a key approach for improving large reasoning models (LRMs) on complex problems. Prior work has mainly studied when to invoke tools, while overlooking how tools are applied. We identify two common patterns: a calculator pattern that uses code for direct computation, and an algorithmic pattern that encodes problems as programs. Misaligned choices often cause failures even when reasoning is sound. We propose a two-stage framework that first builds code competence from both patterns and then aligns pattern selection with teacher preferences. Across challenging math datasets, our pattern-aware method substantially improves both code usage and accuracy, for instance raising Code@1 on MATH500 from 64.0% to 70.5% and on AIME24 from 26.7% to 50.0%. These gains highlight the effectiveness of a pattern-aware approach for tool-integrated reasoning.

Ningning Xu, Yuxuan Jiang, Shubhashis Roy Dipta +1

University of Georgia · University of Maryland, Baltimore County · 3The University of Hong Kong

Tool-Integrated ReasoningLLM Tool UseAlgorithmic ReasoningLLM Mathematical ReasoningLarge Reasoning Models
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Sep 15, 2025

Generative AI performance in core undergraduate mathematics: a curriculum-level case study

Generative artificial intelligence (GenAI) tools such as OpenAI's ChatGPT are transforming the educational landscape, prompting reconsideration of traditional assessment practices. In parallel, universities are exploring alternatives to in-person, closed-book examinations, raising concerns about academic integrity and pedagogical alignment in uninvigilated settings. This study systematically investigates the performance of GenAI on typical mathematics questions from across a first-year mathematics curriculum. Adopting an empirical approach and utilising current examination questions as a proxy for course content, we generate, transcribe, and blind-mark GenAI submissions to eight undergraduate mathematics assessments, spanning the entirety of the first-year curriculum. By combining independent GenAI responses to individual questions, we enable a meaningful evaluation of GenAI performance, both at the level of modules and across the first-year curriculum. We find that GenAI attainment is at the level of a first-class degree, though current performance can vary between modules. Further, we find that GenAI performance is remarkably consistent when viewed across the entire curriculum, significantly more so than that of students in invigilated examinations. Our findings evidence the pressing need for redesigning assessments in mathematics in the era of generative artificial intelligence.

Benjamin J. Walker, Nikoleta Kalaydzhieva, Beatriz Navarro Lameda +1

Department of Mathematics, University College London, Gordon Street, WC1H 0AY, London, UK

Educational AssessmentGenerative AI in EducationLLM Mathematical ReasoningGenerative AI Evaluation
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Aug 12, 2025

An Investigation of Robustness of LLMs in Mathematical Reasoning: Benchmarking with Mathematically-Equivalent Transformation of Advanced Mathematical Problems

Frontier large language models (LLMs) now reach near-ceiling accuracy on standard mathematical-reasoning benchmarks and gold-medal-level performance at the International Mathematical Olympiad. As these benchmarks saturate and their items leak into training data, a high score no longer shows whether a model reasons robustly or which component of its reasoning fails. To evaluate reasoning while keeping results informative and failures diagnosable, we propose GAP (Generalisation-and-Perturbation), a methodology that automatically generates mathematically equivalent variants of existing mathematics problems at scale using two disjoint, interpretable transformations: (1) surface renames, probing the binding between identifiers and latent variable roles, and (2) kernel rewrites, probing whether a high-level proof plan survives a change of mathematical setting. Compared with existing benchmarks, GAP has two key benefits: (1) novel, likely unseen variants mitigate data leakage, and (2) performance across transformation families enables failure diagnosis, each transformation testing a hypothesis about the cause of failure. We instantiate GAP on all 1,051 William Lowell Putnam Competition problems from 1938 to 2024, adding 5,255 unseen variants to form PutnamGAP, a 6,306-item competition-level mathematics corpus and the first public machine-readable dataset from the full Putnam archive. Using PutnamGAP, we evaluated 18 commercial and open-source models spanning sizes and providers. Accuracy drops across all models and variant families, most severely under kernel rewrites. This gap does not close with model strength, suggesting that even the strongest models' dominant weakness is transferring a proof plan to a changed mathematical setting, rather than handling surface changes. Further analysis provides finer failure diagnoses and potentially useful insights for improving LLM reasoning.

Yuren Hao, Xiang Wan, ChengXiang Zhai

Department of Computer Science University of Illinois Urbana–Champaign Urbana, IL, USA · Department of Computer Science Stanford University Stanford, CA, USA

Mathematical ReasoningLLM Mathematical ReasoningMathematical Reasoning Benchmarks
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May 24, 2025

Formally Solving Answer-Construction Problems in Lean

Large language models (LLMs) have achieved remarkable progress in formal mathematical reasoning. Mathematical competition problems fall into two broad types: theorem-proving problems ask for a proof of a fully specified statement, whereas answer-construction problems ask the solver to construct an answer object and prove that it satisfies the stated specification. Existing mathematical reasoning engines mainly target theorem-proving problems, yet answer-construction problems remain less studied. This setting is challenging because model capabilities are misaligned, with general LLMs better suited to answer construction and prover LLMs better suited to proof generation, and because Lean proof checking alone does not rule out inadmissible circular witnesses. To close this gap, we introduce Enumerate-Conjecture-Prove (ECP), a neuro-symbolic framework for solving answer-construction problems in Lean. ECP uses general LLMs to perform bounded enumeration and construct candidate answers, and invokes prover LLMs to produce machine-checked proofs. ECP introduces admissibility checking to ensure that each answer is canonical and does not involve a circular argument. On answer-construction problems from PutnamBench and autoformalized MathArena, ECP formally solves 17/346 PutnamBench instances and 18/75 MathArena instances with admissible answers and proofs, outperforming LLM baselines at aligned inference budgets.

Jialiang Sun, Yuzhi Tang, Ao Li +2

University of Toronto, Toronto, Canada · Georgia Institute of Technology, Atlanta, USA

Lean Theorem ProvingLLM Mathematical Reasoning
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May 22, 2025

EquivPruner: Boosting Efficiency and Quality in LLM-Based Search via Action Pruning

Large Language Models (LLMs) excel at complex reasoning through search algorithms, yet current strategies often suffer from massive token consumption due to redundant exploration of semantically equivalent steps. Existing semantic similarity methods struggle to accurately identify such equivalence in domain-specific contexts like mathematical reasoning. To address this, we propose EquivPruner, a simple yet effective approach that identifies and prunes semantically equivalent actions during LLM reasoning search. We also introduce MathEquiv, the first dataset we created for mathematical statement equivalence, which enables the training of a lightweight equivalence detector. Extensive experiments across various models and tasks demonstrate that EquivPruner significantly reduces token consumption, improving searching efficiency and often bolstering reasoning accuracy. For instance, when applied to Qwen2.5-Math-7B-Instruct on GSM8K, EquivPruner reduced token consumption by 48.1% while also improving accuracy. Our code is available at https://github.com/Lolo1222/EquivPruner.

Jiawei Liu, Qisi Chen, Jianshu Zhang +2

School of Computer Science and Technology, University of Science and Technology of China · Department of Data Science, City University of Hong Kong · iFLYTEK Research

LLM Inference EfficiencySemantic Textual SimilarityLLM Mathematical Reasoning
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May 20, 2025

Reliable Proof Generation with LLMs via Analogical Retrieval and Symbolic Verification: A Case Study in Euclidean Geometry

Large language models (LLMs) struggle with formal domains that require rigorous logical deduction and symbolic reasoning, such as mathematical proof generation. We propose a neuro-symbolic approach centered on the hypothesis that structurally analogous problems often admit similar proofs. As a proof-of-concept, we focus on SAT-level geometry problems. Our approach is two-fold: (1) We retrieve analogous problems and use their proofs to guide the LLM, and (2) a formal verifier evaluates the generated proofs and provides feedback, helping the model fix incorrect proofs. Our complete pipeline substantially improves proof accuracy across model families, achieving 68%-96% accuracy compared with 10%-44% for LLM-only baselines that use neither analogy retrieval nor verifier feedback. When comparing against models with the same verifier feedback and inference budget, analogical guidance improves accuracy from 88% to 96% for GPT-5, 78% to 86% for Claude Sonnet 4.6, 72% to 86% for Gemini-Flash-2.5, and 52% to 80% for OpenAI o1. More broadly, shifting to LLMs that generate provably correct conclusions has the potential to dramatically improve their reliability, accuracy and consistency, unlocking complex tasks and critical real-world applications that require trustworthiness.

Oren Sultan, Eitan Stern, Dafna Shahaf

The Hebrew University of Jerusalem

Automated Theorem ProvingNeuro-Symbolic ReasoningLLM Mathematical Reasoning
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