Mathematical Reasoning

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Overview

Mathematical reasoning asks how AI systems can solve multi-step problems, apply mathematical concepts, construct proofs, and help discover new results. Common approaches include supervised learning and reinforcement learning with verifiable rewards, alongside self-distillation, search, and structured methods that use code, symbolic solvers, or formal proof assistants such as Lean. Work spans arithmetic and competition problems, geometry, statistics, multimodal tasks, and research-level proofs, with benchmarks increasingly testing robustness, multilingual ability, and intermediate reasoning rather than answers alone. Models have improved on many standard tasks, but remain brittle under equivalent problem changes and difficult proof or construction demands; reliable verification, useful exploration, and meaningful measures of reasoning quality remain open challenges.

Latest papers

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155 papers19 papers in the last four weeks, up 19% on the four weeks before. 0.1% of all new papers.

Apr 17, 2026

DenTab: A Dataset for Table Recognition and Visual QA on Real-World Dental Estimates

Tables condense key transactional and administrative information into compact layouts, but practical extraction requires more than text recognition: systems must also recover structure (rows, columns, merged cells, headers) and interpret roles such as line items, subtotals, and totals under common capture artifacts. Many existing resources for table structure recognition and TableVQA are built from clean digital-born sources or rendered tables, and therefore only partially reflect noisy administrative conditions. We introduce DenTab, a dataset of 2{,}000 cropped table images from dental estimates with high-quality HTML annotations, enabling evaluation of table recognition (TR) and table visual question answering (TableVQA) on the same inputs. DenTab includes 2{,}208 questions across eleven categories spanning retrieval, aggregation, and logic/consistency checks. We benchmark 16 systems, including 14 vision--language models (VLMs) and two OCR baselines. Across models, strong structure recovery does not consistently translate into reliable performance on multi-step arithmetic and consistency questions, and these reasoning failures persist even when using ground-truth HTML table inputs. To improve arithmetic reliability without training, we propose the Table Router Pipeline, which routes arithmetic questions to deterministic execution. The pipeline combines (i) a VLM that produces a baseline answer, a structured table representation, and a constrained table program with (ii) a rule-based executor that performs exact computation over the parsed table. The source code and dataset will be made publicly available at https://github.com/hamdilaziz/DenTab.

Laziz Hamdi, Amine Tamasna, Thierry Paquet

LITIS, Normandy, France · Malakoff Humanis, Paris, France

Table QATable Structure RecognitionMathematical ReasoningDocument Image Understanding
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Apr 15, 2026

Correct Prediction, Wrong Steps? Consensus Reasoning Knowledge Graph for Robust Chain-of-Thought Synthesis

Large language models (LLMs) have become increasingly used for various tasks, often coupled with Chain-of-Thought (CoT) prompting to boost accuracy. Recent work has shown that high label-prediction accuracy does not guarantee correct intermediate reasoning, and the causes of reasoning flaws vary from sample to sample, yet existing remedies either focus on a single domain or assume that one flaw type applies uniformly across samples. A simple mitigation method is to provide the model with the correct answer, but we show that this yields no consistent improvement in reasoning quality. This indicates that the problem cannot be fixed by LLMs' awareness of answers, and must instead be addressed through the structure of reasoning. Motivated by this, we propose CRAFT (Consensus Reasoning knowledge graph Aggregation for Flaw-aware Trace synthesis), which aggregates the consensus components shared across multiple candidate reasoning traces to synthesize improved ones. CRAFT consistently improves label-prediction accuracy on both logical and mathematical reasoning benchmarks, outperforming most baselines, while its post-processed traces achieve higher quality under fine-grained benchmark evaluation.

Zipeng Ling, Shuliang Liu, Seonil Son +4

The Hong Kong University of Science and Technology (Guangzhou) · University of Alberta · Alberta Machine Intelligence Institute +3

Reasoning Trace AnalysisLogical ReasoningSelf-Consistency DecodingCoT ReasoningMathematical ReasoningStructured Reasoning+3
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Apr 13, 2026

Lectures on AI for Mathematics

This book provides a comprehensive and accessible introduction to the emerging field of AI for mathematics. It covers the core principles and diverse applications of using artificial intelligence to advance mathematical research. Through clear explanations, the text explores how AI can discover hidden mathematical patterns, assist in proving complicated theorems, and even construct counterexamples to challenge conjectures.

Xiaoyang Chen, Xiaoyang Chen

School of Mathematical Sciences, Tongji University

Mathematical ReasoningLLM Mathematical Reasoning
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Apr 13, 2026

Minimal Embodiment Enables Efficient Learning of Number Concepts in Robot

Robots are increasingly entering human-interactive scenarios that require understanding of quantity. How intelligent systems acquire abstract numerical concepts from sensorimotor experience remains a fundamental challenge in cognitive science and artificial intelligence. Here we investigate embodied numerical learning using a neural network model trained to perform sequential counting through naturalistic robotic interaction with a Franka Panda manipulator. We demonstrate that embodied models achieve 96.8% counting accuracy with only 10% of training data, compared to 60.6% for vision-only baselines. This advantage persists when visual-motor correspondences are randomized, indicating that embodiment functions as a structural prior that regularizes learning rather than as an information source. The model spontaneously develops biologically plausible representations: number-selective units with logarithmic tuning, mental number line organization, Weber-law scaling, and rotational dynamics encoding numerical magnitude (r=0.97r = 0.97, slope =30.6°= 30.6°/count). The learning trajectory parallels children's developmental progression from subset-knowers to cardinal-principle knowers. These findings demonstrate that minimal embodiment can ground abstract concepts, improve data efficiency, and yield interpretable representations aligned with biological cognition, which may contribute to embodied mathematics tutoring and safety-critical industrial applications.

Zhegong Shangguan, Alessandro Di Nuovo, Angelo Cangelosi

Cognitive Robotics Lab, Department of Computer Science, University of Manchester · Smart Interactive Technologies (SIT) Research Laboratory, Department of Computing, Sheffield Hallam University

Embodied CognitionMathematical ReasoningData-Efficient Robot Learning
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Apr 7, 2026

Artificial Intelligence and the Structure of Mathematics

Recent progress in artificial intelligence (AI) is unlocking transformative capabilities for mathematics. There is great hope that AI will help solve major open problems and autonomously discover new mathematical concepts. In this essay, we further consider how AI may open a grand perspective on mathematics by forging a new route, complementary to mathematicallogic, to understanding the global structure of formal proof****s. We begin by providing a sketch of the formal structure of mathematics in terms of universal proof and structural hypergraphs and discuss questions this raises about the foundational structure of mathematics. We then outline the main ingredients and provide a set of criteria to be satisfied for AI models capable of automated mathematical discovery. As we send AI agents to traverse Platonic mathematical worlds, we expect they will teach us about the nature of mathematics: both as a whole, and the small ribbons conducive to human understanding. Perhaps they will shed light on the old question: "Is mathematics discovered or invented?" Can we grok the terrain of these Platonic worlds?

Maissam Barkeshli, Michael R. Douglas, Michael H. Freedman

Meta Superintelligence Labs, Fundamental AI Research · Department of Physics and Joint Quantum Institute, University of Maryland, College Park · Center for Mathematical Sciences and Applications, Harvard University +1

AI Agents for Scientific DiscoveryMathematical Reasoning
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Apr 6, 2026

QED-Nano: Teaching a Tiny Model to Prove Hard Theorems

Proprietary AI systems have recently demonstrated impressive capabilities on complex proof-based problems, with gold-level performance reported at the 2025 International Mathematical Olympiad (IMO). However, the training pipelines behind these systems remain largely undisclosed, and their reliance on large "internal" models and scaffolds makes them expensive to run, difficult to reproduce, and hard to study or improve upon. This raises a central question: can small, open models also be trained to achieve competitive reasoning performance on difficult Olympiad-level math? In this paper, we answer this question by building QED-Nano, a 4B model post-trained for Olympiad-level proofs. Our training recipe has three stages: (1) supervised fine-tuning to imbue good proof-writing styles by distilling from DeepSeek-Math-V2, (2) reinforcement learning (RL) with rubric-based rewards, and (3) expanding RL with a reasoning cache, which decomposes long proofs into iterative summarize-and-refine cycles and enables stronger test-time reasoning. QED-Nano surpasses the proof-generation performance of much larger open models, including Nomos-1 and GPT-OSS-120B, and approaches the performance of proprietary models like Gemini 3 Pro, at a fraction of the inference cost. To support further research on open mathematical reasoning, we release the full QED-Nano pipeline, including the QED-Nano and QED-Nano-SFT models, the FineProofs-SFT and FineProofs-RL datasets, and the training and evaluation code.

LM-Provers, Yuxiao Qu, Amrith Setlur +6

CMU · Hugging Face · ETH Zurich +1

Mathematical ReasoningLLM Mathematical ReasoningEfficient Language Model ReasoningAutomated Theorem ProvingRL for Language Model Reasoning
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Apr 6, 2026

What Makes Good Multilingual Reasoning? Disentangling Traces with Measurable Features

Large Reasoning Models (LRMs) still exhibit large performance gaps between English and other languages, yet much current work assumes these gaps can be closed simply by making reasoning in every language resemble English reasoning. This work challenges this assumption by asking instead: what actually characterizes successful reasoning traces in multilingual settings, and to what extent do English-derived reasoning features genuinely help in other languages? We first define a suite of measurable reasoning features spanning multilingual alignment, reasoning step, and reasoning flow aspects of reasoning traces, and use logistic regression to quantify how each feature associates with final answer accuracy. We further train sparse autoencoders over multilingual traces to automatically discover latent reasoning concepts that instantiate or extend these features. Finally, we use the features to re-rank traces and measure their impact on accuracy at test time. Across two mathematical reasoning benchmarks, four LRMs, and ten languages, we find that most features are positively associated with accuracy, but the strength of association varies considerably across languages and can even reverse in some. Our findings challenge English-centric reward designs and point toward adaptive objectives that accommodate language-specific reasoning patterns, with concrete implications for multilingual benchmark and reward design.

Dayeon Ki, Kevin Duh, Marine Carpuat

University of Maryland · Johns Hopkins University

Multilingual Language Model EvaluationCross-Lingual Reasoning in Language ModelsMathematical ReasoningLarge Reasoning ModelsReasoning Trace Analysis
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Apr 4, 2026

Document-Level Numerical Reasoning across Single and Multiple Tables in Financial Reports

Despite the strong language understanding abilities of large language models (LLMs), they still struggle with reliable question answering (QA) over long, structured documents, particularly for numerical reasoning. Financial annual reports exemplify this difficulty: financial statement analysis often hinges on accurate arithmetic, and analysts derive key indicators by integrating evidence scattered across multiple tables and narrative text. However, existing benchmarks focus largely on single-table settings, leaving cross-table document-level numerical reasoning underexplored. To address this gap, we introduce FinLongDocQA, a dataset for both single-table and cross-table financial numerical reasoning in long-context reports. Evaluating both closed-source and open-source LLMs on FinLongDocQA reveals two bottlenecks: (1) annual reports often exceed 129k tokens, exacerbating the context rot problem for locating relevant tables; and (2) even when relevant evidence is located, LLMs remain prone to errors in multi-step numerical reasoning. We propose FinLongDocAgent, a Multi-Agent Multi-Round Retrieval-Augmented Generation (RAG) approach that iteratively retrieves evidence, performs intermediate calculations, and verifies results across rounds. Experiments highlight the importance of iterative retrieval and verification for reliable numerical QA in long financial documents.

Yi-Cheng Wang, Wei-An Wang, Chu-Song Chen

Department of Computer Science and Information Engineering, National Taiwan University · FinTech Center, National Taiwan University

Numerical Reasoning in Language ModelsFinancial QAMathematical ReasoningRetrieval-Augmented Generation
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Mar 30, 2026

The Long Delay to Arithmetic Generalization

Large language models today still make mistakes on multi-digit arithmetic when they do not have tools, and part of the reason is how numbers are split into tokens. Arithmetic is also a setting where the correct answer to every input and the algorithm that produces it are already known, so training transformers on it lets us study what they learn against a ground truth. These models often fit their training set long before they generalize to held-out inputs, and this delay, known as grokking, decreases as the training set grows. But, how does the representation of numbers change the amount of data a transformer needs to generalize? We study one-step Collatz prediction, in which a transformer reads the digits of an integer in a chosen base and outputs the digits of the next term in its Collatz sequence. We find that the long delay appears only when the training set has few different integers, and that the numeral base determines how many integers are sufficient. In bases where each output digit depends only on a few neighbouring input digits, because the carry between positions depends only on the current digit, a transformer needs several times fewer training integers. We predicted this ordering before training, and input length does not explain it. The same carry structure predicts which bases are easy for a second Collatz map, for integer division, for a decoder-only transformer, and across three model sizes. Finally, a linear probe recovers the low-order residues of the input as accurately from a randomly initialized encoder as from a trained one. Thus, high probe accuracy on such features does not reliably show that a model has learned to compute them. More broadly, we show that the choice of numeral representation affects sample efficiency in ways we can predict from the structure of the task before training.

Laura Gomezjurado Gonzalez

Stanford University

GrokkingNeural Algorithmic ReasoningMathematical Reasoning
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Mar 26, 2026

4OPS: Structural Difficulty Modeling in Integer Arithmetic Puzzles

Arithmetic puzzle games provide a controlled setting for studying difficulty in mathematical reasoning tasks, a core challenge in adaptive learning systems. We investigate the structural determinants of difficulty in a class of integer arithmetic puzzles inspired by number games. We formalize the problem and develop an exact dynamic-programming solver that enumerates reachable targets, extracts minimal-operation witnesses, and enables large-scale labeling. Using this solver, we construct a dataset of over 3.4 million instances and define difficulty via the minimum number of operations required to reach a target. We analyze the relationship between difficulty and solver-derived features. While baseline machine learning models based on bag- and target-level statistics can partially predict solvability, they fail to reliably distinguish easy instances. In contrast, we show that difficulty is fully determined by a small set of interpretable structural attributes derived from exact witnesses. In particular, the number of input values used in a minimal construction serves as a minimal sufficient statistic for difficulty under this labeling. These results provide a transparent, computationally grounded account of puzzle difficulty that bridges symbolic reasoning and data-driven modeling. The framework supports explainable difficulty estimation and principled task sequencing, with direct implications for adaptive arithmetic learning and intelligent practice systems.

Yunus E. Zeytuncu

Item Difficulty EstimationMathematical ReasoningMathematical Reasoning Benchmarks
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Mar 20, 2026

Compression is all you need: Modeling Mathematics

Human mathematics (HM), the mathematics humans discover and value, is a vanishingly small subset of formal mathematics (FM), the totality of all valid deductions. We argue that HM is distinguished by its compressibility through hierarchically nested definitions, lemmas, and theorems. We model this with monoids. A mathematical deduction is a string of primitive symbols; a definition or theorem is a named substring or macro whose use compresses the string. In the free abelian monoid AnA_n, a logarithmically sparse macro set achieves exponential expansion of expressivity. In the free non-abelian monoid FnF_n, even a polynomially-dense macro set only yields linear expansion; superlinear expansion requires near-maximal density. We test these models against MathLib, a large Lean~4 library of mathematics that we take as a proxy for HM. Each element has a depth (layers of definitional nesting), a wrapped length (tokens in its definition), and an unwrapped length (primitive symbols after fully expanding all references). We find unwrapped length grows exponentially with both depth and wrapped length; wrapped length is approximately constant across all depths. These results are consistent with AnA_n and inconsistent with FnF_n, supporting the thesis that HM occupies a polynomially-growing subset of the exponentially growing space FM. We discuss how compression, measured on the MathLib dependency graph, and a PageRank-style analysis of that graph can quantify mathematical interest and help direct automated reasoning toward the compressible regions where human mathematics lives.

Vitaly Aksenov, Eve Bodnia, Michael H. Freedman +1

Mathematical Reasoning
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Mar 19, 2026

Thinking with Constructions: A Benchmark and Policy Optimization for Visual-Text Interleaved Geometric Reasoning

Geometric reasoning inherently requires "thinking with constructions" -- the dynamic manipulation of visual aids to bridge the gap between problem conditions and solutions. However, existing Multimodal Large Language Models (MLLMs) are largely confined to passive inference with static diagrams, lacking the strategic knowledge of when and how to construct effective visual aids. To address this, we present a framework for Visual-Text Interleaved Chain-of-Thought. We first introduce GeoAux-Bench, the first benchmark comprising 4,334 geometry problems that aligns textual construction steps with ground-truth visual updates. Our pilot study reveals two critical insights: (1) interleaved visual-textual aids outperform single-modality counterparts, which cannot losslessly capture geometric synergy; and (2) valid constructions act as entropy reducers, strongly correlating with reduced reasoning perplexity. Building on these findings, we propose Action Applicability Policy Optimization (A2PO), a reinforcement learning paradigm for mastering strategic construction. A2PO employs Adaptive Reward Shaping to regulate the timing and quality of visual aids via counterfactual sampling to distinguish necessary from redundant constructions. Experiments demonstrate our approach enables MLLMs to leverage selective auxiliary constructions, yielding a 3.51% gain over strong baselines. Code and data are available on GitHub.

Haokun Zhao, Wanshi Xu, Haidong Yuan +3

College of Computer Science and Artificial Intelligence, Fudan University · School of ECE, Peking University · School of Software and Microelectronics, Peking University +1

Visual Spatial ReasoningMathematical ReasoningMultimodal CoT ReasoningMathematical Reasoning BenchmarksRL for Language Model Reasoning
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Mar 11, 2026

mAceReason-Math: A Dataset of High-Quality Multilingual Math Problems Ready For RLVR

Reinforcement Learning with Verifiable Rewards (RLVR) has been successfully applied to significantly boost the capabilities of pretrained large language models, especially in the math and logic problem domains. However, current research and available training datasets remain English-centric. While multilingual training data and benchmarks have been created in the past, they were not created with RLVR and current model capability in mind, and their level of difficulty is often too low to provide appropriate training signals for current models. To address this gap, we provide mAceReason-Math, a dataset of high-quality translations of challenging math problems sourced from a corpus specifically curated for RLVR (AceReason-Math). We further take specific care to clean and improve our translations, resulting in a coverage of 14 languages with more than 10,000 samples per language. We release the dataset to facilitate multilingual RLVR research and benchmarking in the research community.

Konstantin Dobler, Simon Lehnerer, Federico Scozzafava +2

Apple · Hasso Plattner Institute & ELLIS Unit Potsdam

LLM Mathematical ReasoningMathematical ReasoningRL for Language Model Reasoning
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Mar 5, 2026

Non-Parametric Structural Priors for Geometry Theorem Prediction

Multi-step theorem prediction is a central challenge in geometry problem solving. Existing neural-symbolic approaches rely heavily on supervised parametric models, which exhibit limited generalization to evolving theorem libraries. In this work, we explore training-free theorem prediction through the lens of in-context learning (ICL). We identify a critical scalability bottleneck, termed Structural Drift: as reasoning depth increases, the performance of vanilla ICL degrades sharply, often collapsing to near zero. We attribute this failure to the LLM's inability to recover latent topological dependencies, leading to unstructured exploration. To address this issue, we propose Theorem Precedence Graphs, which encode temporal dependencies from historical solution traces as directed graphs, and impose explicit topological constraints that effectively prune the search space during inference. Coupled with retrieval-augmented graph construction and a stepwise symbolic executor, our approach enables LLMs to act as structured planners without any gradient-based optimization. Experiments on the FormalGeo7k benchmark show that our method achieves 89.29% accuracy, substantially outperforming ICL baselines and matching state-of-the-art supervised models. These results indicate that explicit structural priors offer a promising direction for scaling LLM-based symbolic reasoning.

Junbo Zhao, Ting Zhang, Can Li +3

School of Artificial Intelligence, Beijing Normal University, Beijing, China · Engineering Research Center of Intelligent Technology and Educational Application, Ministry of Education, Beijing, China · Beijing Key Laboratory of Artificial Intelligence for Education, Beijing, China +1

Mathematical ReasoningAutomated Theorem ProvingSymbolic ReasoningLLM Reasoning with GraphsIn-Context Learning
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Mar 4, 2026

Discovering mathematical concepts through a multi-agent system

Mathematical concepts emerge through an interplay of processes, including experimentation, efforts at proof, and counterexamples. In this paper, we present a new multi-agent model for computational mathematical discovery based on this observation. Our system, conceived with research in mind, poses its own conjectures and then attempts to prove them, making decisions informed by this feedback and an evolving data distribution. Inspired by the history of Euler's conjecture for polyhedra and an open challenge in the literature, we benchmark with the task of autonomously recovering the concept of homology from polyhedral data and knowledge of linear algebra. Our system completes this learning problem. Most importantly, the experiments are ablations, statistically testing the value of the complete dynamic and controlling for experimental setup. They support our main claim: that the optimisation of the right combination of local processes can lead to surprisingly well-aligned notions of mathematical interestingness.

Daattavya Aggarwal, Oisin Kim, Carl Henrik Ek +1

Department of Computer Science and Technology, University of Cambridge.

Autonomous Scientific DiscoveryMathematical ReasoningMulti-Agent SystemsMulti-Agent Reasoning
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Mar 1, 2026

How RL Unlocks the Aha Moment in Geometric Interleaved Reasoning

Solving complex geometric problems inherently requires interleaved reasoning: a tight alternation between constructing diagrams and performing logical deductions. Although recent Multimodal Large Language Models (MLLMs) have demonstrated strong capabilities in visual generation and plotting, we identify a counter-intuitive and underexplored phenomenon. Naively applying Supervised Fine-Tuning (SFT) on interleaved plot-solution data leads to a substantial degradation in reasoning performance compared to text-only baselines. We argue that this failure stems from a fundamental limitation of SFT, which primarily induces distributional alignment: the model learns to reproduce the surface format of interleaved plotting but fails to internalize the causal dependency between the generated plot and reasoning steps. To overcome this limitation, we propose Faire (Functional alignment for interleaved reasoning), a reinforcement learning framework that enforces three casual constraints to move beyond superficial imitation toward functional alignment. Extensive experiments show that Faire induces a qualitative shift in model behavior in which the plotting is effectively internalized, yielding competitive performance on challenging geometric reasoning benchmarks.

Xiangxiang Zhang, Caijun Jia, Siyuan Li +9

ByteDance · Shenyang Institute of Computing Technology, Chinese Academy of Sciences · University of Chinese Academy of Science +2

Visually Grounded ReasoningRL for Language Model ReasoningMathematical ReasoningMultimodal CoT Reasoning
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Feb 26, 2026

The AI Research Assistant: Promise, Peril, and a Proof of Concept

Can artificial intelligence truly contribute to creative mathematical research, or does it merely automate routine calculations while introducing risks of error? We provide empirical evidence through a detailed case study: the discovery of novel error representations and bounds for Hermite quadrature rules via systematic human-AI collaboration. Working with multiple AI assistants, we extended results beyond what manual work achieved, formulating and proving several theorems with AI assistance. The collaboration revealed both remarkable capabilities and critical limitations. AI excelled at algebraic manipulation, systematic proof exploration, literature synthesis, and LaTeX preparation. However, every step required rigorous human verification, mathematical intuition for problem formulation, and strategic direction. We document the complete research workflow with unusual transparency, revealing patterns in successful human-AI mathematical collaboration and identifying failure modes researchers must anticipate. Our experience suggests that, when used with appropriate skepticism and verification protocols, AI tools can meaningfully accelerate mathematical discovery while demanding careful human oversight and deep domain expertise.

Tan Bui-Thanh

Department of Aerospace Engineering & Engineering Mechanics, and Oden Institute for Computational Engineering & Sciences, University of Texas at Austin, Austin, TX, 78712, USA

AI-Assisted Scientific ResearchHuman-AI CollaborationMathematical ReasoningHuman Oversight of AI
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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 18, 2026

Escaping the Cognitive Well: Efficient Competition Math with Off-the-Shelf Models

In the past year, custom and unreleased math reasoning models reached gold medal performance on the International Mathematical Olympiad (IMO). Similar performance was then reported using large-scale inference on publicly available models but at prohibitive costs (e.g., 3000 USD per problem). In this work, we present an inference pipeline that attains best-in-class performance on IMO-style math problems at an average inference cost orders of magnitude below competing methods while using only general-purpose off-the-shelf models. Our method relies on insights about grader failure in solver-grader pipelines, which we call the Cognitive Well (iterative refinement converging to a wrong solution that the solver as well as the pipeline's internal grader consider to be basically correct). Our pipeline addresses these failure modes through conjecture extraction, wherein candidate lemmas are isolated from generated solutions and independently verified alongside their negations in a fresh environment (context detachment). On IMO-ProofBench Advanced (PB-Adv), our pipeline achieves 67.1 percent performance using Gemini 3.0 Pro with an average cost per question of approximately 31 USD. At the time of evaluation, this represented the state-of-the-art on PB-Adv among both public and unreleased models, and more than doubles the success rate of the next best publicly accessible pipeline, all at a fraction of the cost.

Xingyu Dang, Rohit Agarwal, Rodrigo Porto +3

Princeton University · Princeton Language and Intelligence

Language Model Self-VerificationMathematical ReasoningEfficient Language Model ReasoningLLM Answer VerificationMathematical Reasoning Benchmarks
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Feb 18, 2026

DeepVision-103K: A Visually Diverse, Broad-Coverage, and Verifiable Mathematical Dataset for Multimodal Reasoning

Reinforcement Learning with Verifiable Rewards (RLVR) has been shown effective in enhancing the visual reflection and reasoning capabilities of Large Multimodal Models (LMMs). However, existing datasets are predominantly derived from either small-scale manual construction or recombination of prior resources, which limits data diversity and coverage, thereby constraining further gains in model performance. To this end, we introduce DeepVision-103K, a comprehensive dataset for RLVR training that covers diverse K12 mathematical topics, extensive knowledge points, and rich visual elements. Models trained on DeepVision achieve strong performance on multimodal mathematical benchmarks, and generalize effectively to general multimodal reasoning tasks. Further analysis reveals enhanced visual perception, reflection and reasoning capabilities in trained models, validating DeepVision's effectiveness for advancing multimodal reasoning. Data: this url.

Haoxiang Sun, Lizhen Xu, Bing Zhao +5

Alibaba Group · Shanghai Jiao Tong University

Multimodal ReasoningMathematical ReasoningReinforcement Learning with Verifiable Rewards
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Feb 5, 2026

BRIDGE: Bridging Reasoning In Distillation Gap Elimination via Structure-Aware Masking

Chain-of-Thought (CoT) reasoning has significantly improved LLMs' mathematical problem-solving capabilities, but distilling such capabilities into smaller models remains challenging due to the capacity mismatch between verbose teachers and compact students. Directly copying teachers' lengthy reasoning chains causes capacity overload, resulting in truncated outputs or repetitive failure. Existing remedies each sacrifice a critical property of CoT: implicit reasoning methods (e.g., compressing reasoning into hidden states) trade away interpretability and verifiability, while heuristic compression strategies (e.g., random step pruning) destroy logical integrity. To address this, we propose BRIDGE, a curriculum framework that first establishes structural understanding via masked reconstruction, then uses GRPO-based reinforcement learning to guide students in self-discovering the optimal balance between accuracy and brevity, and finally internalizes complex reasoning through teacher-guided rewriting on failure cases. On GSM8K, BRIDGE enables Qwen2.5-3B to achieve 11.29% accuracy improvement and 27.4% token reduction over the original model, outperforming instruction-tuned variants and distillation baselines. Zero-shot transfer experiments on SVAMP and MATH-500 further confirm the generalization of internalized reasoning. Our code and model checkpoints are publicly available at https://github.com/Applied-Machine-Learning-Lab/SDM2026_BRIDGE and https://huggingface.co/bowen0815/BRIDGE.

Bowen Yu, Sheng Zhang, Binhao Wang +8

City University of Hong Kong · Mohamed Bin Zayed University of Artificial Intelligence

CoT DistillationMathematical ReasoningTeacher-Student LearningCoT Compression
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Feb 2, 2026

ProofVerifier: A Scalable, Diversity-Driven Framework for Natural-Language Proof Verification

While large language models (LLMs) have achieved strong performance on mathematical problems with verifiable answers, many advanced problems are proof-based and require evaluating full proofs. However, training such verifiers requires diverse and trustworthy question-proof-check (QPC) examples at scale, which are scarce. To address this challenge, we develop a human-audited, LLM-assisted data pipeline that produces large-scale QPC triplets with limited human effort. By systematically varying problem sources, generation strategies, and generator models, the pipeline creates diverse problem-proof pairs spanning multiple difficulty levels, linguistic styles, and error types. We combine multi-LLM agreement with hierarchical human auditing to obtain accurate proof-correctness labels. Using these data, we train generative proof verifiers and introduce an auxiliary fluency filter together with balanced token weighting to stabilize binary-reward long-form verification RL. Experiments show that our verifier improves proof-judgment accuracy across different proof styles and provides useful guidance for test-time selection. Overall, our results provide a practical data and training framework for natural-language proof verification.

Haotong Yang, Zitong Wang, Shijia Kang +7

Peking University, Beijing, China · Tencent

Synthetic Data GenerationMathematical Reasoning
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Feb 1, 2026

ConPress: Learning Efficient Reasoning from Multi-Question Contextual Pressure

Large reasoning models (LRMs) typically solve reasoning-intensive tasks by generating long chain-of-thought (CoT) traces, leading to substantial inference overhead. We identify a reproducible inference-time phenomenon, termed Self-Compression: when multiple independent and answerable questions are presented within a single prompt, the model spontaneously produces shorter reasoning traces for each question. This phenomenon arises from multi-question contextual pressure during generation and consistently manifests across models and benchmarks. Building on this observation, we propose ConPress (Learning from Contextual Pressure), a lightweight self-supervised fine-tuning approach. ConPress constructs multi-question prompts to induce self-compression, samples the resulting model outputs, and parses and filters per-question traces to obtain concise yet correct reasoning trajectories. These trajectories are directly used for supervised fine-tuning, internalizing compressed reasoning behavior in single-question settings without external teachers, manual pruning, or reinforcement learning. With only 8k fine-tuning examples, ConPress reduces reasoning token usage by 59% on MATH500 and 33% on AIME25, while maintaining competitive accuracy.

Jie Deng, Shining Liang, Jun Li +2

University of Chinese Academy of Sciences, Beijing, China · 2Microsoft, Beijing, China · 3Tongji University, Shanghai, China

Efficient Language Model ReasoningLLM Inference AccelerationMathematical ReasoningCoT CompressionLarge Reasoning Models
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Jan 26, 2026

Teaching Models to Teach Themselves: Reasoning at the Edge of Learnability

RL methods for scaling large reasoning models stall on datasets with low initial success rates, and thus little training signal. We investigate a fundamental question: Can a pretrained LLM leverage latent knowledge to generate an automated curriculum for problems it cannot solve? We explore this with SOAR: An asymmetric self-play framework that uses meta-RL to surface these pedagogical signals. A teacher model proposes synthetic problems for a student model, and is rewarded with its improvement on a subset of hard problems, thus grounding the curriculum in real student progress rather than intrinsic proxy rewards. Our study on the hardest subsets of math benchmarks (0/128 success) reveals three core findings. First, it is possible to realize bilevel meta-RL that unlocks learning under sparse, binary rewards by sharpening a latent capacity of pretrained models to generate useful problems. Second, grounded rewards outperform intrinsic learnability rewards used in prior LLM self-play, reliably avoiding typical instability and diversity collapse modes. Third, the structure and well-posedness of questions are more critical for learning progress than solution correctness. Our results suggest that the ability to generate useful stepping stones does not require the preexisting ability to solve the hard problems, paving a principled path to escape reasoning plateaus without additional curated data

Shobhita Sundaram, John Quan, Ariel Kwiatkowski +3

1MIT · 2Meta FAIR · 3New York University

Self-Play RLRL for Language Model ReasoningCurriculum RLMathematical ReasoningMeta-Reinforcement LearningCurriculum Learning+2
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Jan 11, 2026

DAGGER: Distractor-Aware Graph Generation for Executable Reasoning in Math Problems

Chain-of-Thought (CoT) prompting is widely adopted for mathematical problem solving, including in low-resource languages, yet its behavior under irrelevant context remains underexplored. To systematically study this challenge, we introduce DISTRACTMATH-BN, a Bangla benchmark that augments MGSM and MSVAMP with semantically coherent but computationally irrelevant information. Evaluating seven models ranging from 3B to 12B parameters, we observe substantial performance degradation under distractors: standard models drop by up to 41 points, while reasoning-specialized models decline by 14 to 20 points despite consuming five times more tokens. We propose †DAGGER, which reformulates mathematical problem solving as executable computational graph generation with explicit modeling of distractor nodes. Fine-tuning Gemma-3 models using supervised fine-tuning followed by Group Relative Policy Optimization achieves comparable weighted accuracy on augmented benchmarks while using 89 percent fewer tokens than reasoning models. Importantly, this robustness emerges without explicit training on distractor-augmented examples. Our results suggest that enforcing structured intermediate representations improves robustness and inference efficiency in mathematical reasoning compared to free-form approaches, particularly in noisy, low-resource settings.

Zabir Al Nazi, Shubhashis Roy Dipta, Sudipta Kar

University of California, Riverside · University of Maryland, Baltimore County · Oracle Health AI

Numerical Reasoning in Language ModelsMathematical Reasoning BenchmarksLLM Reasoning with GraphsMathematical Reasoning
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Dec 17, 2025

Stepwise Think-Critique: Interleaved Reasoning and Self-Critique in a Single LLM

Human beings solve complex problems through critical thinking, where reasoning and evaluation are intertwined to converge toward correct solutions. However, most existing large language models (LLMs) treat the reasoning and verification as separate processes: they either generate reasoning without explicit self-checking or rely on external verifiers to detect errors post hoc. The former lacks immediate feedback, while the latter increases system complexity and hinders synchronized learning. Motivated by human critical thinking, we propose Stepwise Think-Critique (STC), an end-to-end trainable framework in which a single LLM emits a structured, step-level critique inline with each reasoning step. STC is trained with reinforcement learning that complements reasoning rewards with a critique-consistency reward derived from final-answer correctness, jointly optimizing reasoning correctness and critique reliability. On five mathematical reasoning benchmarks, STC improves Pass@1 by 7.2% over the 1.5B base model and attains 67.4% step-level critique F1, surpassing seven external process reward models evaluated at their per-dataset oracle thresholds---a step toward LLMs with built-in critical thinking.

Jiaqi Xu, Cuiling Lan, Xuejin Chen +1

University of Science and Technology of China · ⋆This work was done when Jiaqi was an intern at MSRA. · Microsoft Research Asia

LLM Self-CorrectionCoT ReasoningMathematical ReasoningRL for Language Models
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Oct 21, 2025

A Model Can Help Itself: Reward-Free Self-Training for LLM Reasoning

Can a language model improve reasoning by learning from its own imperfect responses, without rewards or teacher-provided solutions? We present Self-evolving Post-Training (SePT), a simple method that alternates temperature-controlled self-generation with next-token likelihood training. Each round uses the updated model to generate new training responses, with one response per prompt by default and no correctness filtering. Across six mathematical benchmarks, SePT improves a temperature-selected no-training baseline by 11.4 and 6.7 AVG points on Qwen2.5-Math-7B and Qwen2.5-7B, respectively, where AVG averages Pass@1, Pass@8 and Pass@32 across benchmarks. We analyze how sampling temperature shapes the learning signal and investigate the value of the resulting responses. Responses from a SePT-trained model improve a student initialized from the original weights, while their reasoning prefixes help an unchanged model complete solutions, even when matched in length to prefixes from a colder initial model. Comparing next-token predictions at identical contexts also reveals changes in token rankings that decoding-temperature adjustment cannot reproduce. Further evaluations across nine starting models, general reasoning and code generation examine broader applicability. Together, these results show that reward-free self-training can improve both a model's predictions and the supervision it provides. Our code is available at https://github.com/ElementQi/SePT.

Mengqi Li, Lei Zhao, Anthony Man-Cho So +2

The Chinese University of Hong Kong, Shenzhen · Shanghai Jiao Tong University · The Chinese University of Hong Kong

Self-Training for Language ModelsMathematical ReasoningLLM Post-TrainingLanguage Model Self-ImprovementLLM Reasoning
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Oct 13, 2025

Advancing the Pareto Frontier of Test-Time Scaling with Multi-Path Plan Aggregation

Test-time scaling seeks to improve reasoning through additional inference-time computation, commonly using solution sampling with majority voting to select the final answer. However, repeated sampling can reproduce correlated errors, limiting the accuracy gains from additional computation. We introduce Multi-Path Plan Aggregation (MPPA), which learns planning and answer selection using separate LoRA adapters on a shared frozen backbone. The planner jointly generates KK short plans, each guiding a separate solution generated by the frozen executor; the aggregator then scores the resulting solutions and selects the final answer. Joint planning encourages complementary approaches suited to the executor's capabilities, while learned selection uses reasoning evidence to identify correct candidates. Alternating optimization improves planning and aggregation through supervised finetuning and reinforcement learning. Across five mathematics and science benchmarks and three backbones, MPPA improves accuracy--token trade-offs compared with training-free and training-based baselines. With three plans, it achieves higher average accuracy than eight directly sampled solutions with majority voting or the same learned aggregator, while reducing output tokens by 47--54%. Evaluation with task-provided subtasks further demonstrates higher final-task accuracy than direct sampling with majority voting or learned aggregation at either the task or subtask level.

Siheng Xiong, Ali Payani, Faramarz Fekri

Georgia Institute of Technology · Cisco Research

Test-Time ScalingLLM ReasoningMathematical ReasoningQuestion Answering
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Oct 10, 2025

StatEval: A Comprehensive Benchmark for Large Language Models in Statistics

Despite rapid advances in large language models (LLMs), statistical reasoning remains underrepresented in existing LLM benchmarks, which often do not reflect the layered, proof-driven nature of real statistical practice. To address this gap, we introduce StatEval, the first large-scale benchmark for statistical reasoning across curricular and research-level settings. StatEval includes over 100,000 curated problems, with 20,000+ foundational questions spanning undergraduate and graduate curricula and 80,000+ research-level proof tasks extracted from leading statistical journals. To construct StatEval, we develop TRACE (Topology and Reasoning-Aware Context Extractor), a multi-agent pipeline with human-in-the-loop validation that converts unstructured academic texts into self-contained theorem-level reasoning tasks. We also propose an Adaptive Process-Based Scoring Pipeline for complex statistical proofs, enabling fine-grained evaluation beyond final-answer matching. Experiments show that while LLMs perform reasonably on foundational tasks, they struggle with rigorous research-level reasoning. Beyond evaluation, StatEval serves as a resource for improving reasoning, as retrieval-augmented generation and domain-specific alignment consistently enhance performance. Together, these results establish StatEval as both a benchmark and an infrastructure for advancing statistical reasoning in LLMs.

Yuchen Lu, Run Yang, Yichen Zhang +6

1Shanghai University of Finance and Economics

LLM EvaluationReasoning EvaluationMathematical 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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