LLM Mathematical Reasoning

LLM: Large Language Model

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19 papers in the last four weeks, up 375% on the four weeks before. 0.2% of all new papers.

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Latest papers 194

Jul 2, 2026cs.CL

TUDUM: A Turkish-Thinking Reasoning Pipeline for Qwen3.5-27B

This paper presents TUDUM (Türkçe Düşünen Üretken Model), a project pipeline for adapting a Qwen-family 27B thinking model toward Turkish reasoning. The central problem is not only to answer Turkish prompts in Turkish, but to make the explicit reasoning trace itself Turkish. A thinking model may translate a Turkish prompt into an English-centered internal or visible scratchpad, solve the problem mostly in English, and only localize the final answer. TUDUM instead treats the generated <think>...</think> block as a trainable behavior. The pipeline starts from the project base checkpoint unsloth/Qwen3.5-27B, applies supervised fine-tuning (SFT) on 15,991 Turkish reasoning examples using LoRA adapters, and then applies GRPO-family reinforcement learning on a proxy-filtered Turkish mathematics environment. The results are mixed. SFT made the model shorter and more consistently Turkish in its reasoning behavior, with large reductions in average response length and thinking exhaustion, but reduced benchmark accuracy. RL recovered some mathematical performance, especially AIME24 at the best early checkpoint, yet did not uniformly improve all benchmarks and did not exceed the base model on the reported Macro-6 average. The contribution is therefore best framed as a technically honest Turkish-thinking reasoning pipeline and evaluation, not as a claim of state-of-the-art Turkish reasoning. The released step-50 model is publicly available.

Baran Bingol, Bahaeddin Turkoglu

Department of Artificial Intelligence and Data Engineering Faculty of Engineering Ankara University Ankara, Türkiye

LLM Fine-Tuning LLM Mathematical Reasoning LLM Reasoning

Jul 2, 2026cs.LG

Geometric Signatures of Reasoning: A Spectral Perspective on Task Hardness

Chain-of-thought (CoT) reasoning enables large language models (LLMs) to solve complex problems by generating intermediate reasoning steps. While much attention has been paid to the length and content of these reasoning chains, far less is known about their internal geometry. We study the \emph{geometry} of CoT trajectories in the hidden state space of transformer models, formalizing each reasoning chain as a discrete curve in

\mathbb{R}^d

and characterizing it through spectral, positional, and kinematic geometric functionals. We introduce the effective dimension

d_ρ

as a measure of trajectory complexity and show theoretically that trajectories with flatter eigenvalue spectra correspond to harder tasks, as they explore more of the hidden dimensions. Lastly, we explore how kinematic features of the trajectory, mean position, positional dispersion, initial and current hidden states, mean velocity, mean speed, and speed dispersion, can be used to predict solution correctness before generation is complete, and may inform future early-stopping strategies. Experimentally, on mathematical reasoning problems from the MATH500 dataset,

d_ρ

achieves

0.93

AUC in distinguishing easy from hard problems, while kinematic features potentially can predict correctness from only the first

20\%

of generated tokens. These correctness signatures transfer across questions of varying difficulty, establishing that the shape of a model's internal reasoning trajectory is a principled window into both task hardness and solution quality.

Aria Masoomi, Mahsa Bazzaz, Adel Javanmard +1

1Northeastern University · University of Southern California · 3Google Research

LLM Evaluation Representation Geometry in Language Models LLM Mathematical Reasoning LLM Reasoning Reasoning Trace Analysis

Jun 30, 2026cs.LG

ISM:Self-Improving Strategy Memory for Continual Mathematical Reasoning

We propose Intelligent Schema Memory (ISM), a self-evolving memory-augmented system that improves mathematical reasoning for a frozen LLM under continual learning with hard episodic resets. ISM maintains a compact, self-refined bank of strategy schemas learned from both successful and failed episodes, with symbolic tools that check intermediate steps and certify answers. Without updating model parameters, ISM outperforms passive, retrieval, and reflection baselines on MATH-Hard and OlympiadBench, using 64% and 86% fewer schemas respectively than the strongest passive baseline. These results show that small, actively maintained, and verified strategy memories can support reliable continual mathematical reasoning under strict episodic isolation. The codebase is available at https://github.com/pdx97/ISM .

Prakhar Dixit, Tim Oates

Department of Computer Science, University of Maryland, Baltimore County, Baltimore, Maryland, USA

Continual Learning for LLMs Memory-Augmented Language Models LLM Memory Mathematical Reasoning LLM Mathematical Reasoning

Jun 29, 2026cs.CL

Are We Measuring Strategy or Phrasing? The Gap Between Surface- and Approach-Level Diversity in LLM Math Reasoning

Diversity in LLM mathematical reasoning is critical for exploration, but common diversity metrics mostly capture surface-level variation rather than differences in how a problem is solved. We address this gap by introducing approach-level diversity: variation in strategies across correct solutions to the same problem. Using a human-calibrated LLM judge framework, we show that prior diversity measures are unreliable proxies for approach-level diversity, and this mismatch carries over to diversity-aware RLVR, where target metrics are preserved while approach-level diversity declines. Investigating when approach-level diversity helps and whether it can be directly induced, we find that approach-diverse candidate sets improve test-time scaling. However, optimizing an LLM judge diversity reward during training causes the policy to exploit judge-specific preferences rather than broaden its approaches, leaving direct optimization of approach-level diversity as an open problem. Together, our work introduces the notion of approach-level diversity and uncovers a systematic divergence between surface- and approach-level signals, marking a step toward LLMs that reason in genuinely diverse, human-like ways.

Sangmook Lee, Minbeom Kim, Jeonghye Kim +3

Seoul National University · KAIST

LLM Evaluation Reasoning Diversity RL for Language Model Reasoning Mathematical Reasoning LLM Mathematical Reasoning

Jun 27, 2026math.HO

The Ramanujan Challenge For AI

To help evaluate the mathematical skills of current AI systems, we present a set of formulas for fundamental mathematical constants. These problems are attractive for AI evaluation because they are concrete and can be checked numerically to arbitrary precision, yet proving them may require non-obvious mathematics. Mathematical constants such as

π

,

e

, Catalan's constant, and special values of the Riemann zeta function have fascinated mathematicians for centuries. The search for formulas evaluating mathematical constants has produced some of the most beautiful mathematics in the field, especially in cases that yield irrationality proofs or fast convergence rates. Ramanujan's legacy is emblematic of this tradition. The list we provide contains two types of problems: formulas whose proofs are known to the authors but will remain encrypted for a short initial period; and formulas that are not yet proven. We are curious to see the achievements of AI in both cases.

Michael Shalyt, Rotem Kalisch, Carsten Schneider +7

1Ramanujan Machine Group, Technion, Haifa, Israel. · 2Research Institute for Symbolic Computation, Johannes Kepler Universität Linz, Linz, Austria. · Department of Mathematics, Computer Science and Statistics, Ghent University, Ghent, Belgium. +1

Mathematical Reasoning Benchmarks LLM Mathematical Reasoning

Jun 27, 2026cs.AI

Self-Supervised Theorem Discovery in a Formal Axiomatic System

Recent artificial intelligence (AI) systems have shown remarkable progress in mathematical reasoning. Many existing approaches, including large language models (LLMs), draw on human prior knowledge in the form of mathematical text, code, or theorem libraries. Although these approaches are highly effective in practice, it remains an open question whether an agent can autonomously discover useful theorems without such human priors. We study this question in a formal axiomatic system by developing an agent that starts from axioms and inference rules alone and gradually grows a library of useful theorems. Concretely, we propose a self-supervised theorem-discovery algorithm that alternates between proof search and useful-theorem extraction, building a theorem library whose entries are reused as lemmas for subsequent proof search. Experiments show that the agent discovers tens of thousands of theorems and finds proofs for human-written benchmark problems, suggesting that its discoveries include theorems meaningful from a human mathematical perspective. Furthermore, the discovered theorems improve LLM proof performance when provided as prompt lemmas, indicating that they can serve as external knowledge for LLM reasoning. Our results provide evidence that useful theorems can emerge from proof search without relying on human-provided theorem libraries. More broadly, they suggest a path toward self-evolving AI systems for mathematics whose discoveries remain formally verifiable.

Kazuki Ota, Takayuki Osa, Tatsuya Harada

The University of Tokyo, Japan · RIKEN AIP, Japan

Automated Theorem Proving Knowledge Augmentation for Language Models LLM Mathematical Reasoning

Jun 24, 2026cs.CL

Riazi-8B: An Urdu Large Language Model for Mathematical Reasoning

Recent LLMs demonstrate strong mathematical reasoning capabilities, but existing gains rely heavily on English-centric training resources and benchmarks. As a result, reasoning performance degrades substantially in low-resource languages such as Urdu, where reasoning-oriented datasets and adapted models remain scarce. Urdu lacks both reasoning-oriented resources and models adapted for multi-step mathematical problem solving, limiting the applicability of recent progress to Urdu-speaking users. We address this gap through Riazi-8B, an Urdu mathematical reasoning model developed through a two-step adaptation process comprising continued pre-training on Urdu Wikipedia and supervised fine-tuning on Urdu Chain-of-Thought data derived from GSM8K. We evaluate Riazi-8B on MGSM-Urdu against existing Urdu instruction-tuned models. Our results show consistent improvements in answer correctness, reasoning quality, response completeness, and Urdu generation. Our findings demonstrate that combining Urdu language adaptation with reasoning-focused fine-tuning is an effective strategy for extending mathematical reasoning capabilities to low-resource languages.

Azher Ali, Ibtsam Haider, Raja Khurram Shahzad +2

School of Electrical Engineering and Computer Science (SEECS), National University of Sciences and Technology (NUST), Islamabad, Pakistan · Department of Communication, Quality Management and Information Systems, Mid Sweden University, Ostersund, Sweden

Numerical Reasoning in Language Models LLM Fine-Tuning Language Modeling Low-Resource Language Processing LLM Mathematical Reasoning

Jun 24, 2026cs.AI

Cliff Tokens: Analyzing Failure Trigger Tokens in LLM Mathematical Reasoning

Large language models reach high accuracy in mathematical reasoning, but individual traces on the same problem diverge; some arrive at the correct answer while others fail. Prior work localizes such failures at the step, chunk, or sentence level, or identifies tokens where failure has already occurred. These approaches leave open which token triggers failure. We introduce the cliff token, a token at which the estimated probability of reaching the correct answer (success probability) drops beyond an adaptive threshold. Across seven models and three mathematical reasoning benchmarks (GSM1K, MATH500, AIME 2025), cliff tokens act as failure triggers. For incorrect traces containing cliff tokens, we compare resampling immediately before and after the first cliff token. Resampling before it shows higher pass@

k

at the same sample count. We further introduce a cliff taxonomy of deterministic, uncertain, and sampled-off cliffs, defined by greedy choice and token entropy. Additionally, we show that the three types differ as training signals. Using single-token preference optimization at cliff positions (Cliff-DPO), we find that uncertain and sampled-off cliffs show larger accuracy gains than deterministic cliffs on three evaluation benchmarks. We release token-level rollout data and source code to enable further analysis without regenerating costly rollouts: https://github.com/beaver-22/Cliff-token

Jaeyong Ko, Jinu Lee, Pilsung Kang +1

Seoul National University · Boston University

Numerical Reasoning in Language Models LLM Reliability LLM Mathematical Reasoning

Jun 20, 2026cs.AI

Human vs Machine Mathematical Difficulty on Project Euler: An Experimental Analysis

We study how the effort and success probability of frontier AI systems scale with human difficulty on problems from Project Euler, an online platform of computational mathematics problems. Our dataset, from the MathArena benchmark, consists of 3840 attempts across 50 problems and 26 model configurations, with problem difficulty measured by the site's public human solve times. Motivated by a proposal of Timothy Gowers, we test a power-law relation

t_{\text{machine}} = a \cdot t_{\text{human}}^b

between generated-token cost per successful answer and human time, and find

b < 1

for 20 of the 25 models with usable fits, including the strongest base models; this operationalization therefore does not support an earlier prediction that machines scale worse than humans with difficulty. We also investigate whether success probability on the tested problems can be modeled by a simple exponential decay

p_{\text{success}} = e^{c t_{\text{human}}}

, predicting a linear relation between

\log p_{\text{success}}

and

t_{\text{human}}

. Using a binning approach for data aggregation we find moderate empirical support (median bin-level

R^2 = 0.92

across the 22 best-covered configurations) for this model. Following METR, we also fit logistic success curves and extract 50% task-length horizons

h_{50}

; the strongest configurations in our 20 April 2026 snapshot reach roughly

2.5

--

4.3

hours on our fastest-five human baseline, with a log-linear fit through the state-of-the-art frontier giving a descriptive doubling time of about

75

~days for the SOTA

h_{50}

.

David Holmes, Johannes Schmitt

Mathematical Reasoning Benchmarks Language Model Scaling Laws LLM Mathematical Reasoning

Jun 17, 2026cs.LG

Hard or Just Unreached? Diagnosing the Sampling Blind Spot in Math-Reasoning Difficulty Estimation

Math and science reasoning benchmarks rely on pass@k, the fraction of sampled chains that reach gold, as the canonical per-example difficulty signal. The same signal drives RL with verifiable rewards, math data curation, synthetic curricula, and verifier training. We show this proxy has a persistent blind spot on its hardest stratum: on the eight free-form math cells we test (GSM8K and MATH across four open-weight models), 10.3-22.9% of the examples that no sampling seed solves in six tries are instead solved at matched compute by a six-chain deterministic regime. These are greedy decoding plus five cheap residual-stream perturbations applied via activation grafting, while greedy alone solves at most 6% on these math cells. Recovery scales with the additional budget, across perturbations whose mechanistic distinctness we verify across all twelve cells (cross-kind fix-set Jaccard <= 0.47 in every setup). Activation grafting is used as an intervention on internal representations, not a decoding method; we use it purely as a diagnostic and diversification tool, and our recovered items show that the pass@k= 0 % stratum is structurally identifiable in the residual stream rather than that the unmodified model reaches them under ordinary inference.

Luca Zhou, Sajel Shah, Emanuele Rodolà +1

1Sapienza University of Rome · 2Not Diamond · 3Paradigma

Mathematical Reasoning Benchmarks LLM Evaluation LLM Mathematical Reasoning

Jun 16, 2026cs.AI

First Proof Second Batch

To assess the ability of current AI systems to correctly solve research-level mathematics problems, we tested several AI systems on a set of ten problems in a broad range of mathematical fields; these problems arose naturally in the research process of the contributors. This document includes the problems, our methodology, and the results of our testing. We provide links to supplementary documents including the human solutions, the AI-generated solutions, and the referee reports and logs for the AI-generated solutions. The ten problems were contributed by the following mathematicians: (1) Dariusz Kalociński and Theodore A. Slaman, (2) Richard Schwartz, (3) Aleksa Milojevic and Benny Sudakov, (4) Larry Guth, (5) Oleg Butkovsky, Jonathan Mattingly, and Lorenzo Zambotti, (6) Joshua Evan Greene and Duncan McCoy, (7) Sucharit Sarkar, (8) Sam Payne and Jidong (Jayden) Wang, (9) Sylvie Corteel and John Lentfer, (10) Srivatsav Kunnawalkam Elayavalli.

Mohammed Abouzaid, Nikhil Srivastava, Rachel Ward +1

Stanford University · University of California, Berkeley · University of Texas at Austin +1

Mathematical Reasoning Benchmarks LLM Mathematical Reasoning Generative AI Evaluation

Jun 16, 2026cs.AI

MathVis-Fine: Aligning Visual Supervision with Necessity via Progressive Dependency-Guided Training for Multimodal Mathematical Reasoning

Chain-of-Thought (CoT) reasoning has extended from purely linguistic domains to multimodal scenarios; however, existing approaches often treat visual inputs as homogeneous or auxiliary signals, failing to capture the intricate and sample-specific dependencies between text and images in mathematical problem-solving. This gives rise to two core issues: first, the supervisory signals for visual content are generalized and coarse-grained, lacking adaptation to the actual necessity of visual information in each sample; second, training feedback becomes inaccurate when visual rewards are uniformly applied without distinguishing the complementary relationships among inputs. These limitations hinder models from achieving precise multimodal reasoning. In this work, we propose a framework for modeling fine-grained visual dependencies in mathematical reasoning. We first construct the MathVis-Fine dataset, augmenting fine-grained visual annotations with visual dependency ratings. Building upon this dataset, we introduce a two-stage progressive visual enhancement training paradigm that balances answer correctness rewards and visual grounding rewards according to the intrinsic visual dependency level of each sample, thereby mitigating reward bias and improving supervision accuracy. Extensive experiments demonstrate that the MathVis-Fine framework effectively enhances visual perception progressively based on visual dependency, offering a more precise training framework for multimodal mathematical reasoning. We will release the dataset upon acceptance.

Wanshi Xu, Haokun Zhao, Haidong Yuan +2

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

Multimodal Reasoning Multimodal Reward Modeling Multimodal CoT Reasoning Visually Grounded Reasoning Fine-Grained Visual Perception LLM Mathematical Reasoning

Jun 16, 2026cs.PL

Visored: A Controlled-Natural-Language Prover for LLM-Generated Mathematics

We present a dependent-type-based prover designed around the way LLMs (and humans) tend to write mathematics, complementing existing systems such as Lean and Rocq. Its core design choices are a surface that imitates mathematical natural language and a rule-driven automation layer that closes the routine steps a textbook would omit, so that an accepted proof can be re-emitted as a checked Lean file. Early experiments suggest that, even without any prover-specific training data, LLMs can learn to use it effectively on the miniF2F benchmark. Lean output excerpts: https://github.com/xiyuzhai-husky-lang/visored/

Xiyu Zhai, Xinyi Chen, Yiping Wang +3

University of Washington · University of Innsbruck

Automated Theorem Proving Lean Theorem Proving LLM Mathematical Reasoning

Jun 16, 2026cs.LG

The Discrete-Log Clock: How a Transformer Learns Modular Multiplication

When small transformers grok modular multiplication, prior work reports that the learned embedding has a "dense" Fourier spectrum requiring all frequencies. This contrasts with modular addition, where only a sparse set of key frequencies suffices. We show this density is an artifact of analyzing in the wrong basis. The natural Fourier transform for multiplication is not the standard additive DFT but the multiplicative character transform, which decomposes functions on the multiplicative group

(\mathbb{Z}/p\mathbb{Z})^*

into its irreducible representations. Applying this transform to a grokked transformer trained on

a \cdot b \bmod 113

, we find the embedding spectrum becomes highly sparse (Gini coefficient 0.58 vs. 0.07 in the additive basis) with only 4 key frequencies carrying significant energy. Furthermore, 96.9% of MLP neurons are cleanly tuned to a single multiplicative frequency, and neuron activation heatmaps reveal 2D-periodic structure when reordered by the discrete logarithm. These results demonstrate the transformer reduces multiplication to addition in discrete-log space, implementing a "Discrete-Log Clock" algorithm analogous to Nanda et al.'s Clock algorithm for addition. The methodology generalizes: matching the analysis basis to the algebraic structure of the task reveals interpretable structure where standard tools see noise.

Huu Danh Nguyen

Stanford University

Transformer Interpretability Frequency-Domain Feature Learning Spectral Methods Modular Arithmetic LLM Mathematical Reasoning

Jun 15, 2026cs.AI

Quantifying Consistency in LLM Logical Reasoning via Structural Uncertainty

Large language models can arrive at the same answer through reasoning paths that are unstable, contradictory, or difficult to rank consistently -- a failure mode especially prevalent in multi-step deductive reasoning. Existing methods assess reliability primarily through output dispersion -- measuring how much sampled answers differ -- but this discards a complementary signal: whether the model can consistently rank competing reasoning candidates. We propose structural uncertainty, a consistency-aware framework derived from the stability of self-preference-induced rankings over sampled reasoning solutions. Given a query, we generate multiple candidate solutions and ask the model to judge pairwise preferences among its own outputs. We aggregate self-preferences into ranking distributions via Bradley-Terry modeling with PageRank, and decompose the signal into two entropy-based components: across-trial ranking instability and within-trial candidate ambiguity. Across five LLMs and eight benchmarks, structural signals provide information complementary to answer dispersion: on logical and mathematical reasoning tasks, the combination improves identification of unreliable instances, while on factual retrieval the structural signal collapses toward uniformity, diagnosing a regime boundary where reasoning-level consistency evaluation is uninformative. The two components relate differently to accuracy: within-trial ambiguity correlates positively with correctness -- consistent with settings where multiple plausible solution paths remain competitive -- while across-trial instability correlates negatively, signaling unreliable reasoning. Structural uncertainty is best understood not as a universal confidence estimator, but as a regime-sensitive evaluator of logical reasoning consistency.

Baishali Chaudhury, Mengdie Flora Wang, Hyunji Hayley Park +3

AWS Generative AI Innovation Center

Reasoning Consistency in Language Models LLM Reliability LLM Uncertainty Estimation LLM Mathematical Reasoning Uncertainty Decomposition LLM Reasoning

Jun 15, 2026cs.AI

The Quality-Utility Paradox: Why High-Reward Data Impairs Small Model Mathematical Reasoning

Knowledge distillation from powerful reasoning models is widely used to improve Small Language Models (SLMs) on mathematical reasoning, often assuming that traces with higher reward model scores provide more useful supervision. We identify a counterintuitive \textbf{Quality-Utility Paradox} in mathematical reasoning distillation. Data refined or synthesized by a stronger Oracle obtains higher perceived quality according to reward models, yet consistently underperforms traces generated by the SLM itself and selected through rejection sampling across Qwen2.5, LLaMA-3, and DeepSeek families. Our analysis shows that Oracle refinement couples logical repair with distributional drift away from the SLM's native reasoning distribution. This drift increases the learner's adaptation cost and can outweigh the benefit of improved reasoning logic. To test this mechanism, we introduce \textbf{Style-Aligned Refinement}, which preserves the native trajectory of the SLM while retaining logical repair from the Oracle. This intervention lowers adaptation cost and restores downstream utility. These findings suggest that effective mathematical reasoning distillation should jointly optimize perceived solution quality and learner-data compatibility, rather than relying solely on reward-model scores. The datasets and code are available at https://github.com/Dracoqhl/Quality-Utility-Paradox.

Haolong Qian, Xianliang Yang, Yinuo ma +6

Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen, China · Microsoft Research Asia, Microsoft, Beijing, China · Faculty of Computer Science and Artificial Intelligence, Shenzhen University of Advanced Technology, Shenzhen, China +1

CoT Distillation Language Model Distillation Rejection Sampling LLM Mathematical Reasoning

Jun 14, 2026cs.IR

Theorem-Grounded Execution Ontologies for Interpretable Machine Reasoning

Large language models have achieved impressive performance on reasoning tasks spanning mathematics, science, programming, and commonsense inference. Despite these advances, their reasoning processes remain largely latent, making them difficult to interpret, verify, replay, debug, and transfer across domains. Existing approaches such as chain-of-thought, tree-of-thoughts, graph-of-thoughts, and tool-augmented reasoning expose intermediate reasoning artifacts but typically lack explicit execution semantics, formal state representations, and verifiable reasoning structures. We introduce Theorem-Grounded Execution Ontologies (TGEO), a framework that models reasoning as an executable state-transition process rather than a sequence of generated tokens. Given an input problem, TGEO identifies relevant theorem families, binds the problem to a domain ontology, discovers semantic objects, instantiates states and operators, constructs predicates and contracts, and synthesizes an executable reasoning graph. The resulting graph provides an interpretable, replayable, and auditable representation of reasoning in which every state transition, operator application, and validation step is explicitly represented. TGEO integrates five architectural components: (1) theorem-grounded reasoning priors, (2) executable ontologies, (3) operator-mediated state transitions, (4) predicate and contract-based execution validation, and (5) architectural auditing and failure localization. We evaluate TGEO on theorem-intensive reasoning tasks derived from mathematical benchmark domains and a curated Golden Execution Suite. Our findings demonstrate the value of executable reasoning representations for interpretable, verifiable, and reproducible AI reasoning systems.

Raghu Anantharangachar

Independent Researcher · Bangalore, India

Interpretable ML Formal Verification LLM Mathematical Reasoning LLM Reasoning

Jun 14, 2026cs.CL

Formalize Once, Edit the Rest: Efficient Lean-Based Answer Selection for Math Reasoning

With large language models (LLMs) increasingly applied to mathematical reasoning, formal proof assistants such as Lean can be leveraged to verify reasoning outputs with machine-checkable rigor, enabling use cases such as answer selection in test-time scaling with K sampled candidate answers. However, employing Lean requires that LLM outputs, originally in natural language, first be formalized. Existing Lean-based answer-selection work uses an autoformalization model to generate a formal statement in Lean for each candidate answer independently, incurring a significant computational cost. We propose BASE, a base-and-edit pipeline that formalizes a single base candidate per problem and derives the remaining K-1 statements by editing the answer expression in place. To facilitate this, we train a rewriter model LEANSCRIBE to localize the answer in the base formalization and generate a reusable edit function for the other K-1 candidates. BASE simultaneously improves selection accuracy and reduces formalization cost - a Pareto improvement that holds on all 12 (dataset, solver) configurations across four benchmarks and three solvers, cutting autoformalizer calls by about 5x at K=8, with the reduction expected to become larger as K grows. Code is available at https://github.com/ucr-rai/base-and-edit.

Ji Feng, Zhouxing Shi

University of California, Riverside

LLM Answer Verification Lean Theorem Proving Autoformalization LLM Mathematical Reasoning

Jun 13, 2026cs.AI

Mask-Proof: An LLM-based Automated Data Curation Pipeline on Mathematical Proofs

Large language models (LLMs) are increasingly capable of mathematical problem solving and can even assist with research-level proofs, yet we still lack a scalable and reproducible way to measure step-level reasoning in long proofs across diverse sources. This evaluation gap limits trustworthy AI assistance in proof-certified scientific progress. Existing evaluations often emphasize final answers or rely on costly expert grading, while end-to-end proof generation remains open-ended and hard to verify automatically. We introduce Mask-Proof, a pipeline that turns real proofs into automatically checkable masked-step tasks. It masks key formula steps, provides the necessary surrounding context, and evaluates model reconstructions with an LLM-based equivalence judge using repeated votes for stability. The resulting Mask-ProofBench contains 292 curated problems across diverse research areas. Experiments with 17 models show that reasoning-enhanced models outperform standard models by 12% to 27%. Our evaluator achieves 96.8% agreement with expert annotators, enabling faithful, reproducible, and comparable measurement of step-level mathematical reasoning. Benchmark, annotations, and code are available at https://github.com/weating/Mask-Proof.

Jierui Zhang, Siyuan Tan, Xinhang Li +8

School of Computer Science Beijing University of Posts and Telecommunications Beijing, China · Graduate College for Engineers Beijing University of Posts and Telecommunications Beijing, China · School of Mathematical Sciences Fudan University Shanghai, China +6

Mathematical Reasoning Benchmarks LLM Evaluation Automated Theorem Proving Training Data Curation LLM Mathematical Reasoning

Jun 12, 2026cs.DL

Failure Modes of Large Language Models on Research-Level Mathematics: A Taxonomy and an Empirical Characterisation

The "First Proof" benchmark [1] posed ten research-level mathematics questions to the strongest publicly available LLMs and found them consistently wrong-not silent, but confidently, fluently wrong. This paper asks why. Working from the per-question post-mortems in First Proof's Appendix A, I identify four failure modes: citation fabrication (F1), premise smuggling (F2), silent problem reformulation (F3), and local-to-global compatibility gaps (F4). I then audit eight one-shot proofs generated by Gemini 2.5 Flash on Questions 1, 2, and 5 of the benchmark, using two instruments built specifically to surface F1 and F2. The central finding is uncomfortable for anyone who sees retrieval-augmented generation (RAG) as the obvious fix: not one of the eight proofs contained a confirmed fabricated citation, yet every single one contained at least one load-bearing claim asserted as a "fundamental result" or "standard argument" with no justification attached. That failure mode-F2, premise smuggling-is invisible to citation verification by design. A premise-audit instrument I introduce flags it at 100% precision (5/5 judge-confirmed flags are true positives) and 50% proof-level recall in this corpus. The taxonomy and the audit together suggest that the right long-term objective is building inference-time pipelines that prevent these failure modes from occurring, not just detecting them after the fact. Index Terms--Large language models, mathematical reasoning, hallucination, premise smuggling, failure-mode taxonomy.

Arnesh Banerjee, Ayushi Bhattacharjee

Dept. of CSE (Data Science) Heritage Institute of Technology Kolkata, India

Mathematical Reasoning Benchmarks LLM Auditing Hallucination in Language Models LLM Mathematical Reasoning

Jun 11, 2026cs.LG

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling

We present MaxProof, a population-level test-time scaling framework for competition-level mathematical proof in the MiniMax-M3 series. M3 first trains three proof-oriented capabilities -- proof generation, proof verification, and critique-conditioned proof repair -- using a defense-in-depth generative verifier engineered for low false-positive rate. These capabilities are merged into a single released M3 model. At test time, MaxProof treats the model as a generator, verifier, refiner, and ranker, searches over a population of candidate proofs, and returns one final proof through tournament selection. With MaxProof test-time scaling, the M3 model reaches 35/42 on IMO 2025 and 36/42 on USAMO 2026, exceeding the human gold-medal threshold on both.

Jiacheng Chen, Xinyu Zhang, Shunkai Zhang +20

MiniMax · The Chinese University of Hong Kong · Fudan University +2

Automated Theorem Proving Inference-Time Scaling Reinforcement Learning with Verifiable Rewards LLM Mathematical Reasoning

Jun 10, 2026cs.AI

Architecture-Aware Reinforcement Learning Makes Sliding-Window Attention Competitive in Math Reasoning

The rapid progress of reasoning and agentic large language models (LLMs) has increased the demand for long-context inference, but self-attention (SA) scales quadratically with context length. To address this, we study SWARR (Sliding-Window Attention with Reinforced Adaptation for Math Reasoning), a practical recipe for adapting SWA models to mathematical reasoning. SWARR has two stages: (1) efficient conversion from a pretrained SA model to SWA with supervised fine-tuning (SFT), which avoids pretraining a new base model, and (2) policy adaptation with reinforcement learning (RL). We find that SWA still underperforms SA after SFT, and we hypothesize that this gap is caused in part by a data-architecture mismatch: most SFT data are prepared for SA models and may contain long-range dependencies that are difficult for SWA to model. Because on-policy RL optimizes self-generated trajectories under the SWA constraint, it can adapt trajectories to better match SWA. Experiments on mathematical reasoning benchmarks show that this recipe substantially narrows the gap between SWA and SA, recovering much of the accuracy lost during SWA conversion while preserving the efficiency benefits of linear-complexity attention. Our central contribution is the empirical finding that RL changes the conclusion one would draw from conversion and SFT alone about SWA's viability for math reasoning.

Kai Liu, Peijie Dong, Xinchen Xie +5

Shanghai Research Institute for Intelligent Autonomous Systems, Tongji University · Hong Kong University of Science and Technology (Guangzhou) · Shanghai AI Laboratory

RL for Language Model Reasoning LLM Inference Acceleration Sliding-Window Attention RL Fine-Tuning LLM Mathematical Reasoning

Jun 9, 2026cs.AI

Evaluating Research-Level Math Proofs via Strict Step-Level Verification

Large Language Models (LLMs) struggle to rigorously verify complex mathematical proofs. Standard global evaluation approaches suffer from "context poisoning," in which superficially plausible statements mask subtle logical flaws, leading to hallucination or over-skepticism. To address this, we shift from global evaluation to strict step-level verification: our framework maintains detailed context for each deduction step and strictly constrains the sources of applied theorems. We evaluate on a carefully curated adversarial diagnostic suite of research-level proofs drawn from the FirstProof challenge. A systematic ablation study demonstrates that these deductive constraints are indispensable, as unconstrained global prompting consistently fails to localize subtle logical errors. Beyond outperforming global evaluation, our approach fundamentally alters the failure taxonomy. Error analysis reveals that, rather than exhibiting severe logical hallucinations, remaining rejections are primarily instances of "pedantic hyper-rigor" stemming from unstated domain conventions, effectively exposing implicit ambiguities within the expert benchmark itself. Our findings suggest that prompting agents to organize their verification notes in a cautious, human-mathematician-like manner can substantially improve their ability to distinguish rigorous proofs from flawed ones, with the potential to strengthen agentic reasoning on frontier mathematical concepts that the base model does not already know well, and to lay a theoretical foundation for future automated proof-review systems. Code and prompts are available at GitHub.

Yifeng Sun

Independent Researcher

LLM Answer Verification Automated Theorem Proving LLM Mathematical Reasoning Reasoning Verification

Jun 9, 2026cs.LG

N-GRPO: Embedding-Level Neighbor Mixing for Enhanced Policy Optimization

The success of Large Language Models in mathematical reasoning relies heavily on the generation of diverse and valid solution paths during the rollout phase. However, current rollout techniques face a fundamental trade-off: token-level sampling often yields redundant trajectories that differ only in rephrasing, while embedding-level methods utilizing random noise frequently disrupt semantic consistency. To resolve this, we introduce N-GRPO, a novel exploration strategy integrated into the Group Relative Policy Optimization (GRPO) framework. Rather than relying on token-level sampling or native embedding-level noise, our approach leverages Semantic Neighbor Mixing. This mechanism dynamically constructs input representations by mixing the embeddings of an anchor token and its nearest semantic neighbors, thereby injecting diversity while strictly adhering to the local semantic manifold. Experimental evaluations on the DeepSeek-R1-Distill-Qwen models across different sizes show that N-GRPO not only achieves consistent improvements over strong baselines on math reasoning benchmarks but also exhibits robust generalization capabilities on out-of-distribution tasks.

Xukun Zhu, Hang Yu, Peng Di +1

Zhejiang University · Ant Group

RL for Language Model Reasoning Group Relative Policy Optimization LLM Mathematical Reasoning

Jun 9, 2026cs.AI

ComBench: A Benchmark for Rigorous Proof Reasoning and Constructive Realization in Olympiad-Level Combinatorics

Combinatorics is central to Olympiad-level mathematical problem solving, requiring deep discrete reasoning, creative constructions, and rigorous structural insight. Recent evidence suggests that even today's strongest frontier models remain uneven on Olympiad combinatorics, revealing a gap in creative mathematical reasoning. We introduce ComBench, an Olympiad-level combinatorics benchmark for evaluating and diagnosing the combinatorial reasoning capabilities of large language models. ComBench contains 100 human-annotated competition-level problems organized around two complementary settings: analysis-centric problems, which primarily require rigorous mathematical arguments, and construction-centric problems, which require explicit constructions in addition to correctness justifications. The evaluation protocol combines rubric-guided proof grading with deterministic construction verification, exposing cases where proof quality and construction validity diverge. Experiments on frontier open- and closed-source models show that ComBench is far from saturated: the strongest model reaches 65.4% overall Avg. and 75.3% overall Best@4. We further find that Rigorous Proof Reasoning and Constructive Realization are distinct capabilities: Kimi-K2.6 trails GPT-5.5 on analysis-centric proof grading but surpasses it on construction-centric Best@4, while Existence and Construction problems remain consistently hardest across representative frontier models.

Shunkai Zhang, Haoran Zhang, Yun Luo +15

2Peking University · 1Shanghai AI Laboratory · 3Shanghai Jiao Tong University +2

Mathematical Reasoning Benchmarks Benchmark Design Mathematical Reasoning LLM Mathematical Reasoning

Jun 9, 2026cs.CL

KCSAT-ML: Probing Reasoning Models with Nationwide-Cohort Human Difficulty

Math reasoning benchmarks have proliferated, yet most lack a per-item difficulty signal grounded in actual human performance. We introduce KCSAT-ML, a decade (2014-2025) of Korean College Scholastic Ability Test (KCSAT; Suneung) mathematics: 664 problems with a 339-item core set carrying official per-item error rates from nationwide cohorts of hundreds of thousands of examinees. We pair the benchmark with Difficulty-aligned Reasoning Gain (DRG): a score-orthogonal metric that asks whether a model's mistakes concentrate on the items humans found hard, or on items humans found easy. Together they expose, across a wide range of VLMs (and LLMs via OCR), three patterns: (i) low-budget accuracy collapses on the high-human-error tail at every model size; (ii) test-time scaling (TTS) raises token use roughly linearly with cohort error rate, while accuracy gains follow a non-monotonic curve; (iii) within a single family, TTS flips between anti-scaling on the hardest items and overthinking on easier ones -- two faces of the same alignment failure. On DRG, models with near-identical accuracy can sit at near-opposite values: one model gets wrong what humans also find hard, while another solves the hardest items yet fails on items humans find easy -- a contrast that aggregate accuracy hides. Our code and dataset builder will be open-sourced at https://github.com/naver-ai/KCSAT-ML.

Sanghee Park, Geewook Kim, Kee-Eung Kim

NAVER Cloud AI · KAIST AI

VLM Evaluation Mathematical Reasoning Benchmarks LLM Evaluation Test-Time Scaling LLM Mathematical Reasoning

Jun 8, 2026cs.AI

Diverse Thinking Schemata Elicit Better Reasoning in Large Language Models

Large reasoning models (LRMs) have attracted increasing attention for their ability to solve complex mathematical problems by generating extended reasoning chains. In this work, we focus on two critical yet underexplored aspects of the reasoning process: reasoning transitions capturing the distinct transitions between reasoning steps and answer candidates reflecting the variety of solution paths produced by the model. We collectively define these two aspects as thinking schemata. We observe a correlation between the diversity of thinking schemata and model performance, which motivates us to enhance diversity as a means to further improve reasoning potential. To this end, we propose Diverse Schemata Policy Optimization (DiScO), a framework that first endows the model with schemata awareness, then encourages diversity through reinforcement learning, and further promotes diverse reasoning at inference time. Experiments on multiple mathematical reasoning benchmarks demonstrate that DiScO consistently outperforms standard group relative policy optimization. Beyond accuracy, human-annotated analyses show that DiScO substantially improves the model's ability to recover from erroneous initial attempts. Overall, our work suggests the important role that diversity of the thinking schemata plays and points to scaling along the diversity dimension as a promising research direction.

Xinyue Liang, Yizhe Yang, Yu Bai +3

School of Computer Science and Technology, Beijing Institute of Technology, Beijing, China

Reasoning Diversity RL for Language Model Reasoning LLM Mathematical Reasoning Large Reasoning Models

Jun 7, 2026cs.AI

Artificial Intelligence for Mathematical Reasoning: An Integrated Survey of Language Models, Neuro-symbolic Systems, and Verified Discovery

Mathematical reasoning has long served as a stringent test of machine intelligence; over the past decade, it has moved from a niche problem within NLP to one of the most consequential AI frontiers. This survey provides a unified account of the field's evolution, from early rule-based math word problem (MWP) solvers and template-driven geometry systems, through neural expression generation and LLM prompting, to contemporary reasoning models, multi-agent systems, neuro-symbolic theorem provers, and verified discovery workflows. We organize the landscape along four axes: (i) informal reasoning over text and diagrams, spanning MWP solving, multimodal geometry, and VLMs; (ii) formal reasoning in proof assistants, including autoformalization, tactic prediction, compiler-guided repair, and proof search; (iii) mathematical discovery, where systems propose constructions, improve bounds, or assist attacks on open problems; and (iv) the inference and training-time techniques, including CoT prompting, tool use, process reward models, and RLVR, that increasingly connect generation with verification. We catalog major benchmarks across grade-school arithmetic, competition mathematics, geometry, formal proving, multimodal and multilingual reasoning, and expert evaluation, and we examine benchmark saturation, contamination, reporting mismatches, and the distinction between pass@1, majority voting, and verifier-assisted pass@

k

. We critically assess failure modes: brittleness under perturbation, reward hacking, multimodal grounding failures, fragile formalization, and the energy cost of reasoning-scale inference. Drawing on recent perspectives from working mathematicians, we identify future directions centered on verified-discovery workflows, reasoning efficiency, and infrastructure to make AI-assisted formalization broadly usable. Companion materials: https://github.com/Starscream-11813/awesome-AI4Math.

Syed Rifat Raiyan, Mohsinul Kabir, Hasan Mahmud +2

Systems and Software Lab (SSL), Department of Computer Science and Engineering, Islamic University of Technology, Gazipur, Bangladesh · Department of Computer Science, University of Manchester, Manchester, United Kingdom

Mathematical Reasoning Benchmarks Automated Theorem Proving Symbolic Reasoning Mathematical Reasoning Neuro-Symbolic Reasoning LLM Mathematical Reasoning

Jun 5, 2026cs.LG

A Comprehensive Anatomy of Human and DeepSeek-R1 LLM Mathematical Reasoning

The emergence of "Aha moments" in large language models, particularly DeepSeek-R1-0120, has raised the question of whether these systems genuinely reason or merely imitate the appearance of reasoning. We conduct a comprehensive empirical comparison between model and human reasoning across all 30 problems from AIME 2025, exhaustively annotating 10,247 reasoning steps into five functional categories: Analysis, Inference, Branch, Backtrace, and Reflection. We find a clear structural difference. Human solutions maintain a compact alternation between analysis and deduction, whereas DeepSeek-R1 frequently revisits intermediate results, performs shallow and often unnecessary verification, and loops through local checks without meaningful logical progress. We describe this as topological mimicry: reproducing the surface form of reasoning without its functional role. Despite this, we identify two signals of genuine reasoning. First, successful traces exhibit stable use of branching and backtracking, while failed traces either underuse or overuse exploratory actions. Second, reflection is only effective when placed within deductive inference; reflections trapped in analysis loops focus on local numerical details while missing global logical errors. These findings suggest that current long-CoT models may be rewarded more for the appearance of reasoning than for genuine deductive progress. We discuss directions for improving evaluation and training, including measuring cross-trace stability, penalising "spinning-wheel" traces, encouraging deeper logical correction, and reallocating inference-time compute toward deduction and backtracking. Overall, reasoning quality depends not simply on how much reflection occurs, but on whether reflection appears consistently and at the appropriate logical scale.

Yuxiang Chen, Jun Wang

UCL Centre for Artificial Intelligence

CoT Reasoning Structured Reasoning Metacognition in Language Models LLM Mathematical Reasoning Large Reasoning Models Reasoning Trace Analysis

Jun 5, 2026cs.CL

From Correctness to Utility: Gain-Based Prefix Evaluation for LLM Reasoning

Reasoning prefixes shape the future trajectory of LLM problem solving, yet existing process reward models usually evaluate them through local step correctness. We argue that correctness is a useful but indirect proxy for the effect we ultimately care about: whether a prefix increases the probability of successful completion. We define this effect as prefix gain, the solve-rate improvement induced by conditioning lightweight student model group on a prefix, and use it to train a Prefix Utility Model (PUM) with a simple pairwise ranking objective. PUM learns outcome-grounded prefix utility and can score both complete trajectories and partial reasoning prefixes. Across Best-of-

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selection, beam search, and reinforcement learning on mathematical reasoning, PUM provides a strong prefix-level supervision signal, especially when candidate pools are large, search budgets increase, or rule-based rewards are sparse. We release all data, models, and code at https://zhiqix.github.io/pum-project-page.

Yuhang Zhou, Yixin Cao, Guangnan Ye

Fudan University · Shanghai Innovation Institute

Process Reward Models RL for Language Model Reasoning LLM Mathematical Reasoning