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.

Jul 13Week of Sep 28

Latest papers 194

Jun 5, 2026cs.LG

RASFT: Rollout-Adaptive Supervised Fine-Tuning for Reasoning

Supervised fine-tuning (SFT) is a prevailing method for adapting large language models to reasoning tasks by imitating offline expert demonstrations, often treating a single expert trajectory as the target behavior. However, reasoning is not simple path imitation: rigidly following one demonstrated solution may overfit to surface forms and suppress the model's own reasoning distribution. We propose Rollout-Adaptive Supervised Fine-Tuning (RASFT), a policy-aware SFT framework that calibrates expert supervision according to problem-level solvability estimated from verified on-policy rollouts. For each problem, RASFT strengthens expert guidance when the current policy struggles, while relaxing rigid imitation and incorporating correct self-generated trajectories when the model already exhibits reliable reasoning behavior. To preserve useful reasoning priors, RASFT further introduces a clipped inverse ratio between the frozen reference model and the current policy to constrain excessive policy drift. Experiments across multiple models on six mathematical reasoning benchmarks and two code reasoning benchmarks show that RASFT achieves better overall performance than SFT, SFT variants, and representative RL methods. The code is available at https://github.com/zjd1sq/RASFT.

Yongliang Miao, Fengyuan Liu, Wei Shi +4

The Chinese University of Hong Kong, Shenzhen · Shanghai Artificial Intelligence Laboratory · New Jersey Institute of Technology +1

Supervised Fine-Tuning LLM Mathematical Reasoning LLM Reasoning

Jun 4, 2026cs.CL

When to Think Deeply: Inhibitory Deliberation for LLM Reasoning

Reasoning Large Language Models can improve problem-solving performance through deliberative inference, but invoking slow reasoning for every input is computationally expensive and often unnecessary. We propose IDPR, a framework for response-conditioned inhibitory deliberation. IDPR first generates a concise intuitive answer and then uses an inhibition controller to decide whether that specific response should be released or suppressed in favor of slow reasoning. Unlike input-only routers, the inhibition controller conditions on the fast answer and fast-side evidence, including confidence, logit margin, parseability, and generation cost. We train the controller from paired fast-slow outcomes and select the inhibition threshold on a held-out validation set under an accuracy-first slow-call budget. On a held-out 5,000-example mathematical reasoning test set, IDPR invokes slow reasoning on only 8.20% of examples and improves accuracy from 47.90% to 48.92%. Under the same slow-call budget, random routing decreases accuracy to 46.76%, while the strongest confidence-based baseline reaches 48.22%. IDPR also achieves the highest corrective precision, showing that response-conditioned inhibition better identifies fast answers that benefit from slow reasoning.

Zhixuan He, Yue Feng

University of Birmingham, United Kingdom

Efficient Language Model Reasoning Inference-Time Compute Allocation LLM Routing Efficient Language Model Inference LLM Mathematical Reasoning

Jun 4, 2026math.HO

Benchmarks in Leipzig

Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the three-day workshop Benchmarks in Leipzig with 35 participants at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. We present the resulting collection of 100~questions. We evaluated these questions in three stages: a single attempt by five state-of-the-art LLMs and their predecessors, followed by a 20-runs-per-model evaluation with three of these models, and finally a 3-run attempt with two heavy-thinking models. After Stage 1, 41 questions remained completely unsolved; after Stage 2, this count dropped to 16; and we concluded Stage 3 with only 2 unsolved questions. This demonstrates that the mathematical reasoning capabilities of LLMs are becoming impressive. In September 2026, we added a fourth stage in which the next generation of models attempted all 100 questions once more, after which only 1 question remains unsolved.

Andrei Balakin, Miklós Bóna, Marie-Charlotte Brandenburg +45

Mathematical Reasoning Benchmarks LLM Evaluation LLM Mathematical Reasoning

Jun 4, 2026cs.AI

Critic-Guided Heterogeneous Multi-Agent Reasoning for Reliable Mathematical Problem Solving

Recent Large Language Models (LLMs) have shown impressive reasoning abilities; but they are still susceptible to hallucinations, intermediate reasoning mistakes, and unreliable reasoning results in complex mathematical reasoning problems. In this study, we introduce a critic-based heterogeneous multi-agent approach to improve the dependability of mathematical reasoning. This framework incorporates several LLM agents of different specialties and employs a critic-driven adaptive learning system to assess and guide the reasoning process based on intermediate feedback. The system adopts a generator-validator framework, with the validator not only determining correctness but also offering critiques to guide regeneration of solutions. This allows for adaptive error correction and prevents error cascading. Our experiments on the GSM8K benchmark show that the proposed method achieves up to 13% accuracy improvement over single-shot and non-critic models. Additionally, findings suggest that heterogeneity and critique reduce the need for large models, allowing smaller models to perform on par. Ablation studies reveal the main performance gains are due to the critic-based feedback loop and not model size. In summary, the proposed approach showcases the benefits of combining heterogeneous multi-agent collaboration and critique to obtain reliable and interpretable reasoning systems.

Muhammad Talha Sharif, Abdul Rehman

dept. of Computer Science · National University of Computer and Emerging Sciences · Islamabad, Pakistan

LLM Self-Correction Multi-Agent Reasoning LLM Mathematical Reasoning LLM Agent Reliability

Jun 4, 2026cs.AI

Evaluation of LLMs for Mathematical Formalization in Lean

Within the past few years, the ability of Large Language Models (LLMs) to generate formal mathematical proofs has improved drastically. We provide a comparison of various LLMs' effectiveness in producing formal proofs in Lean 4 with the goal of assisting those seeking to use LLMs to support their own projects. We utilize both pass@

k

and refine@

k

metrics as the benchmark for our comparison and evaluate on subsets of both miniF2F and miniCTX datasets. Our testing shows that overall, Gemini 3.1 Pro and Claude Opus 4.7 perform best. Gemini 3.1 Pro achieved a 92% success rate on miniF2F via refine@32 whereas Opus 4.7 achieved a 86% success rate on miniCTX via refine@32. When taking cost into account, NVIDIA Nemotron 3 Super and GPT-OSS 120B were the most efficient, with competitive accuracies and average costs of

<\

0.01$ per correct proof.

Tyson Klingner, Drew Bladek, Escher Crawford +6

Math AI Lab, University of Washington, Seattle, WA, USA. · Math AI Lab, University of Washington, Seattle, WA

LLM Evaluation Lean Theorem Proving LLM Mathematical Reasoning

Jun 3, 2026econ.GN

Can AI Refute Economic Theory? Evidence from Beyond the Knowledge Cutoff

Can artificial intelligence (AI) refute economic theory? I document experiments in which I asked several AI models (Gemini, Refine, Claude, and ChatGPT) to check the correctness of four published papers in economic theory, each containing an error that I helped identify or correct. ChatGPT Pro performed best, occasionally constructing counterexamples and corrected proofs, while other models fared worse. However, no model located a true error without substantial human guidance, and data contamination complicates interpretation. I argue that a competent human paired with a frontier model can outperform current peer review, but AI cannot yet refute economic theory on its own.

Alexis Akira Toda

Department of Economics, Emory University.

Human-in-the-Loop AI Automated Peer Review LLM Mathematical Reasoning

Jun 3, 2026cs.CL

Arithmetic Pedagogy for Language Models

We investigate whether methods of human mathematics pedagogy can guide the training of language models toward arithmetic reasoning. Building on the GASING method -- an Indonesian pedagogy that solves basic arithmetic through a left-to-right procedure aligned with the causal order of token generation -- we operationalize each operation as a computational procedure whose execution trace is serialized into natural-language Chain-of-Thought (CoT) supervision. A small GPT-2 decoder (86M parameters) with a syllabic-agglutinative TOBA tokenizer for Indonesian is trained from scratch on this data using only a next-token prediction objective, without reinforcement learning or reward-based optimization. Monitoring training reveals three distinct learning phases, and mechanistic analyses -- attention-masking interventions on the CoT information graph, residual-stream probing, and logit-lens inspection -- show that the model first internalizes a procedural pathway and subsequently develops an associative, ``mental-arithmetic'' capacity that retrieves intermediate results without explicit step-by-step computation. The trained model reaches over 80% accuracy on held-out problems and attains competitive performance against substantially larger language models, indicating that targeted, pedagogically grounded training can yield strong and economical arithmetic capability at small scale.

Andhika Bernard Lumbantobing, Hokky Situngkir

Bandung Fe Institute & Adjunct Science Fellow in InaAI · Adjunct Science Fellow in InaAI · AI Research Center IT Del +1

Numerical Reasoning in Language Models Language Model Pretraining LLM Mathematical Reasoning

Jun 2, 2026cs.AI

CrowdMath: A Dataset of Crowdsourced Mathematical Research Discussions

Large language models have made substantial progress on mathematical reasoning, but existing benchmarks typically evaluate well-specified problems with final answers, step-by-step solutions, or complete proofs. They do not capture collaborative open-problem solving: a setting in which participants propose partial arguments, identify gaps or errors in prior steps, repair flawed reasoning, and gradually synthesize incremental contributions into a proof. We introduce CrowdMath, a dataset of 164 expert-annotated progress chains from the MIT PRIMES--Art of Problem Solving (AoPS) CrowdMath program (2016-2025), a collaborative research initiative whose discussions have led to peer-reviewed publications. Each chain traces a multi-participant forum discussion from an open-problem statement to a completed proof. Posts are labeled by their functional roles in the evolving solution process, including partial progress, proof completion, erroneous reasoning, and error identification. We define evaluation tasks and benchmark six frontier models. Models achieve 83-88% accuracy on next-post prediction, suggesting that they can follow the local flow of mathematical discussion. However, they struggle to identify the functional significance of individual contributions with the best model achieving only 0.42 macro-F1 on post-role classification. CrowdMath exposes a gap between solving well-specified mathematical problems and understanding collaborative mathematical progress as it unfolds.

Sherin Muckatira, Jesse Geneson, Slava Gerovitch +3

University of Massachusetts Lowell · San Jose State University · 3Massachusetts Institute of Technology +2

Mathematical Reasoning Benchmarks LLM Mathematical Reasoning

Jun 2, 2026cs.AI

PyraMathBench: Evaluating and Improving Mathematical Capability in Large Language Models

Despite the pivotal role of numerical reasoning as the cornerstone of mathematical capabilities in large language models (LLMs) across applications, few benchmarks evaluate LLMs by integrating numerical processing and mathematical reasoning, hindering the interpretability of failures in math tasks. We introduce PyraMathBench, a comprehensive hierarchical benchmark with 32,505 questions derived from 7,404 math word problems, spanning 4 key cognitive aspects, 14 subcategories, and 2 modalities. Experiments reveal that LLMs' performance is severely compromised by inadequate numerical computation and weak handling of abstract numerical questions. To address this, we propose the Smart Optimization & Learning-based VErsatile module (SOLVE) and Interactive Relative Policy Optimization (IRPO), which enhance LLMs' numerical-mathematical synergy via efficient tool calls (fuzzy matching and low-quality call rejection). Comparative experiments show Qwen-2.5 achieves a 5.0 score improvement with SOLVE and IRPO training.

Zetian Ouyang, Linlin Wang, Gerard de Melo +1

East China Normal University · Hasso Plattner Institute, University of Potsdam

Numerical Reasoning in Language Models Mathematical Reasoning Benchmarks RL for Language Model Reasoning Mathematical Reasoning LLM Tool Use LLM Mathematical Reasoning

Jun 2, 2026cs.CR

Testing LLM Arithmetic Reasoning Generalization with Automatic Numeric-Remapping Attacks

Large language models achieve strong performance on arithmetic reasoning benchmarks, and one common response to arithmetic brittleness is to delegate computation to code. Yet models are still often used in settings where they must reason directly from natural language, and trustworthy models should solve small-number arithmetic word problems without external tools. Prior work shows that LLMs are sensitive to numerical variation: a model may solve an original problem but fail on structurally similar variants requiring the same reasoning procedure with different numbers. We ask whether this fragility persists under a stricter setting involving small, schema-preserving numeric changes that retain the original reasoning program and avoid large-number stress tests. We introduce an automatic algorithm for generating numeric-remapping attacks on arithmetic word problems. Unlike template-based perturbation methods requiring manual schemas or constraints, our approach derives problem-specific symbolic representations, generates constrained numeric remappings, recomputes gold answers, and realizes transformed questions through deterministic edits guided by LLM-generated edit plans. Stage-wise validation and a high-confidence audit retain reliable attacks, making the pipeline scalable with limited human intervention. We evaluate DeepSeek-R1 (70B), Gemma4 (31B), and GPT-OSS (120B) on GSM8K, MAWPS, and MultiArith. On GSM8K, completed runs show conditional accuracy drops of 12.16 to 25.82 percentage points. MAWPS and MultiArith are far more stable, with most attacked accuracies near or above 98%. These results show that numeric-remapping robustness depends strongly on dataset structure: GSM8K remains sensitive even when reasoning programs are preserved and answers are recomputed, while shorter, more regular datasets are more robust.

Malia Barker, Bishal Lakha, Edoardo Serra +1

Department of Computer Science, Boise State University, Boise, ID, USA · University of L'Aquila, L'Aquila, Italy

Numerical Reasoning in Language Models Mathematical Reasoning Benchmarks Adversarial Attacks on LLMs LLM Reliability Language Model Robustness LLM Mathematical Reasoning

Jun 2, 2026cs.LG

Right Makes Might: Aligning Verified Hidden States Empowers RL Reasoning

Reinforcement Learning from Verifiable Rewards (RLVR) has become the dominant approach for improving mathematical reasoning in large language models, yet current methods reduce each correct rollout to a single reward bit, ignoring the geometric structure shared among their hidden states. Investigating this structure, we find that at the anchor token (the position immediately before the answer marker), correct rollouts converge naturally because they must produce the same answer (cosine similarity ~0.84), yet each retains residual variance from its unique reasoning path. Encouraging full alignment at this point pushes the model to extract a unified "correct decision" representation, reducing sensitivity to which reasoning path was taken. Based on this observation, we propose Hidden-Align, an auxiliary loss function that aligns the last-layer hidden states of correct rollouts at the anchor token during RL training, with zero overhead in both training and inference. On eight mathematical reasoning benchmarks, Hidden-Align improves average pass@1 over the DAPO baseline by 3.8, 6.2, and 5.4 percentage points on Qwen3-1.7B, 4B, and 14B respectively, with consistent pass@k gains across all three scales, supported by ablations on loss type, anchor position, layer depth, and loss weight.

Ziyue Wang, Aomufei Yuan, Yongfu Zhu +10

1Peking University · 2JINGDONG · 3Shanghai Innovation Institute +2

LLM Alignment RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards LLM Mathematical Reasoning

Jun 2, 2026cs.AI

GTBench: A Curriculum-Grounded Benchmark for Evaluating LLMs as Mathematical Research Assistants in Graph Theory

Large language models (LLMs) are increasingly used as self-study assistants in technical disciplines, yet their reliability as mathematical reasoning assistants remains poorly understood. We introduce GTBench, a curriculum-grounded benchmark for evaluating LLMs as mathematical research assistants in graph theory, comprising 63 problems organized into three groups of increasing difficulty: undergraduate definitions and basic properties (Group 1), algorithm tracing and structural reasoning (Group 2), and graduate-level proof construction (Group 3). Problems are sourced from verified academic materials including Diestel's Graph Theory. We evaluate five frontier models -- GPT-5, Claude Sonnet 4.6, Gemini 2.5 Flash-Lite, Llama 3.3 70B, and Mistral Large 3 -- under zero-shot and chain-of-thought prompting, using exact-match and LLM-as-judge evaluation for Groups 1 and 2, and a hybrid human expert and LLM-as-judge protocol for Group 3. Our results reveal a pronounced performance hierarchy: GPT-5 approaches ceiling on Group 1 (95.8% zero-shot) and maintains meaningful accuracy on graduate proofs (82%), while all other models degrade substantially with difficulty, with Llama achieving 0% under human evaluation on Group 3 zero-shot. Failure mode analysis shows that correct algorithm, wrong execution errors dominate Groups 1 and 2, while Group 3 additionally surfaces incomplete reasoning failures and reveals systematic disagreement between human evaluators and the automated judge, particularly on verbose or near-complete proofs (kappa = 0.48-0.83 across human pairs). GTBench provides the first curriculum-grounded evaluation framework for graph-theoretic reasoning in LLMs, with direct implications for the governance of AI tools in mathematical education and scientific research.

Noujoud Nader, Ibrahem Aljabea, Patrick Diehl +1

Louisiana State University, USA · Los Alamos National Laboratory, USA · Texas A&M-Central Texas, USA

Mathematical Reasoning Benchmarks LLM Reasoning with Graphs Graph Reasoning LLM Mathematical Reasoning

Jun 1, 2026cs.AI

eMoT: evolving Memory-of-Thought via Symbolic Anchoring and Memory Corrosion

While Large Language Models (LLMs) achieve impressive performance on multi-step reasoning tasks, their reliability is persistently hindered by critical limitations such as unconstrained hallucinations and poor numerical computation. Fundamentally, these issues arise because standard models treat reasoning as a transient, one-off generation process rather than retaining and refining successful procedural logic. To address these challenges, we propose eMoT (evolving Memory-of-Thought), a unified framework that stabilizes multi-step reasoning by treating reasoning trajectories as dynamic, evolving memories rather than static templates. The framework primarily consists of three interconnected modules: (i) a memory corrosion mechanism that reinforces high-utility reasoning structures while gradually decaying less frequent ones; (ii) a symbolic anchoring engine that utilizes Python for deterministic computation, much like a human uses a calculator; and (iii) a consistency-driven refinement process that aligns neural inference with symbolic outcomes, reducing the accumulation of logical discrepancies. Across multiple reasoning benchmarks, eMoT improves accuracy and solution consistency over standard Chain-of-Thought and structured reasoning baselines.On the traditional task Game of 24, eMoT achieves 100% accuracy, surpassing the baseline by up to 17.6%. Evaluations on mathematical task GSM8K, ASDiv, SVAMP, and MGSM further show consistent gains in multi-step mathematical reasoning. In our evaluation, we achieve superior performance despite utilizing a lightweight backbone model with constrained baseline capabilities. Compared to alternative methods that rely on massively scaled models, our results demonstrate that the performance gains are fundamentally driven by the eMoT framework's reasoning control rather than sheer model size.

Xiang Li, Jiwei Wei, Ke Liu +5

Center for Future Media, University of Electronic Science and Technology of China, Chengdu, China

Memory-Augmented Language Models LLM Mathematical Reasoning LLM Reasoning

Jun 1, 2026cs.CL

Off-the-Shelf LLMs as Process Scorers: Training-Free Alternative to PRMs for Mathematical Reasoning

Selecting the best response from multiple small-model samples using a stronger scorer is a simple inference-time strategy, but fails when the small model has already committed to incorrect reasoning paths. PRM guided search avoids this by scoring candidate continuations during generation, but requires a reward model trained with step-level labels. We propose Chunk-Level Guided Generation, a training-free alternative that uses an off-the-shelf large language model as a process scorer. At each step, a small model samples k fixed-length candidate chunks, while the larger model scores the candidates using likelihoods without generating any text. The selected chunk is committed before the next step, steering generation before errors can propagate. We instantiate this framework with two selection rules: Likelihood-Guided Selection (LGS), which selects the chunk with the highest length-normalized large-model log-probability, and Contrastive-Guided Selection (CGS), which subtracts the small model's log-probability to favor chunks where the large model's preference diverges from the small model's. We show that scoring variable-length reasoning steps with large-model likelihoods is unreliable due to a systematic length bias that persists even after length normalization, and that fixed-length chunks avoid this confound. On GSM8K, MATH, Minerva Math, AMC23, and AIME24 with Qwen2.5-1.5B guided by Qwen2.5-32B and Llama-3.2-1B guided by Llama-3.1-70B, CGS outperforms majority voting by up to 28 pp and, under matched guidance budgets, matches or outperforms Qwen2.5-Math-PRM-72B guided search on most benchmarks without reward-model training. With Qwen2.5-7B guided by Qwen2.5-72B, CGS reaches 81.8% on MATH and 63.6% on Minerva Math at k=16, surpassing majority voting by 4--6 pp. Finally, Chunk-Level Guided Generation produces substantially shorter reasoning traces than PRM guided search.

Atoosa Chegini, Soheil Feizi

Department of Computer Science, University of Maryland

Numerical Reasoning in Language Models Inference-Time Search LLM Inference LLM Mathematical Reasoning

May 30, 2026cs.AI

AXIOM: A Trust-First Neuro-Symbolic Execution Architecture for Verifiable Mathematical Reasoning

We present AXIOM, a trust-first neuro-symbolic execution architecture for natural-language mathematical reasoning. In AXIOM, the language model functions strictly as a canonicalizer: it rewrites informal problem text into a narrow schema consumed by a deterministic Computer-Algebra-System (CAS) pipeline, which derives and verifies the answer or abstains as a first-class output. Routing follows a 1:1:1 alignment between problem-shape regex, schema-specific prompt, and closed-form CAS handler, with 3,100+ such routes shipped and zero LOST_CORRECT regressions across 250+ consecutive ship commits. We report empirical results on 4 MATH categories with a cumulative correctness of 94.36% (2,592/2,747) at 100.00% trust on parseable (zero confident-wrong answers across the full 2,747-record benchmark), all four domains above the per-domain 70/90/70 floor with per-domain trust at 100.0%, and median latency of 1 ms on rule-only handlers (88% of records on the lm-eval arithmetic 20,000-record benchmark). The architecture has served ~30,000 production queries through a public deployment. The contribution we emphasize is not a final accuracy figure but the forward dynamic the architecture establishes: every logged abstain in production is a candidate correct after one ship cycle, since new tasks compose without regressing the registry. The operational discipline behind this property -- math-template bucketing, LOST_CORRECT scan as regression oracle, parseable-first onboarding, and abstain as first-class output -- constitutes a transferable framework for trustworthy neuro-symbolic systems beyond mathematics.

Alessio Bruno

Independent researcher

LLM Answer Verification Symbolic Reasoning Neuro-Symbolic Reasoning LLM Mathematical Reasoning

May 30, 2026cs.AI

KACE: Knowledge-Adaptive Context Engineering for Mathematical Reasoning

Context engineering can improve large language models without updating their weights, but mathematical reasoning exposes a key limitation: feedback accumulated in one growing prompt causes context bloat and limits the amount of learned guidance that can be used. Existing methods often conflate storage, what is learned across runs, with usage, what is included for a particular problem, and therefore inherit this prompt-size ceiling. We introduce Knowledge-Adaptive Context Engineering (KACE), which separates storage from usage through difficulty- and domain-based organization. Offline, a self-reflective learning loop distills training traces into an epistemic tree: a knowledge base of typed cards stratified by problem difficulty and epistemic domain. Each card is assigned to the difficulty-domain node corresponding to the failure from which it originated. At evaluation time, tiered self-consistency with per-tier agreement gates dynamically classifies each problem as easy, medium, or hard. Easy problems exit without retrieved cards, while harder problems retrieve only the matching branch of the tree. This tiered scheme matches or exceeds Best-of-N while using comparable compute, and it classifies problem difficulty with 78 percent pairwise concordance. The main empirical contribution is the construction and use of a difficulty- and domain-stratified knowledge base enabled by tiered self-consistency. On AIME 2025, KACE achieves 62.2 percent accuracy, a 10.4-point absolute gain over fixed Best-of-5 self-consistency at a comparable solver-call budget and a 5.6-point gain over the strongest learned-context baseline, Tiered + GEPA. We also observe consistent gains on MATH-HARD and the verifiable subset of OlymMATH.

Jayant Parashar, Suchendra M. Bhandarkar

School of Computing University of Georgia

Memory-Augmented Language Models Self-Consistency Decoding LLM Mathematical Reasoning

May 29, 2026cs.LG

The Shape of Addition: Geometric Structures of Arithmetic in Large Language Models

Large Language Models exhibit paradoxical fragility in fundamental arithmetic, implying a disconnect between internal computation and discrete output. By analyzing the residual stream geometry during multi-operand addition, we identify the Iso-Raw-Sum Trajectory (IRST), a geometric structure where representations are anchored by semantic digits and modulated by continuous carry fibers. We propose the Noisy Quantization Model to explain this geometry, framing arithmetic errors as Geometric Slippages caused by internal neural noise pushing a continuous, latent Carry Potential across quantization thresholds. This geometric framework further elucidates Probe Versatility, explaining how lightweight probes can disentangle coexisting latent signals (such as ground truth versus hallucination) from a single activation vector. Finally, we validate these insights through a geometric consistency check method that effectively detects and corrects these quantization failures during inference. Our code is available at https://github.com/RL-MIND/Shape-of-Addition.

Liuyuan Wen, Xun Zhu, Lihao Huang +2

State Key Laboratory of Novel Software Technology, Nanjing University, Nanjing 210023, China.

Neural Representation Geometry LLM Interpretability LLM Mathematical Reasoning

May 28, 2026cs.CL

COMPOSE: Composing Future Theorems from Citations and Formal Structure

A plausible future mathematical claim must satisfy two constraints: it should follow the direction of prior work and respect the formal dependencies that constrain what can validly follow. Existing approaches typically model only one of these sources, producing claims that are either weakly grounded or insufficiently motivated. We introduce grounded future mathematical generation, where the goal is to generate a plausible future theorem-like claim for an anchor paper using two complementary sources of context: its scientific citation graph and aligned formal theorem dependency graph. To address this setting, we propose COMPOSE, a dual-graph framework that conditions a language model on both scientific citation context and formal theorem structure. To support this setting, we construct a dataset of 108K paired scientific-formal graph examples from arXiv and Mathlib, together with a benchmark of 47K future papers from 2024--2025. Experiments show that COMPOSE outperforms strong baselines on retrieval to real future papers and achieves the best overall performance under LLM-judge evaluation, producing more grounded and mathematically richer outputs. These results show that future mathematical generation benefits from combining scientific context with formal structure. Project page is available at https://david-busbib.github.io/COMPOSE-page/.

David Busbib, Michael Werman

Hebrew University of Jerusalem

Graph Retrieval-Augmented Generation Evidence-Grounded Generation LLM Mathematical Reasoning

May 28, 2026cs.CL

Knowing What to Solve Before How: Preplan Empowered LLM Mathematical Reasoning

Current plan-based reasoning methods improve large language models (LLMs) by inserting a planning stage before execution, giving rise to the question

\rightarrow

plan

\rightarrow

cot paradigm. While effective, a closer examination reveals an inherent paradigm-level gap: both the planning and its execution stages decide how to solve a problem, while the prior question of what to solve; recognizing the problem type, the applicable tools, and the foreseeable pitfalls; remains entirely implicit. To bridge this gap, we propose PPC (Preplan-Plan-CoT), a framework that introduces an explicit problem-understanding stage, the preplan, yielding a new question

\rightarrow

preplan

\rightarrow

plan

\rightarrow

cot paradigm. Realizing this paradigm requires safeguarding the conceptual integrity of preplan at both ends. Specifically, we design a three-stage synthesis pipeline with a spoiler-score detector that filters out leakage and spoiler failures to build clean preplan supervision, and a composite GRPO reward enforces that the generated plan genuinely follows from the preplan. Experiments across four backbones and five mathematical reasoning benchmarks show that PPC achieves the best results on 39 of 40 metrics, improving maj@16 and pass@16 by +2.23 and +3.06 over the strongest baseline without introducing additional inference token overhead.

Shaojie Wang, Liang Zhang

Hong Kong University of Science and Technology (Guangzhou)

LLM Planning Structured Reasoning LLM Mathematical Reasoning

May 28, 2026cs.AI

DenseSteer: Steering Small Language Models towards Dense Math Reasoning

Large language models (LLMs) demonstrate strong chain-of-thought (CoT) reasoning abilities, while smaller models (<= 3B parameters) significantly underperform on multi-step reasoning tasks. Based on empirical analyses of the Qwen-2.5 model family on math reasoning benchmarks, we find that more proficient reasoning is associated with fewer reasoning steps but higher information density per step, a property we term Dense Reasoning. Motivated by this observation, we propose DenseSteer, a training-free inference-time steering framework that enhances small-model reasoning by modulating internal representations toward dense reasoning patterns. Experiments show that our method yields consistent accuracy improvements without increasing token-level Negative Log-Likelihood, highlighting dense reasoning as an effective structural approach to mathematical problem solving.

Yang Ouyang, Shuhang Lin, Jung-Eun Kim

1North Carolina State University · 2Rutgers University

Small Language Models Activation Steering LLM Mathematical Reasoning

May 28, 2026cs.LG

When RL Suppresses Its Own Vocabulary: Recovering Reasoning Diversity in Puzzle-to-Math Transfer

Reinforcement learning using verifiable rewards (RLVR) improves LLM reasoning, but the conditions under which it transfers across domains -- and why it does so -- remain under-explored. We study cross-domain transfer in a 7B model whose SFT and RL post-training stages use only constraint-satisfaction puzzles, with no mathematics problems in the post-training data. To analyze how transfer emerges, we introduce a reasoning primitive-level framework that combines a 9-class span classifier with motif extraction, allowing us to segment chain-of-thought traces into primitive motifs and track their evolution across training stages and domains. We find that puzzle SFT induces a reasoning-primitive vocabulary, yielding a

+7

pp \texttt{pass@32} gain on OlymMATH-Hard. Vanilla GSPO then composes these primitives into longer compute-verify chains, adding a further

+6

pp. However, this RL stage also suppresses exploratory primitives such as \textit{hypothesize} and \textit{backtrack}. To address this, we introduce a novelty bonus that rewards diverse correct rollouts, using perplexity under the reference model as a signal. This restores recovery primitives during RL and adds a further

+7

pp \texttt{pass@32} relative to vanilla GSPO. Finally, the end-to-end recipe raises the hard-math capability ceiling from

16.0\%

at the OLMo3-7B-Instruct-SFT base to

36.0\%

, without adding any mathematics problems during the SFT or RL stages.

Mayug Maniparambil, Arjun Karuvally, Terrence Sejnowski +1

Fin AI Research · Salk Institute for Biological Studies

RL for Language Models RL for Language Model Reasoning Cross-Domain Transfer Learning Reinforcement Learning with Verifiable Rewards LLM Mathematical Reasoning

May 27, 2026cs.CL

On Compositional Learning Behaviours in Formal Mathematics

Self-evolving scientific agents capable of conquering the hard tail of formal mathematics require Compositional Learning Behaviours (CLBs) -- the capacity to ground and recombine novel symbolic structures in context, beyond mere recombination of prelearned atoms. We propose S2B-LM, an adaptation of the CLB-evaluating Symbolic Behaviour Benchmark that removes numerical processing as a confound and adds chain-of-thought scaffolding to elicit rather than merely probe latent CLB competency. Cross-evaluating ten Lean~4 theorem provers on CLB competency in S2B-LM and miniF2F whole-proof performance, we find correlational and causal evidence of our claim: First, a necessary-condition analysis via quadrant test yields

p=0.004

, with model scale being ruled out as a confound. Second, extracting a CLB-encoding activation direction from DeepSeek-Prover-V2-7B using S2B-LM traces via Contrastive Activation Addition and applying it during miniF2F whole-proof generation on the AIME subset, CLB suppression collapses solve rate from

32.3\%

to

2.9\%

, without loss of coherence, while suppressing a random activation direction of equal magnitude leaves it at

31.9\%

. Together, these results show that CLB competency is necessary but not sufficient for the hard tail of formal mathematical verification.

Kevin Yandoka Denamganaï

University of York

Mathematical Reasoning Benchmarks Formal Verification Lean Theorem Proving LLM Mathematical Reasoning Compositional Generalization

May 27, 2026cs.CL

FABSVer: Faster Training and Better Self-Verification for LLM Mathematical Reasoning

While large language models have made significant progress in mathematical reasoning, they remain unreliable at judging the correctness of their own solutions. Existing approaches that equip models with self-verification typically treat solution generation and verification as two separate tasks, leading to substantially increased training time. In this paper, we propose FABSVer, which fuses these two tasks into a single generation pass, dramatically reducing training overhead while jointly optimizing both capabilities. We further identify a convergence bottleneck both theoretically and empirically: as training progresses, the reward reaches a plateau because the policy is constrained by a fixed reference model. To overcome this, we introduce Dynamic Reference Model Update (DRMU), which raises the reward ceiling and enables sustained reward growth. Extensive experiments on math benchmarks demonstrate that FABSVer achieves superior self-verification and reasoning performance across three model scales, while requiring only 51%--71% of the training time of existing methods. Analysis further reveals distinct learning phases in how models acquire self-verification, and that the gap between verify and answer rewards shrinks noticeably as model size increases.

Haihui Pan, Junwei Bao, Hongfei Jiang +1

Zuoyebang Education Technology

LLM Answer Verification RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards LLM Mathematical Reasoning

May 27, 2026cs.AI

Risk-Controlled Lean-as-Judge for Natural-Language Mathematical Reasoning

Lean is increasingly used to judge natural-language mathematical answers, but its signal is partial: many answers never formalize, and a failed proof may reflect an ill-typed statement or a missing library fact, not a wrong answer. On MATH-500 we show this signal is (i) sharply coverage-dependent, that is the proof-winning answer is correct 96% of the time at high proved coverage but 20% at low, and (ii) sparse and often unfaithful: a 7B autoformalizer proves a class for only 28% of problems, and a manual audit finds only approximately 43% of those proofs faithful. We propose COVCAL, a selector over Lean-trace diagnostics that certifies a finite-sample selective-risk bound on accepted answers or abstains, under two regimes (a conservative Bonferroni bound and a tighter dev-then-cal rule). Feasibility depends on autoformalization coverage: with the 7B formalizer the signal is too sparse and Bonferroni abstains on all 20 bootstrap partitions, whereas a prover-specialized formalizer reaches 79% coverage and flips it to feasible on 17 of 20, accepting approximately 48% of problems at 0.98 accepted accuracy. Since self-consistency alone is already 91% accurate, our contribution is a precise account of when, and with which formalizer, a partial formal signal can be trusted under risk control.

Pauline Bourigault, Xiaotong Ji, Matthieu Zimmer +2

Imperial College London · Huawei Noah’s Ark Lab · UCL Centre for AI

LLM Answer Verification Selective Prediction Lean Theorem Proving Autoformalization LLM Mathematical Reasoning

May 27, 2026cs.CL

ResearchMath-14K: Scaling Research-Level Mathematics via Agents

The frontier of mathematics is defined by problems whose solutions are not yet known. However, whether language models can meaningfully engage with such problems without human intervention remains unclear. A major obstacle is the lack of large-scale research-level math datasets. To this end, we introduce ResearchMath-14k, a set of

14{,}056

problems curated from academic sources via a multi-agent pipeline. ResearchMath-14k spans 11 mathematical domains and ranks above existing math datasets on knowledge, novelty, and procedural difficulty. To our knowledge, it is the largest research-level mathematical problem set available for training. We additionally generate

220

K teacher trajectories through targeted prompting, followed by behavioral filtering. Notably, however, generating correct trajectories is nontrivial at this level, and two LLM judges label only

3.7\%

and

4.3\%

of sampled ResearchMath training trajectories as correct. Nevertheless, across three model families, full-parameter training on ResearchMath improves performance on graduate- and research-level mathematics benchmarks by

2.1

points over the starting checkpoints. In comparison, training on existing datasets such as DASD and Nemotron-SFT-Math-v4 changes performance by

0.0

and

-0.5

points, respectively. Notably, mixing DASD with ResearchMath yields higher scores than token-matched DASD alone on benchmarks covering olympiad short-form (

+2.0

), graduate- and research-level short-form (

+0.8

), graduate- and research-level symbolic (

+2.6

), and proof evaluation (

+5.7

). Further analysis suggests that research-level mathematical content and greater reasoning diversity may help explain why ResearchMath provides complementary supervision to contemporary datasets. We make ResearchMath-14k publicly available for future works on research-level mathematical reasoning.

Guijin Son, Seungyeop Yi, Minju Gwak +3

Seoul National University · OneLineAI · Yonsei University

Supervised Fine-Tuning LLM Mathematical Reasoning

May 26, 2026cs.CL

Beyond Input Understanding: Diagnosing Multilingual Mathematical Reasoning with Directed Acyclic Trace Graphs

Large reasoning models (LRMs) achieve strong mathematical reasoning performance in English, but remain much less reliable in many low- and medium-resource languages. This gap is often explained as a failure to understand non-English problem statements. We show that this view is incomplete: even when the problem is given in English, controlling the model's reasoning language can substantially reduce accuracy, suggesting that language also affects reasoning execution itself. To study this effect, we introduce DATG, a Directed Acyclic Trace Graph framework that maps reasoning traces to language-independent mathematical anchors and dependencies. This allows us to align target-language traces with reference DAGs and measure whether they cover required mathematical nodes, respect dependency edges, and avoid harmful mathematical actions. Experiments on the Qwen3 series across 12 languages show that non-English reasoning often suffers from reduced anchor coverage and weaker dependency fidelity, especially in low-resource languages. Motivated by this diagnosis, we propose Loop-Retry and Formula-Retry, two simple test-time controls targeting DATG-exposed failure modes, and show that they consistently improve target-language reasoning performance in low-resource languages.

Jiaqiao Zhang, Zhoujun Li, Raoyuan Zhao +5

Southwest University · LMU Munich & MCML

Mathematical Reasoning Low-Resource Language Processing LLM Mathematical Reasoning Large Reasoning Models Reasoning Trace Analysis

May 26, 2026cs.CL

ReverseMath: Answer Inversion for Scalable and Verifiable Mathematical Problem Generation

Mathematical reasoning benchmarks are vital for evaluating large language models (LLMs), but many are static and repeatedly exposed through public evaluation and training pipelines, making it difficult to separate genuine reasoning from memorization. Meanwhile, manually constructing new math problems with reliable answers remains costly. We introduce ReverseMath, a scalable method for generating new math problems through answer inversion. Given a problem and its answer, ReverseMath masks a numerical value in the original problem, treats the original answer as a known condition, and rewrites the problem so that the masked value becomes the new answer. The generated problem reverses the original input-output relation, making its answer known by construction. We study ReverseMath for both evaluation and training. For evaluation, paired original/reversed problems reveal substantial behavioral shifts: models sometimes fail on reversed problems and even incorrectly output the original answer, suggesting memorization-like behavior. For training, ReverseMath provides automatically labeled reversed problems as data augmentation for reinforcement learning (RL). Experiments show that including ReverseMath-generated data improves mathematical reasoning performance across multiple benchmarks, demonstrating its value as both an analysis tool and a scalable source of verifiable training data.

Raoyuan Zhao, Yihong Liu, Yupei Du +2

Center for Information and Language Processing, LMU Munich · 2Munich Center for Machine Learning (MCML) · 3Saarland University

Mathematical Reasoning Benchmarks Synthetic Benchmark Generation LLM Mathematical Reasoning

May 26, 2026cs.AI

Reasoning, Code, or Both? How Large Language Models Handle Variations in Math Questions

Large Language Models (LLMs) achieve impressive accuracy on mathematical reasoning benchmarks, yet their performance drops when problems are modified with simple changes like different names or numbers. Code execution methods, which let models generate and run Python code instead of reasoning in natural language, have been proposed as a solution, but their effect on reasoning robustness (the ability to maintain accuracy across problem variations) has not been systematically tested. This study evaluates three approaches on 1,000 problems from the GSM-Symbolic dataset: pure reasoning using chain-of-thought (CoT) prompting, single-shot code execution using Program-Aided Language models (PAL), and iterative code execution using Step-by-Step Coding (SBSC). All three were run on paired original and modified problems using Claude Haiku 4.5. CoT was the most robust method, with an accuracy drop of 1.3 percentage points and 1.8% of problems breaking under perturbation. PAL was the least robust at 1.7 percentage points and 3.1% broke, with SBSC falling in between. Although these differences were not statistically significant (

p = .096

), the directional trend was consistent across all measures, suggesting that code execution, whether single-shot or iterative, does not improve reasoning robustness on grade-school-level problem variations.

Matthew Kutakh

CoT Reasoning Mathematical Reasoning LLM Mathematical Reasoning

May 23, 2026cs.CL

Guarded Repair for Harm-Aware Post-hoc Replacement of LLM Mathematical Reasoning

Post-hoc repair of LLM mathematical reasoning introduces an asymmetric risk: fixing an incorrect reasoning trace is useful, but replacing a trace that was already correct can be harmful. We study this problem under a selective replacement setting, where a system must decide whether a repaired candidate is safer than preserving the original cached trace. We present GuardedRepair, a guarded best-of-N repair framework that diagnoses cached reasoning traces, selectively triggers repair, and accepts answer-changing candidates only when deterministic verification guards support replacement. The framework combines lightweight symbolic checks, surface semantic-risk diagnostics, bounded candidate generation, and conservative acceptance policies. On the full GSM8K test set, where the initial reasoner already achieves 95.60% accuracy, GuardedRepair improves final accuracy to 96.89%, fixing 17 of 58 remaining errors without measured broken-correct cases in the main run. On a weak-reasoner ASDiv setting, accuracy improves from 78.40% to 87.60%. Direct regeneration baselines show that this gain is not explained by stronger-model re-solving alone: re-solving all GSM8K examples lowers accuracy to 93.03% and breaks 47 initially correct answers. Additional analyses show that guarded repair substantially improves the fixed/broken tradeoff, while also revealing that replacement risk is reduced rather than eliminated. These results support viewing post-hoc repair as harm-aware selective replacement rather than unconstrained re-solving.

Haizhou Xia

LLM Self-Correction LLM Mathematical Reasoning

May 21, 2026cs.AI

Advancing Mathematics Research with AI-Driven Formal Proof Search

Large language models (LLMs) increasingly excel at mathematical reasoning, but their unreliability limits their utility in mathematics research. A mitigation is using LLMs to generate formal proofs in languages like Lean. We perform the first large-scale evaluation of this method's ability to solve open problems. Our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars, proved 44/492 OEIS conjectures, and is being deployed in combinatorics, optimization, graph theory, algebraic geometry, and quantum optics research. A basic agent alternating LLM-based generation with Lean-based verification replicated the Erdős successes but proved costlier on the hardest problems. These findings demonstrate the power of AI-aided formal proof search and shed light on the agent designs that enable it.

George Tsoukalas, Anton Kovsharov, Sergey Shirobokov +18

1Google DeepMind1 · 2Aarhus University · 3Google

Automated Theorem Proving AI Agent Evaluation Lean Theorem Proving LLM Mathematical Reasoning Autonomous Research Agents