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

Oct 7, 2026cs.AI

From Expert-Guided Proof Search to Automated Open-Problem Solving

Large language models are increasingly contributing to mathematical research, where progress often depends on efficient proof search, incremental improvements and careful verification. We describe Bolzano, a multi-agent open-source system that uses parallel prover agents with a verifier agent and maintains a human-readable research state. Initial manual use on expert-selected problems yielded 8 results whose proofs were checked by domain experts. Motivated by these case studies, we ran Bolzano without problem-specific human guidance on about 3,800 open problems extracted from four sets of papers, solving about 200 open problems. One experiment used papers accepted to STOC 2026, a top conference in theoretical computer science. There, we answered four questions raised in the papers, as confirmed by their authors.

Adrián Zámečník, Matěj Kripner, Martin Koutecký +5

Computer Science Institute, Charles University · Institute of Formal and Applied Linguistics, Charles University · Department of Applied Mathematics, Charles University +1

Automated Theorem Proving AI Agents for Scientific Discovery LLM Mathematical Reasoning Multi-Agent LLMs

Oct 7, 2026cs.AI

Let the Library Speak: Self-Advertised Method Selection for Formal Proving

LLM-based formal provers can retrieve relevant lemmas and prior proofs, but relevance alone does not say whether a mathematical method can be used on the current theorem. A method has prerequisites, a target, an intended action, and obligations that its use leaves to prove. Methods that look equally related to a theorem may therefore differ substantially in whether they offer a plausible next step. We formulate this as an applicability-aware method-selection problem and introduce self-advertisement: before candidates are ranked, a model generates a problem-specific proposal for each one, stating what part of the goal it targets, what action it would take, and what conditions that action requires. We organize 82 reusable methods from Putnam 2000-2014 as Method Contracts, which pair applicability descriptions with Mathlib anchors, a checked example or scaffold, and expected proof obligations. A single batched call elicits proposals across the library; vague or unsupported proposals are demoted, yielding a ranked shortlist accompanied by inspectable claims about each candidate's use. We analyze when similarity-based representations cannot distinguish methods with different applicability, how errors in applicability estimates affect shortlist quality, and what a checked scaffold guarantees under its stated assumptions. Against lexical, embedding, and embedding-plus-LLM reranking baselines, self-advertisement achieves 95.0% hit@5 on Putnam 2015-2025, compared with 84.2% for the strongest reranker. On IMO ProofBench, it achieves 91.7% compared with 88.3%. These results indicate improved coverage of annotated methods in the retrieved shortlists, particularly on Putnam.

Xiaopeng Yuan, Suijin Wang, Yanli Wang +5

University of Illinois at Urbana-Champaign · Tsinghua University · Imperial College London +1

Automated Theorem Proving LLM Mathematical Reasoning Lean Theorem Proving

Oct 5, 2026cs.CV

CALR: Continuous Anchored Latent Reasoning via Render-of-Thought Compression

Visual latent reasoning compresses rendered derivations into compact intermediate states, reducing textual reasoning overhead. Existing approaches differ in how they represent these states: continuous methods avoid vocabulary constraints, whereas discrete methods improve accuracy through quantization into a finite codebook. Our analysis of representative continuous and discrete systems identifies two functional requirements: answers must rely on latent states, and those states must carry valid, problem-specific reasoning. Continuous latents influence answers despite collapsed reasoning content, whereas discrete latents retain recoverable intermediate reasoning that answer prediction largely bypasses. To address these challenges, we propose Continuous Anchored Latent Reasoning (CALR), which connects latent formation with answer use through functional anchoring. With reference latents from information-balanced compression, CALR couples latent-mediated answer supervision with derivation-level semantic anchoring: the former routes answer supervision through intermediate states, while the latter grounds their decoded content in problem-specific derivations. A parallel-to-autoregressive curriculum develops sequential reasoning by conditioning subsequent latent blocks on generated prefixes. Evaluations on five mathematical reasoning benchmarks across model families show substantial accuracy gains. Under matched budgets, CALR gains 26.0 percentage points over a comparable continuous latent reasoning method. Further analyses show that its latents support answer prediction and carry problem-specific intermediate reasoning.

Zhaoyang Wei, Bowen Jiang, Yanchao Hao +5

University of the Chinese Academy of Sciences · Tencent

Efficient Language Model Reasoning Latent Visual Reasoning Implicit Reasoning in Language Models Continuous Latent Reasoning LLM Mathematical Reasoning Latent Reasoning

Oct 4, 2026cs.LG

When Does Longer Reasoning Help? Predicting Mathematical Reasoning Through Discovery and Execution

Test-time compute can improve mathematical reasoning, but can short-budget runs predict how mathematical reasoning scales with additional compute? We introduce a Discovery--Execution (DE) framework that predicts the aggregate held-out scaling curves through a convolution of strategy discovery and conditional execution. From independent short-budget attempts and oracle-sketch-conditioned runs, the framework estimates cumulative success along held-out reasoning trajectories under alternate compute allocations. We evaluate four models on 35 fresh Olympiad problems and non-geometry problems from IMO-ProofBench Advanced. Under the DE framework, near-saturated execution predicts geometric scaling, as observed for the GPT models. For Claude Opus 4.8, incorporating measured execution substantially improves held-out forecasts over geometric extrapolation across one- and two-arm allocations. As a secondary application, regularized DE (R-DE) decisions to continue or restart yield lower average regret than the best model-specific retrospective policy. Together, these results show that measuring conditional execution provides information about longer reasoning that short-budget success rates do not always capture.

Adib Hasan, Lay Jain, Thanic Nur Samin

Independent Researcher · IU Bloomington

Inference-Time Scaling LLM Mathematical Reasoning

Oct 1, 2026cs.LG

The Missing Primitive: Diagnosing and Repairing Mathematical Reasoning in Large Language Models

While Large Language Models (LLMs) have demonstrated striking capabilities on frontier mathematical problems, it remains unclear whether they possess the structural mathematical understanding underlying their solutions. In this paper, we take a first step toward systematically studying mathematical understanding in LLMs, from diagnosing its distinct capabilities to leveraging these findings to improve post-training. First, we introduce the notion of Mathematical Primitive to probe structural mathematical understanding and propose \hlei{}, a novel benchmark that evaluates mathematical reasoning along four distinct dimensions: Discovery, Generation, Digestion, and Execution. Second, our systematic diagnosis shows that solution accuracy masks distinct capability profiles, primitives unlock substantial latent execution capacity, and Discovery is the dominant bottleneck in mathematical reasoning. Our post-training analysis further shows that discovery-limited failures are particularly amenable to repair. Finally, building on these findings, we introduce \abs{}, a primitive-privileged self-distillation framework that selectively transfers primitive-guided reasoning into the student model. Extensive experiments demonstrate that \abs{} consistently improves mathematical reasoning over baselines across model scales and challenging benchmarks.

Shuo Xing, Zilin Dai, Chengyuan Qian +7

Texas A&M University · Harvard University · Vanderbilt University +3

Mathematical Reasoning Benchmarks Self-Distillation LLM Post-Training LLM Mathematical Reasoning

Oct 1, 2026cs.LG

Function-Structured Reinforcement Learning with Executable Verifiers for Mathematical Reasoning

Algorithmic mathematical reasoning requires reliable decomposition, computation, and aggregation. Final-answer rewards provide limited guidance on intermediate errors, while successful execution does not guarantee mathematical correctness. This work proposes Function-Structured Graph Reinforcement Learning (FSG-RL), connecting subproblem graphs and Python implementations with multi-verifier feedback. The policy first learns to generate code from function graphs through supervised fine-tuning (SFT). Group Relative Policy Optimization (GRPO) then optimizes the policy using answer-gated rewards and span-level credit assignment. The framework also supports teacher supervision and structured memory. A benchmark curated from Grade School Math 8K (GSM8K), MathQA, MATH, and Omni-MATH pairs public function graphs with private verification specifications. Under a unified evaluation protocol, GRPO improves final-answer accuracy from 43.25% to 67.50% and full solution success from 32.25% to 52.25% over SFT. Continued reinforcement learning (RL) with teacher supervision yields additional gains. The gains extend beyond producing correctly formatted code, supporting verifier-guided reinforcement learning for mathematical reasoning. Code is available at https://github.com/ZihanLiummyycc/FSG-RL.

Zihan Liu, Xurong Xie

Department of Computer Science, University College London, UK · Institute of Software, Chinese Academy of Sciences, China

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

Oct 1, 2026cs.AI

ReSolve: Reusing Candidate Reasoning through Selective Generative Moderation

Sampling multiple solutions spends computation on intermediate deductions and unfinished arguments as well as final answers. We introduce ReSolve, a training-free inference procedure that reuses this candidate reasoning through selective generative moderation. An answer-distribution controller invokes a model to examine existing derivations when candidates disagree or lack a parseable answer, then incorporates the generated solution into a bounded loop. Under Hybrid scoring on 130 competition-mathematics problems evaluated with two independently sampled candidate pools, ReSolve obtains 100 and 99 correct answers, compared with 91 and 92 for voting over the same four candidates, with no correct-to-incorrect changes relative to that vote in either pool. Eight-sample self-consistency obtains 94 and 96 correct answers while consuming substantially more tokens; ReSolve uses 46.3% and 47.2% fewer tokens in the two evaluations. A controlled ablation removes visible derivations while retaining answer keys, vote counts, and the per-state output-cap rule, reducing accuracy from 100 to 93 correct despite increasing computation. Selective and always-on Uniform moderation both solve 97 problems, while selectivity reduces moderation tokens by approximately 54% and total pipeline tokens by 6.2%. These results support candidate reasoning as reusable inference computation. They do not establish an accuracy advantage over additional sampling or a distinct benefit from specialized route instructions.

Bangji Yang, Jiajun Fan, Hongba Ma +6

University of Illinois at Urbana-Champaign · Tsinghua University · Rutgers University +2

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

Oct 1, 2026cs.AI

Improving Math Reasoning through Value-guided Informative Search

Reinforcement learning with verifiable rewards (RLVR) has substantially improved the mathematical reasoning capabilities of large language models. Recent work introduces search into RLVR rollouts to increase trajectory diversity, but diversity alone does not ensure that the search-induced rollout policy improves upon the current policy. To address this gap, we propose APIVIS, a training-time framework that adapts finite-budget Gumbel search to chunk-level mathematical reasoning. APIVIS combines direct and searched responses within each rollout group, allowing improvements found by search to produce informative relative rewards. It further applies selective supervision to search-improved tokens, preserving a learning signal when uniform group rewards render GRPO ineffective. We show that exact value-guided selection improves the expected verifier reward at each searched state and that this guarantee extends to the complete rollout policy, with a corresponding approximate guarantee under bounded value-estimation error. Experiments on widely recognized mathematical reasoning benchmarks and different model scales demonstrate substantial improvements over competitive search-based methods, validating the effectiveness of APIVIS.

Shaohuai Liu, Yuning Wu, Haoran Liu +3

Amazon · Department of Electrical and Computer Engineering, Texas A&M University

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

Sep 30, 2026cs.AI

ANI: Adaptive Numerical Injection for Unifying Semantic and Arithmetic Representations in Numerical Reasoning

Precise numerical reasoning with Large Language Models (LLMs) is essential for expanding their applicability to complex real-world tasks. However, text-based tokenization often fragments numbers, significantly hindering precise arithmetic reasoning. Meanwhile, numerical embeddings, despite arithmetic precision, rely on context-agnostic substitution that disregards the semantic role of numbers as identifiers. To combine the complementary strengths, we propose \textbf{ANI (Adaptive Numerical Injection)}, a hybrid framework that governs the selective injection of numerical features based on the semantic context. By employing a context-aware gating mechanism, we selectively inject numerical embeddings (specifically FoNE) into the latent space, explicitly preserving nominal identifiers while enhancing quantitative operands. Through extensive evaluations across various LLMs, we demonstrate that ANI enhances MATH performance by 9.5 points over the official reference model, while maintaining robust performance on general linguistic benchmarks.

Jinsung Jeon, Seung-won Hwang

KAIST InnoCORE LLM · Seoul National University · Korea University

Numerical Reasoning in Language Models LLM Mathematical Reasoning

Sep 30, 2026cs.AI

Budget Boundary Effects in Test-Time Mathematical Reasoning

A cumulative token cap can fall inside a mathematical derivation, forcing a test-time controller to choose between stopping at the cap (strict) and allowing the current attempt to finish (advisory). We measure this boundary choice with paired offline replays of 19,200 public traces: 120 AIME, BrUMO and HMMT problems and two archive configurations of one model. Candidate order and a 16-attempt cap are fixed, and answer selection is blind to reference answers and correctness labels. Three findings emerge. First, at the 4k cap, most advisory accuracy gains replace abstention with a correct answer; strict stopping pays for an unfinished prefix that the completed-only selector cannot use. Second, comparisons along realized cost differ from same-cap comparisons: advisory 4k in low has higher accuracy than strict 8k at comparable mean completion cost, while in high its observed accuracy is 0.42 points below strict 32k using 59% of its mean tokens. These aggregate comparisons do not establish equal-compute superiority or accuracy equivalence. Third, increased candidate coverage does not guarantee higher answer accuracy: a log-probability selector loses accuracy while coverage rises, including after a source-grade consistency repair. Same-cap majority-accuracy differences shrink below 1.3 percentage points at 32k. Budget curves should jointly state the cap, realized cost, eligible candidates, stopping rule and selector information.

Guilin Zhang, Ziqi Tan, Wulan Guo +5

Yantong AI

Mathematical Reasoning Benchmarks Test-Time Optimization Optimal Stopping Inference-Time Compute Allocation LLM Mathematical Reasoning

Sep 29, 2026cs.LG

Beyond Compression: Diagnosing How Post-Training Changes Mathematical Reasoning

Post-training is central to mathematical reasoning in modern large language models (LLMs), but endpoint pass@1 alone underidentifies what has changed. Gains may reflect newly reachable solutions, cheaper sampling of latent solutions, surface robustness, or memorisation. We compare three post-training paths under a common diagnostic readout: our sufficiently trained off-policy distillation trajectories, released Qwen3 off-policy-plus-on-policy distillation endpoints, and a released DeepSeek-Math endpoint trained with Group Relative Policy Optimisation (GRPO). Our probe uses cross-surface pass@K over verbatim prompts, paraphrases, numerical isomorphisms, and translations, plus consistency, distribution-shape, and verified supervised-fine-tuning (SFT) membership analyses. We find two regimes. On easier AMC problems, large-K ceilings are near saturation, so post-training mainly compresses sample cost. On harder AIME problems, post-training expands the large-K ceiling over the base model: sufficient off-policy distillation already raises this ceiling, Qwen3 released endpoints raise it further, and DeepSeek-Math GRPO does not dominate sufficient off-policy distillation at large K. English-dominant distillation improves non-English reasoning but preserves language-tier gaps. A controlled-overfit audit finds limited sensitivity in current SFT-membership probes. Compression is one regime of post-training, not a universal explanation.

Hongyang Li, Yiming Zhu, Xiao Li +3

University of Luxembourg · Seafill Open-Source Community · Université Paris-Saclay

Language Model Distillation RL Post-Training LLM Post-Training LLM Mathematical Reasoning

Sep 27, 2026cs.LG

Fine Until Fine-Tuned: Repeated Solutions Make Reasoning Fragile

Recipes such as s1 and LIMO teach a model to reason with little data by showing it the same thousand or fewer worked solutions many times over. Judged when that training ends, the repetition looks harmless. But reasoning models are often trained again, and we find that repetition leaves their reasoning fragile to that next stage, even when the stage has nothing to do with reasoning. We fine-tuned Qwen3.5-9B-Base on its own correct solutions to competition math problems, either drilling a few hundred of them about eight times each or showing many more once; with the same amount of training, both solve about 95% of held-out problems. A single pass of ordinary instruction tuning leaves the once-trained model where it was, while the drilled one falls to 86.0%, and harsher later stages take it to 59.3% or below. A third model that visited the drilled problems just as often, with a new solution at every visit, was unharmed, so the damage comes from seeing the same texts again rather than from having few problems. The break recurs with a stronger model's traces, in further training runs and on other models and tasks. It is also cheap to undo: the reasoning is suppressed rather than erased, and five updates of reasoning training bring almost all of it back, as does brief training on the reasoning format with almost no mathematics. Fresh solutions prevented the damage, and so did replaying 6.25% of the original solutions in a gentler later stage, so our claim concerns later training without such replay. Sharpening alone does not explain the break, since a model sharpened three-quarters as much without repetition was unharmed. On a skill the base model could not perform within a token budget, repetition mainly cost learning.

Ely Sheikh

Aix-Marseille University

Continual Learning for LLMs Data Repetition in Language Model Training LLM Mathematical Reasoning Catastrophic Forgetting

Sep 23, 2026cs.LG

Order-Invariant Answers, Order-Sensitive Representations in Mathematical Reasoning

Reordering a set of mathematical rules without changing its meaning should preserve the correct answer, but must a model's internal representations stay invariant too? We investigate this question using synthetic multi-step function-composition problems, each presented under multiple rule orderings with the same correct answer. We measure accuracy and permutation signal-to-noise ratio (SNR), which quantifies how distinctly ordering patterns are represented relative to variation across problem instances. Across 16 language models ranging from 1B to 8B parameters, we find a pattern: models that solve reordered problems more accurately represent different rule orderings more distinctly. Layer-averaged permutation SNR is positively rank-correlated with accuracy in every synthetic setting we evaluate, with Spearman correlations reaching 0.86. These findings highlight a distinction between answer invariance and representation invariance: successful mathematical rule composition can accompany distinct internal representations between equivalent rule orderings. This motivates distinguishing answer invariance from representation invariance, and offers a representational perspective on mathematical reasoning beyond answer accuracy alone.

Zhixu Silvia Tao

Operations Research and Financial Engineering Princeton University Princeton, NJ 08540

Mathematical Reasoning Representation Geometry in Language Models Compositional Representation Learning LLM Mathematical Reasoning

Sep 22, 2026cs.CL

Giving Credit Where It's Due: Redundancy-Aware Learning for Efficient Reasoning

Large reasoning models can produce correct yet unnecessarily long reasoning traces. Existing methods improve reasoning efficiency with trajectory-level objectives or local token- and step-level signals, but rarely model inter-step semantic dependencies. This limits their ability to distinguish redundant steps from those that support later deductions, making it harder to shorten reasoning without sacrificing accuracy. We introduce RECAP (REdundancy-aware Credit Assignment via Propagation), which addresses this limitation by assigning credit where it is due based on both a step's downstream role in the reasoning structure and its contribution to solving the problem correctly. We define structural responsibility to capture the step's downstream role by measuring how strongly later reasoning depends on it, using credit propagated backward from the final-answer node through an outcome-independent, LLM-annotated semantic dependency graph. However, a step can have high structural responsibility yet steer the reasoning away from the correct solution. RECAP therefore introduces step efficacy to measure answer-directed progress through changes in gold-answer log-likelihood as each step is added. Together, these signals reshape rollout-level GRPO advantages into step-specific updates. RECAP requires neither a separately trained process reward model nor preconstructed concise trajectories. Across two 7B models and four mathematical reasoning benchmarks, RECAP improves the accuracy-efficiency trade-off. On Qwen2.5-Math-7B, it improves pass@1 by 2.0-3.7 percentage points while reducing reasoning tokens by 8%-31% relative to GRPO across all four benchmarks. Analysis suggests these savings reflect fewer reasoning operations and less dead-end reasoning, rather than more compact expression.

Yuqing Zhou, Hong Wang, Manqing Mao +8

George Mason University · Amazon, Inc

RL for Language Model Reasoning Step-Level Credit Assignment in RL Efficient Language Model Reasoning LLM Mathematical Reasoning

Sep 22, 2026cs.AI

Math Reasoning in LLMs is Organized by Approach, Not Topic

Mathematical reasoning benchmarks are typically organized by topic, but language models may organize their internal computation by reusable reasoning approach instead. In this paper, we investigate whether open math-capable LLMs organize internally by topical sub-skill or by reasoning approach, and we present evidence that the approach is the key. We introduce a generation-replay protocol: a model first generates a solution, after which we replay the exact prompt-plus-generation trajectory and extract activation-importance signatures over the reasoning tokens. We cluster these signatures without supervision across eight models and five mathematical reasoning sources, then evaluate the recovered structure with structural, semantic, and intervention tests. Across all 40 model-source cells, the recovered clusters outperform matched-size random baselines. Two independent frontier-LLM judges find approach-level coherence in 77-82% of real clusters versus 6-11% in within-source controls, and topic-pure clusters usually receive labels finer than the topic itself. In approach-controlled prompting, changing the requested reasoning approach shifts cluster assignment in seven of eight model conditions, whereas paraphrases largely preserve it. These results indicate that math-capable LLMs organize internal mathematical computation by reasoning approach rather than benchmark topic. The implication is that topic-stratified benchmarks and topic-balanced training corpora can still miss the axis that matters: even deliberately topic-balanced corpora may remain imbalanced over reasoning approaches.

Sajad Goudarzi, Samaneh Zamanifard, Moloud Nasiri +1

Clemson University

Mathematical Reasoning Benchmarks LLM Interpretability LLM Mathematical Reasoning

Sep 21, 2026cs.AI

The Endless Exam: Mathematical Constructions from Today's Models toward Superintelligence

We introduce the Endless Exam, a benchmark spanning fourteen parameterised families of mathematical construction problems, with verifiable scores that distinguish progress before and beyond published mathematical frontiers. Each submitted object is checked automatically for validity and assigned a relative quality score against a published frontier or construction baseline, without capping improvements at 1. The benchmark draws long-term challenges from open mathematical problems and generates larger instances by varying their parameters. Compact certificates allow large constructions to be verified without listing every element. Across nine models evaluated on 69 distinct instances, continuous quality scores distinguish performance even though none of the 30 published-frontier references is surpassed. Size-quality curves show how construction quality changes as problem size increases. We release the generators, verifiers, references, model responses and analysis to support continued measurement before and beyond human frontiers.

Muhan Zhang

The University of Texas at Arlington

Mathematical Reasoning Benchmarks Benchmark Construction Synthetic Benchmark Generation LLM Mathematical Reasoning

Sep 16, 2026cs.LG

Sage: Formalization with Semantic Correction

While neural theorem provers have achieved impressive milestones in formal mathematics, they largely operate on the assumption that faithful Lean 4 formal statements are already provided. Translating informal natural language into a formal language is a critical data bottleneck plagued by an "illusion of rigor": standard type-checkers accept statements that compile but drop hypotheses, introduce vacuous truths, or subtly alter mathematical bounds. To resolve this, we introduce Sage (Semantic Agent-Guided Formalization Engine), an agentic framework that replaces monolithic translation with a four-stage decomposed generation pipeline coupled with a dual-signal semantic correction loop. By pairing Lean 4 compiler diagnostics with multi-dimensional semantic feedback, our correction loop enforces mathematical fidelity alongside syntactic validity. By explicitly accounting for the gap between open-ended queries and declarative formal targets, our pipeline prevents models from achieving high formalization rates by guessing unverified answers (exhibiting a 70.9% answer leakage rate in monolithic baselines). Consequently, Sage suppresses leakage to 2.7% while achieving 73.3% pass@4 joint compilation and semantic fidelity on the Omni-MATH without proofs (compared to 42.0% for a fine-tuned Goedel-Formalizer-V2 baseline). Finally, on IMO-Unformalized, a novel frontier of 175 unformalized International Mathematical Olympiad problems, Sage demonstrates effective zero-shot generalization with 87.4% pass@4 verified fidelity compared to just 19.4% for the baseline, winning over 79% of blind pairwise evaluations.

Thomas Hirtz, Farzad Jafarrahmani, Abdelmouksit Sagueni +3

Huawei Lagrange Mathematics and Computing Research Center Paris, France

LLM Self-Correction Lean Theorem Proving Autoformalization LLM Mathematical Reasoning

Sep 15, 2026cs.AI

A Four-Stage Decomposition of Word-Problem Solving and Mechanistic Fragility in LLM Math Reasoning

Large language models solve grade-school math word problems with high accuracy, yet a single irrelevant clause inserted into the problem can collapse it. We reconcile these observations with a mechanistic account. We show that the model's internal computation decomposes into a four-stage sequential pipeline, Schema Abstraction, Operation Planning, Operand Binding, and Computation, each stage producing a distinct intermediate representation in an identifiable band of layers. Using the same scaffold to diagnose distractor-induced failure, we localize the corruption to a single stage, Operation Planning, implemented by a set of attention heads whose causal role we validate bidirectionally. In short, we provide a mechanistic interpretation of math word problem reasoning in LLMs, and their failure when distracted.

Zhongdi Qu, Carla P. Gomes

Cornell University

Mathematical Reasoning Circuits in Language Models Language Model Robustness LLM Mathematical Reasoning

Sep 14, 2026cs.CL

Improving Mathematical Reasoning Capabilities in Large Language Models via Reasoning Process Error Classification

The reasoning ability of large language models (LLMs) is a critical factor for practical LLM-based applications. To investigate the current reasoning capability of LLMs, we clarify the types of errors that arise in LLMs' reasoning processes on mathematical datasets. We focus on problems where LLMs produce an incorrect answer. We define errors in the reasoning process as reasoning errors and manually analyze the features of reasoning errors. We defined and classified 21 error classes and identified the frequently occurring classes among them. Beyond qualitative evaluation, we leverage the evaluation results to improve the reasoning capability. We designed a prompt that explicitly focuses on eight error classes. The experiments demonstrate that this prompt effectively improves reasoning performance. Furthermore, the results suggest that the frequent reasoning errors identified in this paper are common across LLMs of comparable scale.

Runa Yoshida, Kosuke Nishida, Kyosuke Nishida

Human Informatics Labs., NTT, Inc.

LLM Prompting LLM Mathematical Reasoning LLM Reasoning

Sep 12, 2026cs.AI

An Open Recipe for IMO Gold: Training Nemotron for Olympiad Mathematics

We study how model post-training and test-time inference design affect natural-language proof generation for hard olympiad mathematics. Starting from Nemotron 3 Ultra, we train two specialist checkpoints using supervised fine-tuning and reinforcement learning, and evaluate checkpoint choice, verification, and refinement. Based on these findings, we present an open-model test-time-compute pipeline. The system operates entirely in natural language, with no formal prover, external tools, or internet access. Three Nemotron 3 Ultra checkpoints - the general-availability model and two post-trained specialists - power an iterative search that generates, verifies, and refines candidate proofs; a separate high-compute stage then selects each final submission. The system scored 30 out of 42 points at IMO 2026, reaching the gold-medal threshold. We release the two post-trained checkpoints as well as the training data, the training and inference code, the submitted solutions, and Nemotron-IMO-Bench, a new benchmark of 200 novel olympiad-level problems.

Ivan Moshkov, Stephen Ge, George Armstrong +3

NVIDIA

Mathematical Reasoning Benchmarks Test-Time Scaling Language Model Post-Training LLM Mathematical Reasoning LLM Reasoning

Sep 12, 2026cs.AI

Magenta: Closing the Loop Between Mathematical Reasoning and Lean Verification

Most of mathematical knowledge has been communicated through so-called informal use of mathematics and natural language. With large language models (LLMs) being highly adept in using natural language, they achieve strong performance, yet not perfect, in informal mathematical reasoning. Restraining LLMs to informal reasoning misses out on the opportunity to use the discrete verification abilities that machines offer through machine-checkable proofs. In this paper, we bridge the gap between informal and formal reasoning by integrating Lean signals into the informal reasoning process. We introduce Magenta, a training-free agentic pipeline that, given only a natural-language problem, produces an answer, expresses it as a Lean 4 statement, and constructs a machine-checked proof. A statement judge verifies whether the formalisation preserves the original problem, while an error-attribution judge routes failed attempts either to mathematical re-derivation or local Lean repair. Magenta achieves 100% accuracy across all evaluated olympiad benchmarks, including AIME 2025, AIME 2026, and HMMT February 2026. When paired with the open-weight K2-Horizon-7B reasoner, it solves all six IMO 2026 problems. Our analysis shows that statement adjudication is essential for preventing false certificates and that feedback-guided correction outperforms independent resampling on difficult problems.

Joshua Ong Jun Leang, Haonan Li, Zheng Zhao +6

Institute of Foundation Models · Imperial College London · University of Edinburgh +1

Automated Theorem Proving Agentic Reasoning Lean Theorem Proving Autoformalization LLM Mathematical Reasoning

Sep 9, 2026cs.AI

From Symbolic Perception to Logical Deduction: A Framework for Guiding Language Models in Geometric Reasoning

Plane geometry remains a significant challenge in AI, requiring the integration of visual perception and mathematical reasoning. While Large Multimodal Models (LMMs) naturally handle visuo-linguistic inputs, they are often computationally intensive and opaque. We demonstrate that a pure Large Language Model (LLM), when equipped with specialized modules, can rival state-of-the-art LMMs on complex geometry problems. Our framework integrates a Geometric Vision Parser, which translates diagrams into symbolic form, with a Symbolic Solver that performs formal deductions, thereby mitigating hallucinations and promoting interpretable reasoning. To enable rigorous evaluation, we curate a benchmark of challenging problems from the 2025 Chinese Zhongkao examinations, ensuring data novelty and testing deeper deductive skills. Experiments demonstrate that our approach achieves performance comparable to Gemini 2.5 Pro while delivering clearer, human-like solutions.

Weichen Dai, Rafael Medeiros Cabral, Ziyi Shou +4

University of Science and Technology of China

Visual Reasoning Symbolic Reasoning LLM Mathematical Reasoning

Aug 31, 2026cs.AI

HSRM: Hidden-State Reward Models for Test-Time Verification

Large language models can often generate plausible mathematical reasoning traces, but reliably identifying the correct solution among multiple candidates remains a key challenge. Existing test-time reasoning pipelines typically rely on text-based verifiers that re-read each generated solution, making verification an expensive component of inference. Prior work has shown, however, that LLMs often encode correctness-related signals in their internal representations, including awareness of when their own answers are likely to be wrong. Building on this observation, we introduce HSRM, a lightweight hidden-state reward model that verifies candidate solutions by directly reading the generator's internal representations rather than re-processing its text. HSRM extracts hidden states from a frozen generator at reasoning-step boundaries and uses a small Transformer encoder to rank candidates. It is trained from self-generated trajectories with outcome labels, requiring neither human-written process supervision nor a large pretrained verifier. Across four mathematical reasoning benchmarks, HSRM matches or outperforms a 55M-parameter text-only energy verifier in 15 of 16 generator--dataset settings while using only about 2M parameters, providing an efficient alternative to text-only verification by reusing representations already computed during generation.

Xianzhi Li, Xiaodan Zhu

Department of Electrical and Computer Engineering & Ingenuity Labs Research Institute Queen’s University

LLM Answer Verification Efficient Language Model Reasoning Efficient Language Model Inference LLM Mathematical Reasoning

Aug 31, 2026cs.CL

More Capable, Less Faithful: A Multilingual Analysis of Mathematical (Un)Solvability Detection in LLMs

Solvability detection is one of the most challenging aspects of mathematical reasoning for Large Language Models (LLMs). While prior work has studied this capability extensively, these analyses have been limited to English. Consequently, it remains unclear whether multilingual failures arise from differences in internal Solvability Belief or from language-dependent failures to express it. To address this gap, we introduce the first multilingual benchmark of paired solvable and unsolvable mathematical problems, extending ReliableMath to French and Greek. Using this, we train multilingual probes predicting Solvability Belief and analyze the solvability detection capabilities of state-of-the-art LLMs behaviorally, representationally, and in terms of faithfulness. We find that Solvability Belief is encoded as a largely universal, language-agnostic feature, and that higher-resource languages such as English, despite achieving stronger mathematical reasoning performance, exhibit lower solvability-detection faithfulness.

Maria-Eleni Zoumpoulidi, Nikolaos Xiros, Georgios Paraskevopoulos

Institute for Language and Speech Processing, Athena Research Center, Greece

Multilingual Language Model Evaluation LLM Reliability Cross-Lingual Consistency in Language Models LLM Mathematical Reasoning

Aug 30, 2026cs.AI

Mitigating Over-Optimization in PRM-Guided Search in Mathematical Reasoning by Optimizing the Guide

Process reward models (PRMs) provide dense step-level guidance for search-based reasoning, enabling inference-time compute to be allocated toward promising partial solutions. However, recent evidence suggests that PRM-guided search can over-optimize imperfect process rewards, pruning viable trajectories while expanding spurious ones. In this work, we theoretically show that directly leveraging PRM score is vulnerable to verifier noise through an extreme-value effect: non-viable prefixes become more likely to receive spuriously high scores as reasoning depth increase. Therefore, we formulate the PRM-guided search as a robust optimization problem over plausible reward perturbations, termed maximin PRM-guided search, leading to a training-free robust process supervision method that preserves promising alternatives when step-level scores are noisy. Maximin PRM-guided search mitigates this failure mode by reducing sensitivity to over-optimized PRM outliers. Without fine-tuning or online adaptation, maximin search consistently improves the PRM-guided search by 17-35% on average, outperforming outcome- and step-level baselines in 14 out of 16 settings. Our source code is available at https://github.com/tjoo512/maximin-search.

Taejong Joo, Diego Klabjan

Department of Industrial Engineering & Management Sciences Northwestern University

Inference-Time Search Process Reward Models Robust Optimization LLM Mathematical Reasoning

Aug 11, 2026cs.AI

Long-Horizon AI Research for Grothendieck Constant: A Case Study in Human-AI Mathematical Collaboration

AI agents are increasingly used in mathematics research, but it is often unclear how to use them effectively. Towards this, we present an extensive case study of how AI was used to improve bounds on the Grothendieck constant

K_G

, which captures the hardness between combinatorial problems and their continuous relaxations. Specifically, while the precise value of

K_G

is not known, we recently tightened the best known bounds to

\frac{6π}{11} \;\le\; K_G \;\le\; \fracπ{2\log(1+\sqrt2)} - 10^{-4}.

Crucially, these improvements were achieved using an AI research system that could arrive at insights deemed novel by domain experts. We give a detailed discussion of our experience using AI for mathematics research, particularly touching upon its strengths and weaknesses, as well as our experience with creating ideal conditions for AI to arrive at breakthrough insights.

Alan Li, Rahul Saha, Anton Xue +4

Deep Research Agents AI-Assisted Scientific Research Human-AI Collaboration LLM Mathematical Reasoning

Aug 9, 2026cs.AI

MathShikkha: A Controlled Study of Answer-Only and Chain-of-Thought Supervision for Bangla Mathematical Reasoning in Small Language Models

Mathematical reasoning remains challenging in low-resource languages such as Bangla. We study whether teacher-generated Bangla Chain-of-Thought (CoT) supervision provides benefits beyond ordinary supervised fine-tuning. We construct \textsc{MathShikkha}, a Bangla mathematical reasoning dataset with GPT-5.4-generated rationales, and fine-tune four 4B--7B student models under a matched protocol in which answer-only and CoT conditions share data splits, response-only loss masking, decoding, and scoring, differing only in the training target. In-domain, CoT provides no significant improvement over answer-only fine-tuning for three stronger backbones (paired bootstrap 95% CIs include zero; exact McNemar

p \geq 0.17

), despite generating 15--52

\times

more tokens, but significantly improves the weaker 4B model by 18.56 points (

p < 0.0001

). On the larger, contamination-audited BanglaMATH benchmark, this pattern reverses: CoT significantly outperforms answer-only supervision for all four models by 20.1--28.1 points (all

p < 0.0001

). Answer-only fine-tuning also reduces out-of-domain accuracy below the base model for three models, whereas CoT preserves or improves it for all four. A human study with two co-author annotators, external-expert adjudication, and Cohen's

κ= 0.76

--

1.00

finds no significant CoT improvement over the base model on reasoning-content criteria; instead, its measurable effect is target-language adherence and producing inspectable reasoning. Overall, rationale supervision's value depends on backbone capability and distribution shift: in this setting, its main benefits are Bangla adherence, auditable reasoning, and out-of-domain robustness rather than improved in-domain reasoning validity.

Rahma Simin Ali, Jawad Hossain

Independent Researcher · University at Albany

LLM Fine-Tuning Mathematical Reasoning Language Model Robustness LLM Mathematical Reasoning

Aug 8, 2026cs.LG

From token probabilities to calibrated confidence: An empirical study of mathematical question answering

Confidence estimation for large language models (LLMs) aims to estimate the probability that a generated answer is correct, while calibration aligns these estimates with empirical accuracy. Prior work has shown that token probabilities are often overconfident, we investigate whether these readily available signals can nevertheless provide well-calibrated confidence estimation for mathematical question answering. We compare single-pass estimators, which reuse token probabilities from the original generation, with multi-pass estimators, which obtain additional confidence signals through verification or stochastic forward passes. While individual token probabilities can be highly saturated, we find that aggregating token probabilities over the full sequence captures small but consistent differences between correct and incorrect generations, yielding more informative confidence estimates. Multi-pass methods can yield calibrated confidence estimates. We study two such approaches: self-verification through re-prompting, including a lower-cost in-situ variant, and Monte Carlo Dropout, which derives confidence from variation across stochastic forward passes. We further evaluate two post-hoc calibration methods, Platt scaling and isotonic regression, both of which substantially reduce in-domain calibration error. However, their data efficiency varies with dataset difficulty, and the calibration mappings often transfer asymmetrically across datasets and models.

Avery Ma, Lorne Schell, Vin Bhaskara +1

RBC Borealis

LLM Answer Verification Confidence Estimation in Language Models Language Model Calibration LLM Uncertainty Estimation LLM Mathematical Reasoning

Aug 5, 2026cs.CL

Constraint-First Reasoning: A Training-Free Protocol for Exploiting Answer-Space Constraints in Mathematical Problem Solving

Large language models can derive a plausible mathematical object yet still violate explicit requirements--for example, by omitting a modular reduction, returning a non-integer, or using the wrong encoded answer form. We introduce Constraint-First Reasoning (CFR), a training-free two-stage prompting protocol: Stage 1 extracts and summarizes constraints entailed by the problem, and Stage 2 solves while checking intermediate and final results against that summary. Routed-CFR activates the two-stage protocol only when a text-only regex router detects restrictive cues; otherwise it uses direct chain-of-thought (CoT). Across AIME, CMIMC, BRUMO, and AIMO_AMC, the method improves direct CoT on multiple backbones. We further report convention-controlled routing experiments, matched prompting baselines, problem-level paired tests, decoding robustness, constraint-quality audits, total-token accounting, and an OlympiadBench evaluation. These analyses position CFR as a targeted test-time intervention whose benefit depends on recoverable constraints and reliable Stage 1 extraction, rather than as a general-purpose replacement for mathematical reasoning.

Hongbo Ma, Bangji Yang, Yunqian Selina Cheng +3

1Tsinghua University · University of Illinois Urbana-Champaign

LLM Prompting LLM Mathematical Reasoning

Aug 4, 2026cs.CL

Mind the Cap: Output-Budget Regimes Change the Measured Multilingual Reasoning Gap

Multilingual evaluations report accuracy at a single output-token cap, but languages need different numbers of tokens to express the same content, so the cap is a hidden experimental variable. We test whether the native-vs-translate gap on MGSM (German, Thai, Swahili) is a token-budget artifact for Qwen3-8B and Llama-3.1-8B-Instruct under four prompting strategies. The measured gap swings by up to 57 points across budgets, length normalization moves it by up to 38.9 points where the cap binds, and at tight caps normalization can reverse which strategy scores higher. We prospectively froze the sweep's three Qwen peaks and its near-zero value at 1024 and evaluated them on 540,000 independently hard-capped decodes: a second frozen family of six Holm-corrected tests rejects every null. The frozen test at

B^*=1024

still fails to reject because native accuracy has already saturated there; above saturation, the residual difference is a strategy-performance gap, not an identified reasoning deficit. The same truncation channel prices a cost-ordered adaptation ladder: a cross-fitted Thai vocabulary extension closes 0.0 points of the gap at the frozen budget and 4.9 points where 19% of traces still truncate. A third frozen family varies only the announced budget at a fixed enforced cap; announcing 128 rather than 2048 tokens moves Thai native accuracy by 5.1 points, so accuracy is not a function of the enforced cap alone. A correct-emission timing identity computed from one long-cap run matches the three pre-specified MGSM peaks to 0.65 points and, in an exploratory Qwen-only analysis of three further benchmarks, tracks held-out items to 0.92 points, locating the peak exactly in five of seven cells. Treat the output cap as an independent variable and report accuracy across the budget regime, not at a single budget.

Ankit Goyal, Jaideep Ray

Multilingual Language Model Evaluation LLM Evaluation Mathematical Reasoning LLM Mathematical Reasoning