Mathematical Reasoning

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.

Performing deliberate mathematical reasoning in visual contexts is a hallmark of advanced Multimodal Large Language Models (MLLMs) and requires a sophisticated synthesis of perceptual grounding and symbolic logic. However, in the realm of mathematical functions, our investigation reveals a critical modality interference phenomenon: even advanced models, while performing textual computational reasoning, tend to disregard or misinterpret essential visual cues. To address this challenge, we propose Func-R1, which synergistically harmonizes precise visual perception and rigorous logical reasoning. Concretely, built upon an explicitly decoupled architecture, we employ a hierarchical post-training framework to progressively identify critical visual evidence and conduct in-depth theoretical reasoning. Furthermore, the Perception-Aligned Theoretic Optimization (PATO) strategy is proposed to steer policy updating towards internalizing fundamental theoretical properties while dynamically rectifying heterogeneous visual information throughout the reasoning process. Extensive experiments across diverse benchmarks demonstrate that Func-R1 delivers the optimal performance among open-source MLLMs, even surpassing GPT-5 with an 8.4% improvement on MathVerse's function-oriented tasks.

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.

Supervised fine-tuning (SFT) applies a uniform cross-entropy loss to all target tokens, even though different tokens provide unequal learning signals for mathematical reasoning. This uniform treatment can over-sharpen already mastered tokens while amplifying learning pressure on uncertain, low-confidence tokens, leading to suboptimal training dynamics. We propose Trimmed Logit-Gap SFT (TrimSFT), a simple token-level reweighting method that scales the SFT loss according to the logit gap between the gold token and its strongest competitor. TrimSFT trims supervision away from both extremes: tokens already mastered (large logit gap) and tokens weakly supported by the current model (small or negative logit gap), concentrating learning within an intermediate logit-gap region between them. We instantiate this principle with a Gaussian weight centered at margin m with bandwidth τ, requiring no reference model or additional forward pass. We evaluate TrimSFT on six base models from the Llama, Qwen, and DeepMath families across five mathematical reasoning benchmarks. TrimSFT consistently improves over standard SFT, achieving the best average performance on five out of six models, with gains of up to +26.9 points over SFT on MATH500. Further analyses show that the bandwidth τ matters more than the exact margin location, and that half-trim variants that remove supervision pressure from only one side yield inferior trade-offs. A token-level logit-gap distribution analysis suggests that TrimSFT reshapes model confidence in a more balanced way than uniform SFT or monotonic reweighting methods. These results suggest that reasoning SFT can benefit from trimming both extremes rather than treating all tokens uniformly.

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.

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.

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.

In Reinforcement Learning with Verifiable Rewards (RLVR) frameworks for mathematical reasoning tasks, floating-point results are typically evaluated using a tolerance-based reward. However, this strategy suffers from challenges such as difficulty in threshold calibration, unstable training dynamics, and limited accuracy, especially in clinical scenarios. To address these limitations, we propose a knowledge-guided hybrid reward framework (\textsc{MedCalc-R1}). Specifically, we introduce a knowledge verification reward mechanism that enforces explicit generation of computational formulas, which are further validated by an external verifier to enhance interpretability and reasoning reliability. Furthermore, we design a hybrid soft-hard reward scheme combining a hard constraint based on clinical safety thresholds with a soft, precision-sensitive reward that progressively guides learning within the acceptable range. Experimental results demonstrate that our method significantly outperforms existing baselines in both reasoning accuracy and generalization capability, validating the effectiveness and applicability in safety-critical domains.

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.

On-Policy Distillation (OPD) is emerging as a promising alternative to reinforcement learning for LLM post-training, yet its effectiveness in multilingual settings remains underexplored. We study OPD and its advanced variant, On-Policy Delta Distillation (OPD$^2$), for mathematical reasoning in English, Korean, and Japanese. OPD$^2$ improves OPD by using the probability gap between a post-trained teacher and its base model as the learning signal. Experiments with Qwen3 show that OPD$^2$ consistently outperforms the original OPD, with particularly strong improvements in Korean and Japanese, and generally narrows the English-Korean performance gap. We further find that English-only OPD can also increase performance for Korean and Japanese, but often shifts the responses toward English, highlighting the importance of multilingual data to preserving target-language responses.

Preference optimization improves mathematical reasoning in large language models (LLMs), but reliable chosen-rejected pairs usually require verified answers, human annotations, or external reward models. We investigate whether preference supervision can instead be derived from the model's internal representation geometry in a semi-supervised setting. Our analysis shows that reasoning trajectories generated across different mathematical problems form structured global point clouds in which correct and incorrect trajectories exhibit different geometric organization. Based on this observation, we propose Cloud--ScPO, a topology-guided preference-mining framework that uses a small labeled set to construct multiple correct and incorrect reference Clouds. Each trajectory is represented by a mean-pooled hidden state and scored against connectivity-induced components using a component-level soft $k$-nearest-neighbor measure averaged across reference banks. We combine this cross-problem Cloud signal with prompt-level self-consistency: self-consistency determines the answer-level preference direction, while Cloud scoring selects concrete trajectories and filters pairs by their score margin. Experiments on GSM8K and MATH-Numeric across four model settings show that Cloud--ScPO consistently improves over ScPO, with gains of up to 4.49% on GSM8K and 4.19% on MATH-Numeric. Pair-level analyses further show that Cloud--ScPO maintains comparable correctness reliability while more effectively separating informative chosen trajectories from incomplete, repetitive, or otherwise low-quality rejected responses.

Diffusion large language models (dLLMs) have emerged as a promising alternative to autoregressive LLMs, offering efficient generation through block-wise progressive unmasking. However, their strong general-purpose performance does not necessarily translate into reliable mathematical reasoning, where correctness depends on preserving coherent numerical-symbolic reasoning trajectories. In this work, we analyze the decoding trajectories of LLaDA 2.0 and identify a recurring diffusion confidence trap: local token confidence can become misaligned with global reasoning correctness during progressive block decoding. Our analysis reveals two representative failure regimes: sampling-sensitive failures, where correct paths exist but are unstable, and sampling-consistent failures, where repeated sampling converges to repetitive high-confidence but incorrect continuations. Motivated by this observation, we propose Evolutionary Decoding, a training-free test-time scaling framework that views diffusion decoding as an evolutionary process over candidate reasoning states. The framework combines step-wise selection, which preserves useful numerical-symbolic signals and suppresses repetitive patterns, with block-wise mutation, which introduces structured alternatives to escape incorrect high-confidence basins. Experiments on multiple benchmarks show that Evolutionary Decoding improves LLaDA 2.0 over confidence-based decoding, leading to more reliable mathematical reasoning.

Reinforcement learning has emerged as an effective paradigm for enhancing the mathematical reasoning capabilities of large language models. Among existing policy optimization methods, Proximal Policy Optimization (PPO) remains particularly appealing because its learned critic can, in principle, provide token-level credit assignment. However, in mathematical reasoning tasks characterized by long reasoning horizons and sparse outcome rewards, reliable token-level credit assignment remains challenging. The standard critic often fails to accurately evaluate intermediate reasoning states, resulting in noisy advantage estimates and suboptimal policy updates. In this paper, we propose ReDiPPO, a Reference-guided and Discrepancy-aware PPO framework for mathematical reasoning. ReDiPPO introduces a reference-guided critic that uses reference answers as training-time privileged signals to provide more accurate value estimation. Meanwhile, it retains a standard critic and quantifies the token-level reference-standard discrepancy between the standard value estimate and the reference-guided value estimate. This discrepancy serves as an indicator of difficult reasoning states and is used to reweight the corresponding token-level advantages during PPO optimization. Extensive experiments on diverse mathematical reasoning benchmarks demonstrate that ReDiPPO improves value-estimation accuracy and consistently outperforms strong policy optimization baselines, including PPO, DAPO, and GSPO, in final reasoning performance. Our code is available on https://github.com/cii030/ReDiPPO.

Mathematical Word Problems (MWPs) are an important benchmark for evaluating natural language understanding and quantitative reasoning. Despite recent progress in high resource languages, Bengali remains underexplored due to the limited availability of large scale annotated datasets. In this work, we introduce PatiGonit22K, an expanded Bengali MWP dataset containing 22,441 problems, developed by extending the original PatiGonit dataset with a substantially larger collection of complex mathematical problems. The dataset includes both simple and multi operation equations, providing a balanced benchmark for evaluating mathematical reasoning across different difficulty levels. Each problem is carefully translated, annotated, culturally adapted, and verified to ensure linguistic consistency and mathematical correctness. By increasing both the scale and complexity of Bengali MWPs, PatiGonit22K provides a more comprehensive resource for future research on mathematical reasoning and educational NLP applications in low resource languages.

Existing LLM-based theorem provers have achieved impressive results on formal mathematics benchmarks, yet they remain confined to acting as autonomous agents that prove a stated proposition. In this paper, we propose MathCoPilot, a human-in-the-loop system that embodies a new human--AI symbiotic paradigm for mathematical research, in which the mathematician steers the high-level mathematical direction while AI agents carry out the detailed formalization and proof work under continuous human guidance. MathCoPilot unifies three core capabilities: (1) an interactive workbench where the mathematician and AI agents collaborate through a living proof blueprint that decomposes a proof into navigable steps the human can directly inspect, direct, and refine; (2) automated proving skill orchestration with adaptive knowledge base search and Lean-integrated iterative verification; and (3) topic-driven paper retrieval and automated formalization into a verified Lean knowledge base. Using MathCoPilot, we systematically compare four state-of-the-art LLMs, including Gemini3.1Pro, GPT-5.4, and ClaudeOpus4.7, on a FormalMATH subset and on two real PDE theorems requiring deep domain expertise, evaluating their ability to produce verified Lean~4 proofs and to identify errors in deliberately incorrect proofs. Our results show that while current models can handle undergraduate-level problems with high success rates under favorable autoformalization conditions, substantial challenges remain for domain-specific theorems requiring genuine mathematical understanding.

We propose the AIMO Interpretability Challenge, a competition on distinguishing robust from spurious reasoning in frontier mathematical language models based on the models' internal mechanisms. The challenge is motivated by a central limitation of standard reasoning benchmarks: strong final-answer accuracy does not reveal whether a model relies on stable reasoning mechanisms or exploits brittle reasoning shortcuts. Building on AI Mathematical Olympiad (AIMO) problems and submissions, together with resources from the Fields Model Initiative, the competition will provide (1) newly-published olympiad-level math reasoning problems and their symbolic representations, allowing generation of novel functional variants, (2) access to frontier reasoning models, and (3) assessments of models' adversarial robustness on these problems. Participants will use these resources, along with our computing infrastructure support, to develop methods for identifying which models solve problems robustly. Our competition will also create a new, open robustness benchmark and baseline systems, aiming to provide a lasting foundation for standard benchmarking in mathematical reasoning and interpretability. Scientifically, the competition connects interpretability and generalization research around a central question in AI research: can we determine if, and to what extent, the decision-making of frontier AI models is generalizable and thus, reliable?

The evaluation of mathematical reasoning in large language models (LLMs) has predominantly focused on high-resource languages like English. This has created a significant barrier to the equitable development and deployment of AI in linguistically diverse regions such as Bangladesh, where over 230 million people speak Bengali. Despite this global significance, there has been minimal prior work on mathematical reasoning in Bengali and no existing research that systematically benchmarks a perturbated Bengali mathematical dataset, leaving a critical void in assessing model robustness and true comprehension beyond pattern recognition. This study addresses this gap by introducing GSM-Plus-BN, a novel perturbated Bengali mathematical dataset derived from the English GSM-Plus benchmark and verified by human translators. We evaluate six open-source LLMs Qwen3-32B, Llama-3.1-8B-Instant, Llama-3.3-70B-Versatile, Llama-4-Scout-17B-16E-Instruct, GPT-OSS-120B, and GPT-OSS-20B using a benchmark of 9,000 evaluation samples comprising 1,000 seed questions and 8,000 perturbed variants under both Standard Prompting and Chain-of-Thought (CoT) Prompting. Experimental results show that GPT-OSS-20B achieves the highest seed question accuracy of 96.08% under Standard Prompting, while larger models such as Llama-3.3-70B and GPT-OSS-120B demonstrate superior robustness across perturbation types. Furthermore, CoT prompting substantially improves reasoning for most models compared to Standard Prompting, yet a notable performance gap persists across all models relative to their English benchmarks, underscoring the inherent difficulty of perturbed Bengali text. This research makes a foundational contribution by providing GSM-PLUS-BN as a new resource and baseline for future Bengali mathematical reasoning research.

Large language models (LLMs) have achieved remarkable performance on high-school and olympiad-style mathematics, yet their capabilities on advanced mathematics remain poorly understood. Existing benchmarks, however, fall short in both scope and evaluation granularity: they provide limited disciplinary coverage and often rely on final-answer correctness or coarse judgments, leaving the validity of the reasoning process inadequately assessed. To bridge this gap, we introduce AdvancedMathBench, a benchmark suite designed to evaluate advanced mathematical reasoning capabilities. Its core proof-generation benchmark, ProverBench, contains 296 problems spanning undergraduate and doctoral qualifying-exam levels. To provide reliable evaluation of the proofs, we develop a dedicated automatic verification pipeline trained on large-scale expert annotations to produce both correctness verdicts and fine-grained assessments of proof errors, which exhibits strong agreement with human experts on held-out proof trajectories. We further introduce VerifierBench, consisting of 888 model-generated proof trajectories paired with expert ground truth, to evaluate whether models can correctly judge proof validity and provide sound verification rationales. Experiments show that AdvancedMathBench remains challenging for frontier models. On proof generation, the best-performing model, GPT-5.5-xhigh, achieves only 75.8 and 66.1 on the UGD and QE splits, respectively, indicating substantial room for improvement on advanced mathematical proof construction. On proof verification, the best model attains a Balanced F1 of only 65.1, and models generally exhibit low true negative rates, suggesting that critical error detection remains a major bottleneck.

Tree search algorithms enable systematic exploration of the proof space in neural theorem proving. Existing LLM tree search libraries primarily target natural language reasoning and do not provide native integration with formal verifiers, while theorem proving systems often rely on task-specific search implementations. We introduce TreeThink, an open-source Python library for modular, fully asynchronous tree search in neural theorem proving. It integrates established tree search methods with vLLM-based inference pipelines and diverse node evaluation techniques, ranging from lightweight heuristics to neural evaluators. We support Lean~4, Rocq, and Isabelle/HOL alongside natural language. It connects directly to each language's Read-Eval-Print Loop (REPL) server for real-time verification and proof state extraction. We evaluate TreeThink on miniF2F and MATH500, demonstrating cross-language formal proof search, natural language reasoning support, and up to 8.0$\times$ wall-clock speedup from asynchronous execution. Source code is released under the MIT license at https://github.com/GGLAB-KU/treethink , and the library is accessible as a downloadable package at https://pypi.org/project/treethink/ .

Mathematical reasoning benchmarks typically provide all facts needed to solve each problem, while interactive benchmarks often mix reasoning with tools, retrieval, and long-horizon dialogue. We introduce MIRA-Math, a benchmark for a narrower diagnostic capability: solving mathematical problems whose full latent state has a unique answer, but whose solver-facing view is missing exactly one necessary atomic fact. The solver must request the missing information in natural language under a strict budget and then integrate the returned fact into an exact final answer. A fixed constrained LLM responder sees only the dataset-provided atomic fact and must either offer the quoted fact when the request matches it, or decline otherwise. Thus, instance generation, typed hint specifications, validation, and final-answer verification are deterministic, while request metrics are measured under a fixed LLM-mediated responder channel. MIRA-Math contains 2{,}310 generated instances from 22 typed mathematical families spanning algebra, probability, linear systems, discrete structures, signal processing, Markov chains, circuits, interpolation, and numerical boundary-value problems. Experiments across frontier and small models show that request success and final-answer accuracy are separable: models may ask for the right fact yet fail the downstream computation, or fail before obtaining the canonical hint. We release generators, verifiers, prompts, run metadata, and dataset documentation to support reproducible evaluation of minimal information requesting in mathematical reasoning.

Mathematical reasoning has become a central task for evaluating and tuning reasoning Large Language Models (LLMs), yet existing benchmarks remain heavily biased toward high-resource languages, with English and Chinese dominating both pre-training corpora and evaluation suites. The recently released PolyMath (Wang et al., 2025) dataset represents a significant step forward, yet its coverage is still limited to 18 only high-resource languages. To address this gap, we introduce PluraMath, an extension of PolyMath to 18 additional {underrepresented languages spanning 6 language families -- ranging from mid-resource to extreme low-resource settings. We constructed the dataset through a human-curated pipeline, where native speakers thoroughly validated pre-computed translations. Using PluraMath, we then benchmark 27 reasoning LLMs across four model scales -- small, mid-size, large, and closed-source ensembles -- probing the multilingual mathematical reasoning capabilities of state-of-the-art models under diverse linguistic conditions. Our fine-grained analysis confirms a persistent gap in mathematical reasoning performance between high-resource and underrepresented languages, with stronger results largely associated with better instruction-following ability. We fully open-source our dataset, data acquisition pipeline, and evaluation framework, with the goal of lowering the barrier to multilingual benchmark development for underrepresented communities.

Although LLMs have made significant progress in mathematical reasoning, determining whether a mathematical problem is solvable remains a fundamental yet challenging capability. While recent studies have probed internal representations of model solvability beliefs, verbalization has primarily been studied behaviorally rather than as an internal representation, limiting its analysis and manipulation. We address this gap by separately probing representations of solvability knowledge and verbalization, allowing us to disentangle the two within model hidden states. Across multiple LLMs, we show that knowledge and verbalization are encoded as distinct, linearly decodable representations and that fabrication is primarily associated with changes in verbalization rather than the underlying knowledge. Prompting with unsolvability cues reduces fabrication primarily by shifting verbalization, while activation steering demonstrates that these representations can be echanistically manipulated to improve model abstention.

We advance the hypothesis that human mathematical reasoning, constrained by both the undecidability and the computational intractability of even modest logical fragments, relies fundamentally on pattern matching from domains external to pure deduction. The most prolific reservoir of such patterns is the natural world, whose physical laws and biological systems have undergone billions of years of ``pre-computation'' and already exhibit surprisingly innovative solutions. To ground this claim, we trace the history of the Fourier transform and relevant mathematics, from the vibrating string controversy to the hear equation and subsequent formalisms prevalent in mathematics. At each critical juncture, a physics problem forced the acceptance or creation of a mathematical tool that pure formal reasoning failed to anticipate or, worse, human reasoning had resisted. We further survey the landscape of logical complexity, from NP-hard propositional satisfiability to the non-elementary decision-procedures for monadic second-order theories, to demonstrate that even when a logic is decidable, the resources required for worst-case deduction are astronomically prohibitive. We argue that these barriers make physics-inspired pattern matching not just a historical accident but a cognitive necessity. Finally, we draw the consequence for artificial intelligence: if pure reasoning is constitutively insufficient, then any system aiming at human-level mathematical creativity must embed a vast store of cross-domain patterns rather than rely on deduction alone. This furnishes a principled justification for the enormous scale of contemporary large language models.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a promising paradigm for improving mathematical reasoning in language models. Yet most RLVR work rewards only the final answer (outcome-based rewards), leaving the impact of step-level process supervision (process rewards) underexplored especially for small models that lack the capacity to self-correct under sparse feedback. We systematically compare five reward conditions applied to Qwen2.5-0.5B fine-tuned with Group Relative Policy Optimization (GRPO) on GSM8K: a no-RL baseline, process-only, outcome-only, and three hybrid weightings ($λ\in \{0.9, 0.5, 0.1\}$ process weight). Process-only supervision achieves 63.73% test accuracy versus 53.75% for outcome-only, a nearly 10-percentage point gap while yielding reasoning traces with higher step validity and lower deviation from ground-truth chain length. Hybrid rewards generally correlate positively with process weight, with one notable anomaly: the low-process / high-outcome configuration ($λ=0.1$) underperforms pure outcome supervision, suggesting conflicting optimization signals. Error analysis using GPT-4o as a judge reveals distinct failure mode distributions: process models generate structurally inconsistent but arithmetically grounded traces, while outcome models produce concise but derivation-error-prone chains. Our results demonstrate that reward granularity is a first-order design decision for RLVR, with process-level supervision substantially improving both accuracy and trace fidelity in small language models.

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

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

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

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

Large language models (LLMs) reach high accuracy in mathematical reasoning, but individual traces on the same problem diverge; some arrive at the correct answer while others fail. Prior work analyzes failure at the step, chunk, or sentence level, or at tokens where failure has already occurred. Neither identifies the precise token that triggers the shift toward failure. We introduce the cliff token, a token where the token-wise potential drops significantly under an adaptive threshold that scales with the local token-wise potential, based on a one-sided two-proportion z-test. Across seven models and three mathematical reasoning benchmarks (GSM1K, MATH500, AIME 2025), cliff tokens act as failure triggers; deleting the first cliff token and resampling recovers pass@64 to 1.0, while keeping it limits recovery to between 0.71 and 1.00. We further introduce a cliff taxonomy of deterministic, uncertain, and sampled-off cliffs, defined by greedy choice and token entropy. Each type has distinct probabilistic characteristics, and the taxonomy generalizes across model scales. Finally, we validate the taxonomy via single-token preference optimization at cliff positions (Cliff-DPO). Trained on GSM8K, Cliff-DPO improves accuracy across benchmarks by up to +6.6. Optimizing at uncertain and sampled-off cliffs improves reasoning, while deterministic cliffs do not.

Large Language Models (LLMs) are traditionally viewed as autoregressive generators. However, from the perspective of collective computation, they function as high-dimensional Dense Associative Memories that store complex reasoning patterns as latent attractors. In this work, we investigate the energy landscape of mathematical reasoning. We posit that correct reasoning chains correspond to deep, wide attractor basins ("flat minima") in the model's output distribution, whereas hallucinations manifest as sharp, unstable local minima. To exploit this geometry, we introduce a retrieval mechanism based on a Gibbs measure of the trajectory's spectral entropy. By sampling multiple reasoning paths and weighting them by their inverse energy ($P \propto e^{-βE}$), we approximate the equilibrium distribution of the associative memory, effectively ``relaxing'' the system into a robust solution. Empirically, this physics-inspired mechanism improves Microsoft Phi-3.5 performance on GSM8K by 5.38% (84.7% $\to$ 90.1%), demonstrating that inference is better modeled as a dynamic settling process into an attractor basin rather than greedy next-token prediction.

Scaling reinforcement learning for visual mathematical reasoning requires more than generating harder questions: as data volume grows, the reward labels themselves must remain reliable. Yet existing data pipelines scale supervision while trusting the labeller, and policy-side methods assume the underlying answers are already correct. We instead treat scaling as a verifiable data-construction problem and decouple two axes before any policy update: prompt difficulty, expanded by route-specific evolution operators, and answer reliability, enforced by offline hypothesis-test falsification. We instantiate this as VeriEvol, an iterative framework with two extensible components: a type-aware evolution module that rewrites low-difficulty image-question seeds into harder, image-grounded prompts; and HTV-Agent, a verifier that accepts an answer only after multi-source counter-evidence has failed to refute it. The resulting verified data scales in volume, extends by adding evolution routes or verifier channels, and plugs directly into existing GRPO-style RL recipes. On a five-benchmark visual-math suite, scaling evolved SFT data from 10K to 250K samples raises the mean accuracy from 35.42 to 54.73; then, with backbone, SFT initialization, and GRPO recipe held fixed, VeriEvol adds a cumulative +3.88 over an un-evolved RL baseline, of which +1.82 comes from evolved prompts and +2.06 from the HTV-Agent verifier. We release the prompts, data, models, code, and the full verifier trace of every sample, so that downstream work can scale and audit the pipeline rather than only inspect its outputs.

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

Enhancing the formal math reasoning capabilities of Large Language Models (LLMs) has become a key focus in both mathematical and computer science communities in recent years. While significant progress has been made in using state-of-the-art Auto-Regressive (AR) LLMs for formal theorem proving, these models suffer from inherent limitations. Their next-token prediction generation methods may yield suboptimal performance due to the challenges of long-range coherence and the compounding of errors over long sequences. Recent advancements in diffusion LLMs (dLLMs), which generate text through iterative denoising of a multi-token block, offer a promising alternative. However, the application of dLLMs to formal mathematics, where maintaining long-range coherence is critical, remains largely understudied. To address the challenges above, we propose Diffusion-Proof, to the best of our knowledge, the first framework to train and apply dLLMs for formal theorem proving. Our frameworks contain training and inference methods for two models. The first one is dLLM-Prover-7B, which performs whole-proof writing with long-range coherent tactic usage. The second one is dLLM-Corrector-7B, which is a novel large block diffusion-based correction model. It leverages the in-filling capabilities of dLLMs to perform local proof correction using bi-directional information. Extensive experiments demonstrate that Diffusion-Proof relatively significantly outperforms the AR LLM baseline trained under the same dataset. Diffusion-Proof achieves an absolute improvement of 1.61% on ProofNet-Test and 6.14% on MiniF2F-Test benchmarks compare to the baseline. Notably, Diffusion-Proof successfully resolves one IMO problem that more advanced thinking model DeepSeek-Prover-V2-7B could not solve, showcasing the unique advantage of dLLMs in formal theorem proving.

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

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

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

Sharing mathematical content in online forums remains a significant friction point for students and educators: writing raw LATEX is error-prone, standalone optical character recognition tools require platform switching, and current forum software offers no integrated path from a photograph of a formula to a rendered post. We present a unified system that eliminates this friction by embedding an image to LATEX conversion pipeline directly inside a forum posting interface. A user uploads or captures an image of a mathematical expression; the system routes it through the Mathpix OCR API, detects whether the returned output is LATEX or plain text containing inline math, applies the appropriate delimiter normalisation, and renders a live preview in either LATEX or Markdown mode before the post is committed to the database. The architecture is organized in three loosely coupled layers: image processing, rendering, and storage, and supports both desktop and mobile clients. A provisional US patent application has been filed covering the core methods. We describe the full system design, each component in detail, the data schema, and the key technical innovations, and we position the work against existing standalone tools and forum platforms to demonstrate the practical gap it closes. Beyond immediate usability, we argue that a deployed platform of this kind constitutes a continuously growing, community-validated dataset of mathematical problems and step-by-step solutions, a resource that can be used to train and benchmark AI systems for accurate mathematical reasoning

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

While Large Language Models (LLMs) have achieved near-perfect performance in \emph{solving} high-school mathematics, their ability to \emph{evaluate} the diverse reasoning processes of real human students remains under-examined. To bridge this gap, we introduce \textbf{RealMath-Eval}, a rigorously annotated benchmark of 224 real-world exam responses from high schools. Our initial evaluation reveals that even state-of-the-art LLM judges struggle significantly on this task, exhibiting a high Mean Squared Error ($\sim$2.96) against expert human grading. To probe a plausible explanation, we contrast this performance with a control setting where the same judges evaluate synthetic LLM-generated solutions. We identify a stark Evaluation Gap'': judges are considerably more accurate and consistent on synthetic text (MSE $\sim$1.17) but struggle to generalize to authentic student reasoning. Through semantic embedding analysis, we find that synthetic errors suffer from a structural collapse'' into predictable, low-dimensional linear subspaces, whereas human errors form a more diverse error space. Furthermore, generative probability probes suggest that human reasoning involves significantly higher information-theoretic surprisal, indicating that student reasoning transitions are more out-of-distribution for current models. Finally, we find that surface-level style transfer fails to close this gap. Our findings suggest that current LLM evaluation pipelines relying heavily on synthetic data may not adequately capture the diversity of authentic student mathematical reasoning.

Mathematical reasoning has long served as a stringent test of machine intelligence; over the past decade, it has moved from a niche problem within NLP to one of the most consequential AI frontiers. This survey provides a unified account of the field's evolution, from early rule-based math word problem (MWP) solvers and template-driven geometry systems, through neural expression generation and LLM prompting, to contemporary reasoning models, multi-agent systems, neuro-symbolic theorem provers, and verified discovery workflows. We organize the landscape along four axes: (i) informal reasoning over text and diagrams, spanning MWP solving, multimodal geometry, and VLMs; (ii) formal reasoning in proof assistants, including autoformalization, tactic prediction, compiler-guided repair, and proof search; (iii) mathematical discovery, where systems propose constructions, improve bounds, or assist attacks on open problems; and (iv) the inference and training-time techniques, including CoT prompting, tool use, process reward models, and RLVR, that increasingly connect generation with verification. We catalog major benchmarks across grade-school arithmetic, competition mathematics, geometry, formal proving, multimodal and multilingual reasoning, and expert evaluation, and we examine benchmark saturation, contamination, reporting mismatches, and the distinction between pass@1, majority voting, and verifier-assisted pass@$k$. We critically assess failure modes: brittleness under perturbation, reward hacking, multimodal grounding failures, fragile formalization, and the energy cost of reasoning-scale inference. Drawing on recent perspectives from working mathematicians, we identify future directions centered on verified-discovery workflows, reasoning efficiency, and infrastructure to make AI-assisted formalization broadly usable. Companion materials: https://github.com/Starscream-11813/awesome-AI4Math.

Deploying Small Language Models (SLMs) on edge devices requires efficient fine-tuning strategies that adapt models to new tasks without degrading their general capabilities. In this study, we benchmark five sub-1B models (135M-1B) on mathematical reasoning tasks and uncover a critical vulnerability: Full Fine-Tuning (Full FT) actively harms performance in models under 300M parameters, often dropping accuracy below zero-shot baselines. This "negative transfer" makes Parameter-Efficient Fine-Tuning (PEFT) not just an efficiency preference, but a stability requirement. We find that while Low-Rank Adaptation (LoRA) and Weight-Decomposed LoRA (DoRA) perform comparably, their strengths vary by task; DoRA excels in complex reasoning (GSM8K), while LoRA dominates pattern matching (OrcaMath). In particular, Full FT is outperformed by LoRA on aligned models (Qwen2.5-0.5B) and even by simple 5-shot In-Context Learning on the smallest architectures (SmolLM2-135M). Based on these findings, we recommend defaulting to PEFT for all aligned sub-1B models and caution against Full FT for any architecture smaller than 500M parameters to prevent catastrophic forgetting. Reproduction of this work can be found at https://github.com/gulguluu/tiny-slm-finetune-compare.

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 3-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, 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.

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.

Reinforcement learning with verifiable rewards (e.g. GRPO) is now a common way to improve mathematical reasoning in Large Language Models (LLMs). However, current methods usually broadcast one sequence-level advantage to all tokens, or use costly process reward models (PRMs) for step-level supervision. Uniform advantage distribution assumes that all tokens contribute equally to the final reward. This dilutes the gradient signal, since flawed reasoning steps and filler words are updated as strongly as valid logical inferences. To address this, we introduce Gradient-Reweighted Advantage (GRAIL), an intrinsic token-wise advantage reweighting method. GRAIL uses gradient-activation saliency to place more weight on tokens that are more locally sensitive to the final answer. Evaluations across five models from the Qwen3, R1-distilled and OctoThinker families show that GRAIL consistently outperforms GRPO. GRAIL achieved an average improvement of 3.60% in accuracy and 3.05% in Pass@3, demonstrating that fine-grained reasoning alignment can be achieved without process-level supervision.

While Process Reward Models (PRMs) have achieved remarkable success in mathematical reasoning, their application in complex scientific domains-such as biology, chemistry, and physics remains largely unexplored. Scientific problems demand not only logical rigor but also factual consistency and the precise usage of domain-specific tools, areas where current models often suffer from hallucinations and lack of verification. In this paper, we first construct SCIPRM70K, a large-scale dataset featuring Chain-of-Tool trajectories that explicitly interleave reasoning with the execution of scientific tools. Building upon this, we train an efficient reward model called Sci-PRM to provide fine-grained supervision on tool selection, execution accuracy, and result interpretation at each step in one inference. Experiments demonstrate that Sci-PRM significantly enhances foundation models in two key aspects: (1) it enables effective test-time scaling via Best-of-N selection; and (2) when integrated into Reinforcement Learning, it serves as a dense reward signal that mitigates the critical issue of advantage disappearance, allowing the model to break through existing performance ceilings.

For centuries, human mathematicians have written proofs to substantiate their mathematical arguments; yet, the ability to automatically verify the validity of proofs has long been a challenge. Advances in AI systems' ability to generate code and engage in increasingly high-level mathematical reasoning promise to transform people's ability to formalize and thereby verify proofs. While many works focus on benchmarking the current frontier, we instead study how people use these tools. We conduct a mixed-methods analysis into the initial impact of AI on people's formalization workflows: what people claim they want, what they see as the barriers to those visions, and how they actually use and adapt AI in practice. A qualitative survey shows that people's preferences are diverse, but with a general desire for AI assistance in formalization that preserves high-level human control over the proof discovery process. To assess how people actually engage with AI for formalization under such limitations, we conduct a controlled user study in which participants formalize informal math problems and their proofs, with and without AI, across a range of mathematical problems at varying levels of difficulty and domains. Despite limitations of the tools at the time for autoformalization, participants tend to attain higher formalization accuracy when allowed access to AI tools than when formalizing on their own, with most participants flexibly choosing to use multiple different AI tools. Taken together, our work sheds light on the early stages of AI integration into formalization workflows, involving an intimate interplay of human and AI engagement.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Reliable verification of proofs remains a bottleneck for training and evaluating AI systems on hard mathematical reasoning. Fully formal proofs, in languages like Lean, are easy to verify because they are unambiguous and modular. Most proofs, particularly those written by AI systems, have neither property, and translating them into formal languages remains challenging in many frontier math settings. We propose Pseudo-Formalization (PF), a proof format that captures the modularity and precision of formal proofs while retaining the flexibility of natural language. A Pseudo-Formal proof is decomposed into self-contained modules, each stating its premises, conclusion, and proof in natural language. To verify the correctness of a regular natural language proof, an LLM translates it to Pseudo-Formal and then verifies each module independently, an algorithm we call Block Verification (BV). We evaluate PF+BV on two benchmarks spanning olympiad and research-level mathematics, where it pareto-dominates LLM-as-judge baselines on error-finding precision and recall. To support future work, we release our research-level proof verification benchmark ArxivMathGradingBench.

Code has become a standard component of modern foundation language model (LM) training, yet its role beyond programming remains unclear. We revisit the claim that code improves reasoning through controlled pretraining experiments on a 10T-token corpus with fine-grained domain separation. Our findings are threefold. First, when code is restricted to standalone executable programs and Code-NL data are controlled for, code substantially improves programming ability but does not act as a general reasoning enhancer; instead, it competes with knowledge-intensive tasks, especially complex mathematical reasoning. Second, the reasoning gains often attributed to code are better explained by cross-domain structured reasoning traces, such as code-text and math-text mixtures, rather than by executable code alone. Third, increasing the density of structured math-domain samples within a fixed math budget yields substantial gains on difficult mathematical reasoning while largely preserving programming performance, suggesting that cognitive scaffolds offer a targeted way to mitigate cross-domain trade-offs. Finally, routing analyses show that data-composition effects are reflected in expert-activation patterns, providing mechanism-level evidence for competitive and synergistic interactions across domains. Our results clarify which data characteristics transfer across capability dimensions and point to more precise data-centric optimization strategies.

Mathematical reasoning is essential for problem-solving in education, science, and industry, serving as a crucial benchmark for evaluating artificial intelligence systems. As Large Language Models (LLMs) improve their reasoning capabilities, understanding how well they perform mathematical reasoning has become increasingly important. This survey synthesizes recent advancements in mathematical reasoning with LLMs through a structured analysis of datasets, architectures, training strategies, and evaluation protocols. Our systematic review encompasses approximately 120 peer-reviewed studies and preprints, examining the evolution of this research area and providing a unified analytical framework to understand current progress and limitations. Our study particularly introduces a unified taxonomy of mathematical datasets, distinguishing between pretraining corpora, supervised fine-tuning resources, and evaluation benchmarks across varying levels of reasoning complexity. A systematic analysis of reasoning architectures and training strategies, including tool integration, verifier-guided reasoning, and parameter-efficient adaptation, is presented to assess their effects on reasoning robustness and generalization. Moreover, a comparative evaluation of existing metrics highlights the gap between final-answer accuracy and process-level reasoning verification. By synthesizing insights across these areas, our analysis identifies recurring failure modes, such as reasoning faithfulness issues, benchmark biases, and generalization limitations, and outlines key research directions toward improving symbolic grounding, evaluation reliability, and the development of more robust and trustworthy LLM-based reasoning systems.

Frontier AI models and multi-agent systems have led to significant improvements in mathematical reasoning. However, for problems requiring extended, long-horizon reasoning, existing systems continue to suffer from fundamental reliability issues: hallucination accumulation, memory fragmentation, and imbalanced reasoning-tool trade-offs. In this paper, we introduce STAR-PólyaMath, a multi-agent framework that systematically addresses these challenges through meta-level supervision and structured Reasoner-Verifier interaction. STAR-PólyaMath is structured as an orchestrated state machine with nested challenge-step-replan loops, governed by a reasoning-free Python orchestrator that separates control from inference and bounds error propagation through trace-back and re-planning. Our key innovation is a persistent Meta-Strategist that maintains cross-attempt memory and exercises meta-level control by issuing high-level strategic guidance or mandatory directives, so the system can escape unproductive loops rather than stagnate or over-rely on tools. STAR-PólyaMath achieves state-of-the-art results on all eight top-tier competition benchmarks: AIME 2025-2026, MathArena Apex Shortlist, MathArena Apex 2025, Putnam 2025, IMO 2025, HMMT February 2026, and USAMO 2026. It obtains perfect scores on AIMEs, Putnam, and HMMT, and shows its largest margin on Apex 2025, scoring 93.75% compared with 80.21% by the strongest baseline GPT-5.5. Ablation studies show that the gains arise from the framework's orchestration rather than from model-level diversity since removing key components or substituting in mixed backbones consistently weakens performance. Code is available at https://github.com/Julius-Woo/STAR-PolyaMath.