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.
Latest papers 194
Chain-of-Thought (CoT) prompting remains the standard baseline for evaluating models' reasoning abilities. Originally, this technique was introduced to elicit step-by-step reasoning from large language models (LLMs), which would otherwise tend to directly output the final answer. However, many modern LLMs produce CoT-style responses \textit{natively} when presented with reasoning tasks, which made us revisit the effectiveness of standard CoT prompting. We evaluate several modern mid-sized language models on a math problem-solving task and find that models specialized for reasoning achieve better performance in a simple zero-shot setting than when using few-shot CoT examples - significantly surpassing officially reported results at no additional cost (e.g., from 77% to 84% for Mathstral on GSM8K). For the tested general-purpose model, a zero-shot CoT prompt is also sufficient to outperform a few-shot CoT baseline. We attribute this to a `guidance-distraction' tradeoff: standard CoT prompting also demands style adaptation, formatting compliance, and potentially undesired contextualization, which can distract models from the core reasoning task. Our findings suggest that using standard CoT prompting increasingly acts as a source of distraction as models grow stronger.
A Human Audit of OpenAIs AI-Generated Mathematical Proofs
We assess 18 chapter-specific reviews of the ten mathematical results announced by OpenAI on 1 August 2026, alongside review standards, Lean formalizations, subsequent research, and mathematical references. The article audits this review record without claiming a complete reconstruction of all ten proofs. No confirmed substantive mathematical error in a principal result remains in the examined assessments, although review depth varies and some dependencies remain partly checked. Chapter 8 presents the strongest reservation: a specialist review requests major revision of compressed analytic arguments. In Chapter 6, an apparent polarity error was withdrawn after an overbar lost during PDF extraction was recovered from the typeset source. Subsequent research independently reuses the Chapter 3 proof mechanism and confirms that Connes's rigidity conjecture is false, without independently reproducing Chapter 4's stronger infinite-family result. Among the cited follow-ups, Chapter 7 receives the strongest direct theorem-level corroboration through a stronger hardness theorem. Related equality results in Chapter 8 do not verify the analytic inequality proof. Some follow-ups disclose material AI assistance. We argue that confidence should combine formal checking, human reconstruction, independent mathematical use, and a public record supporting correction of both proofs and reviews.
Is More Privileged Information Better? From Solution Traces to Problem-Solving Structure in Self-Distilled Reasoning
On-policy self-distillation (OPSD) improves reasoning by using a privileged view of a model conditioned on reference solutions to supervise a student view that observes only the question. However, the teacher-provided token-level targets may depend on reference-specific information unavailable at inference time. We propose Problem-Space-Guided OPSD (PS-OPSD), which replaces the complete solution with trajectory-grounded guidance describing the initial state, goal conditions, constraints, and a selected state-transition path. The student rollout and OPSD objective remain unchanged. Across three mathematical reasoning benchmarks and model scales ranging from 1.7B to 8B, PS-OPSD achieves the highest aggregate question-only accuracy among the compared methods. Controlled experiments further indicate that guidance relevance and path coherence contribute to these gains, highlighting the representation of privileged information as an important design choice in OPSD.
Cloud-ScPO: Hidden-State Geometry for Semi-Supervised Preference Optimization in LLM Reasoning
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 -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.
Escaping Confidence Trap: Evolutionary Decoding for Mathematical Reasoning in Diffusion LLMs
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.
AMTFV: Agentic Mathematical Tool-Flow Verification for LLM Self-Correction
Large language models have demonstrated strong mathematical problem-solving capabilities, yet reliably verifying their candidate answers remains challenging. Existing representative methods mainly revise outputs through natural-language reflection or assist verification by directly generating verification programs; the former may not reliably support exact computation, whereas the latter prematurely couples mathematical modeling with low-level implementation. We propose AMTFV (Agentic Mathematical Tool-Flow Verification). By introducing Mathematical Tool Flow (MTF) as an interrupt--execute--resume interface, AMTFV decouples verification modeling from concrete execution and supports exact computation through a mathematical toolbox. Specifically, the verification agent first constructs a verification workflow, encodes the mathematical objects and computational intent requiring reliable execution in an MTF request, and sends it to the mathematical toolbox agent. The latter parses the request, generates executable calls, and dispatches them to the backend for exact computation. Tool outputs then support candidate-answer adjudication, answer revision, and verification-workflow revision. We evaluate AMTFV on five challenging mathematical reasoning datasets with seven model configurations from DeepSeek, GPT, and Gemini. Experimental results show that AMTFV outperforms the representative baselines evaluated in this study overall; under an individual model configuration, it improves average accuracy over the strongest baseline by up to 8.3 percentage points, with larger gains on samples of medium and high verification complexity.
Albilich: Steerable Proof-State Orchestration for LLM-Based Mathematical Research with CAS Integration
Large language models can contribute useful ideas to mathematical research, yet long-horizon proof attempts remain difficult to coordinate, evaluate, and reproduce. We present Albilich, an open-source agentic harness for autoresearch in mathematics that combines long-horizon reasoning, computer algebra systems (CAS), literature retrieval, and persistent SQLite-based context management. We evaluate Albilich on the RealMath benchmark (Zhang et al. 2025) and on open problems in group theory from the Kourovka Notebook (Khukhro and Mazurov 2026). It solved 10/10 problems on RealMath with CAS and 9/10 with no CAS. On the Kourovka problems, Albilich produced a counterexample to Problem 21.142 and a proof of a strengthening of Problem20.2. Anablation on Problem 17.91 demonstrates 32.0% token reduction when CAS is enabled. An ablation on Problem 21.142 demonstrates higher verifier-rejection rate and failure to synthesize proof routes in the absence of the advisor agent. These results support Albilich as a human-steerable, CAS-boosted environment for scalable AI-assisted mathematical research.
ReDiPPO: Reference-Guided Value Calibration and Discrepancy-Aware Token Reweighting for 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.
A Reference-Free Score for Detecting Silent Reasoning Failures in Large Language Models
Mathematical chain of thought (CoT) evaluation is commonly reduced to whether the final answer matches a reference. This conflates producing a correct conclusion with producing a valid derivation an invalid chain can accidentally reach the right answer, while a valid calculation can be followed by a transcription error. We call this mismatch the reasoning answer consistency gap. This framework paper introduces the Reasoning Answer Faithfulness Score (RAFS), a reference free, instance level diagnostic of whether an emitted mathematical trace is locally credible, supports its answer, and is stable under resampling and targeted counterfactual interventions. RAFS combines step validity, reasoning to answer entailment and counterfactual sensitivity, answer consensus, and conditional reasoning stability. It evaluates transcript level agreement, not a models private computation and not factual correctness outside the tested mathematical setting. We retain a preregistered, results blind confirmatory study on GSM8K and MATH, with hypotheses, admissibility rules, calibration, and tests fixed before confirmatory outcomes are inspected. A separate feasibility pilot is specified to verify end to end execution and estimate interven tion coverage before that freeze numerical pilot claims are re ported only when trace level artifacts are available. We formalize four reasoning answer outcomes, justify the non compensatory aggregator, instantiate semantic trace distance, quantify compute and abstention tradeoffs, and define verifier independence and power analyses. RAFS is intended to complement mathematical answer accuracy with an auditable warning signal for silent reasoning failures and answer extraction errors
Case study: proving sqrt(2) irrational with LPTP and an LLM
We present the interactions with an LLM (Large Language Model) aiming at proving that the square root of 2 is not a rational number in an LP (Logic Programming) context. We start from a few basic pure logic programming predicate definitions. We rely on the LPTP (Logic Program Theorem Prover) system for stating and proving properties about logic programs. As the proof language of LPTP is based on natural deduction, the proofs are human readable. In our case study, we sketch in LPTP the usual proof showing the irrationality of the square root of 2. Then we describe the interactions we had with the LLM. We end up with a complete formal proof, partially generated by an LLM and fully proof-checked by LPTP.
Mathematical Discovery in the Wild: AI-Guided Proofs in Banach Space Theory
We investigate the capacity of current language models to contribute to mathematical research. In Banach space theory, AI systems generated key ideas and proofs for five new results, which were then verified and refined by humans. We also developed an automated system that searches the literature for open problems and attempts solutions at scale. Our results show both the potential of language models for mathematical discovery and the continuing importance of expert verification.
Explaining and Tuning Transformer-based LLMs in Arithmetic Tasks with Human Strategies
Transformer-based large language models (LLMs) continue to achieve state-of-the-art performance across various natural language processing tasks. However, their subpar performance on seemingly elementary problems, such as basic arithmetic, raises concerns about model reliability, safety, and ethical deployment. In this study, we demonstrate that the performance of a vanilla Transformer model trained on integer arithmetic tasks can be improved using methods effective for human learners. We begin by decomposing the arithmetic task into well-defined subtasks and conducting loss convergence order analysis together with ablation studies for each subtask. Our findings reveal that LLMs exhibit learning patterns similar to those of human learners, with a faster learning speed for simpler subtasks compared to more complex ones. In addition, we successfully improved the accuracy of LLMs by applying problem-solving strategies and cognitive empowerment methods shown to enhance the performance of human learners. This suggests that transformer-based LLMs may share cognitive processes with human learners in arithmetic. Lastly, we provide a comprehensive demonstration of our method's effectiveness, including significant accuracy improvement experiments, visualization verification, and explanation-based analysis to illuminate the intricacies of LLMs in arithmetic learning. In general, this work explores the potential similarities between transformer-based LLMs and human learners, supported by explainable AI (XAI) verifications, ultimately fostering trust in LLMs for critical and high-stakes applications.
Trace-Based On-Policy Distillation for Masked Diffusion Language Models
Diffusion large language models (dLLMs) are a promising alternative to autoregressive generation. However, reasoning-oriented post-training for dLLMs remains challenging. Supervised fine-tuning (SFT) for dLLMs requires dense but often off-policy masked states, while reinforcement learning (RL) relies on sparse rewards or value modeling. This paper proposes \textbf{trace-based on-policy distillation (TOPD)}, a teacher-supervised framework that transfers reasoning ability to a target dLLM without reward estimation. The key idea is to supervise a dLLM on its own denoising trajectory, focusing on the trace-aligned token decisions that form the final response. Specifically, TOPD samples on-policy diffusion trajectories from the target dLLM, obtains teacher token distributions from a teacher model on the corresponding partially denoised states, and updates the target dLLM with a token-level Reverse Kullback-Leibler (Reverse-KL) objective. This design preserves dense teacher supervision while aligning training with the model's own denoising states. On mathematical reasoning benchmarks, TOPD enables SDAR-4B-Chat to match the MATH500 accuracy of its RL-trained counterpart TraDo-4B-Instruct, with gains of +5.7 under static evaluation and +4.5 under dynamic evaluation. Compared with the RL-trained counterpart, TOPD achieves this with 4 fewer rollout rounds, corresponding to an estimated 96.0 to-accuracy model-compute speedup.
Are Arithmetic Heuristic Neurons Form-Invariant? A Mechanistic Analysis of Symbols, Text, and Code in LLMs
Large language models often succeed on one formulation of a problem while failing on an equivalent formulation. Whether these failures arise from distinct internal circuits or different activation states of a shared circuit remains unknown. Recent mechanistic interpretability studies suggest that arithmetic in LLMs emerges from a "bag of heuristics," encoded by a sparse set of MLP neurons that represent distinct arithmetic strategies. We investigate whether arithmetic heuristic neurons are form-invariant across symbolic arithmetic, natural language word problems, and Python code in three Llama-3 models. In each format, we identify arithmetic heuristic neurons using a two-stage pipeline combining attribution patching and activation patching. A compact set of neurons is shared across all three formats, and targeted interventions show this shared circuit is both necessary and sufficient for late-layer arithmetic computation. Transferring the shared neurons' activations from a successful execution in one format to a failed execution in another recovers most incorrect predictions, exceeding 97% for addition and subtraction, indicating that cross-format failures arise from activation states rather than distinct circuits. Moreover, shared neurons consistently belong to the same heuristic families across formats, demonstrating that arithmetic computation in LLMs is largely form-invariant at the neuron level.
CoTu at EXACT 2026: Neuro-Symbolic Reasoning for Transparent Educational QA
Transparent educational question answering asks for answers that are not only correct but explainable, and doing so with small models rules out the reasoning power of the largest proprietary systems. The EXACT 2026 competition poses this problem concretely: open-weight language models of at most 8B parameters, self-hosted, with a natural-language explanation for every answer. It pairs two tasks: logical reasoning over university regulations, and multi-step physics problem solving. We describe the system that team \cotu{} developed to address both, a neuro-symbolic Program-of-Thought pipeline in which a 4B backbone writes a program rather than stating an answer directly: for regulation queries it emits a Z3 encoding whose entailment verdict grounds the deduction, and for physics it emits numerical Python, both wrapped in a shared self-correction loop and a unified explained-JSON output. Answer-type routing, distillation-based task fine-tuning, and a latency-aware serving stack -- SGLang with speculative decoding -- keep the system within the 60-second per-query limit. The system achieved a \textbf{perfect score} on the physics task in both automated selection rounds and obtained the \textbf{highest final-round technical score} of any team -- , combining automated answer evaluation with expert-judged reasoning depth -- with the equally weighted presentation score included, \cotu{} placed 3rd overall. Grounding answers in a symbolic solver yields correct, verifiable deductions at the 4B scale, and the residual difficulty lies in premise selection rather than the deduction itself.
MathCoPilot: An Interactive System for Human-AI Symbiotic Paradigm of Mathematical Research
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.
GSM-Plus-BN: A Perturbation-Based Benchmark for Bangla Mathematical Reasoning in Large Language Models
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.
AdvancedMathBench: A Benchmark Suite for Advanced Mathematical Proof Generation and Verification
Large language models (LLMs) have achieved remarkable performance on high-school and competition-level 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 the reasoning capabilities of LLMs on advanced mathematical proofs. Its core generation benchmark, ProverBench, contains 245 problems spanning undergraduate (UG) and doctoral qualifying-exam (QE) levels. To reliably evaluate these proofs, we develop a dedicated automatic verification pipeline that is trained on large-scale expert annotations, produces both correctness verdicts and fine-grained analyses, and 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 64.5 and 48.9 on the UG and QE splits, respectively. On proof verification, the best model only attains a Balanced F1 of 65.1. Further analysis reveals a notable mismatch between proof generation and verification capabilities across models.
TreeThink: A Modular Tree Search Library for Mathematical Reasoning with LLMs
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 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/ .
ProofCouncil: An LLM Agent for Solving Open Mathematical Problems
Large language models (LLMs) have shown increasing promise in solving open problems in mathematics. However, their performance can be further improved through agentic workflows tailored to real-world mathematical practice. To this end, we introduce ProofCouncil, a mathematical agent that is designed to tackle open problems using an author-critic architecture. ProofCouncil served as a submission to the second batch of FirstProof, a challenge consisting of 10 real-world mathematical problems that agents must solve autonomously. Its submissions for 6 of the 10 problems were judged by the referees to be correct up to at most minor revisions, showing the best performance among participating teams. We also evaluate ProofCouncil on 30 open problems collected from mathematical researchers. Among the 21 solutions that received human feedback, 5 were judged completely correct, 2 more were judged promising pending final verification, and a further 8 contained useful partial progress. In this short paper, we describe the development of ProofCouncil and the agent-building library used to create it, which we release as open source to the community.
From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier
Recent developments in AI for Mathematics (AI4Math), especially Large Language Model (LLM)-driven theorem provers, has achieved remarkable success in formal proof generation for well-defined mathematical problems through Interactive Theorem Proving (ITP) languages. However, current systems remain fundamentally limited in tackling frontier research mathematics, such as discovering new theorems or resolving open conjectures, which are often open-ended, under-specified, and involve multiple layers of abstraction. We argue that the next leap in AI4Math systems requires a decisive shift from predefined problem-solvers to research agents that can address frontier mathematical challenges with rigorous formal mathematical reasoning. In this position paper, we provide a systematic review of the field, covering datasets, auto-formalization, and proof synthesis. More importantly, we identify core limitations of existing systems in serving as mathematical research agents, examining issues across datasets, relational structure, mathematical exploration, tool ecosystem, and human-AI collaboration, outlining a strategic road-map for the future of AI4Math.
MIRA-Math: A Benchmark for Minimal Information Requesting and Mathematical Reasoning
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.
Multiplication Beyond Groups: Stratified Fourier Mechanisms in Transformer Circuits
Transformers have demonstrated a remarkable ability to learn algorithmic reasoning, yet mechanistic analyses have mostly focused on globally invertible operations such as cyclic addition and group composition. In this work, we investigate how small transformers learn modular integer multiplication over composite moduli, a fundamentally non-invertible operation due to the presence of zero-divisors. We propose the monoid extension: a localized generalization of Group Composition via Representation (GCR) that suggests the learned computation does not rely on a single global representation space. Instead, the model partitions the input space into local hierarchical algebraic regions, where group-like structure survives and Fourier mechanisms can be applied. In transformers trained on square-free modular multiplication, we find that embeddings organize around these regions, attention exhibits class-sensitive routing and low-rank write directions, and local character features explain a large fraction of the model's output logits. Our results suggest that representation-theoretic mechanisms previously identified for group operations can extend beyond groups to more general structures.
Evaluating SageMath-Augmented LLM Agents for Computational and Experimental Mathematics
Recent advances in AI for Mathematics have focused largely on autoformalization and theorem proving, leaving the role of Computer Algebra Systems (CAS) in agentic LLM workflows underexplored. We propose a ReAct-style agentic setup that combines LLM reasoning with verifiable feedback from SageMath, together with Context7 for the up-to-date documentation. We evaluate this agentic setup across frontier models for solving research-level mathematical problems from the RealMath benchmark in a setting that emulates a computational-mathematics research loop. We also propose a refinement to the RealMath benchmark by introducing a multi-step post-processing procedure and a multi-stage validation pipeline, both of which improve the quality and reliability of the extracted problem set. Our experiments reveal substantial performance gains from SageMath access across all evaluated models on +9.7pp on average, the gains range from 1.5pp to 27.8pp and narrow the gap between open-weight and closed models. Qwen3.7-Max benefits from SageMath the most, while GPT-5.5 achieves the highest solve rate of and the lowest token usage among tool-enabled configurations. Our findings suggest that CAS-augmented agents represent a promising direction for assisting mathematicians in computational exploration, and we believe that this work is a step towards automated conjecture discovery. The project repository is available online.
Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory
Recent LLM-based mathematical reasoning agents have begun to tackle research-level problems and, in several cases, have contributed to the resolution of open problems. However, scaling and orchestrating such agents effectively remains challenging, due to the difficulty of coordinating parallel proof search while keeping intermediate claims organized and reliable. In this paper, we propose Danus, an orchestration system for research-level mathematical reasoning centered on a shared fact graph as a global memory-management mechanism. Danus consists of a main agent that performs planning and coordination, multiple worker agents that carry out proof search in parallel, and a stateless verifier that checks proposed mathematical claims before they are admitted into the fact graph. Each verified fact is stored together with its proof and logical dependencies, allowing the system to build long arguments incrementally while keeping the shared proof state organized. The main agent periodically summarizes the evolving proof state, redirects workers across promising directions, and supports interaction with human mathematicians through progress reports. We evaluate Danus through six research-level case studies in algebraic geometry, singularity theory, and combinatorics, illustrating how the fact-graph memory mechanism enables Danus to construct long, detailed mathematical proofs. Our results suggest that fact-graph-based orchestration provides an effective route toward scaling mathematical reasoning agents for long-horizon research problems. Danus is open source at https://github.com/frenzymath/Danus.
PluraMath: Extending Mathematical Reasoning Evaluation Beyond High-Resource Languages
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.
Knowledge Knows, Verbalization Tells: Disentangling Latent Directions for Mathematical Solvability in LLMs
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.
SPL: Orchestrating Workflows with Declarative Deterministic-Probabilistic Composition
We present SPL (Structured Prompt Language), a declarative language that composes deterministic and probabilistic computation modes in a single specification. While existing frameworks separate these -- orchestration systems (AutoGen, CrewAI, LangGraph) for LLM calls, symbolic tools (SymPy, SageMath, Lean) for computation -- SPL unifies them. It provides GENERATE/EVALUATE for probabilistic computation and SOLVE/ASSERT for deterministic computation, sharing syntax, variable bindings, and runtime routing. A .spl specification runs unchanged across local nodes (Ollama), cloud APIs (OpenRouter, Anthropic), and distributed grids (Momagrid), with model and verifier selection deferred to invocation time. We validate SPL through an extensive 78-recipe cookbook and a controlled 1,200-run experiment (10 models x 20 problems x 2 arms x 3 repetitions; the 20 problems span 6 difficulty tiers). The solver arm achieves 82-93% machine-verified correctness (sonnet-4-6: 85%, gemma4:e2b: 93%) while the LLM-only arm measures output production without mathematical verification, making the comparison one of verified correctness against unverified fluency. A backend difficulty gradient emerges (SymPy 78%, Sage 54%), and the dominant failure mode is solver_error (kernel-rejected expressions), not format non-compliance.
Detecting Answer-Driven Reasoning in LLM-Based Educational Tutors via Truncated Chain-of-Thought Auditing
Large language model (LLM) tutors often produce fluent step-by-step explanations, but a correct and pedagogically formatted response does not guarantee that the answer was derived from the student-facing problem. In realistic tutoring systems, the model may also have access to teacher notes, answer keys, rubrics, or retrieved solution artifacts. We study whether such private answer information can make tutor explanations answer-driven: the final answer is behaviorally available before the written explanation has justified it. Using Truncated Reasoning AUC Evaluation (TRACE), which probes how early a chain-of-thought prefix can pass a verifier, we evaluate 1000 GSM8K test problems under three paired tutoring contexts: question-only, correct answer-key, and wrong answer-key. At fixed fractions of each generated explanation, we force the model to answer immediately and verify the response against the gold numeric answer. With Qwen2.5-3B-Instruct, answer-key access raises median TRACE AUC from 0.375 to 0.900 and makes the gold answer available at the first 10% prefix in 997 of 1000 cases. The effect remains strong on the 746 examples where both question-only and answer-key explanations end with the correct answer. These results support truncated CoT auditing as a lightweight process-level diagnostic for answer-driven reasoning in math tutoring explanations.
MechMath Agent Team: LLM Driven Agents for Mathematical Research
AI reasoning has become a central focus in contemporary artificial intelligence, largely driven by the success of large language models. However, mathematical research, which is characterized by non-linear derivation paths, rigorous logical requirements, and protracted exploration cycles, poses severe challenges for existing reasoning systems. To overcome these limitations, we present the MechMath Agent Team (MMAT), which is a large language model driven agent designed to serve as a co-pilot throughout the full cycle of mathematical research. We design a tripartite Harness Architecture that decouples system responsibilities into Control, Execution, and Augmentation planes, thereby reconciling rigorous logical control with the agility demanded by open-ended research. Building upon this framework, we instantiate three specialized agents: a Knowledge Base Manager, a Natural Language Prover, and a Formal Language Prover, all operating in a closed loop to produce formally certified mathematical proofs. We evaluate MMAT on open problems in Number Theory, Algebraic Complexity Theory, Differential Algebra, Operator Algebra, and Inequalities. Across a two-month deployment, 11 problems have been solved, demonstrating its capacity to act as a co-pilot throughout the entire research cycle. The contributions are threefold: a general decoupled Harness Architecture for multi-agent mathematical reasoning, its concrete instantiation in the MMAT system, and empirical validation on a diverse suite of open problems.