Mathematical Reasoning

Follow this topic Ask about this topic

Momentum

17 papers in the last four weeks, against 2 the four weeks before. 0.2% of all new papers.

Jul 13Week of Sep 28

Latest papers 113

Oct 7, 2026cs.AI

GAGR-Lab: Evaluating Joint Spatial-Geometric and Analytic Function Reasoning

Joint spatial-geometric and analytic function reasoning requires translating a perceived spatial configuration into a symbolic function whose executed curve satisfies geometric constraints. We present GAGR-Lab, a framework for measuring this capability through Cartesian game scenes, explicit function semantics, and authoritative Rust trajectory execution. It distinguishes spatial perception, metric grounding, geometric relations, function interpretation, function construction, and constrained synthesis. We specify four configurable scene-difficulty presets and a prospective 24-cell diagnostic design, while reporting only the subset actually evaluated. A bounded pilot of one hosted model (Llama 3.2 11B Vision Instruct) using two API credentials as execution replicas yields 72 balanced games with 432 attempts, 429 valid provider responses, and no target hits; exploratory ordinary-function prompt variants also fail to hit, while the structured localization interface yields no scoreable outputs. A privileged analytic search control independently succeeds on 600 directional cases from 300 generated scenes, with exact repeatability and 1,200 successful vertical-reflection or translation checks. The framework separates serving reliability, symbolic compliance, and geometric success, and preserves exact model-visible inputs and realized paths. A staged protocol outlines diagnostic calibration, held-out replication, multi-model comparison, and paired robustness tests. The contribution is an operational research framework with an executed pilot and a clearly identified prospective study plan; the full difficulty matrix and comparative model results remain untested.

Jingyao Zhang, Yun Li, Lu Han

Spatial Reasoning Benchmarks Mathematical Reasoning Visual Spatial Reasoning Constraint Satisfaction

Oct 7, 2026cs.LG

A Drosophila Whole-Connectome Network Can Learn Human-Designed Cognitive Tasks

Can a biological wiring diagram serve as a useful computational substrate beyond the behaviors for which it evolved? We use the publicly released MaleCNS v1.0 connectome, reconstructed from a single adult male Drosophila specimen, as the fixed recurrent topology of an artificial network. We train separate models for bounded addition and for a controlled grounded relational language task built from a fixed 100-word lexicon. In both models, one scalar is learned per anatomical edge. The anatomical graph reaches 92.77% mean accuracy on held-out addition, compared with 67.93% for directed degree-preserving rewires. On the strict paired language endpoint, which matches original and order-reversed scenes to their corresponding descriptions, it reaches 61.59% across four fixed interfaces, compared with 44.17% for matched rewires. At the canonical interface, it ranks first in a fixed 21-graph comparison. On the matched 48-group intervention subset, shuffling task-defined sensory features reduces its score from 60.94% to 19.27%. Together, these results show that higher-order MaleCNS wiring provides a reusable inductive bias for bounded addition and grounded relational language.

Joonghui Cho, Minchan Kang, Daeshik Kim

Korea Advanced Institute of Science and Technology (KAIST)

Mathematical Reasoning

Oct 5, 2026cs.AI

Teaching a Minimalist Machine to Discover Recursive Programs for Arithmetic

Humans can often acquire and synthesize complex, recursive concepts from minimal experience. Leveraging cognitive insights, we propose the Minimalist Machine, a framework for inductive program synthesis designed to model such conceptual learning. The system uses a compact relational subset of Prolog: Programs are searched within a fixed schema of body-free facts and two-body conjunctive Horn clauses. Recursion is not defined by a dedicated metarule. Instead, it emerges when a target predicate is reused inside the body of a learned clause. Inspired by a primary school curriculum, the model is taught through a human-curated, sequential introduction of new concepts in arithmetic. Starting from initially empty knowledge base, it first acquires simple structural predicates, then successor-based state transformations, and finally recursive programs for addition, subtraction, multiplication, and division. Ultimately, this approach yields the fully transparent, inductive reasoning trace necessary for human-like conceptual learning.

Dominik Magiera, Christiane Wiebel-Herboth, Frank Jäkel

Centre for Cognitive Science & Institute of Psychology, Technische Universität Darmstadt, Germany · Honda Research Institute Europe GmbH, Offenbach am Main, Germany

Cognitive Modeling Program Synthesis Mathematical Reasoning Inductive Logic Programming

Oct 5, 2026cs.LG

Transfer-Stratified On-Policy Distillation for RL-Improved Reasoning Teachers

Reinforcement learning can substantially improve a reasoning teacher, but it is unclear which of those improvements survive when the teacher supervises a smaller on-policy student. We study this question in mathematical reasoning by comparing teacher lineages before and after GRPO, multiple student scales, direct GRPO, and several on-policy distillation objectives. The central finding is that transfer is structured rather than scalar: teacher strength alone does not make dense distillation competitive, while an RL-improved teacher creates useful but metric-dependent student gains. This motivates Transfer-Stratified On-Policy Distillation (TS-OPD), which screens training problems by the joint sampled success of the student and teacher, routes acquisition problems to gated forward KL, routes consolidation problems to gated reverse KL, and adds an entropy brake to protect sampled coverage. Across the main comparison, TS-OPD is the strongest student objective for macro average correctness with the GRPO-improved teacher, while pass@K remains more mixed. Ablations show that the gains come from routing and token gating rather than skipping problems. These results support a transfer-aware view of OPD: stronger teachers help when the supervision direction and token budget match the student's observed ability, not merely because the teacher endpoint is stronger.

Xiaoyu Chen, Bo Shao, Tiangang Zhu +5

Institute of Software, Chinese Academy of Sciences · Microsoft · Work done during an internship at Microsoft +1

RL for Language Model Reasoning Selective Knowledge Distillation Mathematical Reasoning On-Policy Distillation

Oct 4, 2026cs.LG

E $^2$ -OPSD: Taming Entropy Overshoot in On-Policy Self-Distillation

On-policy self-distillation (OPSD) provides dense token-level supervision without a second model: one network acts as teacher with the reference solution and as student with only the problem. We identify a specific failure mode of this recipe. During training, student token entropy rises past the teacher's and remains elevated, a pattern we call entropy overshoot. We trace it to both sides of distillation. The reference-conditioned teacher is confident along its answer-directed reasoning path, but this confidence transfers poorly to student-generated prefixes, making its supervision overly tied to answer-specific cues rather than reusable reasoning patterns; meanwhile, the forward KL used by OPSD continually diffuses the student's predictive distribution without pulling it back. We introduce E

^2

-OPSD to address both causes. Exemplar-guided teaching replaces the current answer with a retrieved solved neighboring problem, providing transferable reasoning guidance without revealing the destination and better matching student-reachable states. Entropy-aware distillation uses the student-teacher entropy gap to determine the direction and strength of each token's correction. E

^2

-OPSD improves math reasoning by up to 4.3 points in mean@16 over OPSD, while out-of-domain evaluations show gains over the corresponding base models of up to 4.9 points in mean@16 and 5.5 points in pass@8. Despite these gains, E

^2

-OPSD remains simple, requiring no additional forward passes or networks.

Yifei Liu, Minghao Fang, Xinyu Gu +6

The Chinese University of Hong Kong · Zhejiang University · Shanghai Artificial Intelligence Laboratory +5

On-Policy Self-Distillation Mathematical Reasoning Uncertainty-Aware Knowledge Distillation Knowledge Distillation

Oct 3, 2026cs.LG

Questioning the Questions: Sustaining Self-Evolution in Reasoning Models

Self-evolving reasoning models learn from their own generated questions, yet repeated self-training can lead to performance collapse. In this paper, we investigate why performance deteriorates over successive rounds and how to sustain self-evolution. Our analysis identifies two recurring quality problems in self-generated questions: invalid questions and repeated variants of the same mathematical questions. First, invalid questions become more prevalent across rounds, and answer-consistency filtering further increases their proportion in training data. Second, existing question diversity controls based on lexical similarity can miss mathematically equivalent questions expressed in different ways, which leads to question diversity collapse in later training rounds. Building on these findings, we introduce R-Quest, which uses question validity and novelty feedback to guide self-evolution. We first train the solver to recognize and reject invalid questions, then use its judgments to guide questioner rewards and filter solver training data. To avoid question repetition, we use a frozen base model to compare sampled question pairs and provide novelty feedback. Empirically, our method consistently achieves the highest average performance on 12 benchmarks in mathematical reasoning, general-domain reasoning, and code generation across two model families. Additionally, R-Quest maintains stable performance gains over ten rounds of self-evolution, peaking in the final round and outperforming R-Zero by 17.32 points.

Jinyuan Li, Chengsong Huang, Langlin Huang +4

Washington University in St. Louis · University of Michigan, Ann Arbor

Language Model Self-Improvement Self-Training for Language Models Text Generation Mathematical Reasoning

Sep 30, 2026cs.CL

SCB: SpeechConversationBench for Evaluating Multi-Turn Reasoning in Speech-to-Speech Models

Speech-to-speech systems must solve tasks whose requirements emerge across conversational turns. We introduce SpeechConversationBench (SCB), a focused evaluation of spoken mathematical reasoning using 103 sharded GSM8K problems. The framework compares the original problem delivered in one turn (full), its concatenated information shards delivered together (concat), and incremental spoken disclosure across turns (sharded). We report final-answer accuracy for four commercial speech systems and LEGO, a proprietary speech pipeline developed internally by the SCBX Innovation Lab team with explicit conversational context management. Relative to concat, sharded accuracy decreases by 5.0-25.3 percentage points across the four commercial systems. LEGO achieves 77.5 percent accuracy in all three conditions, compared with 76.6 percent sharded accuracy for GPT-4o Realtime. The two single-turn baselines distinguish sensitivity to problem reformulation from the additional challenges introduced by incremental spoken interaction.

Kanpat Vesessook, Saksorn Ruangtanusak

SCBX R&D · SCB DataX, SCBX Group

Numerical Reasoning in Language Models Mathematical Reasoning Multi-Turn Dialogue Evaluation

Sep 30, 2026cs.CL

Better Supervision Is Nearby: Neighborhood On-Policy Self-Distillation

On-policy self-distillation (OPSD) trains mathematical reasoning models using a privileged teacher that sees a reference solution and supervises student-sampled prefixes. Standard OPSD uses one fixed parameter setting at every state, but nearby settings may offer additional supervision. We find that local parameter perturbations reveal complementary reference-aligned corrections under the same reference context. Different experts supply these corrections at different reference positions. Their pool covers more such positions than the unperturbed privileged teacher. We introduce Neighborhood OPSD (N-OPSD) to turn these corrections into supervision at student-visited states. Offline, greedy selection builds a compact pool of frozen experts by rewarding filtered reference-token gains beyond the pool's current best at each position. The highest-peak expert need not provide the best training target. Online routing therefore separates the anchor direction from its level of support. MaxPeak selects the anchor token, and quantile selection chooses among experts whose top token matches it. The student learns from the chosen expert's full next-token distribution through the clipped forward-KL objective inherited from OPSD. We evaluate on AIME 2024, AIME 2025, and HMMT February 2025. Across three independent runs per method, Neighborhood OPSD improves the three-benchmark Average@12 over OPSD by 2.75, 1.67, and 1.94 points on Qwen3-1.7B, 4B, and 8B, respectively. Student-prefix continuations support using the pool beyond the reference trajectories used for selection. Matched ablations support filtered reference-token gains as a selection criterion. Accounting for overlap within the pool and routing by state further improve student accuracy. Inference uses only the distilled student.

Xincheng Wei, Yifan Ding, Yoshua Li +6

The Chinese University of Hong Kong, Shenzhen · Meituan, LongCat Team · University of Toronto +1

On-Policy Self-Distillation Mathematical Reasoning Language Model Distillation

Sep 30, 2026cs.LG

T-Router: Learning Thalamic Routing for Reasoning with Parameter-Efficient Reinforcement Learning

Parameter-efficient reinforcement learning aims to improve reasoning with a compact trainable interface to a pretrained model. We introduce the Thalamic Router (T-Router), which concentrates adaptation on the reuse of completed computations. A compressed, addressable bank preserves block changes; a depth-recurrent controller conditions their selection and relative-scale writeback. This coupling gives thalamic context-dependent routing a concrete computational form: learn which earlier contributions a receiving layer uses, and with what influence. Correctness rewards train the interface while preserving backbone parameters and layer order. On an 8.95B-parameter backbone, T-Router allocates 41.73M parameters (0.466% of the backbone) and achieves 83.64 +/- 1.16 MathAvg after GSM8K RL, compared with 73.79 +/- 1.83 for full-parameter GRPO across three evaluation rounds. At a comparable parameter budget and with matched retries, it exceeds LoRA's 77.28 +/- 1.95 MathAvg, improving all three task families and raising mean AIME accuracy from 48.33 to 60.56. Capacity-controlled comparisons favor addressable block changes and recurrent context; separate search training extends the interface to tool-mediated reasoning. These results establish controlled computation reuse as an effective route to parameter-efficient reasoning reinforcement learning.

Liuxian Ma, Jiale Dai, Jiaqi Li +1

College of Artificial Intelligence, Tsinghua University · State Key Laboratory of General Artificial Intelligence, School of Intelligence Science and Technology, Peking University · Beijing Institute for General Artificial Intelligence

Reinforcement Learning RL for Language Model Reasoning Mathematical Reasoning Parameter-Efficient Fine-Tuning

Sep 29, 2026cs.LG

Overcoming Scaling Limits in On-Policy Self-Distillation for LLM Reasoning

On-policy self-distillation (OPSD) trains a student to match a privileged teacher distribution along its own sampled trajectory. Standard OPSD applies this supervision to unverified student rollouts while conditioning the teacher on privileged context, typically a reference solution. We separate these roles in a factorial analysis and find that scaffold correctness has a stronger effect on downstream accuracy than context correctness. Unverified scaffolds create an imitation gap because the teacher can use information unavailable to the student. This gap shrinks with model scale, yet OPSD continues to supervise mostly unverified trajectories. In contrast, verified scaffolds remain effective even when the teacher is conditioned on the student's own unsuccessful rollout. Based on this finding, we introduce OASIS, which retains the OPSD objective but supervises mostly verified by label on-policy trajectories and replaces written solutions with unverified model-generated attempts as the teacher context. OASIS therefore requires only final-answer labels. Across Qwen3-1.7B, 4B, and 8B on AIME 2024, AIME 2025, and HMMT 2025, OASIS improves over the base model by 3.2--3.8 points on average, while OPSD's gain falls from 3.05 points at 1.7B to 0.14 at 8B. At 8B, OASIS improves over OPSD by 3.05 points, showing that verified on-policy scaffolds preserve the effectiveness of self-distillation as models scale.

Md. Ismail Hossain, Humaira Kousar, Isidora Chara Tourni

North South University · KAIST · Andria Labs

On-Policy Self-Distillation Mathematical Reasoning Self-Supervised Knowledge Distillation Knowledge Distillation

Sep 28, 2026cs.AI

Mathematics for and by human cognition: A resource-rational search for bottlenecks in problem-solving

Human cognitive constraints are generally viewed as limiting factors in problem-solving. We argue that these constraints can instead play a critical role in driving advances in mathematics and beyond. We propose a theory of mathematical abstraction as a resource-rational search for bottlenecks in problem-solving. Bottlenecks arising from cognitive constraints create pressure to restructure existing knowledge, potentially giving rise to novel formalisms with applications beyond the problems that originally motivated them. Drawing on episodes from the history of mathematics, we illustrate how such bottlenecks can drive the development of novel abstractions and examine how cognitive constraints and affective responses shape this process. Finally, we discuss the implications of this account for machine mathematical discovery and argue that incorporating human-like constraints may facilitate the discovery of useful mathematical abstractions.

Sneha Aenugu

Salk Institute of Biological Studies La Jolla, CA

Mathematical Reasoning Bounded Rationality

Sep 27, 2026cs.LG

Towards Eliminating Catastrophic Forgetting in the Curriculum Learning of Math Reasoning Tasks

Curriculum learning has found broad application across numerous domains. Nevertheless, its effectiveness is intrinsically curtailed by catastrophic forgetting, driven by the shifts in model parameter distributions between curriculum tasks. In this paper, we investigate the phenomenon of catastrophic forgetting in this training paradigm, building on the established efficacy of curriculum learning. Our theoretical analyses of parameter update dynamics demonstrate that catastrophic forgetting in curriculum learning stems from the divergence of task optima, which is generally essential to the faster convergence of curriculum learning; therefore, forgetting cannot be completely eliminated. Based on this finding, we augment the training process and propose IV-EWC, which incorporates Elastic Weight Consolidation (EWC) into the curriculum learning objective to curb catastrophic forgetting in mathematical reasoning, a prototypical curriculum learning scenario. IV-EWC employs the influence function to construct a representative validation set from the curriculum's training data, which is used to drive dynamic regularization during training. We further present an extended theoretical analysis to show that EWC-based regularization methods mitigate catastrophic forgetting in curriculum learning, thereby providing theoretical support for IV-EWC. Empirical evaluations on three backbone models and three benchmarks indicate that curriculum learning exhibits catastrophic forgetting. IV-EWC alleviates this issue, reducing forgetting by 162% on average relative to vanilla curriculum learning and yielding positive backward transfer, as evidenced by improved performance on easier tasks after subsequent training on challenging tasks.

Zengyan Yang, Yangyang Wu, Kai Huang +3

Zhejiang University · Ant Group

Elastic Weight Consolidation Mathematical Reasoning Curriculum Learning Catastrophic Forgetting

Sep 27, 2026cs.CL

Quizzing the Translation: A Prover-Grounded Evaluation Metric for NL $\rightarrow$ FOL

A standard pipeline for symbolic reasoning over natural-language problems translates them into first-order logic and invokes a theorem prover. The translation step is the bottleneck: swap "every" for "some" and every inference that follows is corrupted. Yet today's metrics often score more broken translations higher than less broken ones, because BLEU, BERTScore, and Smatch++ reward surface overlap that the worst errors happen to preserve. We introduce SIV, which derives two kinds of probes from the target formula and uses a theorem prover to verify the candidate translation against each. Positive probes are statements the candidate must entail, which detect translations that drop content; contrastive probes are statements the candidate must not entail, which detect translations that assert more than the original. On a controlled pool of perturbed FOLIO translations, the severity of the error accounts for 80% of SIV's score variance, compared with at most 17% for any prior metric. Across six error classes on a disjoint pool, SIV scores the reference above the perturbed candidate in over 99% of pairs. Because each probe is labeled with what it tests, the failure pattern also supplies a labeled error trace, recovering the perturbation class at macro-F1 0.638, nearly double the score-only baseline. On 434 expert-audited real LLM translations, SIV attains the top AUC, uniquely detects and grades expert-labeled major errors, and abstains, rather than mis-scoring, on out-of-vocabulary translations.

Pu Suo, Ali Emami

Emory University

Language Model Error Detection Mathematical Reasoning Semantic Parsing

Sep 27, 2026cs.AI

Reasoning on the Simplex: Geometric Fixed-Point Models

Looped reasoners spend test-time compute by iterating a weight-tied map, but a small residual does not mean the state is a fixed point when that map lives in unconstrained latent space. We propose Geometric Fixed-Point Reasoning (GFPR), in which the iterated state is the prediction itself: a field of categorical beliefs on a product of simplices, whose argmax is the answer at every step. Because the state is a belief, task structure can be imposed through compact convex relaxations, either as structured readouts or directly in the recurrent state; in the latter case the update remains a continuous self-map, so a fixed point exists for any parameters. At about 7M parameters, GFPR reaches 95.1% exact match on Sudoku-Extreme, 92.0% on Maze-Hard, and 100% sequence accuracy on S_5 length 128, above the published FPRM numbers at the same scale. The same update also trains a 201M language model on FineWeb-Edu in which each site is a distribution over the vocabulary; with 24 Picard steps it is above GPT-2 small on four zero-shot multiple-choice tasks and above GPT-2 medium on ARC-Easy.

Talgat Daulbaev, Ilya Glazkov, Maxim Rakhuba +1

Applied AI Institute · LigandPro · HSE University +1

Mathematical Reasoning Recursive Language Models

Sep 23, 2026cs.LG

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

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

Zhixu Silvia Tao

Operations Research and Financial Engineering Princeton University Princeton, NJ 08540

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

Sep 22, 2026cs.LG

Train Where the Quantized Model Goes: On-Policy Distillation for Low-Bit Reasoning

Quantization-aware distillation (QAD) restores much of the short-form question-answering performance lost to sub-3-bit quantization, yet leaves mathematical and code reasoning substantially impaired. Long generations often degenerate into repetitive loops, exhausting the decoding budget without completing a solution. We trace this gap to quantization-amplified exposure bias: QAD trains on fixed corpus prefixes, while quantization-induced deviations compound along the model's own autoregressive trajectories. To address this mismatch, we introduce an on-policy distillation (OPD) stage that places teacher supervision where the quantized model actually goes. Starting from a QAD checkpoint, the student generates through the quantized forward path used at deployment and receives feedback from a frozen full-precision teacher on its own prefixes, combining dense token-level guidance with task-verifier rewards. Across four models at 2.79 and 1.88 effective bits, OPD raises average BF16 performance retention from 35% to 70% on MATH-500 and from 66% to 91% on HumanEval while preserving short-form performance, with reasoning gains substantially exceeding those of continued teacher-forced QAD in matched-budget comparisons. By coupling QAD's stable low-bit initialization with OPD's on-policy reasoning recovery, our framework provides a comprehensive sub-3-bit solution that preserves broad capabilities while restoring long-form reasoning.

Yuanteng Chen, Zhilei Liu, Peisong Wang +7

Institute of Automation, Chinese Academy of Sciences · School of Artificial Intelligence, University of Chinese Academy of Sciences · Zhongguancun Academy +2

LLM Quantization Quantization-Aware Training Code Generation Mathematical Reasoning On-Policy Distillation Low-Bit Quantization

Sep 16, 2026cs.CL

Voice of Reason: Reinforcement Learning for Spoken Math

Speech language models enable richer spoken interactions between humans and machines than cascaded systems, allowing access to paralinguistic information and lower latency. However, their accuracy on mathematical reasoning benchmarks has lagged behind those of text models. Reinforcement learning (RL) with verifiable rewards has been instrumental in extending text models' capabilities for solving complex problems and limiting hallucinations. In this work, we explore applying RL to the GLM-4-Voice speech model (Zeng et al., 2024) to bridge the gap between textual and spoken mathematical problem solving. We first adapt the model to the domain using supervised fine-tuning on synthesized spoken question-answering data. We then show that, even without extra reasoning tokens, RL improves the accuracy on GSM8K beyond levels previously achieved for speech models only with supplementary reasoning traces. When combined with existing streaming reasoning techniques, we show further gains to 74.8% free-form accuracy. This establishes a new state-of-the-art for mathematical spoken abilities with speech-native models.

Timothée Weisselberger, Edouard Graves, Alexandre Défossez

Kyutai Paris, France · Kyutai, Gradium Paris, France

Numerical Reasoning in Language Models RL for Language Model Reasoning Mathematical Reasoning Speech Language Models

Sep 15, 2026cs.AI

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

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

Zhongdi Qu, Carla P. Gomes

Cornell University

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

Sep 15, 2026cs.CY

AI and Human Approaches to Mathematical Problem Solving

AI systems have begun to report solutions, disproofs, and substantive advances on long-standing mathematical problems, raising questions about whether they approach research in the same way as mathematicians. This study compares public AI research accounts with the human literature on 11 such problems. The human corpus contains 58 papers that directly addressed the same mathematical targets later reported by AI sources as resolved, disproved, or substantially advanced; 31 within-problem comparisons were constructed from these materials. Six validated text-based measures capture problem resolution, method articulation, uncertainty and boundary specification, successor-question generation, generality, and cross-disciplinary integration. AI accounts place greater emphasis on resolving the focal problem and connecting ideas across fields. Human papers devote significantly more attention to explaining methods, specifying assumptions and limitations, and identifying questions for subsequent research. No precise difference is detected in generality. The estimated directions remain unchanged when each mathematical problem is removed in turn. The findings reveal two distinct research profiles: AI accounts concentrate on closing and recombining problems, whereas mathematical papers more extensively document the procedures, limits, and research opportunities through which results become cumulative knowledge. Evaluating research AI therefore requires attention to the organization of inquiry, not only whether a target is solved.

Yang Ding

University of Edinburgh Business School, Edinburgh EH8 9JS, UK. · Centre for Technomoral Futures, Edinburgh Futures Institute, University of Edinburgh, Edinburgh EH3 9EF, UK. · Manchester Institute of Innovation Research, Alliance Manchester Business School, University of Manchester, Manchester M13 9PL, UK.

AI-Assisted Scientific Research Mathematical Reasoning

Sep 13, 2026cs.CL

Func-R1: Incentivizing Mathematical Function Reasoning in Multimodal Large Language Models

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.

Mingze Yin, Xiaohan Wang, Dian Li +8

College of Computer Science and Technology, Zhejiang University · Tencent 3AI Thrust, HKUST (GZ) · Zhongguancun Academy +3

Multimodal Large Language Models Mathematical Reasoning VLM Reasoning RL for VLMs

Sep 11, 2026cs.AI

Beyond Generation and Accuracy: Diagnosing and Enhancing Visual Chain-of-Thought for Geometry Problem Solving

While multimodal reasoning has advanced rapidly, solving complex geometry problems critically hinges on active visual assistance, such as constructing auxiliary lines, spurring the rise of Visual Chain-of-Thought (VCoT). However, existing evaluations typically assess visual generation quality and final answer accuracy in isolation, failing to examine whether intermediate visual aids are geometrically valid, effectively utilized in subsequent reasoning, or causally responsible for task success. To bridge this gap, we introduce GeoVAD-Bench, a diagnostic benchmark that pairs a fine-grained five-dimensional trajectory diagnosis covering perception, auxiliary quality, utilization, deductive reasoning, and final correctness with controlled No-Aux, Auto-Aux, and GT-Aux intervention settings to systematically isolate intermediate error modes, the causal gains of visual aids, and the resulting autonomy gap. Our findings reveal that while high-quality auxiliary aids offer substantial theoretical gains for geometric problem solving, autonomous generation is frequently hampered by compounding errors across geometric perception, faithful visual manipulation, visual-state grounding, and deductive reasoning. Guided by these diagnostic insights, we establish a specialized data construction pipeline encompassing geometric perception, diagram editing, and interleaved visual-textual reasoning trajectories, and develop a progressive SFT and multimodal RL training framework. The resulting model, GeoWeave-8B, outperforms the base model by +25.3% in final geometric accuracy and achieves a +30.4% gain in process average across the four intermediate diagnostic dimensions.

Zhitong Dong, Jicai Pan, Yingguo Gao +3

1Key Laboratory of New Generation Artificial Intelligence Technology, Southeast University Group · 2Ant

Mathematical Reasoning Multimodal CoT Reasoning Multimodal Model Evaluation Visual CoT

Aug 31, 2026cs.CL

Reactivating Test-Time Scaling for Plane Geometry Problem Solving

Plane geometry problem (PGP) solving has become a critical benchmark for multimodal reasoning because it requires accurate visual perception and precise multi-step symbolic deduction. Although test-time scaling (TTS) has demonstrated remarkable success in general mathematical reasoning, it fails to scale effectively under the symbolic-program paradigm for plane geometry. We identify two key obstacles: limited reasoning diversity induced by rigid symbolic programs and insufficient explicit visual grounding before symbolic deduction. To address these issues, we propose Multi-Trace Synthesis (MTS), which converts each symbolic program into heterogeneous reasoning traces, including executable Python scripts and CoT-augmented variants. We further propose Perception-Augmented (PA) training, which parses diagrams into structured semantic clauses before deduction, and Consensus-Guided Multi-Trace Ensemble (CG-MTE) for efficient self-adaptive inference. Experiments on three geometry benchmarks show that our method consistently improves PGP-solving across model scales and achieves strong performance against both general-purpose MLLMs and specialized geometry solvers. Under test-time scaling, CG-MTE achieves comparable accuracy to high-budget self-consistency while reducing sampling cost by up to 8x. Code and data are publicly available at https://github.com/Jason8Kang/ReTTS-PGPS.

Xiaoqiang Kang, Shengen Wu, Maizhen Ning +5

School of Advanced Technology, Xi’an Jiaotong-Liverpool University · The Hong Kong University of Science and Technology (Guangzhou) · Hithink Research +2

Spatial Reasoning Benchmarks Inference-Time Scaling Symbolic Reasoning Mathematical Reasoning

Aug 12, 2026cs.AI

OEIS Open: How many conjectures can language models turn into theorems?

We construct OEIS Open, a benchmark based on 492 open mathematical conjectures from the OEIS, formalized in Lean by Tsoukalas et al. Whereas these conjectures had previously been attempted only with a bespoke agent, our open-source evaluation code runs any generic language model (LM) against them, and is secure against LM cheating attempts. We find that LMs equipped with a minimal set of tools resolve 147 of these conjectures with a budget of $50 per attempt, scoring 30% on OEIS Open. OEIS Open Lite is a random subset of 100 conjectures for cheaper evaluation. When evaluated with a budget of $200 per attempt, the best current LM scores 44% on OEIS Open Lite. Giving LMs access to the mathematics literature via 476,000 papers from arXiv did not increase performance on OEIS Open Lite, and nor did using more sophisticated agent loops. The conjectures covered in this work are of uncertain mathematical significance, and most have likely received little previous attention. Nevertheless, our results show that LMs can resolve open research conjectures autonomously and at modest cost.

Tom Adamczewski

Epoch AI

LLM Evaluation Automated Theorem Proving Mathematical Reasoning

Aug 9, 2026cs.AI

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

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

p \geq 0.17

), despite generating 15--52

\times

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

p < 0.0001

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

p < 0.0001

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

κ= 0.76

--

1.00

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

Rahma Simin Ali, Jawad Hossain

Independent Researcher · University at Albany

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

Aug 4, 2026cs.CL

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

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

B^*=1024

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

Ankit Goyal, Jaideep Ray

Multilingual Language Model Evaluation LLM Evaluation Mathematical Reasoning LLM Mathematical Reasoning

Jul 30, 2026cs.AI

MECA: A Mechanism-Centered Agent for Constructing Well-Specified and Valuable Mathematical Conjectures

Automatically constructing well-specified and valuable mathematical conjectures remains a central challenge in AI-assisted mathematical discovery. Many existing open problems and conjectures are often too broad, underspecified, or difficult to connect to plausible proof or refutation strategies. We view a mathematical mechanism as a structure or reasoning principle that connects the assumptions of a candidate problem to its target conclusion, such as an inequality, invariant, decomposition, or reduction to an intermediate claim. We present MECA (MEchanism-centered Conjecture Agent), a multi-agent framework that constructs conjectures by jointly developing candidate statements and their supporting mechanisms. Explorer agents propose mechanisms, test how they apply, and revise the candidate conjecture accordingly, while critic agents assess their mathematical validity and research value. Their feedback guides changes to the assumptions, scope, and conclusion. Through this process, MECA transforms broad research directions into precise conjectures with substantive mathematical support while retaining a clearly identified unresolved core. We evaluate MECA in two complementary settings. First, we compare it with a generate-and-revise baseline on reconstructing preselected target-paper conclusions from target-conditioned but article-blind source materials. Second, we construct 100 semi-open problems from literature-derived seeds and existing open problems and evaluate them through independent proof and refutation attempts by automated provers. Our results indicate that mechanism-centered refinement produces well-specified and research-worthy conjectures that remain challenging for current automated provers.

Wentao Long, Yunfei Zhang, Chenyi Li +1

Fudan University · Qingdao University · Peking University

AI Agents for Scientific Discovery Mathematical Reasoning

Jul 24, 2026cs.CL

PatiGonit22K: A Comprehensive Dataset for Solving Complex Bengali MWPs

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.

Swastika Kundu, Azizul Hakim Fayaz, Tashreef Muhammad

Department of Computer Science and Engineering Ahsanullah University of Science and Technology, Dhaka, Bangladesh · Department of Computer Science and Engineering Southeast University, Dhaka, Bangladesh

Mathematical Reasoning Benchmarks Mathematical Reasoning Low-Resource Language Processing

Jul 21, 2026cs.LG

Off-Context GRPO: Learning to Reason on Hard Problems using Privileged Information

Reinforcement learning with verifiable rewards (RLVR) improves reasoning in large language models. Yet, typical RLVR approaches fail on difficult problems: when a model cannot generate any correct solutions, it receives \textit{zero} learning signal. Providing privileged guidance during training, such as solution prefixes, can help overcome this learning cliff by steering the model towards {correct solutions with non-zero reward}. {We call these rollouts \textit{off-context}: they are generated from a training prompt that contains privileged guidance, while the target objective is defined by the original prompt without that guidance.} {We introduce} Off-Context GRPO (OC-GRPO), a minimally modified variant of GRPO that uses guided rollouts but applies an importance-corrected objective to steer the update back toward the original unguided objective, avoiding the mismatch that destabilizes uncorrected guided training. Empirically, our algorithm achieves a 3.9% absolute improvement (13.8% relative gain) over vanilla GRPO on average across standard mathematical reasoning benchmarks with negligible additional cost.

Priyank Agrawal, Ankur Samanta, Shervin Ghasemlou +4

1Meta AI · 2Columbia University · *Work done at Meta

Reinforcement Learning RL for Language Model Reasoning Group Relative Policy Optimization Mathematical Reasoning Reinforcement Learning with Verifiable Rewards

Jul 20, 2026cs.LG

MADA-RL: Multi-Agent Debate-Aware Reinforcement Learning for Parameter-Efficient Reasoning in Compact Models

Large language models achieve strong reasoning performance, but often at prohibitive training cost - a challenge that is especially acute for compact models (

\leq 4 \, \mathrm{B}

parameters) trained under limited budgets. We introduce MADA-RL, a post-training framework that specializes compact models into generator and critic roles and trains them with a debate-aware learning signal, fine-tuning only a small subset of parameters via LoRA adapters. Our central contribution is a counterfactual critic advantage: a dynamic, role-conditioned baseline that redefines the critic's advantage as its reward minus the generator ensemble's per-instance accuracy. This explicitly optimizes critics to improve over generator consensus rather than to merely reproduce a correct answer, yielding more targeted credit assignment than static mean-reward normalization. At deployment, the specialized agents are composed in a lightweight multi-round protocol. Across five mathematical reasoning benchmarks, MADA-RL raises the accuracy of the DeepSeek-R1-Distill-Qwen-1.5B model from

39.9 \, \%

to

41.9 \, \%

(

+2.0

points,

p < 0.001

) using

16

times fewer trainable parameters than fully fine-tuned baselines, placing it on the accuracy-trainable-parameter Pareto front. It approaches, but does not surpass, the strongest baselines (DeepScaleR, STILL-3), which are trained on substantially larger datasets; we analyse this gap and the associated inference-time cost directly. A controlled study isolates the source of MADA-RL's gains: the counterfactual advantage produces the highest critic improvement rate of any model evaluated, indicating that trained critics learn to correct generator errors rather than to imitate them.

Martino M. L. Pulici, Cuong Xuan Chu, Evgeny Kharlamov +3

Bosch Center for Artificial Intelligence, Germany · LMU Munich, Germany · University of Oslo, Norway +2

RL for Language Model Reasoning Multi-Agent Debate Efficient Language Model Reasoning Mathematical Reasoning Multi-Agent LLMs

Jul 16, 2026cs.CL

Gold-Guided Programmatic Distillation for Financial Reasoning over Hybrid Tables and Text

Financial question answering over hybrid tabular and textual data may require multi-source reasoning and precise numerical computation. While large language models (LLMs) can generate intermediate reasoning steps, natural-language rationales remain prone to arithmetic errors, making them an unreliable supervision source for distillation. Building on programmatic distillation, we develop an approach that transfers reliable numerical reasoning from a large teacher model to a compact student using execution-verified Python programs instead of free-form textual rationales. It leverages gold derivations to guide teacher-side program synthesis and retains only programs that execute correctly and produce the gold answer, ensuring high-quality supervision. We further introduce an iterative recovery stage that revisits teacher-failed examples, enabling the student to recover and incorporate newly verified programs into training. Experiments on TAT-QA show that our framework is highly effective for hybrid financial reasoning. Our best 7B student achieves 87.00 EM / 87.18 F1 on the test set, substantially outperforming the 72B teacher (78.46 EM) as well as traditional and strong LLM-based baselines, including TAGOP and TAT-LLM. These results demonstrate that execution-verified programmatic distillation provides an effective and extensible framework for training smaller models to perform reliable numerical reasoning.

Yun Dong, Erica Zhao, Elana Chen

College of Computing · Georgia Institute of Technology · Department of Management Science & Engineering +2

Table QA Financial QA LLM-Based Program Synthesis Mathematical Reasoning Language Model Distillation Finance