Numerical Reasoning in Language Models

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

Large language models reach high accuracy in mathematical reasoning, but individual traces on the same problem diverge; some arrive at the correct answer while others fail. Prior work localizes such failures at the step, chunk, or sentence level, or identifies tokens where failure has already occurred. These approaches leave open which token triggers failure. We introduce the cliff token, a token at which the estimated probability of reaching the correct answer (success probability) drops beyond an adaptive threshold. Across seven models and three mathematical reasoning benchmarks (GSM1K, MATH500, AIME 2025), cliff tokens act as failure triggers. For incorrect traces containing cliff tokens, we compare resampling immediately before and after the first cliff token. Resampling before it shows higher pass@$k$ at the same sample count. We further introduce a cliff taxonomy of deterministic, uncertain, and sampled-off cliffs, defined by greedy choice and token entropy. Additionally, we show that the three types differ as training signals. Using single-token preference optimization at cliff positions (Cliff-DPO), we find that uncertain and sampled-off cliffs show larger accuracy gains than deterministic cliffs on three evaluation benchmarks. We release token-level rollout data and source code to enable further analysis without regenerating costly rollouts: https://github.com/beaver-22/Cliff-token

AI systems based on artificial neural networks are being developed with aspirations of pushing the boundary of human mathematical knowledge. A key question for these systems is how much they can reach beyond their training data. Mathematical discovery requires a strong form of out of distribution generalization; the ability to hypothesize genuinely new - and potentially logically more powerful - mathematical structures. It has been hypothesized that language abilities support such generalizations in human cognition. In this work, we use simple arithmetic as a case study for examining how modern AI models could expand their mathematical horizons, evaluating whether these models can independently discover the concept of "zero". We show that (1) language models of a GPT-2 size are unable to perform this generalization at test time regardless of language pretraining, but (2) models can improve substantially after training on tens or hundreds of examples of zero. Additionally, we find that language pretraining reduces the number of required examples by approximately $50\%$, showing that language abilities can scaffold mathematical discovery in neural models.

Irrelevant numbers in a prompt can shift language model judgments, producing anchoring effects in numerical reasoning. We study where this anchor-sensitive signal is carried inside language models using a controlled multiple-choice setup with shared answer options. We define a logit-difference metric comparing the correct answer option with the answer option corresponding to the anchor, and validate that it tracks behavioral anchoring. Using attribution-based circuit localization on 7B--8B Qwen and Llama base and instruction-tuned models, we find that edge-level methods recover this signal more faithfully than node-level methods. Low- and high-anchor circuits transfer strongly within a model, suggesting shared pathway structure across anchor direction. However, sparse transfer across base and instruction-tuned variants is less reliable, indicating that post-training changes which pathways matter most. Overall, our results provide a mechanistic account of how anchoring-related decision signals are carried inside language models.

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

Quantities with measurement units, such as 110 cm and 1.2 m, require language models (LMs) to combine a numeral with a symbolic unit scale. Here, we study how LMs compare such quantities in controlled settings spanning several unit systems. We find that accuracy degrades near the comparison boundary, where small changes in value determine the correct answer. The resulting errors are systematic: linear surrogate models predict LM preferences from numerical-difference and unit-scale-difference cues, and causal interventions on subspaces aligned with these variables shift model's output. The results suggest that LMs compare quantities through a bag of heuristics over numerals and units, rather than first converting both expressions to an exact shared-scale representation.

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

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

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

Structured prompts require integrating components according to task-relevant relations. How a network implements this integration is often hard to judge in language or vision, where those relations are rarely specified precisely enough to define a candidate internal algorithm. Arithmetic offers a cleaner setting. We study a Transformer trained on base-digit extraction: given $N$, $B$, and $D$, it must report the coefficient of $B^D$ in the base-$B$ expansion of $N$. The closed-form solution, $\lfloor N/B^D \rfloor \bmod B$, provides explicit candidate algorithmic intermediates. Across three seeds, the model reaches 99.83% exact-answer accuracy on held-out number-base intersections, establishing reliable task competence. Linear probes decode the intermediates, making staged arithmetic computation plausible. Causal tests then separate representation from use: within the localized route from the stream with $D$ as input to the output positions, behavior depends on early $D$-selective communication, independent of $N$ and $B$. Relatedly, a sparse circuit search finds mostly separate $N$, $B$, and $D$ routes that combine late rather than the staged route suggested by the probes. Thus, the model represents the intermediates that make the closed-form solution plausible, but the identified localized causal route does not transmit them to the output stream. This case shows that probe-based conclusions can diverge sharply from causal observations, even when explicit algorithmic hypotheses are available.

Do language models preserve the ordinal meaning of intensity words when those words must produce numeric actions? I study a researcher-constructed scale of 10 English degree modifiers, from slightly to drastically, informed by the Quirk et al. degree-modifier taxonomy, in a controlled resource-allocation environment where Claude Haiku receives a natural-language instruction, produces a numeric allocation, and a deterministic backend converts that allocation into a measurable outcome. The only variable that changes between runs is the intensity word or the starting system state, isolating their effects on the model's numeric output. Across 6,620 runs at T=0.0 and T=0.7, three patterns emerge. First, the model compresses 10 intensity words into 5 distinct median outputs: four lower-tier words all map to the same value, while stronger words break into higher regimes (Spearman rho = 0.845, p < 0.001). Second, when the current system state is supplied as context, separate Kruskal-Wallis tests show that grouping by starting allocation captures far more rank-based variance than grouping by word (epsilon-squared baseline = 0.782 vs. epsilon-squared word = 0.079), and lexical differentiation collapses to zero as the system approaches capacity. Third, near feasibility limits the model exhibits three behavioral modes: weak words hedge with small adjustments, strong words abstain entirely, and the word drastically pushes to the local ceiling. These patterns persist across temperature, with stochastic sampling broadening distributions but not restoring ordinal distinctions between words. In this model and domain, the model's numeric interpretation of vague intensity words is compressed, state-dependent, and discontinuous near operational boundaries.

LLMs have demonstrated exceptional proficiency in a wide range of NLP tasks. However, a notable gap remains in practical data analysis scenarios, particularly when LLMs are required to process long sequences of unstructured documents, such as news feeds or, as specifically addressed in this paper, social media posts. To empirically assess the effectiveness of LLMs in this setting, we introduce a question-based evaluation framework comprising 470 manually curated questions designed to evaluate LLMs' semantic understanding and reasoning abilities over aggregated text data. We apply our benchmark on diverse Twitter datasets covering various NLP tasks, including sentiment analysis, hate speech detection, and emotion recognition. Our results reveal that the performance depends heavily on input scale and the complexity of the data sources, declining noticeably in multi-label or target-dependent scenarios. In addition, as task complexity increases, performance drops progressively from basic semantic existence identification to more demanding operations such as comparison, counting, and calculation. Furthermore, as the input size grows beyond 500 instances, we identify a common limitation across LLMs, particularly Open-weights models: performance degrades substantially, especially on numerical tasks. These findings highlight critical architectural bottlenecks in current LLMs for performing rigorous quantitative analysis over large text collections.

Chain-of-thought (CoT) prompting is necessary for arithmetic in small language models, yet shuffling its steps preserves most performance. What does CoT contribute if not logical sequencing? In three 1-3B instruction-tuned LMs on GSM8K, we isolate the answer-readout stage via prefix completion and identify a positional shortcut: the model copies whichever number occupies the trailing position before the answer delimiter, regardless of intermediate reasoning. Gold-answer presence accounts for 54-92 pp of accuracy (89-92% of each model's teacher-forcing ceiling); even on incorrect items, the final answer matches the last CoT number 95-96% of the time. The copy channel takes precedence over retained-context completion: replacing the trailing number with a wrong value collapses accuracy to near-zero despite correct intermediates, yet removing it recovers 5-32 pp above that floor--even single-step arithmetic the model can otherwise perform is suppressed when a copyable number is present. Qwen and Llama copy novel distractors 87-95% of the time; Gemma gates selectively. Head-level ablation implicates architecture-specific head sets; the effect replicates on GSM-Symbolic. On non-arithmetic BBH tasks, shuffle retention drops sharply; at 7-8B, content-selective gating emerges. Step-level faithfulness evaluations risk conflating positional answer transport with genuine computation--a failure mode for CoT-based oversight.

Number prediction stands as a fundamental capability of large language models (LLMs) in mathematical problem-solving and code generation. The widely adopted maximum likelihood estimation (MLE) for LLM training is not tailored to number prediction. Recently, penalty-driven approaches, e.g., Number Token Loss and Discretized Distance Loss, introduce an inductive bias of numerical distance but induce over-sharpened and over-flattened digit distributions, respectively. In this paper, we make an in-depth analysis on LLM numerical learning, and show that existing numerical learning methods conceptually follow a criterion-distance formulation, where the criterion term represents optimization pattern and the distance term instills geometric prior. Consequently, we present Digit Entropy Loss (DEL) for auto-regressive numerical learning, which reformulates the conventional unsupervised entropy optimization in three key designs: leveraging digit conditional probability and binary cross-entropy to guide the entropy optimization into a supervised manner; deprecating the distance term to bypass the issue of numerical distance; and generalizing the integer-based numerical learning to floating-point number optimization, enabling more accurate number prediction. Our DEL formulation can incorporate integers, decimals, and decimal points, expanding the learning objective from a single digit to the floating-point number domain. Experiments conducted on seven mathematical reasoning benchmarks with four representative LLMs, including CodeLlama, Mistral, DeepSeek, and Qwen-2.5, demonstrate that DEL consistently outperforms its counterparts in both overall prediction accuracy and numerical distance. Source codes are at https://github.com/PolyU-VCLab/DEL

Tool-integrated reasoning (TIR) has emerged as a dominant paradigm for mathematical problem solving in language models, combining natural language (NL) reasoning with code execution. However, this interleaved setup has three key limitations: code often acts as a post-hoc verifier, intermediate NL computations are error-prone, and NL and code play overlapping rather than clearly distinct roles. We propose ThinC (Thinking in Code), a framework in which code itself serves as the reasoner rather than as a tool invoked by NL. A ThinC trajectory begins with a brief NL planning step, after which all reasoning unfolds through code blocks connected only by their execution outputs. We distill 12.2k code-centric trajectories from a teacher model and train ThinC-1.7B and ThinC-4B with supervised fine-tuning followed by reinforcement learning. ThinC-4B consistently outperforms every TIR baseline on five competition-level math benchmarks and even surpasses the much larger Qwen3-235B-A22B-Thinking. Further analysis shows that ThinC reasons through code: 99.2% of its final answers are grounded in interpreter output, and the model recovers reliably from code execution failures without intermediate NL reasoning. Our code and models will be released soon.

Large language models often fail at simple counting tasks, even when items to count are in the prompt. We investigate whether this failure occurs because transformers do not represent counts internally, or because they cannot convert representations to the correct output tokens. Across three model families: Pythia, Qwen3, and Mistral, ranging from 0.4B to 14B parameters, we find evidence for the second explanation. Linear probes recover the correct count from intermediate layers with $R^2>0.99$, showing that the information is present. However, the internal directions that encode counts are nearly orthogonal to digit-token output-head rows ($|\cos| \leq 0.032$). In other words, the model stores the count in a form that the digit logits do not naturally read out. We localize this failure with two interventions. Updating only the digit rows of the output head (36,864 parameters) substantially improves constrained digit prediction (60.7--100.0% on four tasks), but it does not fix unconstrained generation (0%); we do not claim that digit-row repair fixes open-ended text. By contrast, small LoRA on attention Q/V (7.67M parameters) improves upstream routing and achieves 83.1%$\pm$7.2% in true greedy autoregressive generation (deployable fix). Logit-lens at layer 35 (entity counting; correct-digit rank): (i) median over 3 seeds drops from order-$10^4$ to 1; (ii) seed 42 shows $54{,}332 \to 838$ (median top-1 while one seed stays far below). Norm, logit-lens, and cross-task analyses generalize the bottleneck to counting, addition, and list length; nulls on MMLU and GSM8K and limited DROP transfer. These results identify counting failure as a geometric readout bottleneck, not an internal-representation failure: the model knows the count but the output pathway is misaligned with tokens needed to express it.

Does structure in representations imply structure in computation? We study how Llama-3.1-8B reasons over cyclic concepts (e.g., "what month is six months after August?"). Even though Llama-3.1-8B's representations for these concepts are circularly structured, we find that instead of directly computing modular addition in the period of the cyclic concept (e.g., 12 for months), the model re-uses a generic addition mechanism across tasks that operates independently of concept-specific geometry. First, it computes the sum of its two inputs using base-10 addition (six + August=14). Then, it maps this sum back to cyclic concept space (14->February). We show that Llama-3.1-8B uses task-agnostic Fourier features to compute these sums--in fact, these features have periods that respect standard base-10 addition, e.g., 2, 5, and 10, rather than the cyclic concept period (e.g., 12 for months). Furthermore, we identify a sparse set of 28 MLP neurons re-used across all tasks (approximately 0.2% of the MLP at layer 18) that can be partitioned into disjoint clusters, each computing the sum for a Fourier feature with a different period. Our work highlights how an interplay between causal abstraction and feature geometry can deepen our mechanistic understanding of LMs.

Standard subword tokenization methods fragment numbers inconsistently, causing large language models (LLMs) to lose positional and decimal structure - a primary driver of errors in arithmetic and scientific reasoning. We introduce Triadic Suffix Tokenization (TST), a deterministic scheme that partitions digits into three-digit triads and annotates each triad with an explicit magnitude marker. Critically, the scheme defines a fixed, one-to-one mapping between suffixes and orders of magnitude for the integer part (thousands, millions, billions, etc.) and a parallel system of replicated markers for fractional depth (tenths, thousandths, millionths, etc.). Unlike approaches that rely on positional inference, this method provides a consistent gradient signal, which should ensure stable convergence. Two implementation variants are proposed: (1) a vocabulary-based approach that adds at most 10,000 fixed tokens to an existing vocabulary, covering 33 orders of magnitude ($10^{-15}$ to $10^{18}$); and (2) a suffix-marker approach that uses a small set of special tokens to denote magnitude dynamically. Both variants preserve exact digits while making order-of-magnitude relationships transparent at the token level. While we focus on 3-digit groups (Triadic), the framework is inherently scalable to any group size for precise vocabulary optimization. Furthermore, it allows for linear vocabulary expansion to accommodate arbitrary precision and range. TST is architecture-agnostic and can be integrated as a drop-in preprocessing step. Experimental validation is deferred to future work.

Large Language Models (LLMs) are increasingly being explored for clinical question answering and decision support, yet safe deployment critically requires reliable handling of patient measurements in heterogeneous clinical notes. Existing evaluations of LLMs for clinical numerical reasoning provide limited operation-level coverage, restricted primarily to arithmetic computation, and rarely assess the robustness of numerical understanding across clinical note formats. We introduce ClinicNumRobBench, a benchmark of 1,624 context-question instances with ground-truth answers that evaluates four main types of clinical numeracy: value retrieval, arithmetic computation, relational comparison, and aggregation. To stress-test robustness, ClinicNumRobBench presents longitudinal MIMIC-IV vital-sign records in three semantically equivalent representations, including a real-world note-style variant derived from the Open Patients dataset, and instantiates queries using 42 question templates. Experiments on 17 LLMs show that value retrieval is generally strong, with most models exceeding 85% accuracy, while relational comparison and aggregation remain challenging, with some models scoring below 15%. Fine-tuning on medical data can reduce numeracy relative to base models by over 30%, and performance drops under note-style variation indicate LLM sensitivity to format. ClinicNumRobBench offers a rigorous testbed for clinically reliable numerical reasoning. Code and data URL are available on https://github.com/MinhVuong2000/ClinicNumRobBench.

Large language models are trained and evaluated on quantitative reasoning tasks written in clean, emotionally neutral language. However, real-world queries are often wrapped in frustration, urgency or enthusiasm. Does emotional framing alone degrade reasoning when all numerical content is preserved? To investigate this, a controlled emotion translation framework is developed that rewrites problems into emotional variants while preserving all quantities and relationships. Using this framework, Temper-5400 (5,400 semantically verified emotion-neutral pairs) is constructed across GSM8K, MultiArith, and ARC-Challenge, and evaluated on eighteen models (1B to frontier scale). Two core results emerge: First, emotional framing reduces accuracy by 2-10 percentage points even though all numerical content is preserved. Second, neutralizing emotional variants recovers most of the lost performance, showing both that the degradation is tied to emotional style rather than content corruption and that neutralization can serve as a lightweight inference-time mitigation. Non-emotional paraphrases cause no such degradation, implicating emotional content rather than surface-level changes. Beyond emotion specifically, the benchmark construction procedure offers a controllable instrument for stylistic translation and robustness evaluation.

Recent alignment work on Large Language Models (LLMs) suggests preference optimization can improve reasoning by shifting probability mass toward better solutions. We test this claim in a resource-constrained setting by applying GRPO with LoRA to SLMs (up to 3B) for math reasoning on GSM8K and MATH datasets with difficulty-stratified analyses. As problem difficulty increases, accuracy plateaus, revealing a capacity boundary: GRPO primarily reshapes output preferences without reliably improving hardest-tier solving. Consistent with this, training GRPO only on lower-difficulty problems matches full-dataset accuracy across difficulty tiers while using only ~45% training steps, indicating diminishing returns from harder samples in this regime. We also find a cross-dataset generalization effect: GSM8K-trained GRPO achieves higher accuracy on the numeric subset of MATH than MATH-trained GRPO, exceeding it by ~5% at 1.5B and by ~3% at 3B. We show that the best achievable gains depend strongly on the base model's prior reasoning competence and the dataset's difficulty profile.

Despite the strong language understanding abilities of large language models (LLMs), they still struggle with reliable question answering (QA) over long, structured documents, particularly for numerical reasoning. Financial annual reports exemplify this difficulty: financial statement analysis often hinges on accurate arithmetic, and analysts derive key indicators by integrating evidence scattered across multiple tables and narrative text. However, existing benchmarks focus largely on single-table settings, leaving cross-table document-level numerical reasoning underexplored. To address this gap, we introduce FinLongDocQA, a dataset for both single-table and cross-table financial numerical reasoning in long-context reports. Evaluating both closed-source and open-source LLMs on FinLongDocQA reveals two bottlenecks: (1) annual reports often exceed 129k tokens, exacerbating the context rot problem for locating relevant tables; and (2) even when relevant evidence is located, LLMs remain prone to errors in multi-step numerical reasoning. We propose FinLongDocAgent, a Multi-Agent Multi-Round Retrieval-Augmented Generation (RAG) approach that iteratively retrieves evidence, performs intermediate calculations, and verifies results across rounds. Experiments highlight the importance of iterative retrieval and verification for reliable numerical QA in long financial documents.

Charts are ubiquitous in scientific and financial literature for presenting structured data. However, chart reasoning remains challenging for multimodal large language models (MLLMs) due to the lack of high-quality training data, as well as the need for fine-grained visual grounding and precise numerical computation. To address these challenges, we first propose DuoChart, a scalable dual-source data pipeline that combines synthesized charts with real-world charts to construct diverse, high-quality chart training data. We then introduce CharTool, which equips MLLMs with external tools, including image cropping for localized visual perception and code-based computation for accurate numerical reasoning. Through agentic reinforcement learning on DuoChart, CharTool learns tool-integrated reasoning grounded in chart content. Extensive experiments on six chart benchmarks show that our method consistently improves over strong MLLM baselines across model scales. Notably, CharTool-7B outperforms the base model by +8.0% on CharXiv (Reasoning) and +9.78% on ChartQAPro, while achieving competitive performance with substantially larger or proprietary models. Moreover, CharTool demonstrates positive generalization to out-of-domain visual math reasoning benchmarks.

Reinforcement Learning (RL) substantially improves the reasoning capabilities of language models, but most existing RL fine-tuning approaches rely entirely on ground-truth verifiable rewards and thus labeled datasets with verifiable answers. To overcome this, we propose a RL framework for reasoning distillation that leverages continuous, LLM-based rewards. Our method employs an efficient mechanism that computes a continuous CoT reward (CCR) directly from a single-token logit of a judge LLM, evaluating the student model's reasoning trajectory. This formulation provides an effective and scalable online training signal that can be applied to large volumes of unlabeled data. We demonstrate that, when paired with a strong judge, simply using CCR achieves performance comparable to that of ground-truth or pseudo-label verifiable rewards, and even surpasses them as the amount of unlabeled data increases. Furthermore, we find that combining them in a semi-supervised setup is highly synergistic: verifiable rewards help stabilize the CCR, while CCR improves the generalizability of verifiable rewards to related tasks. We also provide a comprehensive empirical comparison of various reward sources across multiple model architectures and dataset sizes. Our results show that this semi-supervised approach consistently enhances mathematical reasoning, yielding an absolute improvement of 5-10% on multiple reasoning tasks.

Large language models often default to step-by-step computation even when efficient numerical shortcuts are available. This raises a basic question: do they exhibit number sense in a human-like behavioral sense, i.e., the ability to recognize numerical structure, apply shortcuts when appropriate, and avoid them when they are not? We introduce SenseMath, a controlled benchmark for evaluating structure-sensitive numerical reasoning in LLMs. SenseMath contains 4,800 items spanning eight shortcut categories and four digit scales, with matched strong-shortcut, weak-shortcut, and control variants. It supports three evaluation settings of increasing cognitive demand: Shortcut Use (whether models can apply shortcuts on shortcut-amenable problems); Applicability Judgment (whether they can recognize when a shortcut is appropriate or misleading); and Problem Generation (whether they can generate new problem items that correctly admit a given type of shortcut). Our evaluation across five LLMs, ranging from GPT-4o-mini to Llama-3.1-8B, shows a consistent pattern: when explicitly prompted, models readily adopt shortcut strategies and achieve substantial accuracy gains on shortcut-amenable items (up to 15%), yet under standard chain-of-thought prompting they spontaneously employ such strategies in fewer than 40% of cases, even when they demonstrably possess the requisite capability. Moreover, this competence is confined to the Use level; models systematically over-generalise shortcuts to problems where they do not apply, and fail to generate valid shortcut-bearing problems from scratch. Together, these results suggest that current LLMs exhibit procedural shortcut fluency without the structural understanding of when and why shortcuts work that underlies human number sense.

Although language models demonstrate remarkable proficiency on mathematical benchmarks, it remains unclear whether this reflects true mathematical reasoning or statistical pattern matching over learning formal syntax. Most existing evaluations rely on symbolic problems grounded in established mathematical conventions, limiting insight into the models' ability to construct abstract concepts from first principles. In this work, we propose Math Takes Two, a new benchmark designed to assess the emergence of mathematical reasoning through communication. Motivated by the hypothesis that mathematical cognition in humans co-evolved with the need for precise communication, our benchmark tests whether two agents, without prior mathematical knowledge, can develop a shared symbolic protocol to solve a visually grounded task where the use of a numerical system facilitates extrapolation. Unlike many current datasets, our benchmark eschews predefined mathematical language, instead requiring agents to discover latent structure and representations from scratch. Math Takes Two thus provides a novel lens through which to develop and evaluate models with emergent numerical reasoning capabilities.

Scaling test-time compute via extended reasoning has become a key paradigm for improving the capabilities of large language models (LLMs). However, existing approaches optimize reasoning under fixed or uniformly sampled token budgets, ignoring the fundamental mismatch between problem difficulty and allocated compute. This leads to overthinking on easy problems and underthinking on hard ones, resulting in suboptimal token efficiency across diverse reasoning scenarios. In this paper, we propose Budget-Adaptive Curriculum Reasoning (BCAE), a unified framework that jointly optimizes reasoning quality and token efficiency through three synergistic components: (1) a budget-conditioned unified policy that embeds the token budget as a continuous conditioning signal, eliminating the need for decoupled thinking and summarization strategies; (2) a curriculum-aware budget scheduler that adaptively shifts the training budget distribution from easy to hard problems based on real-time learning progress; and (3) a truncation-aware dense reward mechanism that provides fine-grained credit assignment at intermediate reasoning steps via process-level verification. We further introduce Budget-Conditioned Advantage Estimation (BCAE), a novel variance reduction technique that conditions the advantage baseline on the sampled budget, yielding more stable policy gradients. Experiments on mathematical reasoning benchmarks (MATH, GSM8K, AIME, and Minerva Math) demonstrate that BACR consistently outperforms other strong baselines across all token budgets, achieving up to 8.3% accuracy improvement under tight budgets while reducing average token consumption by 34% compared to unconstrained reasoning.

Arithmetic benchmarks are often reduced to a single held-out score, but that score can conflate qualitatively different failures. We study a controlled minimal GPT trained on exhaustive 2-digit addition, where all local digit transitions are already present in training, and ask why 3-digit generalization still fails. The failure is staged. First, there is a layout barrier: a learned absolute-position model collapses under a pure 3-digit layout shift, and mixed-layout exposure is the only intervention that materially weakens this barrier. Second, after layout repair, the hundreds position behaves like a carry flag rather than a semantic hundreds digit; targeted carry probes reverse the relevant logit margin, whereas a matched extra-data control does not. Third, after carry repair, the main remaining bottleneck is conditional recomposition: high-conditioned tail data outperforms a matched control, high-only data, and tail-only data on all true-3-digit suites, and the same ordering reappears in a larger 2-layer bridge experiment. The residual errors after recomposition are then overwhelmingly tens-only, and a separate 10-seed late-stage study shows that a sign-aware tens repair raises exact match on the hardest thousands-carry suite from 0.664 to 0.822. We therefore provide an experimentally testable decomposition of arithmetic OOD failure into layout, carry-semantics, recomposition, and late tens-residual stages.

While Large Language Models (LLMs) achieve high performance on standard mathematical benchmarks, their problem-solving abilities depend on the context and textual formatting. We introduce the Robust Reasoning Benchmark (RRB), a pipeline of 13 deterministic textual perturbations applied to AIME 2024 and AIME 2025. Evaluating 8 state-of-the-art models, we find that frontier models are largely resilient, with the notable exception of Claude, which categorically refuses many transformed prompts. Open-weights reasoning models exhibit a range of failure modes under structural noise (cognitive thrashing, tokenization breakdown, and reasoning collapse), with up to 54% average accuracy drops across perturbations and up to 100% on some. We further study one of these failure modes in isolation: attention dilution caused by the model's own chain-of-thought. By tasking models with solving multiple independent mathematical problems sequentially within a single context window, we identify Intra-Query Attention Dilution. Open-weights models ranging from 7B to 120B parameters exhibit accuracy decay on subsequent problems, suggesting that intermediate reasoning steps progressively pollute standard dense attention mechanisms. We argue that in order to achieve reliable reasoning, future architectures need to integrate explicit contextual resets within models' own chain-of-thought, leading to open research questions regarding the optimal granularity of reasoning tasks.

Large Language Models (LLMs) have demonstrated remarkable proficiency in English mathematical reasoning, yet a significant performance disparity persists in multilingual contexts, largely attributed to deficiencies in language understanding. To bridge this gap, we introduce Translation-Augmented Policy Optimization (TAPO), a novel reinforcement learning framework built upon GRPO. TAPO enforces an explicit alignment strategy where the model leverages English as a pivot and follows an understand-then-reason paradigm. Crucially, we employ a step-level relative advantage mechanism that decouples understanding from reasoning, allowing the integration of translation quality rewards without introducing optimization conflicts. Extensive experiments reveal that TAPO effectively synergizes language understanding with reasoning capabilities and is compatible with various models. It outperforms baseline methods in both multilingual mathematical reasoning and translation tasks, while generalizing well to unseen languages and out-of-domain tasks.