Numerical Reasoning in Language Models

We present ThermoQA, a benchmark of 293 open-ended engineering thermodynamics problems in three tiers: property lookups (110 Q), component analysis (101 Q), and full cycle analysis (82 Q). Ground truth is computed programmatically from CoolProp 7.2.0, covering water, R-134a, and variable-cp air. Six frontier LLMs are evaluated across three independent runs each. The composite leaderboard is led by Claude Opus 4.6 (94.1%), GPT-5.4 (93.1%), and Gemini 3.1 Pro (92.5%). Cross-tier degradation ranges from 2.8 pp (Opus) to 32.5 pp (MiniMax), confirming that property memorization does not imply thermodynamic reasoning. Supercritical water, R-134a refrigerant, and combined-cycle gas turbine analysis serve as natural discriminators with 40-60 pp performance spreads. Multi-run sigma ranges from +/-0.1% to +/-2.5%, quantifying reasoning consistency as a distinct evaluation axis. Dataset and code are open-source at https://huggingface.co/datasets/olivenet/thermoqa

Large language models (LLMs) generalize smoothly across continuous semantic spaces, yet strict logical reasoning demands the formation of discrete decision boundaries. Prevailing theories relying on linear isometric projections fail to resolve this fundamental tension. In this work, we argue that task context operates as a non-isometric dynamical operator that enforces a necessary "topological distortion." By applying Gram-Schmidt decomposition to residual-stream activations , we reveal a dual-modulation mechanism driving this process: a class-agnostic topological preservation that anchors global structure to prevent semantic collapse, and a specific algebraic divergence that directionally tears apart cross-class concepts to forge logical boundaries. We validate this geometric evolution across a gradient of tasks, from simple mapping to complex primality testing. Crucially, targeted specific vector ablation establishes a strict causal binding between this topology and model function: algebraically erasing the divergence component collapses parity classification accuracy from 100% to chance levels (38.57%). Furthermore, we uncover a three-phase layer-wise geometric dynamic and demonstrate that under social pressure prompts, models fail to generate sufficient divergence. This results in a "manifold entanglement" that geometrically explains sycophancy and hallucination. Ultimately, our findings revise the linear-isometric presumption, demonstrating that the emergence of discrete logic in LLMs is purchased at an irreducible cost of topological deformation.

Large Language Models (LLMs) have demonstrated great capabilities across diverse natural language tasks; yet their ability to solve abstraction and optimization problems with constraints remains scarcely explored. In this paper, we investigate whether LLMs can reason and optimize under the physical and operational constraints of Optimal Power Flow (OPF) problem. We introduce a challenging evaluation setup that requires a set of fundamental skills such as reasoning, structured input handling, arithmetic, and constrained optimization. Our evaluation reveals that SoTA LLMs fail in most of the tasks, and that reasoning LLMs still fail in the most complex settings. Our findings highlight critical gaps in LLMs' ability to handle structured reasoning under constraints, and this work provides a rigorous testing environment for developing more capable LLM assistants that can tackle real-world power grid optimization problems.

Neural tree search is a powerful decision-making algorithm widely used in complex domains such as game playing and model-based reinforcement learning. Recent work has applied AlphaZero-style tree search to enhance the reasoning capabilities of Large Language Models (LLMs) during inference, but we find that this approach suffers from a scaling failure: on GSM8K and Game24, accuracy drops as the search budget increases. In this paper, we present ReSCALE, an adaptation of Gumbel AlphaZero MCTS that replaces Dirichlet noise and PUCT selection with Gumbel sampling and Sequential Halving, restoring monotonic scaling without changes to the model or its training. ReSCALE reaches 58.4% on GSM8K and 85.3% on Game24 at budgets where the baseline degrades. Ablations confirm that Sequential Halving is the primary driver of the improvement.

How do transformer language models represent magnitude? Recent work disagrees: some find logarithmic spacing, others linear encoding, others per-digit circular representations. We apply the formal tools of psychophysics to resolve this. Using four converging paradigms (representational similarity analysis, behavioural discrimination, precision gradients, causal intervention) across three magnitude domains in three 7-9B instruction-tuned models spanning three architecture families (Llama, Mistral, Qwen), we report three findings. First, representational geometry is consistently log-compressive: RSA correlations with a Weber-law dissimilarity matrix ranged from .68 to .96 across all 96 model-domain-layer cells, with linear geometry never preferred. Second, this geometry is dissociated from behaviour: one model produces a human-range Weber fraction (WF = 0.20) while the other does not, and both models perform at chance on temporal and spatial discrimination despite possessing logarithmic geometry. Third, causal intervention reveals a layer dissociation: early layers are functionally implicated in magnitude processing (4.1x specificity) while later layers where geometry is strongest are not causally engaged (1.2x). Corpus analysis confirms the efficient coding precondition (alpha = 0.77). These results suggest that training data statistics alone are sufficient to produce log-compressive magnitude geometry, but geometry alone does not guarantee behavioural competence.

Large Language Models (LLMs) achieve strong performance on many reasoning benchmarks, yet these evaluations typically focus on isolated tasks that differ from real-world usage in task-oriented dialogue (TOD). In this setting, LLMs must perform reasoning inherently while generating text and adhering to instructions on role, format, and style. This mismatch raises concerns about whether benchmark performance accurately reflects models' reasoning robustness in TOD setting. We investigate how framing reasoning tasks within TOD affects LLM performance by introducing BOULDER, a new dynamic benchmark covering eight travel-related tasks that require arithmetic, spatial, and temporal reasoning with both commonsense and formal aspects. Each problem is presented in both isolated and dialogue-based variants, enabling controlled comparison while mitigating data contamination. Experiments on eight LLMs reveal a substantial and consistent performance gap between isolated and dialogue settings. Through ablations and qualitative analysis, we show that this gap is largely driven by the multi-turn nature of dialogue, with additional effects from role conditioning and tool-use requirements. Our results highlight the need to evaluate LLM reasoning in realistic interactive scenarios.

Financial analysis requires reasoning over heterogeneous signals, including company fundamentals from regulatory filings and trading signals derived from market prices. However, existing financial benchmarks focus almost exclusively on regulatory filings, neglecting market dynamics and cross-signal interactions. We introduce FinTradeBench, a benchmark of 1,400 questions covering NASDAQ-100 stocks over a ten-year window. To holistically evaluate financial reasoning, FinTradeBench includes questions in three reasoning categories: fundamentals-focused, trading-signal-focused, and hybrid questions that require cross-signal reasoning. Our benchmark starts with 150 expert-authored seed questions, which are then scaled to 1400. To ensure reliable evaluation, we create a Core400 subset in which every item is expert-reviewed. In zero-shot (No-RAG) response generation, we observe that all models perform substantially worse on trading-focused questions compared to fundamental-focused questions. To identify failure causes, we evaluate LLMs under controlled settings including retrieval-based baselines, tool-assisted computation, oracle conditions, and time-series-specialized pipelines. LLMs' capabilities to do computation and evidence selection contribute most to reasoning failures. Additionally, retrieval significantly improves trading-signal reasoning but does not improve fundamental reasoning, where identifying relevant signals remains the dominant failure mode. Finally, rather than relying on a small, fine-tuned model for time-series reasoning, it is better to summarize the time-series data and leverage LLMs' reasoning capabilities.

Generative retrieval (GR) models encode a corpus within model parameters and generate relevant document identifiers directly for a given query. While this paradigm shows promise in retrieval tasks, existing GR models struggle with complex queries in numerical contexts, such as those involving semantic reasoning over financial reports, due to limited reasoning capabilities. This limitation leads to suboptimal retrieval accuracy and hinders practical applicability. We propose ReasonGR, a framework designed to enhance multi-step semantic reasoning in numerical contexts within GR. ReasonGR employs a structured prompting strategy combining task-specific instructions with stepwise reasoning guidance to better address complex retrieval queries. Additionally, it integrates a reasoning-focused adaptation module to improve the learning of reasoning-related parameters. Experiments on the FinQA dataset, which contains financial queries over complex documents, demonstrate that ReasonGR improves retrieval accuracy and consistency, indicating its potential for advancing GR models in reasoning-intensive retrieval scenarios.

Reinforcement learning with Verifiable Rewards (RLVR) has emerged as a powerful paradigm for eliciting reasoning capabilities in large language models, particularly in mathematics and coding. While recent efforts have extended this paradigm to broader general scientific (STEM) domains, the complex interplay between supervised fine-tuning (SFT) and RL in these contexts remains underexplored. In this paper, we conduct controlled experiments revealing a critical challenge: for general STEM domains, RL applied directly to base models is highly sample-inefficient and is consistently surpassed by supervised fine-tuning (SFT) on moderate-quality responses. Yet sequential SFT followed by RL can further improve performance, suggesting that the two stages play complementary roles, and that how training data is allocated between them matters. Therefore, we propose DeReason, a difficulty-based data decoupling strategy for general reasoning. DeReason partitions training data by reasoning intensity estimated via LLM-based scoring into reasoning-intensive and non-reasoning-intensive subsets. It allocates broad-coverage, non-reasoning-intensive problems to SFT to establish foundational domain knowledge, and reserves a focused subset of difficult problems for RL to cultivate complex reasoning. We demonstrate that this principled decoupling yields better performance than randomly splitting the data for sequential SFT and RL. Extensive experiments on general STEM and mathematical benchmarks demonstrate that our decoupled curriculum training significantly outperforms SFT-only, RL-only, and random-split baselines. Our work provides a systematic study of the interplay between SFT and RL for general reasoning, offering a highly effective and generalized post-training recipe.

Decoding strategies play a central role in shaping the reasoning ability of large language models (LLMs). Traditional methods such as greedy decoding and beam search often suffer from error propagation, while sampling-based approaches introduce randomness without adequate robustness. Self-consistency improves reliability by aggregating multiple rollouts, but incurs significant computational overhead. We propose an entropy-guided decoding framework that introduces token-level adaptivity into generation. At each step, the model computes the entropy of the token distribution, identifies high-uncertainty positions, and selectively branches on these vulnerable points. A dynamic pool of partial rollouts is maintained and expanded until solutions are completed, concentrating computation where uncertainty is greatest and avoiding unnecessary exploration in confident regions. To enable efficient termination, we apply a rollout-level Entropy After </Think> (EAT) stopping criterion by performing entropy evaluation after the full reasoning trace, rather than incrementally at every step. Experiments on GSM8K, AMC2023, and their perturbed variants demonstrate that our method achieves consistently strong accuracy. Notably, on smaller LLMs, performance is comparable to GPT-5 while operating at a fraction of the cost.

We introduce SciTaRC, an expert-authored benchmark for question answering over scientific tables that targets composite, multi-step reasoning. To enable fine-grained diagnostic analysis beyond end-task accuracy, SciTaRC pairs each question with a manually constructed reasoning plan and explicit complexity metrics. State-of-the-art models fail on at least 23% of these questions, while highly capable open-weight models like Llama-3.3-70B collapse on 65.5% of the benchmark. Error analysis shows that, in zero-shot settings, failures are driven primarily by question comprehension, where models misinterpret the scientific query and derive the wrong reasoning objective. To determine whether overcoming this gap is sufficient, we use the structured plans to decouple strategy formulation from execution. Surprisingly, providing oracle step-by-step plans yields only limited gains and fails to eliminate the performance gap. This reveals a substantial execution bottleneck: both natural language and code-based methods struggle to reliably carry out long-horizon computational chains over structured data. Ultimately, SciTaRC serves as a rigorous diagnostic testbed for studying both planning and execution in scientific table reasoning.

Chain-of-thought (CoT) prompting enables reasoning in language models but requires explicit verbalization of intermediate steps. Looped transformers offer an alternative by iteratively refining representations within hidden states. This parameter efficiency comes at a cost, as looped models lack the storage capacity of deeper models which use unique weights per layer. In this work, we investigate transformer models that feature both adaptive per-layer looping, where each transformer block learns to iterate its hidden state via a learned halting mechanism, and gated memory banks, that provide additional learned storage. We find that looping primarily benefits mathematical reasoning, while memory banks help recover performance on commonsense tasks compared to parameter and FLOP matched models. Combining both mechanisms yields a model that outperforms an iso-FLOP baseline, with three times the number of layers, across math benchmarks. Analysis of model internals reveals layer specialization: early layers learn to loop minimally and access memory sparingly, while later layers do both more heavily.

While Large Language Models (LLMs) can accelerate text-heavy tasks in alternative investment due diligence, a gap remains in their ability to accurately extract and reason over structured tabular data from complex financial spreadsheets. Progress is held back by the lack of real industry fund portfolio datasets for benchmarking, as private equity data rooms are confidential. To address this, we introduce FinSheet-Bench, a benchmark of synthetic financial portfolio data modeled on real private equity fund structures, designed to evaluate LLM performance on text-serialized spreadsheet question answering and numeric reasoning tasks. Our evaluation of ten model configurations from OpenAI, Google, and Anthropic on financial spreadsheets, including complex layouts, fund dividers, and multi-line column names, reveals that no standalone model achieves error rates low enough for unsupervised use in professional finance applications. The best-performing model, Gemini 3.1 Pro, achieves 82.4% accuracy across twenty-four evaluation files of varying complexity and structural layout (approximately 1 error per 6 questions), followed by GPT-5.2 with reasoning at 80.4%, Claude Opus 4.6 with thinking at 80.2%, and Gemini 3 Pro at 80.2%. Performance degrades substantially on larger, more complex spreadsheets: the largest spreadsheet (152 companies, 8 funds) yields an average accuracy of just 48.6% across all models, compared to 86.2% on the easiest evaluation file. These difficulty patterns are consistent across all ten models, indicating that they reflect LLM limitations rather than idiosyncratic model weaknesses. Reliable financial spreadsheet extraction will likely require architectural approaches that separate document understanding from deterministic computation.

Large pre-trained models (LMs) and Large Language Models (LLMs) are typically effective at capturing language semantics and contextual relationships. However, these models encounter challenges in maintaining optimal performance on tasks involving numbers. Blindly treating numerical or structured data as terms is inadequate -- their semantics must be well understood and encoded by the models. In this paper, we propose CONE, a hybrid transformer encoder pre-trained model that encodes numbers, ranges, and gaussians into an embedding vector space preserving distance. We introduce a novel composite embedding construction algorithm that integrates numerical values, ranges or gaussians together with their associated units and attribute names to precisely capture their intricate semantics. We conduct extensive experimental evaluation on large-scale datasets across diverse domains (web, medical, finance, and government) that justifies CONE's strong numerical reasoning capabilities, achieving an F1 score of 87.28% on DROP, a remarkable improvement of up to 9.37% in F1 over state-of-the-art (SOTA) baselines, and outperforming major SOTA models with a significant Recall@10 gain of up to 25%.

Standard Retrieval-Augmented Generation (RAG) architectures fail in high-stakes financial domains due to two fundamental limitations: the inherent arithmetic incompetence of Large Language Models (LLMs) and the distributional semantic conflation of dense vector retrieval (e.g., mapping "Net Income" to "Net Sales" due to contextual proximity). In deterministic domains, a 99% accuracy rate yields 0% operational trust. To achieve zero-hallucination financial reasoning, we introduce the Verifiable Numerical Reasoning Agent (VeNRA). VeNRA shifts the RAG paradigm from retrieving probabilistic text to retrieving deterministic variables via a strictly typed Universal Fact Ledger (UFL). We mathematically bound this ledger using a novel Double-Lock Grounding algorithm. Coupled with deterministic Python execution, this neuro-symbolic routing compresses systemic hallucination rates to a near-zero 1.2%. Recognising that upstream parsing anomalies inevitably occur, we introduce the VeNRA Sentinel: a 3-billion parameter SLM trained to forensically audit candidate using a single-token inference budget with optional post-hoc reasoning. To train the Sentinel, we steer away from traditional hallucination datasets in favour of Adversarial Simulation, programmatically sabotaging financial records to simulate Ecological Errors. The compact Sentinel consequently outperforms 70B+ frontier models in error detection. Through Loss Dilution phenomenon in Reverse-CoT training, we present a novel Micro-Chunking loss algorithm to stabilise gradients under extreme verdict penalisation, yielding a 28x latency speedup without sacrificing forensic rigor.

Large language models are increasingly applied to materials science, yet fundamental questions remain about their reliability and knowledge encoding. Evaluating 25 LLMs across four materials science tasks -- over 200 base and fine-tuned configurations -- we find that output modality fundamentally determines model behavior. For symbolic tasks, fine-tuning converges to consistent, verifiable answers with reduced response entropy, while for numerical tasks, fine-tuning improves prediction accuracy but models remain inconsistent across repeated inference runs, limiting their reliability as quantitative predictors. For numerical regression, we find that better performance can be obtained by extracting embeddings directly from intermediate transformer layers than from model text output, revealing an ``LLM head bottleneck,'' though this effect is property- and dataset-dependent. Finally, we present a longitudinal study of GPT model performance in materials science, tracking four models over 18 months and observing 9--43% performance variation that poses reproducibility challenges for scientific applications.

Discrete diffusion language models (dLLMs) generate text by iteratively denoising a masked sequence. However, standard dLLMs condition each denoising step solely on the current hard-masked sequence, while intermediate continuous representations are discarded after sampling and remasking. We term this bottleneck the Information Island issue: continuous information remains isolated within individual denoising steps and fails to propagate across the trajectory. This bottleneck is especially harmful for reasoning, which requires intermediate reasoning state to be preserved and updated across many denoising steps. To address this limitation, we introduce MetaState, a lightweight recurrent augmentation that equips a frozen dLLM backbone with persistent, fixed-size working memory. MetaState comprises three modules with a shared time conditioner: a cross-attention Mixer that reads backbone activations into memory slots, a GRU-style Updater that integrates information across steps, and a cross-attention Injector that writes the updated memory back into the backbone. We train these modules with a dedicated $K$-step unrolling pipeline to learn multi-step dynamics. MetaState adds only ${\sim}0.6\%$ trainable parameters while keeping the backbone frozen, and consistently improves reasoning performance over frozen baselines on mathematical reasoning and code generation benchmarks, with an average gain of 4.5 percentage points across all evaluations.

Recent breakthroughs in large language models (LLMs) have led to notable successes in complex reasoning tasks, such as mathematical problem solving. A common strategy for improving performance is parallel thinking, in which multiple reasoning traces are generated and the final prediction is made using aggregation schemes like majority voting or best-of-$N$ decoding. However, two key challenges persist. First, multi-sample decoding incurs substantial inference latency, especially for long-form outputs. Second, effective mechanisms for reliably assessing the correctness of individual reasoning traces are still limited. To address these challenges, we introduce One-Token Verification (OTV), a computational method that estimates reasoning correctness in a single forward pass during generation. OTV is activated by a learnable token and integrated into the LLM via low-rank adaptation to probe internal reasoning signals through the key-value cache, supporting token-level correctness estimation at any stage of generation without disrupting primary reasoning. Experiments on mathematical reasoning benchmarks demonstrate that OTV consistently surpasses existing verifiers. Additionally, OTV reduces token usage by up to $90\%$ through correctness-guided early termination, prioritizing shorter, more reliable solutions.

Large language models (LLMs) have shown great promise for autonomous driving. However, discretizing numbers into tokens limits precise numerical reasoning, fails to reflect the positional significance of digits in the training objective, and makes it difficult to achieve both decoding efficiency and numerical precision. These limitations affect both the processing of sensor measurements and the generation of precise control commands, creating a fundamental barrier for deploying LLM-based autonomous driving systems. In this paper, we introduce DriveCode, a novel numerical encoding method that represents numbers as dedicated embeddings rather than discrete text tokens. DriveCode employs a number projector to map numbers into the language model's hidden space, enabling seamless integration with visual and textual features in a unified multimodal sequence. Evaluated on OmniDrive, DriveGPT4, and DriveGPT4-V2 datasets, DriveCode demonstrates superior performance in trajectory prediction and control signal generation, confirming its effectiveness for LLM-based autonomous driving systems.

We present a method for generating training data for reinforcement learning with verifiable rewards to improve small open-weights language models on mathematical tasks. Existing data generation approaches rely on open-loop pipelines and fixed modifications that do not adapt to the model's capabilities. Furthermore, they typically operate directly on word problems, limiting control over problem structure. To address this, we perform modifications in a symbolic problem space, representing each problem as a set of symbolic variables and constraints (e.g., via algebraic frameworks such as SymPy or SMT formulations). This representation enables precise control over problem structure, automatic generation of ground-truth solutions, and decouples mathematical reasoning from linguistic realization. We also show that this results in more diverse generations. To adapt the problem difficulty to the model, we introduce a closed-loop framework that learns modification strategies through prompt optimization in symbolic space. Experimental results demonstrate that both adaptive problem generation and symbolic representation modifications contribute to improving the model's math solving ability.

We study three-digit addition in Meta-Llama-3-8B (base) under a one-token readout to characterize how arithmetic answers are finalized after cross-token routing becomes causally irrelevant. Causal residual patching and cumulative attention ablations localize a sharp boundary near layer~17: beyond it, the decoded sum is controlled almost entirely by the last input token and late-layer self-attention is largely dispensable. In this post-routing regime, digit(-sum) direction dictionaries vary with a next-higher-digit context but are well-related by an approximately orthogonal map inside a shared low-rank subspace (low-rank Procrustes alignment). Causal digit editing matches this geometry: naive cross-context transfer fails, while rotating directions through the learned map restores strict counterfactual edits; negative controls do not recover.

We present ArabicNumBench, a comprehensive benchmark for evaluating large language models on Arabic number reading tasks across Eastern Arabic-Indic numerals (0-9 in Arabic script) and Western Arabic numerals (0-9). We evaluate 71 models from 10 providers using four prompting strategies (zero-shot, zero-shot CoT, few-shot, few-shot CoT) on 210 number reading tasks spanning six contextual categories: pure numerals, addresses, dates, quantities, and prices. Our evaluation comprises 59,010 individual test cases and tracks extraction methods to measure structured output generation. Evaluation reveals substantial performance variation, with accuracy ranging from 14.29% to 99.05% across models and strategies. Few-shot Chain-of-Thought prompting achieves 2.8x higher accuracy than zero-shot approaches (80.06% vs 28.76%). A striking finding emerges: models achieving elite accuracy (98-99%) often produce predominantly unstructured output, with most responses lacking Arabic CoT markers. Only 6 models consistently generate structured output across all test cases, while the majority require fallback extraction methods despite high numerical accuracy. Comprehensive evaluation of 281 model-strategy combinations demonstrates that numerical accuracy and instruction-following represent distinct capabilities, establishing baselines for Arabic number comprehension and providing actionable guidance for model selection in production Arabic NLP systems.

Recent work on test-time scaling for large language model (LLM) reasoning typically assumes that allocating more inference-time computation uniformly improves correctness. However, prior studies show that reasoning uncertainty is highly localized: a small subset of low-confidence tokens disproportionately contributes to reasoning errors and unnecessary output expansion. Motivated by this observation, we propose Thinking by Subtraction, a confidence-driven contrastive decoding approach that improves reasoning reliability through targeted token-level intervention. Our method, Confidence-Driven Contrastive Decoding, detects low-confidence tokens during decoding and intervenes selectively at these positions. It constructs a contrastive reference by replacing high-confidence tokens with minimal placeholders, and refines predictions by subtracting this reference distribution at low-confidence locations. Experiments show that CCD significantly improves accuracy across mathematical reasoning benchmarks while substantially reducing output length, with minimal KV-cache overhead. As a training-free method, CCD enhances reasoning reliability through targeted low-confidence intervention without computational redundancy. Our code will be made available at: https://github.com/bolo-web/CCD.

Complex problems, whether in math, logic, or planning, are solved by humans through a sequence of steps where the result of one step informs the next. In this work, we adopt the perspective that the reasoning power of Transformers is fundamentally limited by a fixed maximum number of steps along any latent path of computation. To address this, we introduce Turbo Connection (TurboConn), a novel architecture that overcomes the fixed-depth constraint by routing multiple residual connections from the higher-layer hidden states of each token $t$ to the lower layers of token $t+1$. Fine-tuning pre-trained LLMs with our method not only yields accuracy gains of 0.9% to over 10% on benchmarks like GSM8K, Parity, and multi-step arithmetic, but also demonstrates that the density of these backward connections is critical; our dense interaction significantly outperforms "sparse" alternatives that only pass a single hidden state or vector. Notably, TurboConn can be integrated into pre-trained LLMs to overcome task-specific plateaus: while a fine-tuned Qwen-3-1.7B achieves only 53.78% on Parity, adding our architectural modification enables the model to reach 100% accuracy, all without the necessity to retrain the full model from scratch or sophisticated curriculum learning. Our results provide strong empirical evidence that the depth of the computational path is a key factor in reasoning ability, also offering a new mechanism to enhance LLMs without significantly affecting generation latency.

Large language models (LLMs)-based chatbots are increasingly being adopted in the financial domain, particularly in digital banking, to handle customer inquiries about products such as deposits, savings, and loans. However, these models still exhibit low accuracy in core banking computations-including total payout estimation, comparison of products with varying interest rates, and interest calculation under early repayment conditions. Such tasks require multi-step numerical reasoning and contextual understanding of banking products, yet existing LLMs often make systematic errors-misinterpreting product types, applying conditions incorrectly, or failing basic calculations involving exponents and geometric progressions. However, such errors have rarely been captured by existing benchmarks. Mathematical datasets focus on fundamental math problems, whereas financial benchmarks primarily target financial documents, leaving everyday banking scenarios underexplored. To address this limitation, we propose BankMathBench, a domain-specific dataset that reflects realistic banking tasks. BankMathBench is organized in three levels of difficulty-basic, intermediate, and advanced-corresponding to single-product reasoning, multi-product comparison, and multi-condition scenarios, respectively. When trained on BankMathBench, open-source LLMs exhibited notable improvements in both formula generation and numerical reasoning accuracy, demonstrating the dataset's effectiveness in enhancing domain-specific reasoning. With tool-augmented fine-tuning, the models achieved average accuracy increases of 57.6%p (basic), 75.1%p (intermediate), and 62.9%p (advanced), representing significant gains over zero-shot baselines. These findings highlight BankMathBench as a reliable benchmark for evaluating and advancing LLMs' numerical reasoning in real-world banking scenarios.

Large Language Models (LLMs) are increasingly deployed in applications that interact with the physical world, such as navigation, robotics, or mapping, making robust geospatial reasoning a critical capability. Despite that, LLMs' ability to reason about GPS coordinates and real-world geography remains underexplored. We introduce GPSBench, a dataset of 57,800 samples across 17 tasks for evaluating geospatial reasoning in LLMs, spanning geometric coordinate operations (e.g., distance and bearing computation) and reasoning that integrates coordinates with world knowledge. Focusing on intrinsic model capabilities rather than tool use, we evaluate 14 state-of-the-art LLMs and find that GPS reasoning remains challenging, with substantial variation across tasks: models are generally more reliable at real-world geographic reasoning than at geometric computations. Geographic knowledge degrades hierarchically, with strong country-level performance but weak city-level localization, while robustness to coordinate noise suggests genuine coordinate understanding rather than memorization. We further show that GPS-coordinate augmentation can improve in downstream geospatial tasks, and that finetuning induces trade-offs between gains in geometric computation and degradation in world knowledge. Our dataset and reproducible code are available at https://github.com/joey234/gpsbench

Large language models (LLMs) have achieved strong results in mathematical reasoning, and are increasingly deployed as tutoring and learning support tools in educational settings. However, their reliability for students working in non-English languages, especially low-resource languages, remains poorly understood. We examine this gap by evaluating mathematical reasoning in Sinhala and Tamil -- two languages widely used in South Asian schools but underrepresented in artificial intelligence (AI) research. Using a taxonomy of six math problem types, from basic arithmetic to complex unit conflict and optimization problems, we evaluate four prominent large language models. To avoid translation artifacts that confound language ability with translation quality, we construct a parallel dataset in which each problem is independently authored in Sinhala and Tamil by native speakers, and in English by fluent speakers, all with strong mathematical backgrounds. Our analysis demonstrates that while basic arithmetic reasoning transfers robustly across languages, complex reasoning tasks show significant degradation in Tamil and Sinhala. The pattern of failures varies by model and problem type, suggesting that strong performance in English does not guarantee reliable performance across languages. These findings have direct implications for the deployment of AI tools in multilingual classrooms, and highlight the need for language-specific evaluation before adopting large language models as math tutoring aids in non-English educational contexts.

Test-Time Training offers a promising way to improve the reasoning ability of large language models (LLMs) by adapting the model using only the test questions. However, existing methods struggle with difficult reasoning problems for two reasons: raw test questions are often too difficult to yield high-quality pseudo-labels, and the limited size of test sets makes continuous online updates prone to instability. To address these limitations, we propose TTCS, a co-evolving test-time training framework. Specifically, TTCS initializes two policies from the same pretrained model: a question synthesizer and a reasoning solver. These policies evolve through iterative optimization: the synthesizer generates progressively challenging question variants conditioned on the test questions, creating a structured curriculum tailored to the solver's current capability, while the solver updates itself using self-consistency rewards computed from multiple sampled responses on both original test and synthetic questions. Crucially, the solver's feedback guides the synthesizer to generate questions aligned with the model's current capability, and the generated question variants in turn stabilize the solver's test-time training. Experiments show that TTCS consistently strengthens the reasoning ability on challenging mathematical benchmarks and transfers to general-domain tasks across different LLM backbones, highlighting a scalable path towards dynamically constructing test-time curricula for self-evolving. Our code and implementation details are available at https://github.com/XMUDeepLIT/TTCS.

Large language models have made substantial progress in mathematical reasoning. However, benchmark development for multilingual evaluation has lagged behind English in both difficulty and recency. Recently, GSM-Symbolic showed a strong evidence of high variance when models are evaluated on different instantiations of the same question; however, the evaluation was conducted only in English. In this paper, we introduce MGSM-Pro, an extension of MGSM dataset with GSM-Symbolic approach. Our dataset provides five instantiations per MGSM question by varying names, digits and irrelevant context. Evaluations across nine languages reveal that many low-resource languages suffer large performance drops when tested on digit instantiations different from those in the original test set. We further find that models robustness in HRL setting do not necessarily translate to LRL. Moreover, proprietary models, such as Gemini 2.5 Flash and GPT-4.1 are less robust to digit, whereas Gemini 3.0 Pro is more robust. Among open models, GPT-OSS 120B and DeepSeek v3 show stronger robustness. Based on these findings, we recommend evaluating each problem using at least five digit-varying instantiations to obtain a more robust and realistic assessment of math reasoning.

Large language models (LLMs) have achieved impressive proficiency in basic arithmetic, rivaling human-level performance on standard numerical tasks. However, little attention has been given to how these models perform when numerical expressions deviate from the prevailing conventions present in their training corpora. In this work, we investigate numerical reasoning across a wide range of numeral scripts and formats. We show that LLM accuracy drops substantially when numerical inputs are rendered in underrepresented scripts or formats, despite the underlying mathematical reasoning being identical. We further demonstrate that targeted prompting strategies, such as few-shot prompting and explicit numeral mapping, can greatly narrow this gap. Our findings highlight an overlooked challenge in multilingual numerical reasoning and provide actionable insights for working with LLMs to reliably interpret, manipulate, and generate numbers across diverse numeral scripts and formatting styles.