Arithmetic

Large language models are unreliable at arithmetic, which is a problem for clinical calculators where a single numerical error changes the recommendation. The standard response is to hardcode each calculator as a validated function, one at a time. We test an alternative: the model does not calculate. Instead, it writes case-specific Python that a restricted local executor runs as a deterministic solver, and the model's task reduces to deciding how to use it. We evaluate this Program-Solve interface on MedCalc-Bench Verified (1,100 cases, 55 calculators) against direct model arithmetic and a hand-written 22-calculator library, using Qwen2.5-7B and Qwen2.5-32B-AWQ, after auditing the benchmark's formulas against current clinical guidelines and flagging 16 of 55 with version, use or coefficient concerns. With formulas and gold variables supplied and both routes reading the whole note, handing off to the solver is not a reliable advantage at 7B (75.31% against 72.02%, a paired +3.29 points with a 95% calculator-cluster interval of [-3.49, 10.38]) but is one at 32B (90.53% against 83.47%, +7.05 [0.47, 14.60], clear of zero). The hand-written library is exact on its 440 supported cases but abstains elsewhere (40.0% overall). Adding an executor thus helps some open-weight models more than others even under matched formula, variable and note access, and is not a substitute for verified formulas or reliable variable extraction either way.

Task arithmetic composes skills by adding weight displacements, and merged models are then judged on benchmark suites. We measure when that composition is functionally additive, and find that the answer depends as much on how the model is prompted as on which tasks are merged. Across two-dimensional composition surfaces -- five model settings from 0.5B to 8B, two families, LoRA and full fine-tuning -- pairwise non-additivity is real, seed-stable, and transfers in coarse order to unseen task pairs: all eight preregistered sign predictions held. But it is input-conditioned everywhere we measured: the same merged model that shows a six-point interaction contrast on code prompts shows none on math prompts, and wrapping the identical code prompts in the instruction template the adapters were trained on collapses the contrast twenty-fold, from +6.9 to +0.3 points -- while re-serializing them in an untrained chat template leaves it intact (+12.5), falsifying our own preregistered prediction. Execution benchmarks (pass@1) inherit the training-format wrapper's blindness. Weight-space composition therefore supports coarse, input- and format-conditioned functional statements -- not a universal merging-performance predictor, and not one that training-format evaluations can see.

Low-precision formats usually optimize scalar fidelity while inheriting conventional product arithmetic. We introduce CurveFP, a block-scaled family that distributes magnitudes across interleaved logarithmic curves. Uniform curve indices make every nonzero product an exact sign and integer-index update, while a rational radix exposes the finite phase schedule required for accumulation. We instantiate the algebra as CurveFP8 E4C3/E5C2 for training and CurveFP7 E3C3 for compact inference. On four 7B-9B models, CurveFP7 beats tensorwise FP8 perplexity with one fewer element bit and stays within 1.32% of native quality. CurveFP8 lowers error in all 36 paired training-GEMM comparisons. Across three matched 3B-token pretraining triplets, it reaches mean BF16-inference perplexity 22.5366 versus 22.5407 for FP8 and has a lower format penalty in every seed. Downstream evaluation shows transfer parity and a consistent WikiText-103 gain. In a preliminary 4x4 Nangate45 spatial accelerator tile, CurveFP8 uses one fewer product register and 4.6% less area than timing-closing FP8 at 500 MHz. These results support CurveFP as a numerical and arithmetic co-design, while leaving system-level efficiency to future study.

Transformer-based large language models (LLMs) continue to achieve state-of-the-art performance across various natural language processing tasks. However, their subpar performance on seemingly elementary problems, such as basic arithmetic, raises concerns about model reliability, safety, and ethical deployment. In this study, we demonstrate that the performance of a vanilla Transformer model trained on integer arithmetic tasks can be improved using methods effective for human learners. We begin by decomposing the arithmetic task into well-defined subtasks and conducting loss convergence order analysis together with ablation studies for each subtask. Our findings reveal that LLMs exhibit learning patterns similar to those of human learners, with a faster learning speed for simpler subtasks compared to more complex ones. In addition, we successfully improved the accuracy of LLMs by applying problem-solving strategies and cognitive empowerment methods shown to enhance the performance of human learners. This suggests that transformer-based LLMs may share cognitive processes with human learners in arithmetic. Lastly, we provide a comprehensive demonstration of our method's effectiveness, including significant accuracy improvement experiments, visualization verification, and explanation-based analysis to illuminate the intricacies of LLMs in arithmetic learning. In general, this work explores the potential similarities between transformer-based LLMs and human learners, supported by explainable AI (XAI) verifications, ultimately fostering trust in LLMs for critical and high-stakes applications.

Low-precision neural networks are attractive for resource-constrained hardware, but fixed-point arithmetic introduces failure modes that are often hidden by idealised quantisation models. In particular, two's-complement overflow wrapping can corrupt hidden activations by changing both their magnitude and sign, leading to unstable numerical error propagation and severe accuracy degradation. This paper proposes a Lyapunov-stabilised quantisation framework for low-precision neural networks operating under hardware-style wrapping arithmetic. The hidden-state energy is monitored through a layerwise Lyapunov function, and a monotone projection is applied to enforce bounded and non-increasing state evolution across depth. The method is evaluated on MNIST using a compact patch-based transformer under post-training quantisation and quantisation-aware training with fixed-point bit-widths from 4 to 16 bits. Monte Carlo results show that unconstrained wrapped quantisation-aware training collapses to near-chance accuracy across 6-16 bits, with activation overflow rates exceeding 11%. In contrast, the proposed monotone Lyapunov projection suppresses activation overflow to below 0.012% and restores stable low-precision learning, achieving 86.55% accuracy at 12 bits. These results demonstrate that Lyapunov-based state control can act as a hardware-aware stabilisation mechanism for reliable fixed-point neural inference and training.

Current benchmarks for evaluating large language models (LLMs) in medical calculation are largely based on simplified settings, where each patient case corresponds to a single calculator and the required tool is explicitly specified in the query. However, real clinical scenarios often require multiple calculators for joint evaluation, nested-scale calculation, and fuzzy queries that do not directly specify the target calculator. To this end, we propose a new medical calculation benchmark, MedCalc-Pro, which covers three progressively challenging task settings: single-calculator, multi-calculator, and nested-calculator calculation settings. MedCalc-Pro contains 2,268 real-world clinical cases, covering 77 medical calculators across 14 clinical departments. Meanwhile, to address the limited performance of existing frameworks and methods in complex clinical scenarios, we further propose a more generalizable agent framework that supports multi-tool selection and nested-tool calling, while suppressing parameter error propagation through structured validation and evidence review. We conduct systematic comparisons across open-source, closed-source, and medical-specialized LLMs, and the results show that our framework achieves the best performance across all three task settings. This work provides a new benchmark and method for evaluating and applying LLMs in challenging medical calculation scenarios.

This paper presents an energy-efficient hardware acceleration of the convolutional layers in the U-Net architecture for image segmentation, implemented on FPGA. While digit-serial arithmetic, particularly most-significant-digit-first (MSDF) techniques, offers a compact hardware footprint, it suffers from initial latency before producing the first output digit. This delay accumulates in cascaded operations like multiplication followed by addition, where each unit introduces its own startup overhead. To overcome this, we propose a merged multiply-add (MMA) architecture that fuses these operations into a unified pipeline. Instead of incurring separate delays, the MMA introduces a single streamlined latency per iteration, shorter than the combined latency of conventional cascaded units, resulting in enhanced throughput and efficiency. The MMA units are designed to process spatial input depths in parallel, achieving significantly higher performance than both standalone MSDF-based and conventional designs. We evaluate the proposed design using U-Net as a target application. Despite operating at a lower frequency than a CPU, the FPGA-based accelerator achieves up to an order of magnitude higher energy efficiency, delivering up to $15.14$ GOPS/W compared to $1.93$ GOPS/W for CPU-based inference. The design also shows approximately $9\times$ reduction in energy consumption compared to MSDF-based FPGA implementations. These results highlight the efficacy of the merged arithmetic approach for resource-constrained, latency-sensitive edge applications in medical imaging and computer vision.

Open Vocabulary Action Recognition (OVAR) enables the recognition of novel actions by leveraging vision-language representations, overcoming the limitations of traditional closed-set approaches. However, achieving robust performance in real-world scenarios typically requires domain-specific fine-tuning, which is often costly and raises privacy and regulatory concerns. In this work, we propose an alternative paradigm that bypasses target-domain training and recombines knowledge from existing datasets and models. Leveraging model merging and task arithmetic, we extract and combine task vectors from models fine-tuned on diverse public OVAR datasets. We show that, in out-of-distribution settings, the resulting merged model achieves superior zero-shot generalization to the pre-trained base model. Code is available at https://github.com/omaymaMoussadek/robust-ovar

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

Model merging (MM) has gained significant attention as a cost-effective approach to integrate multiple task-specific models into a unified model. However, recent work reveals that MM is highly susceptible to backdoor attacks. Existing defenses based on task arithmetic often fail to eliminate backdoors without substantially degrading clean-task performance, owing to their reliance on direct parameter-space editing. To address this gap, we propose Linear Feature Path Minimization (LFPM), a backdoor mitigation framework for model merging, which introduces an anti-backdoor task vector into the backdoored merged model. Unlike prior approaches, LFPM formulates the backdoor robustness of the merged model from a unified feature-space perspective under the Cross-Task Linearity (CTL) framework, which leverages the approximate linearity of features across tasks. This perspective guides the optimization of the anti-backdoor task to suppress backdoors while preserving clean-task performance. Furthermore, we introduce an effective optimization mechanism based on gradient accumulation and loss path-integral, ensuring robust backdoor suppression along the interpolation path. Extensive experiments demonstrate that LFPM consistently exhibits strong robustness against backdoor attacks in both full fine-tuning and Parameter-Efficient Fine-Tuning (PEFT) settings.

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.

Simulating large molecular systems comprising thousands of atoms requires highly scalable methodologies. While modern Density Functional Theory (DFT) codes exhibit linear scaling, solving the associated large, sparse generalized eigenproblems remains a critical computational bottleneck on exascale architectures. In the context of the LimitX project, we propose a data-driven framework to accelerate these calculations. By shifting the machine learning target from discrete eigenvalues to the coefficients of an interpolating Chebyshev polynomial, and by comparing both all-atom and fragment-based structural representations, we successfully overcome the dimensionality constraints of large-scale spectral prediction. We investigate three machine learning models (Kernel Ridge Regression, Graph Neural Networks, and Random Forests) trained on a novel 2 TB dataset of protein dimers. The predicted spectra provide initial guesses that effectively bypass early Self-Consistent Field (SCF) iterations in BigDFT. Ultimately, these spectral predictors will be deployed to dynamically optimize upcoming rational filter-based eigensolvers, such as FrASE, which is currently in initial development.

Large language models are able to compose skills in order to perform complex tasks, many of which might not have been seen during training. The details of how exactly this composition occurs remain elusive. In this paper, we study a mechanism for compositional generalization in transformers by considering a simple controlled setting involving variable assignment and modular addition. By partitioning our training data into disjoint sets, we observe that small transformers are able to generalize to previously unseen combinations of variables and numbers. Our mechanistic analysis shows that the same ``modular addition'' MLP module is used whether the inputs are given directly or indirectly through a separate variable assignment mechanism. We also analyze the training dynamics from an empirical lens, which reveals three phases of learning: first, modular addition is learned, then the structure required for variable assignment, and finally a refinement phase where the model generalizes to some hard sequences not seen in training. Finally, we provide a theoretical framework to explain how compositionality emerges from training dynamics. These results suggest that compositional generalization can be a natural consequence of the compositionality of internal mechanisms in~transformers.

Large Language Models exhibit paradoxical fragility in fundamental arithmetic, implying a disconnect between internal computation and discrete output. By analyzing the residual stream geometry during multi-operand addition, we identify the Iso-Raw-Sum Trajectory (IRST), a geometric structure where representations are anchored by semantic digits and modulated by continuous carry fibers. We propose the Noisy Quantization Model to explain this geometry, framing arithmetic errors as Geometric Slippages caused by internal neural noise pushing a continuous, latent Carry Potential across quantization thresholds. This geometric framework further elucidates Probe Versatility, explaining how lightweight probes can disentangle coexisting latent signals (such as ground truth versus hallucination) from a single activation vector. Finally, we validate these insights through a geometric consistency check method that effectively detects and corrects these quantization failures during inference. Our code is available at https://github.com/RL-MIND/Shape-of-Addition.

Mathematical reasoning is a hallmark of human intelligence, requiring logical deduction, symbolic manipulation, and abstract thinking. Recent multimodal large language models (MLLMs) have demonstrated strong performance on geometry problems through multi-step reasoning. To better emulate human problem-solving, intermediate steps can incorporate auxiliary visual constructions, such as additional lines or points, which improve geometric interpretation and educational clarity. In this work, we introduce the GeoMathCode, where programmatic representations serve as intermediate visual outputs. We further conduct an in-depth analysis of the underlying reasoning geometry. Experimental results show that reasoning and code generation steps can be disentangled in the latent space, while supervised fine-tuning (SFT) makes the reasoning manifold more structured and informative. Moreover, hierarchical syntactic code structures emerge as disentangled latent subspaces, and contain more mathematical symbolic information than visual representations.

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.

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.

Highly over-parameterized models can simultaneously memorize noisy labels and generalize well, yet how these behaviors coexist remains poorly understood. In this work, we investigate the underlying mechanisms of this coexistence using modular arithmetic tasks under heavy label noise. Through extensive experiments on two-layer neural networks, we find that larger models tend to generalize better under appropriate optimization and model configurations, while noisy labels are memorized faster than clean data. Over-parameterized models internally form a generalization structure, but its expression in the output is suppressed by the need to fit noisy labels. Remarkably, even with 80% label noise, near-perfect test accuracy can be achieved by extracting this internal structure using frequency-based methods. We further propose a task-agnostic method to partition networks into generalization and memorization components. Although this subnetwork improves generalization, it is limited compared with frequency-based extraction, indicating that the generalization structure is distributed across neurons and motivating the development of new tools to retrieve generalizable knowledge from over-parameterized networks.

Advanced driver-assistance systems (ADAS) require neural compute engines that deliver low-latency inference under strict power and area constraints. Posit arithmetic is attractive for such accelerators because it provides high numerical fidelity at low precision, but its variable-length regime encoding increases encode/decode cost and exposes the datapath to large regime-field fault effects. This paper presents EULER-ADAS, a SIMD-enabled logarithmic bounded-Posit neural compute engine for energyefficient and reliability-aware ADAS acceleration. The proposed datapath combines bounded-regime Posit representation, stageadaptive logarithmic mantissa multiplication with bit truncation, and a SIMD-shared quire accumulation path supporting Posit- (8,0), Posit-(16,1), and Posit-(32,2) execution. The unified architecture enables 4xPosit-8, 2xPosit-16, or 1xPosit-32 operation without duplicating precision-specific hardware. FPGA implementation shows that the proposed configurations reduce LUT count by up to 41.4%, delay by up to 76.1%, and power by up to 71.9% relative to exact Posit neural compute engines, while achieving up to 10x lower energy-delay product than radix-4 Booth-based Posit multipliers. In 28-nm CMOS, the bounded variants occupy 0.013-0.016 mm2 , consume 19.8-22.1 mW, and operate at up to 1.84 GHz. Application-level evaluation across image-classification, ADAS, and edge-inference workloads shows that the evaluated Posit-16 and Posit-32 configurations remain within about 1.5 percentage points of FP32 accuracy. A TinyYOLOv3 prototype on Pynq-Z2 achieves 78 ms latency at 0.29 W and 22.6 mJ/frame, demonstrating the suitability of EULERADAS for low-power real-time ADAS inference.

We analyse the computational power of transformer encoders as sequence-to-sequence functions on vectors. We show that average hard attention can be used to simulate arithmetic circuits if they are given as an input to an encoder. The circuit families that can be simulated this way have constant depth while using unbounded addition, binary multiplication and sign gates. The transformers we use have arithmetic circuits instead of feed-forward networks. With typical average attention the functions they compute are also computed by the same class of circuit families. Our results hold for transformers over the reals, rationals and any ring in between the two.

In classification tasks, models may rely on confounding variables to achieve strong in-distribution performance, capturing spurious features that fail under distribution shift. This shortcut behavior leads to substantial degradation in out-of-distribution settings. Task arithmetic offers a potential solution by removing unwanted signals via subtraction of secondary model updates, but it typically requires full fine-tuning, which is computationally expensive. Prompt tuning provides a parameter-efficient alternative by adapting models through a small set of trainable virtual tokens. Task arithmetic on the resulting prompts presents an appealing alternative to operations on entire models, but the extent to which this approach can limit reliance on spurious features remains to be established. In this work, we study whether composing soft prompts through task arithmetic improves robustness to confounding shifts. We propose Hybrid Prompt Arithmetic (HyPA), which combines task prompts with linearized confounder prompts to counteract spurious correlations. Across multiple benchmarks, HyPA consistently improves the robustness-performance trade-off relative to prompt-arithmetic baselines under distribution shift. We further analyze how HyPA affects hidden representations and find evidence consistent with it mitigating confounding either by reducing the influence of confounder signals on predictions or by suppressing them in the representation. These results establish HyPA as a parameter-efficient and promising approach for improving robustness under confounding shifts in the evaluated setting.

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.

Multimodal LLMs can accurately perceive numerical content across modalities yet fail to perform exact multi-digit multiplication when the identical underlying arithmetic problem is presented as numerals, number words, images, or in audio form. Because existing benchmarks often lack systematically paired instances across modalities, it remains difficult to compare genuine arithmetic limits within and across model families. We therefore introduce a controlled multimodal multiplication benchmark that factorially varies digit length, digit sparsity, representation (e.g., numerals vs. number words), and modality (text, rendered images, audio), with paired instances from a reproducible generator. We also define arithmetic load, C, as the product of the total and non-zero digit count as a compact, mechanistically motivated proxy for operation count. Across evaluations, accuracy falls sharply as C grows, often nearing zero by C > 100. Indeed, C remains predictive of performance across modalities and models, with R-squared often > 0.5, nearing the value from more complex measures of arithmetic load that count the number of intermediate arithmetic steps. A separate perception-versus-computation decomposition shows that multimodal degradation is primarily computational rather than perceptual: on matched-perception checks, models are near-perfect (> 99%) across modalities, even when multiplication accuracy drops. Beyond measuring when models fail, we ask which procedures they are predisposed to follow. We introduce a forced-completion loss probe that scores heuristic-specific reasoning prefixes--including columnar multiplication, distributive decomposition, and rounding/compensation. Here, decomposition is favored in both text and vision modalities; heuristic-specific LoRA adapters produce near-orthogonal updates yet degrade accuracy, indicating the base model maintains a well-tuned internal router.

Modern text-to-image (T2I) models amplify harmful societal biases, challenging their ethical deployment. We introduce an inference-time method that reliably mitigates social bias while keeping prompt semantics and visual context (background, layout, and style) intact. This ensures context persistency and provides a controllable parameter to adjust mitigation strength, giving practitioners fine-grained control over fairness-coherence trade-offs. Using Embedding Arithmetic, we analyze how bias is structured in the embedding space and correct it without altering model weights, prompts, or datasets. Experiments on FLUX 1.0-Dev and Stable Diffusion 3.5-Large show that the conditional embedding space forms a complex, entangled manifold rather than a grid of disentangled concepts. To rigorously assess semantic preservation beyond the circularity and bias limitations of of CLIP scores, we propose the Concept Coherence Score (CCS). Evaluated against this robust metric, our lightweight, tuning-free method significantly outperforms existing baselines in improving diversity while maintaining high concept coherence, effectively resolving the critical fairness-coherence trade-off. By characterizing how models represent social concepts, we establish geometric understanding of latent space as a principled path toward more transparent, controllable, and fair image generation.

Large language models (LLMs) have demonstrated impressive capabilities, yet their internal mechanisms for handling reasoning-intensive tasks remain underexplored. To advance the understanding of model-internal processing mechanisms, we present an investigation of how LLMs perform arithmetic operations by examining internal mechanisms during task execution. Using early decoding, we trace how next-token predictions are constructed across layers. Our experiments reveal that while the models recognize arithmetic tasks early, correct result generation occurs only in the final layers. Notably, models proficient in arithmetic exhibit a clear division of labor between attention and MLP modules, where attention propagates input information and MLP modules aggregate it. This division is absent in less proficient models. Furthermore, successful models appear to process more challenging arithmetic tasks functionally, suggesting reasoning capabilities beyond factual recall.

Tool-integrated reasoning (TIR) has become a key approach for improving large reasoning models (LRMs) on complex problems. Prior work has mainly studied when to invoke tools, while overlooking how tools are applied. We identify two common patterns: a calculator pattern that uses code for direct computation, and an algorithmic pattern that encodes problems as programs. Misaligned choices often cause failures even when reasoning is sound. We propose a two-stage framework that first builds code competence from both patterns and then aligns pattern selection with teacher preferences. Across challenging math datasets, our pattern-aware method substantially improves both code usage and accuracy, for instance raising Code@1 on MATH500 from 64.0% to 70.5% and on AIME24 from 26.7% to 50.0%. These gains highlight the effectiveness of a pattern-aware approach for tool-integrated reasoning.

Task arithmetic, representing downstream tasks through linear operations on task vectors, has emerged as a simple yet powerful paradigm for transferring knowledge across diverse settings. However, maintaining a large collection of task vectors introduces scalability challenges in both storage and computation. We propose Task Vector Bases, a framework compressing $T$ task vectors into $M < T$ basis vectors while preserving the functionality of task arithmetic. By representing each task vector as a structured linear combination of basis atoms, our approach supports standard operations such as addition, negation, as well as more advanced arithmetic ones. The framework is orthogonal to other efficiency-oriented improvements in task arithmetic and can be used in combination with them. We provide theoretical analysis showing that basis compression retains addition generalization guarantees and enables principled unlearning, with error bounds depending on reconstruction quality. Empirically, our proposed basis construction methods consistently outperform heuristic basis construction baselines and, in some cases, even surpass the performance of full task vector collections across diverse downstream applications while reducing storage and computational requirements. The code is available at https://github.com/uiuctml/TaskVectorBasis.

We present ABACUS, a unified vision-language model that jointly addresses object counting, crowd counting, referring-expression counting, and count-faithful image generation within a single 3B-parameter model. ABACUS introduces three contributions: density-aware adaptive zooming paired with an objectness map from multi-head self-attention decomposition to spatially ground count predictions; a boundary-aware count policy trained via GRPO with nested local, boundary, and global rewards to eliminate over- and undercounting at crop boundaries; and a cycle-consistent GRPO strategy in which the frozen understanding branch scores generated candidates on count-deviation and aesthetic quality, closing the understanding-generation synergy gap without any external critic or annotation. ABACUS achieves state-of-the-art results across seven benchmarks spanning object counting (FSC-147, CARPK), crowd counting (ShanghaiTech A/B), referring-expression counting (REC-8K), count-faithful generation (CoCoCount, T2I-CompBench, GenEval), and count reasoning (CountQA), surpassing both task-specific specialists and larger generalist models. Project page is at https://mondalanindya.github.io/pages/ABACUS.