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arXiv2026

Circular Chromatic Numbers, Signability, Relation Algebras, and Network Satisfaction Problems

In this paper, we characterize finite graphs with circular chromatic number less than 3 in terms of the existence of certain signings ($\mathbb Z_2$-labellings studied in the context of signed graphs). In fact, we construct a signed graph which is universal for all such signings -- called anti-triangle-signings in this paper -- of finite $\overline{K_3}$-free graphs, and is closely related to the generic circular triangle-free graph studied by Bodirsky and Guzmán-Pro. Moreover, our universal structure gives rise to a representation of the relation algebra $56_{65}$. We then use this representation to show that the network satisfaction problem described by this relation algebra belongs to NP. This concludes the full classification of the existence of a universal square representation, as well as the complexity of the corresponding network satisfaction problem, for relation algebras with at most four atoms.

math.CO↗

arXiv2026

An algebraic proof of Colombo's difference-power determinant conjecture

Let $n\ge2$ be even, let $λ=(λ_1,\ldots,λ_n)\in\mathbb{R}^n$ have pairwise distinct coordinates, and define the difference-power matrix \[ A_d(λ) := \bigl[(λ_r-λ_s)^d\bigr]_{r,s=1}^n, \qquad d\in\mathbb{N}. \] In 1928, Colombo proved that $\det A_{n-1}(λ)\ne0$---and hence $\det A_{n-1}(λ)>0$---and that $\operatorname{rank} A_d(λ)=d+1$ for $0\le d<n-1$. He conjectured that \[ \det A_d(λ)\ne0 \qquad\text{for every } d\ge n-1. \] For even $d$, the conjectured nonsingularity follows from previously published results on distance-power matrices. The remaining open cases were therefore the supercritical odd exponents $d\ge n+1$. We prove nonsingularity for all these odd exponents, thereby completing Colombo's conjecture. Consequently, \[ \operatorname{rank} A_d(λ)=\min\{n,d+1\} \qquad(d\in\mathbb{N}). \] Our proof converts a hypothetical kernel vector into a real binary form having more projective real linear factors, counted with multiplicity, than its real Waring length permits.

cs.LG↗

arXiv2026

An Algebraic Framework for Data Systems: Classification and Optimization of Algebraic Structures for Data Organization and Coding

Modern data systems are commonly studied through computational and information-theoretic methods, while their algebraic properties remain largely unexplored. This paper introduces a mathematical framework for modelling data systems using algebraic structures drawn from group theory and coding theory. The central result is an axiomatic classifier: the coding-theoretic capability of a finite algebraic structure (group, ring, or field) is shown to be determined by its axiom signature (the set of algebraic axioms it satisfies), with each axiom acting as a gate that enables a specific capability (inverses enable the algebraic Hamming metric; commutativity enables syndrome decoding via quotient groups; field structure enables MDS codes via polynomial evaluation). Three types of results are presented. Proved: the axiomatic classification theorem; that ACID transactions form a monoid under sequential composition (not a group); that schema-preserving transformations form a finite group whose orbits, counted by Burnside's lemma, yield exact deduplication of equivalent configurations; and that every integrity-preserving bijection of a data system defines an algebraizable equivalence class under a finite group action. Demonstrated: explicit code constructions over $\mathbb{F}_5$, $\mathbb{Z}_4$ (yielding codes inaccessible to classical $\mathbb{F}_q$-linear theory), and $(2^U, \triangle)$ (yielding group-theoretic anomaly detection for set-valued data). Proposed: an encoding efficiency measure Ef and an optimization functional $Φ$ with tunable weights, whose induced ranking is verified computationally to be consistent with the axiomatic classification for the structures studied.

math.RA↗

arXiv2026

Representation Redundancy and Structural Complexity in Finite-Field Inversion

The representation chosen for a mathematical operation can affect both its algebraic form and its empirical learning difficulty. We study this phenomenon for inversion over $\mathbb F_{2^n}$, with field elements expressed in varying ordered $\mathbb F_2$-bases. We prove that two ordered bases induce the same coordinate inversion map if and only if they belong to the same Galois orbit. Since every orbit has size $n$, the correspondence between ordered bases and distinct inversion maps is exactly $n$-to-one. We then analyze three Boolean formulations of inversion. The reference formulation has algebraic degree $n-1$ and joint ANF leap $1$, the mixed representation formulation has degree $2(n-1)$ and joint ANF leap $2$, and the complete raw formulation has degree at most $3(n-1)$ and joint ANF leap at least $n$. Exhaustive computations agree with the theoretical results and bounds in the cases considered. Controlled experiments with multilayer perceptrons show the same ordering in learning difficulty, while Galois orbit redundancy provides only a limited generalization benefit under the tested conditions. These results show that exact redundancy among representations can coexist with changes in Boolean structure and learning behavior when the representation is exposed as part of the input.

cs.LG↗

arXiv2026

When Teacher Guidance Misleads: Reward-Aligned On-Policy Distillation

On-policy distillation (OPD) has recently emerged as a popular post-training paradigm for large language models (LLMs), providing an efficient way to transfer the knowledge and capabilities of teacher models into student models. However, teacher guidance on student-generated prefixes is not always reliable. Training should optimize the model to generate responses that are more likely to be correct, or equivalently, to get higher outcome rewards. But during OPD, the teacher model may provide guidance that discourages the student from moving toward correct trajectories or moves the student toward incorrect ones, which is misaligned with outcome reward. Such misaligned guidance is unreliable, as it would mislead the optimization process and ultimately degrade model performance. To mitigate misaligned teacher guidance, we propose Reward-Aligned On-Policy Distillation (RA-OPD). The key insight is to keep only trajectories whose induced updates move the student toward correct trajectories or discourage the student from moving toward incorrect ones. Specifically, for each sampled trajectory, RA-OPD checks whether its trajectory-level distillation return is consistent with its outcome reward and then filters out the misaligned trajectories. RA-OPD selects more reliable trajectories to improve student model performance without requiring additional computational cost. We evaluate RA-OPD on math and code benchmarks using models from the Qwen3 family and the DeepSeek-R1 family. Across seven math benchmarks and three code benchmarks, RA-OPD significantly outperforms standard OPD and other tested OPD variants.

cs.AI↗

arXiv2026

Deep Reinforcement Learning for Reach-Avoid-Stay Problems

Reach-Avoid-Stay (RAS) tasks are essential in applications where systems must safely reach a target set and remain within it under all bounded disturbances. Existing approaches either struggle to compute the maximal robust RAS set, the set of all states from which the RAS task is achievable, or are limited in handling general dynamic systems. To address these challenges, this paper proposes a two-step deep reinforcement learning framework that jointly learns the maximal robust RAS set and the corresponding control policy. The first step identifies the maximal robust control-invariant set within the target set and derives a policy that ensures the system remains within it. The second step computes the maximal robust reach-avoid (RA) set using this invariant set as the target, and it is proven that this RA set is equivalent to the maximal robust RAS set. Leveraging this result, a switching policy is constructed from the two step-wise policies, which constitutes a valid policy guaranteeing completion of the RAS task. Simulation results demonstrate that the proposed framework (1) computes the exact maximal robust RAS set in the absence of training errors, yielding the least restrictive RAS policy, and (2) identifies the RAS set with high accuracy while outperforming baseline methods on RAS tasks.

eess.SY↗

arXiv2026

A.X K2 Technical Report

We introduce A.X K2, a 688B-parameter Mixture-of-Experts (MoE) language model trained from scratch as a high-performance foundation for \emph{agentic} applications. Trained on approximately 8.5T tokens---fewer than its predecessor, A.X K1---on a smaller but higher-quality mixture with substantially expanded agentic and software-engineering data, it nonetheless improves over A.X K1 across the board, by over 30 percentage points on some benchmarks, reflecting large gains in token efficiency. To support long contexts efficiently, we introduce Sparse Gated Attention (SGA), which combines sparse attention with gated attention, and adopt Gated Norm (GN) to stabilize large-scale training. SGA is trained natively at 128K through a \emph{sparse} indexer warmup that optimizes the indexer against its own sparse top-$k$ selection rather than the dense attention distribution, making adaptation markedly cheaper: each query reads only 2,048 positions, yet long-context quality is unchanged and A.X K2 scores 94.6 on RULER out to 256K. The outlier suppression of GN in turn keeps 4-bit NVFP4 serving within one point of FP8 accuracy. A simple yet effective Think-Fusion recipe further lets users switch between thinking and non-thinking modes within a single unified model. Extensive evaluations show that A.X K2 performs competitively against strong open-weight baselines, matching or exceeding them on math and Korean-language benchmarks.

cs.AI↗

arXiv2026

Dataset Protection via Watermarked Canaries in Retrieval-Augmented LLMs

Retrieval-Augmented Generation (RAG) has become an effective method for enhancing large language models (LLMs) with up-to-date knowledge. However, it may pose a risk of copyright infringement, as IP datasets may be incorporated into the knowledge database by malicious Retrieval-Augmented LLMs (RA-LLMs) without authorization. To protect the rights of the dataset owner, an effective dataset membership inference algorithm for RA-LLMs is needed. In this work, we introduce a novel approach, \textit{CanaryTrace}, to safeguard the ownership of text datasets and effectively detect unauthorized use by RA-LLMs. Our approach preserves the original data completely unchanged while protecting it by inserting specifically designed canary documents into the IP dataset. These canary documents are created with synthetic content and embedded watermarks to ensure uniqueness, consistency, and statistical provability. During the detection process, unauthorized usage is identified by querying the canary documents and analyzing the responses of RA-LLMs for statistical evidence of the embedded watermark. Our experimental results demonstrate high query efficiency, detectability, and consistency, along with minimal perturbation to the original dataset, all without compromising the performance of the RAG system.

cs.CR↗

arXiv2026

Resolution-Aware Experimental Design under Partial Identifiability

Experimental design is commonly framed as choosing the experiment expected to provide the most information. Under partial identifiability however, persistent nuisance uncertainty can make the same observation carry different structural meanings. We introduce Resolution-Aware Experimental Design (RAED), which selects an experiment by the smallest expected nonempty structural candidate set achievable subject to false-exclusion control. We prove an exact cross-nuisance aliasing separation: an experiment can be preferred by structural and full-latent information gain, average classification, and nuisance-marginalized informativeness while having arbitrarily poorer valid structural resolution. RAED nevertheless preserves the expected ordering under a genuine composite Blackwell comparison. To make this criterion operational, we develop a learned score-based implementation with finite-sample nuisance-average and positive-tail calibration, and characterize a rare-tail sample-complexity obstruction. Under constrained sensing, two subsurface-flow benchmarks exhibit genuine RAED--expected-information-gain (EIG) experiment-selection disagreements, with the clearest and largest held-out resolution differences in WCA. In a fluvial benchmark, tail protection changes the selected physical experiment and replaces hard-region false exclusions primarily with explicit ambiguity. In a mechanistic methane-oxidation benchmark, a prospectively specified 5\% false-exclusion tolerance also yields a nontrivial finite-sample population guarantee for tail-sensitive nuisance risk, with 95\% joint confidence across all three structural families.

cs.LG↗

arXiv2026

SkyShare: Constellation-wide Sky Sharing for LEO-Radio Astronomy Coexistence

Rapidly growing low-Earth-orbit (LEO) constellations increasingly operate in the spectrum shared with radio astronomy services (RAS), creating escalating interference risks for sensitive scientific observations. Existing mitigation mechanisms rely on reactive beam steering, or avoidance near observatories but fail to account for aggregate sidelobe emissions-leading to residual interference and substantial, unnecessary capacity loss. We present SkyShare, a constellation-wide sky-sharing system that enables predictive, interference-aware spot beam scheduling to protect radio astronomy while preserving network coverage. SkyShare integrates high-fidelity orbital prediction with International Telecommunication Union (ITU)-compliant Equivalent Power Flux Density (EPFD) modeling, and real-time observatory data via Operational Data Sharing (ODS) to jointly optimize beam-cell assignments over observation windows. To make constellation-scale coordination tractable, we introduce a concept of EPFD-budgeted Region-of-Interest(RoI) that bounds residual sidelobe interference while confining optimization to a minimal, provably sufficient set of cells. Building on RoI, we formulate LEO-RAS coexistence as a scalable scheduling problem and design SkySched, a flow-based algorithm that is optimal in special cases and yields scalable near-optimal solutions in the general NP-hard setting. SkyShare operates entirely in the control plane and requires no satellite hardware changes. Using real Starlink constellation geometries, we evaluate SkyShare across 25 single-dish, Ku-band RAS sites worldwide. Compared to Starlink boresight avoidance, SkyShare reduces unserved cells by up to 90.68% while remaining within EPFD limits.

cs.NI↗

arXiv2026

GPU-Accelerated Astrodynamics World Models for Spacecraft Rendezvous and Proximity Operations

World models are an emerging paradigm in representation learning in which an agent jointly learns state-action dynamics and observation models from offline trajectory data, enabling multi-step planning and trajectory prediction with uncertainty estimates. They have shown strong results in robotics and game environments, but, to the best of our knowledge, have not previously been applied to the space domain. This paper introduces a world model-based approach to cooperative and non-cooperative spacecraft rendezvous and proximity operations. First, we introduce an open-source, JAX-based International Space Station (ISS) docking environment supporting parallel GPU simulation of spacecraft orbit and attitude dynamics, generating the thousands of state-action transitions that world model training requires. Second, we introduce Out-of-this-World-Model, a transformer-based world model that encodes relative kinematic states and body-fixed camera imagery into a latent state and predicts its evolution under commanded thrusts and torques using one-step flow matching. It produces a distribution over future observations, capturing stochastic dynamics and per-timestep uncertainty, and outperforms DreamerV3-style posterior-correction baselines with fewer trainable parameters and hyperparameters. Third, we apply the approach to a capsule autonomously docking with the ISS under keep-out-zone constraints, demonstrating improved sample efficiency and task performance over reinforcement learning baselines (53% versus 29% docking success across ports), better out-of-distribution generalization (on held-out ports the world model more than doubles baseline success, 40% versus 17%), and detection of anomalous objects encountered during approach with 98% classification accuracy. We open-source the simulation environment and model architecture to enable further study of this paradigm.

cs.RO↗

arXiv2025

Provably Efficient Reward Transfer in Reinforcement Learning with Discrete Markov Decision Processes

In this paper, we propose a new solution to reward adaptation (RA) in reinforcement learning, where the agent adapts to a target reward function based on one or more existing source behaviors learned a priori under the same domain dynamics but different reward functions. While learning the target behavior from scratch is possible, it is often inefficient given the available source behaviors. Our work introduces a new approach to RA through the manipulation of Q-functions. Assuming the target reward function is a known function of the source reward functions, we compute bounds on the Q-function and present an iterative process (akin to value iteration) to tighten these bounds. Such bounds enable action pruning in the target domain before learning even starts. We refer to this method as "Q-Manipulation" (Q-M). The iteration process assumes access to a lite-model, which is easy to provide or learn. We formally prove that Q-M, under discrete domains, does not affect the optimality of the returned policy and show that it is provably efficient in terms of sample complexity in a probabilistic sense. Q-M is evaluated in a variety of synthetic and simulation domains to demonstrate its effectiveness, generalizability, and practicality.

cs.LG↗

arXiv2026

Step Back to Move Forward: Reflection-Aware Preference Optimization for Visual Generation

Diffusion models have become the mainstream paradigm for modern visual generation and have substantially advanced multimedia content synthesis, especially in text-to-image and text-to-video tasks. To further align such generative models with human preferences, reinforcement learning (RL) has recently shown strong potential as a post-training strategy. Nevertheless, existing policy gradient-based methods often explore inefficiently, making them vulnerable to local optima that may degrade semantic faithfulness and visual realism. To address these challenges, we present Reflection-Aware GRPO (RA-GRPO), a new RL-based preference alignment framework for diffusion generative models. The core idea is to improve "forward" generation by incorporating "backward" reflection during optimization. We first introduce Diffusion Reflection, which rectifies intermediate sampling trajectories by inverting the diffusion process with a weak estimator, guiding latent states toward higher-probability regions of the true data manifold. Furthermore, we introduce Counterfactual Path Synthesis to implicitly distill these rectified trajectories into the policy, enabling the model to internalize the benefits of search-based exploration without incurring inference-time overhead. Extensive experiments on T2I and T2V models demonstrate that RA-GRPO significantly outperforms existing methods, particularly in mitigating reward hacking and improving generalization. The method remains architecture-agnostic and integrates seamlessly with standard pipelines, suggesting a promising direction for stable preference alignment.

cs.CV↗

arXiv2026

Are Economists Open to AI? A Text-as-Data-as-Survey Approach via Language Models

Traditional surveys yield comparable measures but are costly to field, difficult to reconstruct retrospectively, and often ill-suited to fast-moving or sensitive topics. While large-scale internet text is often noisy and weakly structured. To bridge this gap, we introduce Text-as-Data-as-Survey (TaDaS). TaDaS employs Reference-Anchored Semantic Reparameterization (RAS) to project unstructured main text into survey-like evidence, leveraging structured auxiliary text as semantic anchors. Applying TaDaS to 1.25 million Economics Job Market Rumors posts linked with 53,585 top economics and finance publications, we track economists' evolving research sentiment toward AI. Cross-sectionally, AI-related research discussions are less open, with openness and curiosity declining rapidly at first years. Over time, however, economists have become increasingly open and curious, with a notable shift around 2018. Ultimately, TaDaS provides a scalable, non-reactive method to extract longitudinal insights from digital archives, unlocking diverse applications across industry and academia.

cs.CE↗

arXiv2026

A Tri-Agent Framework for Evaluating and Aligning Question Clarification Capabilities of Large Language Models

Large Language Models (LLMs) are increasingly deployed in interactive systems where understanding user intent precisely is paramount. A key capability for such systems is effective question clarification, especially when user queries are ambiguous or underspecified. This paper introduces a novel tri-agent framework for the robust evaluation of an LLM's ability to engage in clarifying dialogue. Our framework comprises three distinct LLM-based agents: (1) a Question Clarifying Agent (QCA), the system under evaluation, tasked with identifying ambiguities and posing clarifying questions; (2) a Respondent Agent (RA), designed to simulate human user responses, potentially including irrelevant or challenging replies; and (3) an Evaluator Agent (EA), an LLM-as-a-judge, which assesses the quality of the dialogue based on a comprehensive set of metrics. We detail a methodology for synthetic data generation in the supply chain domain as an example. We propose metrics evaluating ambiguity handling, question quality, dialogue efficiency, language appropriateness, and final intent alignment. We also briefly discuss the validation of the EA against human judgments. This work provides a structured approach to benchmark, validate, and improve the clarification capabilities of conversational LLM applications.

cs.CL↗

arXiv2026

ScoreMix: Synthetic Data Generation by Score Composition in Diffusion Models Improves Recognition

Synthetic data generation is increasingly used in machine learning for training and data augmentation. Yet, current strategies often rely on external foundation models or datasets, whose usage is restricted in many scenarios due to policy or legal constraints. We propose ScoreMix, a self-contained synthetic generation method to produce hard synthetic samples for recognition tasks by leveraging the score compositionality of diffusion models. The approach mixes class-conditioned scores along reverse diffusion trajectories, yielding domain-specific data augmentation without external resources. We systematically study class-selection strategies and find that mixing classes distant in the discriminator's embedding space yields larger gains, providing up to 3% additional average improvement, compared to selection based on proximity. Interestingly, we observe that condition and embedding spaces are largely uncorrelated under standard alignment metrics, and the generator's condition space has a negligible effect on downstream performance. Across 8 public face recognition benchmarks, ScoreMix improves accuracy by up to 7 percentage points, without hyperparameter search, highlighting both robustness and practicality. Our method provides a simple yet effective way to maximize discriminator performance using only the available dataset, without reliance on third-party resources. Paper website: https://parsa-ra.github.io/scoremix/.

cs.CV↗

arXiv2026

Star-Fusion: A Multi-modal Transformer Architecture for Discrete Celestial Orientation via Spherical Topology

Reliable celestial attitude determination is a critical requirement for autonomous spacecraft navigation, yet traditional "Lost-in-Space" (LIS) algorithms often suffer from high computational overhead and sensitivity to sensor-induced noise. While deep learning has emerged as a promising alternative, standard regression models are often confounded by the non-Euclidean topology of the celestial sphere and by the periodic boundary conditions of Right Ascension (RA) and Declination (Dec). In this paper, we present Star-Fusion, a multi-modal architecture that reformulates orientation estimation as a discrete topological classification task. Our approach leverages spherical K-Means clustering to partition the celestial sphere into K topologically consistent regions, effectively mitigating coordinate wrapping artifacts. The proposed architecture employs a tripartite fusion strategy: a SwinV2-Tiny transformer backbone for photometric feature extraction, a convolutional heatmap branch for spatial grounding, and a coordinate-based MLP for geometric anchoring. Experimental evaluations on a synthetic Hipparcos-derived dataset demonstrate that Star-Fusion achieves a Top-1 accuracy of 93.4% and a Top-3 accuracy of 97.8%. Furthermore, the model exhibits high computational efficiency, maintaining an inference latency of 18.4 ms on resource-constrained COTS hardware, making it a viable candidate for real-time onboard deployment in next-generation satellite constellations.

cs.CV↗

arXiv2026

CMRVision: A Foundation Model for Cardiac MR Image Analysis

Cardiac magnetic resonance (CMR) imaging provides complementary information on cardiac anatomy, function, and tissue characterization across multiple sequences and views. In this work, we investigate foundation model pretraining for 2D CMR and introduce CMRVision, a CMR-specific foundation model trained using DINOv3-style self-supervised learning on a multi-center, multi-sequence cohort of 36 million CMR images. We systematically evaluate architectural and training design choices for domain-specific pretraining. CMRVision is evaluated on two downstream tasks: multi-task segmentation across cine, late gadolinium enhancement (LGE), and mapping sequences, and cine view classification. Our experiments show that CMR-specific pretraining, smaller patch sizes, and patch-level objectives consistently improve downstream performance. Across a multi-task segmentation benchmark, CMRVision achieved the strongest overall performance, outperforming prior natural-image (NI), medical-image, supervised, and CMR foundation model baselines. Improvements were modest but consistent across structures and sequences, with Dice scores ranging from 0.940-0.967 for LV and 0.855-0.905 for myocardium, and reaching 0.929 for RV, 0.920 for LA, and 0.931 for RA. The largest gains were observed for myocardium segmentation in LGE and mapping images. In a zero-shot segmentation task on unseen LGE long-axis views, the model achieved an average Dice score of 0.692, demonstrating cross-view generalization. For cine view classification, CMRVision achieved the highest average accuracy (0.906), compared to prior methods reported in the literature. These results highlight the potential of CMRVision to support robust and generalizable cardiac MRI analysis across multiple sequences and views.

cs.CV↗