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arXiv2026

Order-Disorder in Fe-Si Alloys: Implications for Seismic and Transport Properties of Earth's Inner Core

Understanding the structure and dynamics of Earth's inner core is essential for constraining its composition, thermal evolution, and seismic behavior. Silicon is widely considered a major light element in the core. Here, we use first-principles molecular dynamics and thermodynamic modeling to investigate the structural, elastic, and transport properties of iron-silicon (Fe-Si) alloys under inner-core conditions. By computing the Gibbs free energies of multiple solid phases, including ordered B2, hexagonal close-packed (hcp), face-centered cubic (fcc), and body-centered cubic (bcc) Fe-Si, we construct a phase diagram relevant to inner-core pressures and temperatures. We identify a pronounced miscibility gap between hcp and cubic Fe--Si phases, with phase coexistence over approximately 6--11 wt\% Si at 6000 K. The cubic phase exhibits a strong order-dependent elastic response, with disordered bcc showing a large intrinsic shear anisotropy (22.9\% at 6000 K), in contrast to the nearly isotropic hcp phase (0.6\%). Meanwhile, disordered bcc Fe--Si produces shear wave velocities ($V_S=3.73$ km$\cdot$s$^{-1}$) and Poisson's ratios comparable with seismological constraints. In addition, disordered bcc Fe--Si exhibits a lower thermal conductivity than hcp Fe--Si, providing a quantitative constraint on phase-dependent electronic heat transport. These results demonstrate that chemical ordering strongly controls the stability and physical properties of cubic Fe--Si alloys, and suggest that metastable or localized cubic Fe--Si phases may contribute to the structural and transport complexity of Earth's inner core.

cond-mat.mtrl-sci↗

arXiv2026

Age Optimal Sampling and Routing under Intermittent Links and Energy Constraints

Links in practical systems, such as satellite--terrestrial integrated networks, exhibit distinct delay distributions, intermittent availability, and heterogeneous energy costs. These characteristics pose significant challenges to maintaining timely and energy-efficient status updates. While link availability restricts feasible transmission routes, routing decisions determine the actual delay and energy expenditure. This paper tackles these challenges by jointly optimizing sampling and routing decisions to minimize monotonic, non-linear Age of Information (AoI). The proposed formulation incorporates key system features, including multiple routes with correlated random delays, stochastic link availability, and route-dependent energy consumption. We model the problem as an infinite-horizon Constrained Semi-Markov Decision Process (CSMDP) with a hybrid state--action space and develop an efficient nested algorithm, termed Bisec-\textsc{ReaVI}, to solve this problem. We analyze the structural properties of the solution and reveal a well-defined jointly optimal policy structure: (i) For general monotonic penalty functions, the optimal sampling policy is a piecewise linear waiting policy with at most $N$ breakpoints given $N$ routes; and (ii) under a derived Expected Penalty Ordering condition, the optimal routing policy is a monotonic threshold-based handover policy characterized by at most $\binom{N}{2}$ thresholds. Numerical experiments in a \textit{satellite--terrestrial} integrated routing scenario demonstrate that the proposed scheme efficiently balances energy usage and information freshness, and reveal a counter-intuitive insight: \textit{even routes with higher average delay, higher delay variance or lower availability can still play a critical role in minimizing monotonic functions of AoI}.

cs.IT↗

arXiv2026

Velocity-space turbulent cascade in the near-Sun solar wind: first insights from the Parker Solar Probe mission

In space plasmas, the rarity of collisions leads to complex structures in the velocity space where a turbulent cascade of the velocity distribution function fluctuations is thought to occur. Previous studies have explored this phenomenon using the Hermite decomposition of the ion velocity distribution function (VDF) in both magnetosheath data and numerical simulations. In this work, we investigate the Hermite spectrum of the ion VDFs measured by Parker Solar Probe in the inner heliosphere. We analyze a superalfvénic stream at a radial distances of $R \approx 28 R_{sun}$ and a subalfvénic at $R \approx 11 R_{sun}$, the former characterized by a prevalence of VDFs with suprathermal beams (also known as hammerhead). The Hermite analysis is also compared with various proxies of energization and dissipation, in order to establish a connection between turbulent cascades in real space and those in the velocity space. A qualitative agreement between the energization proxies and the Hermite analysis is observed. The results are suggestive of the presence of a dual cascade in real and velocity space.

astro-ph.SR↗

arXiv2026

How Does RL Post-training Induce Skill Composition? A Case Study on Countdown

While reinforcement learning (RL) successfully enhances reasoning in large language models, its role in fostering compositional generalization (the ability to synthesize novel skills from known components) is often conflated with mere length generalization. To this end, we study what RL post-training teaches about skill composition and how the composition structure affects the skill transfer. We focus on the Countdown task (given n numbers and a target, form an expression that evaluates to the target) and analyze model solutions as expression trees, where each subtree corresponds to a reusable subtask and thus can be viewed as a "skill." Tracking tree shapes and their success rates over training, we find: (i) out-of-distribution (OOD) generalization to larger n and to unseen tree shapes, indicating compositional reuse of subtasks; (ii) the order of skill acquisition depends upon the composition structure---models master shallow balanced trees (workload is balanced between subtasks) before deep unbalanced ones, with persistent fragility on right-heavy structures for a fixed composition depth. Our diagnostic reveals what is learned, in what order, and where generalization fails, clarifying how RL-only post-training induces OOD generalization beyond what standard metrics such as pass@k reveal. To study the implications of our findings for real-world tasks, we present a preliminary study on formal theorem proving.

cs.LG↗

arXiv2026

Hilbert's Basis Theorem for generalized nonassociative Ore extensions

We introduce a broader class of nonassociative Ore extensions that unifies and generalizes several earlier constructions, taking subadditivity of polynomial degree under multiplication as a guiding principle. We prove generalizations of Hilbert's Basis Theorem for this class, showing that left and right Noetherianity follow from corresponding degree-reduction properties. Applications include flipped nonassociative Ore extensions, mutations, cochain twists, and depolarizations of Poisson Ore extensions.

math.RA↗

arXiv2026

Evaluating A/B Testing Methodologies via Sample Splitting: Theory and Practice

We develop a theoretical framework for sample splitting in A/B testing environments, where data for each test are partitioned into two splits to measure methodological performance when the true impacts of tests are unobserved. We show that sample-split estimators are generally biased for full-sample performance but consistently estimate sample-split analogues of it. We derive their asymptotic distributions, construct valid confidence intervals, and characterize the bias-variance trade-offs underlying sample-split design choices. We validate our theoretical results through simulations and provide implementation guidance for A/B testing products seeking to evaluate new estimators and decision rules.

econ.EM↗

arXiv2026

MarkTune: Improving the Quality-Detectability Trade-off in Model-Embedded LLM Watermarking

Language model watermarking schemes fall into two broad categories: inference-time methods, which modify the decoding process at generation time, and model-embedded methods, which encode a secret signal directly into the model weights. Inference-time watermarking methods incur additional inference overhead and are not applicable in the increasingly prevalent open-weight model settings. In contrast, model-embedded watermarks introduce no additional inference latency-generation uses the standard sampling pipeline-and are particularly well-suited to open-weight settings. However, existing model-embedded approaches such as GaussMark face a fundamental quality-detectability trade-off: achieving strong detection power typically requires weight perturbations that noticeably degrade generation quality. Our key insight is that regularizing toward a high-quality reference distribution decouples quality preservation from proximity to the base model, enabling a better quality-detectability trade-off. Building on this insight, we introduce MarkTune, a theoretically grounded on-policy fine-tuning framework that uses the GaussMark detection statistic as a reward while preserving generation quality through cross-entropy on high-quality human text or KL divergence to a stronger oracle model. Empirically, MarkTune substantially improves the quality-detectability frontier of GaussMark, approaching the detectability of strong inference-time schemes while preserving generation quality and downstream task performance. Marktune also remains robust under the evaluated text-editing and open-weight attacks and generalizes across datasets, retaining substantial detection power on corpora unseen during training. Code is available at https://github.com/zhyzmath/MarkTune-LLM-Watermarking.

cs.LG↗

arXiv2026

OFF Newton-type Methods for Unconstrained Optimization with Inexact Derivatives

We study objective-function-free (OFF) Newton-type methods for smooth unconstrained optimization with inexact gradients and Hessians. We propose two methods for first- and second-order stationarity, coupling explicit step-size rules with derivative-accuracy and linear-solver residual conditions to guarantee sufficient decrease without evaluating the objective function. The OFF regularized Newton method finds an $\varepsilon$-first-order stationary point in $\mathcal{O}(\varepsilon^{-2})$ iterations. Under strong convexity, a local Newton variant converges with $Q$-order at least $1+θ$, where $θ\in (0, 1]$. For second-order stationarity, we combine regularized Newton steps with approximate eigenvectors associated with the minimum eigenvalue. With high probability, the resulting method finds an $(\varepsilon, \sqrt{L_h\varepsilon})$-second-order stationary point in $\mathcal{O}(\varepsilon^{-3/2})$ iterations, using a total of $\widetilde{\mathcal{O}}(\varepsilon^{-7/4})$ approximate gradient evaluations and Hessian-vector products. For finite-sum problems, we develop subsampling schemes that yield best-iterate convergence in expectation in the nonconvex first-order setting and local $R$-linear convergence with high probability in the strongly convex setting, followed by $Q$-order at least $1+θ$ once full sampling is reached. The subsampled second-order method achieves a high-probability $\widetilde{\mathcal{O}}(\varepsilon^{-7/2})$ total sample-operation bound, counting component gradient evaluations and component Hessian-vector products.

math.OC↗

arXiv2026

Softly Symbolifying Kolmogorov-Arnold Networks

Kolmogorov--Arnold Networks (KANs) offer a promising path toward interpretable machine learning: their learnable activations can be studied individually while collectively fitting complex data accurately. In practice, however, trained activations often lack *symbolic fidelity*, learning pathological decompositions with no meaningful correspondence to interpretable forms. We propose Softly Symbolified Kolmogorov--Arnold Networks (S2KAN), which integrate symbolic primitives directly into training. Each activation draws from a dictionary of symbolic and dense terms, with learnable gates that sparsify the representation. Crucially, this sparsification is differentiable, enabling end-to-end optimization, and uses an optimization objective inspired by minimum description length. S2KAN can discover interpretable activation functions while retaining dense splines for flexible approximation. We introduce a quantitative measure of symbolic fidelity, assessing how closely learned activations can be approximated by short expressions from a specified symbolic language. Across real-world prediction tasks, dynamical-system forecasting, and synthetic symbolic benchmarks, S2KAN generally achieves higher symbolic fidelity and often far smaller models, while maintaining competitive predictive accuracy on real-world data and low prediction errors on most symbolic benchmark configurations. We also examine the models' learned responses on scientific data, including the contributions of their symbolic components. Together, these results show that incorporating symbolic structure during training can yield compact, accurate KAN models whose learned responses can be examined through their symbolic components.

cs.LG↗

arXiv2026

LLM-Auction: Generative Auction towards LLM-Native Advertising

The commercialization of LLM applications is the next frontier in online advertising, with LLM-native advertising emerging as a promising paradigm by integrating ads into LLM-generated content. However, classic mechanisms are no longer applicable in this setting where the auction object is shifted from discrete ad slots to distributions over LLM outputs, and existing methods are impractical in industrial scenarios due to ignored externalities or high inference costs. To address these issues, we propose LLM-Auction, the first learning-based generative auction mechanism that integrates auction and generation. By formulating the allocation as preference alignment between LLM outputs and a mechanism objective that balances advertisers' value and user experience, we optimize the LLMs to inherently model allocation externalities without extra inference cost. Theoretically, we identify the allocation monotonicity and continuity of the optimal allocation rule of LLM-Auction, and prove that a simple first-price payment rule exhibits favorable incentive properties under this allocation rule. Furthermore, we build an LLM-as-a-judge simulation environment for quantitative evaluation, and experiments demonstrate that LLM-Auction achieves the state-of-the-art allocation efficiency while approximately satisfying key mechanism properties.

cs.GT↗

arXiv2026

Verification of Unknown Dynamical Systems via Autoencoder Latent Space

Formal verification provides a powerful framework for proving that dynamical systems satisfy their specifications. However, these techniques face scalability challenges in high-dimensional settings, as they often rely on state-space discretization which grows exponentially with dimension. Learning-based approaches to dimensionality reduction have shown great potential to alleviate this problem. However, ensuring correctness of latent space verification results remains an open question. In this work, we provide a formal approach to reduce the dimensionality of systems via convex autoencoders and learn the dynamics in the latent space through a kernel-based method. We then construct a finite abstraction from the learned model in the latent space and guarantee that the abstraction contains the true behaviors of the original system. We show that the verification results in the latent space can be mapped back to the original system. Finally, we demonstrate the approach on multiple systems, including a 26D system controlled by a neural network, showing significant scalability improvements.

cs.LG↗

arXiv2026

CountZES: Counting via Zero-Shot Exemplar Discovery and Selection

Zero-shot object counting (ZOC) aims to count instances of unseen categories using only a class name. Existing exemplar-based methods discover candidate exemplars through random image patches or open-vocabulary detector proposals. However, random patch sampling often yields poorly delineated instances, while detector-based discovery may produce multi-instance regions and, in dense scenes, provide only a few or even no suitable single-instance candidates. Thus, exemplar selection is constrained by candidate quality and coverage, while counting itself remains highly sensitive to exemplar choice. To this end, we propose CountZES, an inference-only framework that jointly addresses candidate discovery and exemplar selection through density-guided candidate generation and complementary exemplar-selection mechanisms. Starting from a text-grounded anchor, CountZES progressively discovers and refines visual exemplars through three stages: Detection-Anchored Exemplar (DAE), Density-Guided Exemplar (DGE), and Feature-Consensus Exemplar (FCE). DAE establishes a reliable single-instance anchor, DGE leverages density responses to expand candidate discovery beyond the detector-constrained pool and select a statistically consistent exemplar, and FCE promotes representativeness through feature-level consensus. Together, they yield complementary exemplars balancing semantic grounding, count consistency, and feature representativeness. CountZES requires no ZOC-specific optimization and adopts a modular design that supports different pretrained components. Extensive evaluations across natural, aerial, and medical domains demonstrate strong cross-domain generalization and superior performance among inference-only ZOC methods.

cs.CV↗

arXiv2026

High-dimensional normal approximations for sums of Langevin Markov chains

Consider the well-known Langevin diffusion on $\mathbb{R}^d$ $$\mathrm{d} X_t = -\nabla U(X_t)\,\mathrm{d} t + \sqrt{2}\mathrm{d} B_t, $$ and its Euler-Maruyama discretization given by $$X_{k+1}=X_k-η\nabla U(X_k)+\sqrt{2η}ξ_{k+1},$$ where $η$ is the step size. Under mild conditions, the Langevin diffusion admits $π(\mathrm{d} x)\propto \exp(-U(x))\mathrm{d} x$ as its unique stationary distribution. In this paper, we mainly study the normal approximation of the normalized partial sum $$ W_n = η^{1/2} n^{-1/2} \left( \sum_{i=0}^{n-1} X_i- \int_{\mathbb{R}^d} x\,π(\mathrm{d} x) \right).$$ To the best of our knowledge, this work provides the first dimension-explicit convergence rates in high-dimensional settings. Our main tool is a novel upper bound for the 1-Wasserstein distance $W_1(W,γ)$ via the exchange pair approach, where $W$ is any random vector of interest and $γ$ is a $d$-dimensional standard normal random vector.

math.PR↗

arXiv2026

Hall's Exact Uncertainty Relation and the Geometry of Krylov-Space Operator Dynamics

Krylov representations of operator dynamics encode two inequivalent motions: probability transport along the discrete Krylov chain and motion of the normalized amplitude vector on the associated sphere. We lift Hall's exact uncertainty relation to Liouville space and show that, for the fixed Krylov-basis measurement, its classical Fisher information locally attains the quantum Fisher information. The amplitude vector therefore follows an exact constant-speed spherical trajectory. We construct its moving frame, identifying the first Lanczos coefficient with the speed and the higher coefficients with successive bending. Geodesic curvature controls the leading departure from Mandelstam-Tamm saturation. Separately, we compare cumulative and pointwise bounds on Krylov-index transport, which depends on the full Lanczos sequence.

quant-ph↗

arXiv2026

Operator-in-the-Loop Rolling Dispatch for Damage-Assessment Crews Under Partial Storm Information

Natural disasters often inflict severe damage on distribution grids. Rapid, reliable damage assessment (DA) is essential for storm restoration, yet most optimization work targets repair dispatch after faults are identified. This paper presents a production, rolling horizon DA crew allocation system deployed across multiple U.S. states in Eversource Energy's service territory and used during live storms. The method implements a sequential k-job assignment policy per available crew, executed on a fixed cadence and on operators' control. The objective jointly prioritizes critical facilities and customer impact while controlling travel time on the actual road network via the Google Maps API. A key constraint is the absence of live crew GPS; we infer crew locations from the last confirmed DA site and robustify travel estimates for staleness, yielding stable recommendations without continuous tracking. The operator remains in the loop with controls to limit churn and to publish a feasible plan. Using data from the March 7 New Hampshire storm with 90 moderate outages and seven DA crews, we observe shorter time to first assessment, fewer revisits with reduced distance traveled. To our knowledge, this is among the first multi-state enterprise integrated deployments to treat DA crews as a first-class optimized resource in storm restoration.

eess.SY↗

arXiv2026

AI-Guided Computational Design of a Room-Temperature, Ambient- Pressure Superconductor Candidate: Grokene

We introduce Grokene, a novel two-dimensional superlattice derived from graphene, which was identified through an AI-guided materials discovery workflow utilizing a large language model. Grokene is predicted to exhibit ambient-pressure, room-temperature superconductivity, with computational simulations revealing a high electron-phonon coupling constant and a substantial logarithmic-averaged phonon frequency (~1650 K), leading to a mean-field critical temperature of approximately 325 K. Full isotropic Eliashberg solutions further support a critical temperature around 310 K, underscoring its strong potential for room-temperature superconductivity. However, the strict two-dimensional nature of Grokene introduces phase fluctuations, limiting the observable superconducting transition to a Berezinskii-Kosterlitz-Thouless (BKT) temperature of about 120 K in monolayers. To elevate TBKT toward room temperature, strategies such as few-layer stacking, substrate or gate engineering, and optimization of superlattice structure and doping levels are proposed. Our integrated workflow, combining AI-driven materials discovery with advanced many-body theories (DFPT/EPW, Eliashberg, and RPA), provides a systematic and reproducible framework for exploring novel superconductors. We suggest that experimental synthesis and comprehensive characterization of Grokene will be essential to assess these computational predictions and to explore routes toward practical superconductivity under ambient pressure.

cond-mat.supr-con↗

arXiv2026

AlignDrive: Aligned Lateral-Longitudinal Planning for End-to-End Autonomous Driving

Practical autonomous driving requires models that generalize by reasoning through spatial-temporal possibilities to exclude unsafe outcomes. While state-of-the-art (SOTA) methods use parallel planning architectures, they fail to explicitly couple speed decisions with agent behavior along the driving path, leading to suboptimal coordination. To address this, we propose a cascaded framework that transforms longitudinal planning from an independent prediction task into a path-conditioned reasoning process. On the model side, we introduce an anchor-based regression design that conditions longitudinal prediction on the lateral drive path, and reformulate longitudinal planning as 1D displacement prediction along the path. This reduces geometric uncertainty and sharpens the model's focus on interaction-driven dynamics. On the data side, we introduce a planning-oriented data augmentation strategy that simulates rare safety-critical events by programmatically inserting agents and relabeling longitudinal targets to enforce collision avoidance. Evaluated on the challenging Bench2Drive benchmark, our method achieves SOTA performance with a driving score of 89.07 and a success rate of 73.18%, demonstrating significantly improved coordination and safety. Further evaluation on Fail2Drive confirms strong generalization to rare edge cases where parallel formulations typically fail. Project page:https://yanhaowu.github.io/AlignDrive/.

cs.RO↗

arXiv2026

A modern perspective on Tutte's homotopy theorem

We begin with a review of Tutte's homotopy theory, which concerns the structure of a certain graph associated to a matroid (together with some extra data). Concretely, Tutte's path theorem asserts that this graph is connected, and his homotopy theorem asserts that every cycle in the graph is a composition of ``elementary cycles'', which come in four different flavors. We present an extended version of the homotopy theorem, in which we give a more refined classification of the different types of elementary cycles. We explain in detail how the path theorem allows one to prove that the foundation of a matroid (in the sense of Baker--Lorscheid) is generated by universal cross-ratios, and how the extended homotopy theorem allows one to classify all algebraic relations between universal cross-ratios. The resulting ``fundamental presentation'' of the foundation was previously established in [Baker--Lorscheid], but the argument here is more self-contained. We then recall a few applications of the fundamental presentation to the representation theory of matroids. Finally, in the most novel but also the most speculative part of the paper, we discuss what a ``higher Tutte homotopy theorem'' might look like, and we present some preliminary computations along these lines. The appendices include modern language translations of Tutte's proofs of the path theorem and the homotopy theorem (the latter is a contribution by Juš Kocutar).

math.CO↗