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arXiv2026

Topological preservation under aggregation of quantale-valued metric spaces

Aggregation functions provide a general mechanism for combining multiple inputs into a single output and constitute a fundamental tool in information fusion. Beyond the classical numerical setting, aggregation can also be considered at the level of mathematical structures, where the main question is to determine which properties of the original structures are preserved by the aggregation process. In this paper, we study the preservation of topology within a unified framework based on continuous quantales and quantale-enriched categories. We introduce strongly preserving functions, for which the natural topology induced by a $\mathscr V$-metric space coincides with that induced after aggregation. We decompose this property into its upper and lower versions and show that upper strong preservation is equivalent to Scott continuity at the unit of the domain quantale, whereas lower strong preservation is characterized in terms of the fibre of the unit and the Scott upper semicontinuity of the inverse multifunction. Combining these results, we obtain a complete characterization of strongly preserving functions, together with analogous results for separated, symmetric, and separated and symmetric $\mathscr V$-metric spaces. Finally, we specialize the general theory to functions from $\mathsf P_+^{\,I}$ to $\mathsf P_+$, obtaining characterizations of strongly extended quasi-pseudometric and pseudometric aggregation functions on Cartesian products and on common underlying sets, together with corresponding results for extended quasi-metric and metric aggregation functions on Cartesian products.

math.GN↗
arXiv2026

Diagnosing the Explicit-to-Applied Theory of Mind Gap in LLMs

As large language models are deployed in high-stakes roles such as therapy, education, and personal assistance, their social cognition must be assessed reliably. Current Theory of Mind (ToM) benchmarks report aggregate scores that conflate two distinct competencies: explicit ToM (recognizing what someone believes) and applied ToM (using that belief to predict behavior). Building on this distinction, we investigate the gap and its recoverability using four structurally diverse benchmarks. Aggregate reporting obscures uneven competence: on SimpleToM, Llama-3.1-8B scores 51.5 percent overall while recognizing mental states at 99.4 percent and predicting behavior at only 28.9 percent. To assess recovery, we compare structured prompting, matched in-context learning, and our proposed Belief-State Test-Time Training (BSTTT), which adapts lightweight parameters on answered support questions through action reconstruction. Across Qwen2.5-7B, Llama-3.1-8B, and GPT-oss-20B, recovery depends on the model, task, and intervention. On SimpleToM, reminders combined with BSTTT improve Llama's behavior prediction by 40.8 percentage points; reminders account for most of the overall improvement, with BSTTT adding 1.3 to 5.5 points beyond reminders across models. Contextual examples often outperform weight updates, while BSTTT remains stronger in specific model-benchmark settings. Temporal belief tracking remains difficult despite partial recovery. These findings show that aggregate ToM scores can overstate applied competence while hiding performance recoverable through targeted assistance. We recommend decomposed reporting of explicit and applied accuracy alongside intervention comparisons that establish which failures are recoverable and under what conditions.

cs.AI↗
arXiv2026

Safety-constrained position tracking of a flexible transmission system

We consider safety-constrained position tracking for a flexible transmission system consisting of a torsional shaft driven by a motor and connected to a gravitationally loaded inertia. The coupled PDE-ODE dynamics admit a semilinear formulation in which the linear input-output system is impedance passive and the gravitational torque enters as a bounded, globally Lipschitz perturbation. We establish an energy dissipation inequality for the nonlinear model and show that a position-funnel controller with supervised damping admits a unique global closed-loop trajectory. In particular, the motor-angle tracking error remains strictly inside a prescribed time-varying funnel, which can be chosen to enforce safety bounds on the motor angle.

math.OC↗
arXiv2026

SCOPE: Compiling Natural-Language Policies into Neurosymbolic Predicate Programs

Policy enforcement for LLM agents has shifted from single-turn content moderation to compound, application-specific policies evaluated over multi-turn transcripts and tool-call trajectories. Fine-tuned guardrail models are efficient but degrade on compound policies, and while LLM-as-a-judge approaches achieve higher accuracy, they incur high inference cost and tend to produce unstable predictions. Neurosymbolic methods offer a natural alternative, decomposing the policy into narrow neural decisions composed under explicit logic. However, existing symbolic layers lack the required expressiveness and are optimized against fixed data rather than the policy itself. To address this, we introduce SCOPE, which compiles a policy into an executable program of neural predicates executed by small models, deterministic extractors, and code that composes their outputs. The program is optimized jointly with its training data, using adversarial generation of counterexamples that target where the program deviates from the policy. We evaluate SCOPE on a benchmark of real-world policy tasks against prompt optimization, task decomposition, and workflow generation methods, achieving the highest average accuracy across tasks and models. Programs optimized with the weakest executor transfer to other models without further optimization. This allows them to be deployed in production on inexpensive models, where they reach accuracy comparable to frontier reasoning models. Unlike these monolithic models, SCOPE produces substantially more stable predictions, and each verdict is interpretable through the predicate outputs that produced it. Additionally, we validate the key components of our method through an ablation study. SCOPE thus enables reliable, interpretable, and efficient policy enforcement at production scale.

cs.AI↗
arXiv2026

A unified quantitative approach to Chiti and Kohler--Jobin inequalities

We obtain two quantitative results related to classical rearrangement inequalities for the first Dirichlet eigenvalue. The first one is a quantitative version of Chiti's reverse Hölder inequality. More precisely, for every $1\le q<p<\infty$, we prove that the deficit controls, up to translations, the $L^q$ distance between a first eigenfunction of a bounded domain and the first eigenfunction of the ball having the same first eigenvalue. The second result concerns the Kohler--Jobin inequality and gives a quantitative estimate in terms of the $L^\infty\!\to L^2$ distance between the corresponding Dirichlet resolvents. In both cases, the main idea is to find a suitable superlevel set of the first eigenfunction which is almost optimal for the Faber-Krahn inequality.

math.AP↗
arXiv2026

AstroBench: Benchmarking Free-Floating Capture and Space Robotic Manipulation

Capturing a loose object inside a spacecraft requires a robot to approach a target that can translate and rotate, and whose motion changes when contact is made. These conditions motivate evaluation beyond manipulation of supported objects. We introduce AstroBench, an open benchmark for free-floating capture and space robotic manipulation with a fixed-base arm. The task suite combines four initial-motion tracks with sample-storage operations based on MELFI hardware and cap and captive-screw servicing based on the Robotic Refueling Mission. The two workcells provide nine task definitions and 29 task-target combinations. A metadata-filtered Objaverse inventory supplies 8,414 object instances for asset expansion, alongside five cabin visual contexts and versioned task instructions. We report two acquisition-focused experiments using existing checkpoints. A frozen $π_{0.5}$ DROID joint-position policy captures 2 of 200 static targets and 4 of 200 moving targets across twenty object identities. A cube-specific ACT policy captures 8 of 10 static targets and 6 of 10 moving targets in a separate experiment. Each experiment retains its recorded observation, action and capture configuration. The results document difficult direct transfer for the tested generalist checkpoint and successful task-specific capture behavior, while identifying concrete settings for studying motion-aware perception and contact-sensitive control. Code, task definitions and editable workcell assets are available at https://github.com/AstroUpLab/AstroBench.

cs.RO↗
arXiv2026

A General-Purpose Dual-Path Interaction Framework for Multi-View Medical Image Segmentation: Alleviating Data Volume Dependency in Small-Sample Regimes

The performance of medical image segmentation depends on a large amount of annotated data, and most segmentation models experience a significant drop in performance in low-sample-size scenarios. Existing semi-supervised and weakly supervised methods rely on specific network architectures, while traditional multi-view-assisted segmentation methods require strictly paired samples, which limits their clinical applications. To address these issues, we propose DPIF, a general-purpose plug-and-play dual-path interaction framework compatible with arbitrary encoder-decoder segmentation networks, which leverages cross-view anatomical complementarity to reduce annotation reliance. DPIF first performs Sinkhorn optimal transport-based cross-view distribution alignment to establish global semantic correspondence without sample-level pairing. It then adopts conditional mutual information-guided feature decoupling to enable bidirectional knowledge transfer only within the shared semantic space, suppressing view-specific noise. Finally, persistent homology-based topological regularization encodes anatomical invariants as unsupervised priors to mitigate overfitting. Experiments were conducted using 24 baseline models on a private multicenter echocardiography dataset and two public datasets. The results show that DPIF-enhanced models outperform single-view baseline models when trained on comparable amounts of data. Notably, most dual-path variants trained on 40\% - 60\% of the data achieve performance comparable to that of single-view models trained on the full dataset. This demonstrates that DPIF provides a general-purpose and scalable solution for data-scarce multi-view medical image segmentation problems.

cs.CV↗
arXiv2026

Dual Exam: Controlled Evaluation of Multimodal Agents and World Models

World Models (WMs) can support Multimodal Agents (MAs) through simulated experience, while Agent trajectories can inform World Model refinement. Evaluating this interaction requires separate evidence about action selection and state evolution. We introduce Dual Exam, which tests Agents under fixed environment response mechanisms and World Models under fixed action interventions across physical, digital, social, and scientific domains. The Agent track compares 10 configurations on 1,088 items in 15 reporting categories. The World Model track contains 400 cases across eight scenarios; 15 configurations have full coverage and one covers six scenarios. GPT 6 Astra leads the case-weighted Agent Index at 31.3, and GPT-Image-2 leads the full-coverage WM Overall at 21.7. These scores use different tasks and metrics and are not directly comparable. A 150-case pilot of one coupled pairing records zero task successes and motivates component-level diagnosis. Results reveal uneven task performance and failures to preserve task-relevant state. Their value for predicting coupled success or sustained reciprocal learning remains to be established.

cs.AI↗
arXiv2026

Scale-free localization in pseudo-Hermitian quantum walk

Scale-free localization is a distinctive phenomenon in non-Hermitian systems, where specific eigenstates, known as scale-free localized states (LSs), possess localization lengths that scale linearly with the system size. Here, we show that a single non-Hermitian defect in one-dimensional quantum walks (QWs) induces scale-free localization. Driven by non-Hermiticity, the QW system undergoes a pseudo-Hermitian (PH) phase transition and the PH symmetry breaking is accompanied by the emergence of scale-free localization. As non-Hermiticity is further increased, the system undergoes a second phase transition, marked by the emergence of two pairs of bound states. Crucially, we identify the critical points of both transitions using localization properties. Furthermore, we extend our investigation to two-dimensional QWs, where a single non-Hermitian defect also induces scale-free localization, thereby demonstrating the universality of this phenomenon. Our work provides a valuable perspective for further exploring defect physics and a type of novel LS in non-Hermitian quantum systems.

quant-ph↗
arXiv2026

On global Chern-Calabi flows on compact complex surfaces

In this paper, we propose the Chern-Calabi Flow for extremal metrics in conformal Hermitian geometry. After the existence of extremal metric, we consider the existence of global flows to the Chern-Calabi flows for extremal metrics on compact complex surfaces with Gauduchon metrics. To achieve this goal, we use the Adams inequality and the Gronwall inequality method to obtain fourth order derivative norm estimates. We also study the convergence of the global solution to the Chern-Calabi flow and its limit is the solution of the Chern-Calabi extremal metric problem. We propose a fourth order elliptic problem and derive the Bochner formula relevant to the elliptic estimates for the Chern-Calabi heat flow.

math.DG↗
arXiv2026

Federated Differentially Private Optimization via Low-Rank Preconditioning

We consider differentially private (DP) optimization in the federated setting, where a DP estimate of the feature covariance matrix serves as a Hessian approximation for generalized linear models. Existing central DP methods for covariance estimation have communication cost that grows quadratically in the data dimension, while existing baselines with linear communication cost are only locally private and degrade as the number of clients increases. As a remedy, we propose a federated DP-PCA framework that achieves linear communication cost while providing central DP guarantees under secure aggregation. It consists of a DP rank selection step, a DP power method, and a noisy release of the covariance projected onto the selected low-dimensional subspace. We give error bounds for this covariance estimate and for the resulting low-communication preconditioned DP gradient descent. Empirically, our proposed DP optimization method matches a centralized baseline with quadratic communication cost at a fraction of the communication.

cs.LG↗
arXiv2026

Norm-Aware Product Formulas for Hamiltonian Simulations

With the target of efficiently solving the quantum-mechanical Schrödinger equation on future quantum computers, first- and second-order product formulas for the time-evolution operator are proposed, designed to exploit hierarchies in the operator norms of the individual Hamiltonian components. Inspired by the THRIFT product formula, these decomposition schemes achieve a suppression of the leading error term by a factor of alpha<1 with respect to the standard Lie--Trotter--Suzuki splitting of the same order, where alpha quantifies the norm ratio between the dominant and subdominant parts of the Hamiltonian. Unlike THRIFT, the proposed formulas retain the general product-formula structure, making them applicable to Hamiltonians whose terms act on an arbitrary number of qubits. The advantages of this approach are demonstrated through numerical experiments on engineered two-dimensional spin-1/2 XY model systems and isolated molecules within the double-factorization framework.

quant-ph↗
arXiv2026

Conditional Mean Flows for Frequency-Domain Scattered Wavefield Generation

Frequency-domain wavefield modeling underpins seismic imaging and inversion, where repeated Helmholtz solutions become costly as velocity models, source locations, and frequencies vary. Neural operators learn mappings between physical fields, and their accuracy can decline outside the training distribution. Diffusion models learn wavefield distributions and generally require repeated sampling evaluations to produce accurate fields. A conditional mean flows model is introduced for complex scattered-wavefield generation. The analytical background wavefield serves as a condition alongside the velocity model, supplying spatially organized information about the source location and frequency. Learning the scattered wavefield avoids the point-source singularity in the target and supports generalization across these physical conditions. The model estimates the average transport velocity over a finite interval connecting Gaussian noise to the scattered field. This representation captures the accumulated displacement and permits both full-interval one-step generation and interval-partitioned sampling. Trained on OpenFWI, the model is evaluated on in-distribution and out-of-distribution velocity structures, frequency generalization, and inference efficiency. In most evaluated settings, one-step mean flows sampling achieves lower wavefield error than multi-step diffusion. Because each sampling step requires another network evaluation, accurate one-step generation also reduces inference time. These results indicate that mean flows can provide accurate and efficient frequency-domain scattered-wavefield generation.

physics.geo-ph↗
arXiv2026

Extended-regularized lattice Boltzmann method

Regularized lattice Boltzmann methods improve numerical stability by projecting the nonequilibrium distribution onto its hydrodynamic second-order Hermite sector and filtering the remaining kinetic modes. We introduce a separate relaxation time for the complete complementary kinetic subspace. The resulting extended-regularized lattice Boltzmann method (E-RLBM) interpolates continuously between the Bhatnagar-Gross-Krook and regularized collision operators while retaining a two-rate implementation. A Chapman-Enskog expansion shows that the isothermal Navier-Stokes equations are recovered with kinematic viscosity $ν=c_s^2(τ-Δt/2)$, independently of the kinetic relaxation time $τ_{\rm nh}$, whereas $τ_{\rm nh}$ changes higher-order truncation errors. For linear transverse shear, its first effect is a fourth-derivative term whose coefficient depends on the direction of propagation relative to the lattice. This fourth-order coefficient is affine in $τ_{\rm nh}$. For linear shear on D2Q9 and $τ_{\rm nh}$ between $τ$ and $Δt$, it lies between the corresponding BGK and RLBM values in every propagation direction. Stability maps for the decaying Taylor-Green vortex and flow past a rectangular obstacle show that an appropriate kinetic relaxation substantially enlarges the stable parameter domain. In a Taylor-Green convergence test with consistent initialization, all three operators converge at second order, and their error differences follow the hyperviscous coefficient for diagonal shear waves: the E-RLBM error is about 6% above the BGK error and about 40% below the RLBM error; in a double shear layer at Re = 80000, the E-RLBM follows the RLBM convergence curve. The proposed collision operator therefore provides a low-cost means of controlling nonhydrodynamic modes without introducing a full multiple-relaxation-time basis.

physics.comp-ph↗
arXiv2026

The classification of smooth 3-fold flops

This paper establishes the analytic classification of smooth, simple, 3-fold flops, and in the process provides both an intrinsic characterisation and a classification of their contraction algebras.

math.AG↗
arXiv2026

Directionally Correct Views in Portfolio Optimization

Portfolio optimization techniques often use historical data to estimate parameters for Markowitz-type or ML models. Effective allocation requires a way to align the available data with conditions expected over the investment horizon. Inspired by the Black-Litterman framework, we ask whether incorporating expert views leads to better portfolio allocations. In a simple Markowitz setting, we show that even a small amount of directionally correct information can materially improve investment decisions. We also discuss how expert views can be incorporated in end-to-end portfolio models using entropy-based ideas. Somewhat surprisingly, our empirical results indicate that simpler Markowitz-type model perform as well as an end-to-end model, once directionally correct experts are incorporated.

q-fin.PM↗
arXiv2026

When Forgotten Facts Return: Relearning-Aware Unlearning of Knowledge Graph Embeddings via Tangent Kernels

Approximate unlearning answers a deletion request on a knowledge-graph embedding by editing entity embeddings instead of retraining from scratch. These embeddings keep being updated as new facts arrive. Under this ordinary continued training, forgotten facts recover more than in the exact-unlearning baseline, a model retrained without them and given the same continued training. We trace this relearning to entity embeddings shared by forgotten and new facts. To our knowledge, this is the first study of relearning in knowledge-graph embeddings. Relearning-robust methods developed for deep networks do not account for the graph structure underlying this recovery. We propose Kernel-Anticipated Unlearning (KAU), which uses the tangent kernel to model how continued training changes forgotten facts' scores through these shared parameters. KAU expresses this first-order response in closed form and differentiates through it inside the forgetting loss. The objective steers edits toward directions that continued training reverses less. We compare KAU with state-of-the-art knowledge-graph unlearners and relearning-robust methods under identical continued training, measuring leakage beyond the exact-unlearning baseline. Across three backbones, KAU reduces the strongest knowledge-graph unlearner's leakage by 46-63% on FB15k-237 and 50-82% on CoDEx-L, at higher retained utility.

cs.LG↗
arXiv2026

A precise TESS ephemeris for the short-period binary TIC-259864042

This paper presents a precise eclipse-timing analysis of TIC 259864042 (= EC 02406--6408, Gaia DR3 4647004122914240640) using eleven sectors of TESS photometry spanning 2018--2025 (7.1 yr). The system exhibits a single deep eclipse with a V-shaped profile at phase 0 and no detectable counterpart at phase 0.5. Four independent period-search methods converge to a common period agreeing to within 7 parts per million. An independent reduction with the SHERLOCK pipeline, using its own 120-second TESS photometry, confirms the same period at the same precision. A joint linear ephemeris fit gives P = 0.46048676 $\pm$ 0.0000001 d (11.059 h) and T_0 = 2458325.70381 $\pm$ 0.00002 BJD, with a residual RMS of 0.87 min across the 7.1-yr baseline. Per-sector ephemeris fits show residual medians within $\pm$0.42 min of zero, confirming that no cycle slip has occurred across the large TESS data gaps. The period is stable at the level of $Δ$BIC = +5.21 (in favour of the constant-period model); an upper limit of $|\dot{P}| \le 1 \times 10^{-9}$ d d$^{-1}$ is placed on any secular period change. A PHOEBE 2.4 fit to the Sector 27 light curve yields $i = 80.8^{\circ} \pm 0.5^{\circ}$, $R_1 = 0.21 \pm 0.01\ R_{\odot}$, $R_2 = 0.25 \pm 0.02\ R_{\odot}$, $q = 0.49 \pm 0.07$, consistent with a grazing detached geometry and a mid-M-dwarf companion. The TESS light curve alone cannot distinguish between the two temperatures 25100 K and 31000 K. The paper discusses the system in the context of the sdB eclipsing binary population, and notes that while the current data are consistent with a short-period sdB+M-dwarf architecture, a phase-resolved spectrum is needed to independently confirm the sdB identification.

astro-ph.SR↗