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arXiv2026

One Evaluation Is Enough: Inversion-Free Image Editing with Trajectory Correction

Real-image editing commonly relies on inversion and iterative sampling, while efficient one-step editors often stabilize a large update by estimating and averaging editing fields at multiple noise levels. We ask a stricter question: is a single temporal field evaluation sufficient for reliable inversion-free editing? Our answer rests on a previously underused duality in a variance-preserving (VP) one-step model: one source-target query at a shared noisy state simultaneously determines a local angular semantic tangent and a clean endpoint displacement. We integrate the tangent over the analytically known remaining VP angle to obtain a nominal one-step transport, and use the endpoint displacement to correct its long-horizon extrapolation through a constrained projection. This transport-first, correction-second construction yields Endpoint-Calibrated Transport (ECT), a closed-form, training-free editor requiring only one noise level, two text conditions packed into one denoiser batch, and one latent update. ECT requires no image inversion, trajectory integration, second temporal observation, or iterative optimization, and can be directly applied to pretrained epsilon-prediction VP one-step generators. Across the reported benchmark settings, ECT delivers stable gains in editing quality within an extremely short editing time.

cs.CV↗
arXiv2026

TutorCaseBench: A Case-Grounded Benchmark for Evaluating LLM Tutors in Supporting Students with Challenging Behaviors

LLMs are increasingly used as AI tutors, yet existing benchmarks overlook the challenging behaviors common among students with learning and behavioral difficulties. We introduce TutorCaseBench, a benchmark that grounds AI-tutor evaluation in qualitative case studies from pedagogy, educational psychology, and special education. Our pipeline converts documented student categories into case-specific scenarios, tutor-student dialogues, and rubrics grounded in source evidence. The benchmark contains 923 dialogues across five challenge types, evaluated through open-ended response generation (1,560 items) and multiple-choice selection (2,537 items). Across 11 state-of-the-art LLMs, even the strongest model satisfies a rubric criterion in only 55.8% of open-ended cases, while models are better at selecting appropriate strategies than generating situated responses. These results reveal a gap between intervention recognition and context-specific adaptation, especially for open-weight models. All data are publicly available at https://github.com/kixlab/TutorCaseBench.

cs.CL↗
arXiv2026

Arithmetic Conditions for the Invertibility of Circulant Matrices over Finite Fields

This paper investigates arithmetic conditions, over finite fields, for the invertibility of circulant matrices whose first row is $(a_0,\ldots,a_{n-1})$, in terms of linear constraints on the entries $a_0,\ldots,a_{n-1}$. The coefficients of such linear constraints are determined by field traces arising from Frobenius orbits of roots of unity, and under additional assumptions they are given by Ramanujan sums reduced modulo the characteristic of the given field. These results provide finite-field analogues of the corresponding criteria over $\mathbb{Q}$, where the relevant Ramanujan-sum linear constraints can be formulated without such additional assumptions, while also showing some differences caused by reduction modulo the characteristic. The constraints can often determine invertibility using aggregate information about the entries of a row, rather than the particular pattern of the entries. We also discuss the polynomial gcd criterion, which gives necessary and sufficient conditions for invertibility and yields further consequences for special families of circulant matrices.

math.CO↗
arXiv2026

Radial Schur multipliers on trees and quantitative Fourier--Schur separation

We study ordinary and completely bounded norms of radial Schur multipliers on trees. On the homogeneous tree of countably infinite degree, we show that boundedness implies complete boundedness. The argument realizes arbitrary matrix amplifications of a shifted Toeplitz symbol inside rectangular corners of the original multiplier. It also applies to some locally finite trees of unbounded degree. Consequently, the completely bounded radial multiplier theorem of González-Pérez, Parcet, Pérez García and Wang has no general counterpart with ordinary norms. We also determine the optimal order of the comparison constant for positive definite even symbols. For each $1 < p < \infty$ with $p \neq 2$, we construct radial symbols on finitely generated nonabelian free groups, supported in the ball of radius $N$, whose completely bounded Herz--Schur norms remain bounded as $N \to \infty$, whereas their ordinary Fourier multiplier norms grow at least like $N^{|1-\frac{2}{p}|}$. For each fixed $0 < \vartheta < π$, we also determine the exact order $t^{-|1-\frac{2}{p}|}$ of both the ordinary and completely bounded Fourier multiplier norms of the damped radial phases $s \mapsto \mathrm{e}^{(-t+\mathrm{i}\vartheta)|s|}$ as $t \to 0$. Finally, we compute the local amenability modulus of free balls, we bound the Fourier--Schur ratio of free balls from above and below in terms of this modulus, and we show that its restriction to radial symbols grows polynomially in the radius.

math.FA↗
arXiv2026

Light pulse propagation through a dielectric slab. II. Two roads to the same field: numerical cross-validation of local (FDTD-ADE) and non-local (retarded integral-equation) solutions

Paper I of this series (arXiv:2610.29336) solves, via the retarded Green's functions, the local differential equations governing light propagation in a plane-parallel dielectric slab, obtaining a pair of retarded integral equations that represent, at the continuum level, the microscopic multiple-scattering picture underlying the local formulation. The two formulations are thus proven to be analytically equivalent. The local system of differential equations is then solved numerically with a finite-difference time-domain scheme (FDTD-ADE). This report closes the loop by solving the retarded integral equations directly, without passing through the local formulation, and comparing the result with the FDTD-ADE solution under the same physical and material conditions. The two independently implemented, structurally different numerical schemes agree within 0.02-1.3% for the reported reflected and transmitted energy fractions; the absorbed fraction, computed independently in both schemes from the Lorentz damping term rather than merely inferred from 1-R-T, agrees closely between the two methods, to the precision reported. Both solutions are also compared with an exact reference, the Airy formula weighted over the pulse spectrum: the direct solution reproduces it to within 0.3%, and its Richardson extrapolation to all digits shown. The agreement persists for a few-cycle pulse and for a sub-wavelength slab, where naive fixed-point iteration of the integral equations is not a practical convergent strategy. A complementary set of movie-style, nine-frame visualizations is also presented, illustrating the same equivalence directly in the field rather than only in integrated observables. To the author's knowledge, a direct numerical cross-validation of these two formulations in this genuinely time-dependent regime has not previously been reported.

physics.optics↗
arXiv2026

RECAP: Looking Once Is Not Enough for Vision-Language Reasoning

Long chain-of-thought reasoning improves vision-language models (VLMs), but it also exposes a temporal grounding failure: as generation unfolds, models may drift from image evidence while reinforcement learning still provides only a final-answer reward. We study this gap through visual revisit, a temporally localized reactivation of image evidence during reasoning. Across three VLM backbones, revisit peaks are predictive of correctness and causally linked to performance: masking them degrades accuracy more than masking non-peaks, while injecting revisit-like peaks into failed rollouts partially restores correct answers. Control analyses show that this signal is not explained by raw attention strength, uncertainty, logit margin, or hidden-state norm. We propose RECAP, a GRPO-compatible training method that turns detached revisit traces into a credit-assignment signal. RECAP uses revisit-conditioned advantage estimation, dynamic discounting, and gated value prediction to propagate final rewards through visually important reasoning steps, without extra rewards, additional rollouts, or inference-time changes. Across Qwen2.5-VL-7B, InternVL3-8B, and LLaVA-OV-7B, RECAP consistently improves over GRPO and VPPO, yielding $+6.1$--$+6.5$ HallusionBench gains over the SFT base while preserving reasoning and text-only performance. Results on larger LoRA-tuned models, Chinese zero-shot benchmarks, visual perturbations, and human evaluation further demonstrate robust improvements in image-grounded reasoning.

cs.CV↗
arXiv2026

Central limit theorem for the Smoluchowski--Kramers approximation

We establish the central limit theorem (CLT) for Smoluchowski--Kramers approximation of Langevin equations with state-dependent, possibly non-symmetric friction and multiplicative noise. To the best of our knowledge, this is the first such result, even in the case of constant friction and additive noise. The result differs in an essential way from classical averaging CLTs and reveals a new two-scale fluctuation phenomenon. The normalized difference process retains a persistent fast component and is generally not tight in $C([0,T];\mathbb{R}^d)$, thus the usual continuous-path CLT cannot hold. We derive an exact decomposition of the normalized difference into an accumulated slow fluctuation and a rapidly relaxing velocity correction. The auxiliary slow fluctuation converges to a linear diffusion driven by both the original Brownian motion and an additional independent Brownian noise whose effective covariance is characterized explicitly through a quadratic Poisson corrector. For every fixed $t>0$, the fast correction converges conditionally to its local Gaussian equilibrium. Consequently, the fixed-time fluctuation limit is identified as a Gaussian mixture: $$ \frac{\bar X_t-X_t^\varepsilon}{\sqrt{\varepsilon}} \Longrightarrow \bar Y_t+N^{1/2}(\bar X_t)ξ, $$ where $\bar X_t$ is the overdamped limit, $\bar Y_t$ is the limiting slow fluctuation, $N$ is the covariance of the terminal velocity correction and solves a Lyapunov equation, and $ξ$ is a standard Gaussian random variable independent of $(\bar X_t,\bar Y_t)$. A quantitative convergence rate is also provided. The result fills the remaining gap in the asymptotic theory of the overdamped approximation at the central limit scale, complementing the existing law of large numbers, moderate deviation and large deviation results.

math.PR↗
arXiv2026

Exceptional point structure and chiral state transfer in a rate-operator-engineered $\mathcal{PT}$-symmetric Bose-Hubbard dimer

The rate operator quantum jumps (ROQJ) is a general method to unravel a master equation describing the dynamics of an open quantum system into stochastic pure state trajectories. By post-selecting on no jumps occurring, one obtains a trajectory generated by a non-Hermitian Hamiltonian. The ROQJ allows for versatile modification of this non-Hermitian evolution, with each non-Hermitian Hamiltonian corresponding to a different continuous measurement scheme. We apply the ROQJ to modify the non-Hermitian dynamics of the open three-particle Bose-Hubbard dimer model under environmental non-local dephasing to construct a $PT$-symmetric non-Hermitian Hamiltonians with a general two parameter $PT$-symmetric imaginary diagonal part, which generalizes the well-known gain-loss Bose-Hubbard dimer. We characterize the symmetry structure of the spectrum for different on-site interaction strengths. For vanishing on-site interaction, we find the unbroken $PT$-symmetry region enclosed by order two exceptional lines of different character and four order four exceptional points, two of which are not found in the usual gain-loss Bose-Hubbard dimer. Additionally, we find regions with degenerate imaginary parts of the eigenvalues, also not found in the gain-loss Bose-Hubbard dimer. In this parameter space, we use time-dependent rate operator transformations to investigate state transfer under slowly varying parameter loops. Loops encircling an order four exceptional point exhibit chiral state transfer. Because the loops cross exceptional lines, the final states are superpositions of eigenstates. Chiral state transfer is also found with non-zero homogeneous on-site interaction when the loop crosses over broken $PT$-symmetry regions of different character. Using inhomogeneous on-site interactions, we find direction dependent state transfer between eigenstates without crossing or encircling exceptional points.

quant-ph↗
arXiv2026

A homological approach to Dax invariants

We present a homological construction of Dax invariants in 4-manifolds that is intrinsic to the topological category and requires neither smoothing nor generic representatives. Applications include an extension of the invariant of Cha-Kim to neat disks without assuming local flatness, and a topological definition of the Dax map for arc spaces that yields a short alternative proof of the topological Dax isomorphism theorem.

math.GT↗
arXiv2026

Semi-Linear Gaussian Bayesian Networks: Enhancing Flexibility in Node Dependencies via penalized B-Splines

Bayesian networks for continuous variables often rely on restrictive assumptions: linear Gaussian models lack flexibility to capture nonlinear dependencies, while non-parametric approaches provide greater flexibility at a high computational cost. This creates a gap for modeling scenarios where variables are approximately Gaussian but exhibit nonlinear relationships. We introduce Semi-Linear Gaussian Bayesian Networks (SLGBN), a novel class of continuous Bayesian networks designed to bridge this gap, preserving Gaussian conditional distributions while introducing nonlinear conditional mean functions. The proposed framework provides a unified probabilistic framework combining linear Gaussian and nonlinear Gaussian conditional models within the same Bayesian network Nonlinear conditional mean functions are estimated using generalized additive models (GAMs) with penalized B-splines. We present parameter estimation procedures for both types of nodes, as well as a greedy hill-climbing algorithm for learning the network structure under the SLGBN framework. To assess its performance, we compare SLGBN with other continuous Bayesian network models, including semiparametric Bayesian networks (SPBN) and Linear Gaussian Bayesian networks (LGBN). Experimental results evaluate both goodness of fit and the ability to recover the underlying network structure. The results show that, in the presence of nonlinear Gaussian relationships, SLGBN achieves better fit and improved structure recovery compared to alternative models, particularly when combined with the Extended Bayesian Information Criterion (EBIC*) score. Furthermore, by retaining Gaussian conditional distributions, SLGBN offers computational advantages over fully semiparametric approaches.

cs.AI↗
arXiv2026

ProTrav: Probabilistic Traversability with Elevation Uncertainty Propagation for Risk-aware Off-road Navigation

Reliable traversability estimation is essential for autonomous navigation on unstructured terrain. State-of-the-art approaches typically evaluate traversability from deterministic terrain elevation, thereby overlooking spatial correlations in the terrain and failing to account for how elevation uncertainty propagates through nonlinear traversability mappings. We propose ProTrav, a probabilistic terrain representation that uses function-space sampling to generate spatially correlated terrain realizations and propagate elevation uncertainty to traversability cost. First, a Sparse Variational Gaussian Process (SVGP) reconstructs a probabilistic elevation field from observations. Then, posterior function-space sampling generates spatially correlated terrain realizations, preserving spatial dependencies between neighboring locations while propagating elevation uncertainty through geometric features into traversability cost. Consequently, a distribution of traversability cost at arbitrary location is established, where the mean and standard deviation quantify the expected traversability cost and associated uncertainty, respectively. A risk term is defined by moment-based Upper Confidence Bound (UCB) and incorporated into a Model Predictive Path Integral (MPPI) controller for risk-aware navigation. Simulation experiments demonstrate navigation success rates exceeding 90\% on different terrains, compared with below 40\% for a baseline, while reducing attitude variations. Furthermore, the proposed approach is validated on a robot traversing real-world off-road terrain.

cs.RO↗
arXiv2026

Discovering supernovae in the Virgo Cluster during the LSST era

In the era of unprecedentedly wide, deep and high-cadence sky surveys the chance for discovering new explosive transients in nearby, rich galaxy clusters within hours after explosion is expected to increase dramatically. In this paper we focus on the Virgo cluster ($D \simeq 16.5$ Mpc), and estimate the expected numbers of new, very young supernovae in different sub-structures and galaxy types of the cluster during the proposed lifetime (~10 years) of the Legacy Survey of Space and Time (LSST) by the Vera Rubin Observatory. These predictions will be tested observationally during the ongoing Shadow-LSST discovery and follow-up program led by University of Arizona in the upcoming years.

astro-ph.GA↗
arXiv2026

TRIAD: Benchmarking Omni-Modal Ambiguity in Multimodal Large Language Models

Ambiguity is a central feature of natural communication: a sentence, image, or sound may remain underspecified until evidence from the other channels is considered. Yet current multimodal benchmarks mostly test perception, recognition, or reasoning over already determinate inputs, leaving unclear whether omni-modal large language models can resolve ambiguity distributed across text, vision, and audio. We introduce TRIAD, a benchmark for omni-modal ambiguity resolution. Each item is a text--image--audio triplet paired with a question and a set of answer options, and is constructed so that the full triplet determines a unique answer while removing any one modality makes the set of consistent options non-unique. TRIAD contains $327$ hand-written queries in $86$ scene groups and covers $18$ ambiguity categories across the three modalities. Evaluating $14$ omni-modal systems reveals that strong item-level performance does not translate into robust scene-level disambiguation: the best model still trails humans by over $20$ points in group accuracy. Removing answer options widens the gap further, driving model group accuracy to the single digits while humans remain robust. Leave-one-out and transcript/description counterfactuals further show that current systems often underuse audio cues when text and image are present. TRIAD reframes omni-modal evaluation around disambiguation rather than recognition, and provides a taxonomy, benchmark, and diagnostic protocol for measuring this capability.

cs.CL↗
arXiv2026

Reverse-KL Policy Distillation: Statistical Guarantees and Limits

Reverse Kullback-Leibler (KL) distillation has shown empirical promise for training language models, but its statistical guarantees and limits remain poorly understood. In particular, the effects of two design choices remain unclear: Does access to the teacher's full action distribution provide an advantage over observing only the probability it assigns to the sampled action? Must training trajectories come from the student (online distillation), or can teacher-generated trajectories (offline distillation) suffice? Our work takes a first step toward understanding the theory underlying these questions. We study the sample complexity of online reverse-KL distillation under two feedback models: the teacher reveals either its full action distribution or only the probability of the student-generated action. We establish minimax bounds for the former and show that substantially more data may be required for the latter. In contrast, offline reverse-KL distillation can attain the same minimax rate whether the teacher reveals its full distribution or only its probability of the teacher-generated action. The key distinction is that teacher-generated actions themselves reveal the teacher's preferences, while queries at student-selected actions can leave those preferences unobserved. This highlights an additional statistical challenge for online distillation compared to offline distillation. These statistical guarantees do not follow simply from optimizing standard distillation objectives: our lower bounds show that common online objectives can fail when candidates are evaluated only on past student-generated trajectories. We develop trajectory-loss and importance-ratio truncation techniques together with policy-space learning to achieve optimal online distillation guarantees.

cs.LG↗
arXiv2026

Bipartite solutions of the set-theoretic Yang--Baxer equation

We introduce a new class of unitary set-theoretic Yang--Baxter solutions, called \emph{bipartite} solutions. This class, which is purely combinatorial in nature, forms a full subcategory of $\mathcal{LYB}$, the category of Lyubashenko type solutions of Agore, Chirvasitu and the author, and is particularly amenable to explicit classification results. We give an intrinsic necessary and sufficient condition for a solution of the Yang--Baxter equation to be isomorphic to a bipartite solution, we settle the isomorphism problem within this class and, as an application, we determine their automorphism groups. If $X$ is a finite set with $|X| = n$, the number of isomorphism classes of bipartite solutions on $X$ is $\lfloor n/2 \rfloor + 1 + \sum_{m=2}^{n} \bigl( o(m, n-m) - 1 \bigr)$, where $o(m, k)$ denotes the number of orbits of the symmetric group $S_m$ acting by conjugation on the $k$-element multisets of elements of $S_m$. The paper also offers a comparative study: for small $|X| = n$ we count, up to isomorphism, all set-theoretic Yang--Baxter solutions on $X$, together with five distinguished families of solutions. Throughout the range we were able to compute, the class $\mathcal{LYB}$ accounts for a large share of all bijective and of all non-degenerate solutions, strikingly close to the total.

math.QA↗
arXiv2026

Fit-error corrected auxiliary master equation approach for low-temperature nonequilibrium dynamical mean-field theory

The auxiliary master equation approach (AMEA) is a nonperturbative impurity solver for steady-state nonequilibrium dynamical mean-field theory (DMFT). It replaces the physical hybridization function by that of a small auxiliary open quantum system, whose parameters are fitted to reproduce it, and then solves the resulting many-body problem essentially exactly. AMEA's accuracy is thus limited by the quality of this fit: since the fastest-varying part of the hybridization is set by the distribution function, whose Fermi edges sharpen at low temperature, the fit tends to overestimate the temperature, especially for weak reservoir coupling. For the auxiliary hybridization itself, however, AMEA's self-energy is essentially exact. We correct for the fit error by estimating, via an approximate solver applicable to arbitrary hybridization functions, the ratio between the self-energies of the physical and auxiliary hybridizations, and using it to extend AMEA's self-energy from the auxiliary to the physical case. This fit-error correction becomes exact whenever either the fit or the approximate solver is exact. We benchmark the scheme using the nonequilibrium Kajueter-Kotliar iterated perturbation theory (IPT), generalized here to spin-polarized solutions, as the approximate solver; other approximate solvers, such as the noncrossing approximation, can be used as well. Applied to an Anderson impurity model with a spin-polarized bath, and within DMFT to a Hubbard layer weakly coupled to two metallic reservoirs, the correction largely removes the spurious heating from the AMEA fit and from insufficient thermalization by the reservoirs, while preserving the correct description of the Mott insulator, which IPT alone cannot fully recover.

cond-mat.str-el↗
arXiv2026

Bifurcations from Weight Space: JEPAWG on Parametric Dynamical Systems

Trained network weights are increasingly treated as data, but there is rarely ground truth about what a population of models should encode, which hampers systematic validation of weight-space methods. This makes parametric dynamical systems a natural test-bed: the dynamics change qualitatively at submanifolds of parameter space that are, in well-studied families, known analytically or numerically. Fine-tuning a forecasting network at each point of a parameter grid yields a model zoo indexed by a physical parameter, against which weight-space diagnostics can be checked. We build such model zoos for four dynamical systems and analyse them with JEPAWG, a recently proposed joint-embedding hypernetwork built around a learned parameter--weight representation. It was first introduced for neural samplers for lattice field theories, where its latent geometry located the phase transition and estimated the 2D Ising critical exponent. Applied to dynamical systems, JEPAWG is competitive on forecasting with a hypernetwork trained directly for that task, despite being supervised on the parameter--weight correspondence rather than on forecasts. Beyond forecasting, the norm of the parameter-derivative of the decoded weights locates Hopf points and curves, tracks a period-doubling cascade closely enough to give a qualitative estimate of the Feigenbaum constant, and detects a global bifurcation, which has no eigenvalue signature and is therefore invisible to linearisation. As such, JEPAWG provides interpretable insight into the dynamics that is not available from other hypernetworks. Together, these results indicate that JEPAWG is a workable instrument for reading physical structure off a population of trained networks.

cs.LG↗