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arXiv2026

Characteristic Functions of Parahoric Character Sheaves

We establish a Jordan decomposition formula for the characteristic functions of the character sheaves on parahoric subgroups defined in \cite{INY25}. For a sufficiently large $q$, these functions are $(-1)^{\dim G}$ times the corresponding deep level Deligne--Lusztig characters, extending \cite{Lu90} to positive depth. We also prove the orthogonality for the generalized deep level Green functions and the character sheaves functions. Moreover, we obtain an explicit expression of characteristic functions of simple character sheaves. As an application, we present a sheaf-theoretic expression to the multiplicity of the $G^{F^2}$ character restricted to $G^{F}$.

math.RT↗
arXiv2026

Uni-LaDiR: Latent Diffusion Unifies Multimodal Reasoning

Multimodal models increasingly think with different modalities such as images, 3D point clouds, and robot states, not just text. Yet each modality is still encoded into its own representation space, creating a modality-switching gap whenever reasoning moves from one modality to another. In this paper, we introduce Uni-LaDiR (Unified Latent Diffusion Reasoner), a framework that unifies different modalities into a shared latent space for multimodal reasoning. A unified encoder maps teacher reasoning steps from different modalities into latent thought tokens in a shared space, trained to extract the information needed for later reasoning steps and the final output. A diffusion reasoner, trained jointly with the encoder, generates these tokens at inference without teacher reasoning steps. Across eleven vision-language model (VLM) benchmarks and two vision-language-action (VLA) suites, Uni-LaDiR achieves relative gains over the strongest baselines of 7.3% on four mathematical and logical VLM benchmarks and 6.1% on RLBench manipulation tasks. Controlled comparisons show increasing gains as more teacher modalities are unified. These results suggest that unification improves multimodal reasoning by weaving it into a single thread, where the model predicts successive thoughts in a common representation space.

cs.LG↗
arXiv2026

Thermodynamic formalism and multifractal analysis of Birkhoff averages for parabolic rational maps

In this paper, we study the multifractal analysis of Birkhoff averages for parabolic rational maps. We establish a conditional variational principle and prove the real analyticity and strict monotonicity of the Birkhoff spectrum, as well as the existence and uniqueness of the measure attaining the supremum in the conditional variational principle, on a certain region. To this end, we prove the existence and uniqueness of an expanding equilibrium measure and the real analyticity of the pressure function on a suitable domain. For parabolic systems, our approach using thermodynamic formalism provides a unified framework for establishing the conditional variational principle and investigating finer properties of the Birkhoff spectrum, including its real analyticity, strict monotonicity, and the existence and uniqueness of a measure attaining the supremum on a certain region.

math.DS↗
arXiv2026

The Last Picard Rank 1 Double--Mirror Calabi--Yau Pair?

We consider a certain pair of families of Picard rank 1 Calabi--Yau threefolds, that have appeared earlier in mathematical literature in unrelated contexts: the family $\mathcal{X}$ of degree 33 threefolds in $G(2,6)$ (constructed by Miura) and the family $\mathcal{Y}$ of arithmetically Gorenstein degree 21 threefolds in $\mathbb{P}^8$ (constructed by Schenck--Stillman--Yuan). After establishing a natural geometric correspondence between their general members, we go on to show that any pair of corresponding threefolds in these families satisfy certain classical dualities. We moreover discover that their geometries may be related by a mathematical gauged linear sigma model, using which we prove that they are derived equivalent. This settles a conjecture of Miura, who predicted the existence of non-trivial Fourier--Mukai partners to members of $\mathcal{X}$, based on a study of its mirror moduli. This conjecture was also formulated later by Gerhardus--Jockers, in a physical context. In fact, starting from the other family $\mathcal{Y}$, we conjecturally arrive at the same mirror moduli. Thus, such a pair is a new, and quite possibly the last, addition to the small list of deformation families of non-birational, double-mirror Calabi--Yau threefolds having Picard rank 1.

math.AG↗
arXiv2026

FlowSGS: Improving Flow Matching Priors for Inverse Imaging with Stochastic Interpolants

Flow matching has emerged as the state-of-the-art generative model and has been used for plug-and-play (PnP) priors to solve inverse problems in computational imaging. However, existing flow-based inverse solvers assume linear forward models and/or make simplifying approximations in posterior sampling. To circumvent these problems, we introduce FlowSGS, a flow-based posterior sampling method using Split Gibbs Sampling (SGS) to decompose the posterior into a likelihood step and a prior step. Specifically, we sample from the likelihood step using Langevin dynamics and leverage the Stochastic Interpolants (SI) framework to integrate a pretrained flow model into the prior step. We provide a form for the prior step that uses SI's reverse-time SDE, and show connections to previous PnP methods. Moreover, with the aid of the flow prior's straight probability paths and a novel timestep correction technique for the reverse-time SDE, FlowSGS requires fewer network evaluations in its prior step than plug-and-play diffusion samplers. Our experiments show state-of-the-art performance on a range of inverse problems. For the first time, we provide an experiment on a nonlinear inverse problem (Fourier phase retrieval) for flow-based inverse solvers.

cs.CV↗
arXiv2026

Workspace Models: Lightweight Robotic Memory via Saliency-Driven Supervision

Complex robotic manipulation tasks frequently require a long-term memory of past events and actions. As conditioning on full histories renders policies prone to spurious correlations and degrades performance, many approaches to policy memory involve compressing historical information through expensive VLM queries in-the-loop to process only task-salient information. In this paper, we propose an alternative approach in which computationally intensive VLM queries are made during train-time to learn a lightweight latent memory that can be efficiently queried at deployment time. Our representation, which we call the \textbf{workspace token}, is trained by (1) using a VLM to identify current and historical information necessary for completing a task, then (2) distilling these into the workspace token using a set-reconstruction decoder loss. In both simulation and hardware, we show that the workspace token can be used as a drop-in replacement for observations during deployment, enabling policies to solve memory-intensive tasks without the need for VLM reasoning in-the-loop, in effect serving as a \textbf{latent harness} for distilling a stronger reasoning models ability to solve long-horizon tasks to a reactive robotic policy. We further demonstrate that the workspace tokens are not only more lightweight, but also lead to better policy performance compared to conditioning policies on explicit modalities like curated past image frames, motivating a ``latent'' approach to history curation and reasoning model harnesses more broadly.

cs.RO↗
arXiv2026

Wild frieze patterns over the integers

We study frieze patterns over the integers that are allowed to have wild entries. We introduce the quiddity number as a new invariant. The quiddity number is then used to classify strongly connected components of the directed graph $Γ_{2,n}(\mathbb{Z})$. Furthermore, we show that every finite simple directed graph arises as an induced subgraph of a directed graph $Γ_{2,n}(\mathbb{Z})$ for $n$ sufficiently large.

math.CO↗
arXiv2026

Just-Infinite Loops and Loop Algebras

Let $F$ be a field and let $L$ be a loop. We call $L$ just-infinite if it is infinite and every nontrivial normal subloop has finite index, and we call the possibly nonassociative loop algebra $F[L]$ just-infinite if it is infinite-dimensional and every nonzero two-sided ideal has finite codimension. We first prove that just-infiniteness of $F[L]$ always implies just-infiniteness of $L$. Next, using the Chein construction, we show for every infinite group $G$ that $M(G,2)$ is just-infinite if and only if $G$ is just-infinite, and that $F[M(G,2)]$ is just-infinite if and only if $F[G]$ is just-infinite. We extend the algebraic equivalence to the generalized Moufang doubles $M(G,*,g_0)$ whenever $G$ is infinite and nonabelian. Finally, we construct a single locally finite, residually finite, nonassociative Moufang loop $L$ for which $F[L]$ is residually finite-dimensional, locally finite-dimensional, and just-infinite over every field, and we explain why infinite nonassociative RA loops cannot be just-infinite.

math.RA↗
arXiv2026

Explicit Constructions of Maximum-Cardinality Families of Plateaued Functions with Pairwise Disjoint Walsh Supports

Families of plateaued Boolean functions with pairwise disjoint Walsh supports are useful in secondary constructions of cryptographic Boolean functions. Of particular interest are maximum-cardinality families whose members admit no nonzero linear structures. To the best of our knowledge, the previously known general construction attaining both properties is spectral (Hodžić et al., IEEE Trans. Inf. Theory 65(9): 5865--5879, 2019). In that work, explicit algebraic normal forms are not generally provided, and no general method is established for prescribing a common algebraic degree for all family members. In this paper, we present two new explicit algebraic constructions within a unified framework, one based on linear functions and the other on partially linear functions with bent components. Let $p\geq 2$ and $q\geq 0$ satisfy $q<2^p-p-1$, and set $m=p+q$. Both constructions yield maximum-cardinality families of $2^{q+1}$ $(q+1)$-plateaued Boolean functions with pairwise disjoint Walsh supports. No member admits a nonzero linear structure, and every member has an explicit generalized Maiorana--McFarland representation. The first construction produces functions in $m+p+1$ variables and realizes any prescribed common algebraic degree $3\leq d\leq p+1$, provided that $q<\sum_{i=2}^{d-1}\binom{p}{i}$; its maximum attainable degree $p+1$ is optimal. The second construction produces functions in $n+p+1$ variables, where $n>m$ and $n-m$ is even, and realizes any prescribed common algebraic degree $3\leq d\leq p+(n-m)/2$, provided that $q<\sum_{i=2}^{\min\{d-1,p\}}\binom{p}{i}$; its maximum attainable degree $p+(n-m)/2$ is next-to-optimal.

cs.IT↗
arXiv2026

A Geodesic Route toward Holography beyond AdS: Tessellating the Schwarzschild Black Hole

In vacuum anti-de Sitter (AdS) space, the geometry admits a perfect tessellation by a geodesic network. This tessellation characterizes the background exactly, and a partial entanglement entropy (PEE) tensor-network toy model of AdS/CFT can be defined on it. It has so far been unclear whether this construction can be extended beyond vacuum AdS. In this paper we take the first step in this direction. We show that the exterior region of the Schwarzschild black hole and its Einstein--Rosen bridge are perfectly tessellated by a specific geodesic gas emitted from the boundary, so that Crofton reconstruction holds in these regions. We then prove that such a geodesic tessellation and the Crofton reconstruction it defines extend to more generic Riemannian manifolds. This allows us to construct a PEE tensor-network model of holographic duality on more generic geometric backgrounds. In the case of the PEE tensor-network model for the Schwarzschild background, we can concretely realize the Bekenstein--Hawking entropy, the Ryu--Takayanagi formula and the ER=EPR proposal. Our approach opens a new route towards holographic toy models beyond AdS/CFT.

hep-th↗
arXiv2026

A Novel Path-Tracking Algorithm for Automated Tractor-Trailer Forward and Backward Maneuvers

Fully autonomous tractor--trailer systems are increasingly deployed in logistics, agriculture, and industrial environments, where precise and robust path-tracking capabilities are essential. However, the articulation between the tractor and the trailer introduces additional nonlinearities and significantly complicates lateral and longitudinal control, particularly during reversing maneuvers. This paper introduces a novel path-tracking algorithm specifically designed for articulated vehicles with a single trailer. The proposed method combines a lateral control law applied at the trailer level with a short-horizon predictive adjustment of the tractor steering angle, ensuring stable convergence toward the desired path in both forward and backward motion. The approach is geometry-based and requires no per-vehicle calibration or training. Simulation studies in a high-fidelity physics simulator demonstrate the ability of the controller to match or outperform classical and state-of-the-art methods in terms of accuracy, stability, and robustness to disturbances.

cs.RO↗
arXiv2026

Real quadratic fields and finite quantum dilogarithms I

We prove that Stark--Shintani ray class invariants (Stark units) associated to real quadratic fields are algebraic numbers. These invariants are given by special values of Faddeev's modular quantum dilogarithm, introduced by Garoufalidis--Kashaev--Zagier. Our main discovery is that special values of the modular quantum dilogarithm satisfy an explicit overdetermined system of polynomial equations, matching a variation on the defining equations of Andersen--Kashaev's notion of a quantum dilogarithm on a product of two cyclic groups. We give two and a half proofs that this system of equations defines a zero-dimensional variety. The simplest follow from an uncertainty principle for finite Fourier transform and $2$-adic valuation bounds. The last proof is more involved and shows finite quanatum dilogarithms can be used to categorify fusion rings introduced by Izumi, and the algebraicity of the special values then follows by Ocneanu's rigidity theorem. As a byproduct, we obtain an explicit infinite family of irrational near-group fusion categories. As a further application, we prove a family of quadratic relations for Stark units recently conjectured by Appleby, Flammia, and Kopp motivated by Zauner's conjecture about SIC-POVMs (complex equiangular lines).

math.NT↗
arXiv2026

Abstention and Noise Filtering: Two Missing Primitives of Softmax Attention

Softmax attention has two structural gaps. A head cannot abstain, because its weights sum to one, so it outputs something even when nothing is relevant. Nor can it filter what it reads, because its output is a weighted average of value vectors, passing interference as faithfully as signal. We call these missing primitives abstention and noise filtering. Recent studies report that gating the value pathway improves pretraining but attribute the gain to different causes. We show that a value gate partly supplies both primitives, which unifies the reported causes as views of one gain. We give each primitive its own mechanism in matched models of 10M to 350M parameters and measure what each contributes. The gain from gating is almost entirely abstention at 10M, whereas by 350M filtering contributes as much as abstention, so what a study observes depends on its scale. The two benefits are largely additive, with a small overlap. A gate determined by each value alone leaves the attention sink in place, whereas a query-controlled mechanism removes it. Injecting interference into the value reads shows that abstention and filtering protect against it in distinguishable ways. The same patterns appear in pretrained models up to 20B parameters.

cs.LG↗
arXiv2026

On different notions related to APN mappings

An APN mapping $F:\mathbb{F}_{2^n}\to \mathbb{F}_{2^n}$ is a polynomial characterized by the non-vanishing property on 2-flats. In this work, we analyze notions that are closely related to this property. To understand which $k$-flats of $\mathbb{F}_{2^n}$ remain flats under $F$, we study the $k$-breaking. The function $x^{-1}$ has been studied in the past in this context---we extend this study to general mappings and characterize the 2-breaking of APN functions. Recently, two generalizations of the APN property have been introduced: $k$-strongly non-normality and $k$-th-order sum-freedom. Sum-freedom generalizes the non-vanishing property of APN functions to higher dimensional flats. We provide in-depth observations of the relations between the breaking property, strongly non-normality and sum-freedom. We introduce a fourth concept called $k$-strongly breaking, which implies the breaking property. We derive several structural results for both notions and give a characterization of a subclass of APN functions in terms of the 2-strongly breaking property. We propose a different perspective of the non-vanishing property via a natural character transformation, which is closely related to the sum-of-square indicator of the components of $F$. We derive a precise value for the total sum of the sum-of-square indicators of $F$. With this approach, we provide a simple answer to Open Problem 4 in IEEE Trans. Inf. Theory 52(9): 4160-4170, 2006. Moreover, it allows us to explore balancedness properties of polynomials, one of which characterizes component-wise APNness, for odd $n$, and provides a natural extension to any dimension. We show that Dillon's APN permutation and the Gold functions satisfy a related property, termed $k$-balanced, which is presented under our framework.

cs.IT↗
arXiv2026

Rethinking Vision Architectures with Gated Linear Attention and KAN

Vision Transformers devote most of their parameters to MLPs for channel mixing, but still rely on quadratic multi-head self-attention for token interactions. While linear attention fixes the complexity problem, bringing it down to O(N), it is usually just paired with the same fixed-activation MLP as before. Kolmogorov-Arnold Networks take a different approach, placing learnable univariate functions on the edges instead. However, existing vision KANs either retain standard attention or remove attention entirely, so the two ideas have not been effectively combined. We introduce LKAT (Linear Kolmogorov-Arnold Transformer) to close this gap: an isotropic ViT-style encoder that couples chunk-wise Gated Linear Attention with a two-layer KAN feed-forward block, backed by an I/O-aware fused RBF-KAN kernel to make radial-basis grid functions efficient in practice. Under a shared DeiT-style training recipe, LKAT-B outperforms ViT-B/16, ViT-5-B, and Mixer-B/16 on ImageNet-100, while Tiny, Small, and Base variants scale consistently on CIFAR-10/100. ImageNet-100 pretraining also transfers effectively to CIFAR fine-tuning, suggesting that gated linear attention and KAN-based radial basis functions provide complementary inductive biases for mid-scale visual representation learning. Code: https://github.com/mehizelali/linear-kan-transformer

cs.CV↗
arXiv2026

The $ϕ$-conjugation of quaternionic matrices and generalized Autonne-Takagi factorization

Let $ϕ$ be a quaternion of modulus $1$. In this article, we study some topics related to $ϕ$-conjugation for quaternionic matrices, including $ϕ$-Hermitian matrices, $ϕ$-conjugate normal matrices, unitary $ϕ$-congruence and $ϕ$-HSH decomposition (decomposition of a $ϕ$-Hermitian matrix and a skew $ϕ$-Hermitian matrix). In particular, we generalize the Autonne-Takagi factorization of quaternion $ϕ$-Hermitian matrices for all unit quaternion $ϕ$. This gives an affirmative answer to a problem proposed by R. Horn and F. Zhang in the paper ``A generalization of the complex Autonne-Takagi factorization to quaternion matrices, Linear Multilinear A. 60: 1239--1244, 2012''.

math.RA↗
arXiv2026

$\boldsymbol{i}$-conjugate for quaternionic matrices and related properties

Motivated by the result that a complex $n\times n$ matrix $A$ being unitarily equivalent to a real matrix, we extend the conclusion to the quaternion skew field in this paper, we present a necessary and sufficient condition for that a quaternion $n\times n$ matrix $A$ is unitarily equivalent to a complex matrix. To state the truth more clearly, we put forward the concept which we call $\boldsymbol{i}$-conjugate. Furthermore, we study the concepts related to $\boldsymbol{i}$-conjugate and their properties, such as unitary $\boldsymbol{i}$-congruence, $\boldsymbol{i}$-conjugate normality and $\boldsymbol{i}$-Hermicity in $M_{n}(\mathbb{H})$ as generalizations of the conventional unitary congruence, conjugate normality and Hermicity of matrices in $M_{n}(\mathbb{C})$. Finally, we present a new type of polar decomoposition of quaternion matrices.

math.RA↗
arXiv2026

The interplay between active galactic nucleus photoionization, radio jet, and star formation in the z $\sim$ 3.5 radio galaxy 4C +03.24

High-redshift radio galaxies (HzRGs) are among the most powerful radio sources, and are associated with the most massive galaxies and dense environments at redshifts z $\gtrsim$ 1. They are ideal laboratories for studying how active galactic nucleus (AGN) events can shape the evolution of galaxies, as intense radiation, jets, and star formation can be observed simultaneously in these galaxies. We present JWST/NIRSpec integral field spectroscopy ($\sim$ 1.6 kpc spatial resolution) of the 4C +03.24 system, a powerful HzRG at z $\sim$ 3.5 with a bolometric luminosity of $\sim 10^{47.6}$ erg s$^{-1}$. We identified kinematically disturbed regions in the warm ($\sim 10^4$ K) ionized gas by decomposing the emission-line spectra into multiple Gaussian components, which is crucial to avoid overestimating the outflow properties. The outflow power peaks at $\sim$ 2 kpc away from the nucleus, with a corresponding low kinetic coupling efficiency of $\sim 8_{-5}^{+7} \times 10^{-3}$ %. A combined analysis of the rest-frame optical and ultraviolet (from VLT/MUSE and HST imaging) continua revealed an extended emission (spanning $\sim$ 14 kpc), which we interpret as partially tracing star-forming regions. With a clearly delineated bipolar morphology, we show that the AGN photoionization dominates the ionization of the interstellar medium along the radio jet axis. The [C II]$λ$158$μ$m emission gap in this region might be direct evidence of negative AGN feedback. We also discuss a possible scenario where 4C +03.24 could be situated in an overdense environment experiencing multiple galaxy interactions and the possibility of jet-induced star-formation.

astro-ph.GA↗