Papers

We construct Anderson-type operators in the full subcritical regime via a SPDE argument. Our work encompasses perturbations by a Gaussian noise of the fractional Laplacian on the torus or on the full space, and of the harmonic oscillator on the full space. In every case, we are able to derive an optimal estimate on the difference between the solution and the deterministic heat flow, which allows us to get a Weyl law for the fractional case on the torus and for the harmonic oscillator case, and so in the full subcritical regime. For the non-translation-invariant operator, the main novelty is that we develop a general argument in order to prove how to renormalise the operator with some constants. The SPDE argument uses the flow approach introduced by Duch. To construct the operators, we use the Hille-Yosida theorem for the fractional Laplacian on the torus and the harmonic oscillator, and the Klein-Landau theorem for the fractional Laplacian in the full space.

Fréchet regression provides a framework for predicting random object-valued responses by minimizing the conditional expected squared distances. We propose semi-supervised kernel ridge Fréchet regression (SS-KRFR) to exploit unlabeled covariates when labeled samples are scarce. The method learns kernel spectral features from both labeled and unlabeled covariates, then utilizes labeled responses to estimate the conditional Fréchet function by kernel ridge regression. We derive prediction error bounds that distinguish the roles of labeled and unlabeled samples, and establish polynomial convergence rates under suitable conditions. Under additional kernel assumptions, a sharper analysis relaxes the sample size requirements for learning the features. We also develop hybrids that combine features, predictions, or both from models fitted with and without unlabeled covariates, establishing uniform stability over mixing weights in Hadamard spaces at common tuning and a separate rate guarantee for prediction mixing. Simulations and a real-data application show semi-supervised gains and improvements over competing Fréchet regression methods in several settings.

In 1999, A. Smoktunowicz solved a long-standing problem and constructed a simple nil ring. We present a substantial simplification of this construction found by Aristotle (Harmonic) with the assistance of ChatGPT.

We study quantum state tomography when the unknown state lies in a trace-distance ball of radius $R$ around a known reference state $\rho$. For arbitrary collective measurements, we determine the minimax expected risk and confidence radius up to universal constants, uniformly over the dimension, center spectrum, radius, and sample size. Write $\Gamma=d(\mathrm{Tr}\sqrt\rho)^2+d^2R$. For $d\ge2$, $0<R\le1$, and $n\ge1$ copies, the expected risk is $\Theta(\min\{R,\sqrt{\Gamma/n}\})$. At failure probability $0<\delta\le1/3$, the minimax confidence radius is \[ \Theta\!\left(\min\left\{R,\sqrt{\frac{\Gamma+\log(1/\delta)}{n}}\right\}\right). \] For $\varepsilon\le c_0R$ with a universal $c_0>0$, the copy complexity is $\Theta((\Gamma+\log(1/\delta))/\varepsilon^2)$. For the lower bound we use a weighted trace-norm witness to retain all spectral scales in a square-root perturbation family. For the upper bound, we give a direct spectral and concentration analysis of a modified existing purification-based procedure.

We consider multivariate Poisson autoregressive processes evolving on random networks generated by a Stochastic Block Model. This framework extends existing network Poisson autoregressions by incorporating latent community structure and allowing for non-Markovian dependence, both excitatory and inhibitory. We establish a mean-field approximation and prove that the finite system converges to its limit at the optimal rate $N^{-1/2}$ in the quenched setting with overwhelming probability, both on average and in uniform norm. Our analysis yields quantitative error bounds that are explicit in the time horizon and, under suitable stability conditions, remain polynomial or uniform in time, thereby avoiding the exponential growth typically associated with Grönwall-type arguments. As a consequence, the mean-field approximation is shown to hold uniformly over any time horizon. We further demonstrate how the approximation can be exploited to reduce the computational complexity of parametric inference from $\mathcal{O}(N^2T^2)$ to $\mathcal{O}(NT^2)$. The results provide a theoretical foundation for the analysis and inference of count-valued processes on large random networks with community structure.

Let $F$ be a non-Archimedean local field of characteristic zero, let $D/F$ be the quaternion division algebra, and put $G_n = GL_n(D)$ and $H_n = Sp_n(D)$. We prove Prasad's revised conjecture on tempered representations with symplectic period. If $\rho$ is a unitary $H_r$-distinguished supercuspidal representation and $\delta_m(\rho)$ denotes the generalized Steinberg representation associated to $\rho$, set $B_0(\rho) = \rho, B_m(\rho) = \delta_{m+1}(\rho) \times \delta_m(\rho), m \geq 1.$ We prove that an irreducible tempered representation $\pi$ of $G_n$ is $H_n$-distinguished if and only if $\pi$ is isomorphic to the product over $i$ of $B_{m_i}(\rho_i),$ where each $\rho_i$ is a unitary distinguished supercuspidal representation.

By making use of partial information decomposition, we analyse information transmission through a local processor which receives two streams of integrated inputs. We revise previously stated conditions for contextual modulation that are based on the components of partial information decomposition, and introduce a condition for the presence of signal integration. Exact results are obtained for information-theoretic measures and the components of partial information decomposition using the Ibroja method. A general class of activation functions is considered along with a particular divisive normalisation transfer function. These results are applied with several activation functions, of additive, multiplicative, subtractive and divisive forms, with graphical illustration, and the conditions for modulation and integration are tested in each case. The revised information-theoretic conditions are also applied to noisy data obtained from a realistic simulation of a model layer 5b pyramidal cell in three distinct regimes defined by experimental design and characterised by the values of the input basal and apical amplitudes. Evidence for driving basal input with modulating apical input, driving apical input with modulating basal input, and of apical-basal signal integration is presented.

We study a log-Hardy-Littlewood-Sobolev (log-HLS) inequality under constraints and a naturally induced Wasserstein flow. The constrained log-HLS inequality is dual, in a variational sense, to the constrained Trudinger-Moser-Onofri-Aubin inequality. The corresponding gradient-flow formulation leads to a parabolic-elliptic Patlak-Keller-Segel (PKS) system whose global dynamics exhibit a dichotomy between global existence and finite-time blow-up depending on the initial mass. Remarkably, the constraint raises the critical mass threshold of the flow by a factor of two compared with the unconstrained setting. This provides a distinct mechanism for suppressing blow-up from those arising in related formulations of the PKS-system, where blow-up is mitigated through additional effects such as a forcing term, coupling with a fluid equation, or the ambient flow generated by harmonic perturbations of the Newtonian potential.

We develop an algebraic-dynamical correspondence for type-I oscillatory blow-up solutions of autonomous ODEs with asymptotically quasi-homogeneous vector fields. Periodic solutions of the balance law arising from asymptotic expansions of blow-ups are related to periodic orbits on the horizon for the desingularized vector field. Unlike the stationary case, the correspondence involves a positive periodic scaling function and a nontrivial time reparametrization. We further establish the correspondence of linearized structures: distinguished phase and scaling directions generate invariant subbundles, while the remaining characteristic multipliers are identified through associated quotient bundles. These results yield a criterion for the existence of type-I periodic blow-up solutions based on periodic solutions of the balance law and their characteristic multipliers, without explicitly constructing compactifications or periodic orbits at infinity. The theory also clarifies the difference between stability information in the balance law and in dynamics at infinity. Two examples illustrate the correspondence.

We determine the sharp uniform constant in the quartic stability estimate for the logarithmic Sobolev inequality on finite cycles. We equip $C_{n}=\mathbb{Z}/n\mathbb{Z}$ with the uniform probability measure, and we let $\mathcal{E}_{n}$ be the Dirichlet form of the simple random walk on $C_{n}$, normalized so that its spectral gap is one. Frank and Ivanisvili recently proved the sharp logarithmic Sobolev inequality $2\mathcal{E}_{n}(u)\geq \mathrm{Ent}(u^{2})$ for all $n\geq 4$. We prove that $2\mathcal{E}_{n}(u)-\mathrm{Ent}(u^{2})\geq \frac{1}{3}\Vert u-1\Vert _{2}^{4}$ for all $n\geq 4$ and all nonnegative $u$ with $\Vert u\Vert _{2}=1$. This is the sharp form, uniformly in $n$, of the quartic stability estimate of Xie and Zhang, who obtained the constant $\frac{1}{12}$. The same estimate holds on the circle and for the word-length energy on $C_{n}$. Among these models, the four-cycle is the only extremal one: on $C_{4}$ the constant $\frac{1}{3}$ is sharp but not attained, while on $C_{n}$ with $n\geq 5$ and on the circle the sharp constant is at least $\frac{1}{3}+c$, where $c>0$ does not depend on $n$. We also give explicit upper bounds for these sharp constants. Our second main result is a stability estimate for the cubic Sobolev inequality of Frank and Ivanisvili, which is the main step in their proof. It gives the quantitative version of this inequality expected by Frank and Ivanisvili, with the optimal powers of $\Vert u-1\Vert _{2}$: four on $C_{n}$ with $n\geq 5$ and on the circle, and six on $C_{4}$. Two key new ingredients of the proofs are a bound for the first Fourier coefficient of $u$ in terms of its mean, which holds because $u$ is nonnegative, and a sharp stability estimate for Gross's two-point logarithmic Sobolev inequality, with sharp constants in both the fourth and the sixth powers of the distance.

In this paper, we construct a series of smooth affine varieties of dimension $n \geqslant 3$ that are generically flexible but not flexible. Moreover, these varieties admit a nontrivial divisor invariant under the subgroup of special automorphisms.

Generative foundation models are attracting interest for their ability to produce desired outputs from demonstrations given at inference time, without updating parameters. However, since a few demonstrations cannot uniquely identify the intended task, the challenge is how to learn and sample from an output distribution that reflects this task uncertainty. In this work, we theoretically analyze how a Diffusion Transformer (DiT), pretrained across diverse tasks, learns and generates predictive distributions for a new query from demonstrations. We first show that the natural target to generate from finite demonstrations is not an output derived from estimating a single task, but rather a predictive distribution that captures the task uncertainty remaining after observing the demonstrations. We then prove that a DiT can learn this predictive distribution through score estimation, using attention to aggregate information from demonstrations and diffusion to generate samples. Owing to this property, with sufficient pretraining resources and diffusion sampling steps, the resulting DiT achieves the minimax optimal rate over a Hölder class of test-time tasks. These results imply that DiT acts as a statistically grounded in-context generator capable of generating distributions adapted to new tasks while retaining the uncertainty inherent in finite demonstrations.

Consider a sample of size $N$ from a two-layer hierarchical Pitman–Yor model, and let $\xi(N)$ denote the random number of clusters generated at the first layer of the hierarchy. We study moderate deviation principles for partition statistics evaluated at the random sample size $\xi(N)$. In particular, we establish moderate deviation principles for the total number of clusters $K_{\xi(N)}$ and for the frequency count $M_{l,\xi(N)}$, the number of upper-level clusters represented by exactly $l$ first-level clusters. Our analysis combines fixed-sample-size small-tilt asymptotics with a moderate deviation principle for the random index $\xi(N)$. A uniform small-tilt argument, together with exponentially weighted tail estimates, allows us to pass from deterministic sample sizes to the hierarchical random-index setting. We obtain explicit deviation speeds and good rate functions on a family of intermediate scales between the typical growth order $N^{\alpha_1\alpha_2}$ and the linear scale $N$. The resulting rate functions exhibit a common power-law structure governed by the product $\alpha_1\alpha_2$, revealing a multiplicative effect of the two levels of the hierarchy.

Approximate Message Passing (AMP) algorithms are attractive as they are computationally efficient and simultaneously admit a precise characterization in terms of the low-dimensional "state-evolution" recursion. In this work, we establish quantitative universality for AMP with centered rank-one sensing matrices $Z_i=(x_ix_i^\top-I_d)/\sqrt d$, where $x_i$ are independent standard Gaussian vectors. These matrices arise in quadratic regression and learning quadratic neural networks. Their normalized vectorizations have isotropic covariance in dimension $p=d(d+1)/2$, but strongly dependent coordinates. For linear observations with Gaussian noise and prescribed smooth, bounded spectral denoisers, we compare rank-one sensing with a covariance-matched isotropic Gaussian ensemble. At a fixed iteration horizon, with $n\leq Cp$ and uniform bounds on the normalized signal energy, initialization, coefficients, and first four denoiser derivatives, normalized signal overlaps, cross-time overlaps, and smooth linear spectral statistics differ by $O(d^{-1/2})$ in every fixed $L^m$. This yields universality of the normalized mean-squared error, fixed spectral moments, and empirical spectral distributions. Whenever the Gaussian observables admit a joint state-evolution limit, the same predictions hold under rank-one sensing. The proof uses columnwise Lindeberg replacement, leave-one-out analysis, smooth spectral expansions, and high-moment concentration. For prescribed smooth spectral denoisers, the result establishes the AMP universality conjectured by Maillard et al. (arXiv:2408.03733).

The corpus callosum (CC) is a major white-matter structure and a well-established marker of brain aging, but most studies quantify it using scalar summaries that discard its boundary geometry. We present a Riemannian shape-space framework \cite{fletcher2013geodesic-regression} for analyzing age-related morphological change in the midsagittal CC, applied to the OASIS-1 cohort. Each contour is represented by $128$ landmarks and embedded into Kendall shape space, where translation, rotation, and scale are removed. We derive a multivariate geodesic regression with exact Riemannian gradients and use the fitted age-velocity field to localize age-related deformation to five anatomical sub-regions. In the cognitively normal cohort ($n = 252$), geodesic regression outperforms the Euclidean linear benchmark ($R^2 = 0.1355$ vs.\ $0.1216$). Regional energy is anterior-dominant: the Genu carries $46.1\%$ and the Anterior Body $23.3\%$ of total age-related shape change, together accounting for $\sim 69\%$ despite comprising only $\sim 41\%$ of landmarks. Signed projections confirm the ordering (Genu $r = 0.562$; Anterior Body $r = 0.507$). In contrast, age explains less than $0.5\%$ of shape variance in Alzheimer's disease ($n = 88$), indicating that the disease disrupts the healthy aging trajectory. A tangent-space classifier achieves an age-group AUC of $0.791$ from the 2D contour alone, exceeding a recent volumetric benchmark ($0.67$).

Let $\mu$ be a doubling Radon measure in $\mathbb R^d$ with support $X$. We prove a local and quantitative form of the Bennewitz–Lewis theorem in this setting: if a Radon measure $\nu$ satisfies the Bennewitz–Lewis condition on a bounded relatively open set $V\subset X$ with a small enough constant, then $\nu|_V\ll\mu|_V$ and its density satisfies a reverse Hölder inequality on every compact subset $K\subset V$. Along the way we develop the dyadic toolkit these arguments run on — triple cubes defined through neighbours, the basis $\mathcal{D}\cup3\mathcal{D}$, and a Whitney decomposition — and we collect the Muckenhoupt-type classes of weights over that basis. The same machinery yields a local Gehring lemma for enlarged balls: both theorems are deduced from a single bootstrapping lemma, stated once for $p=1$ and $p>1$.

An addition formula is a closed-form formula for reproducing kernels of orthogonal polynomials of several variables; the latter are kernels of orthogonal projection operators. We derive addition formulas for two families of orthogonal polynomials, which consist of wrapped products of classical Gegenbauer polynomials on the unit ball with either classical Jacobi polynomials on the simplex or product Laguerre polynomials, each of which is given explicitly as a multilayer of integrals of a classical orthogonal polynomial of one variable. These formulas provide a powerful tool for analysis and allow us, in particular, to reduce a substantial portion of analysis on the Fourier orthogonal expansions on the domains to that of one variable.

We develop and analyse a conforming piecewise affine finite element method for the Grad–Mercier equation, a nonlocal elliptic model for magnetic equilibria whose source depends on the global rank of a flux function. The principal difficulty is that the unique continuum equilibrium has a maximum plateau of prescribed positive area. The solution therefore samples the rank law only above the plateau mass, while perturbations that separate equal values may distribute this mass over several numerical ranks and thereby alter the non-strict rank source. We exploit the continuum well-posedness structure to replace the rank law by an equivalent clipped, nonnegative law, and replace Lebesgue measure by a mass-lumped nodal measure. Equal nodal values are assigned a common rank interval, thereby retaining the full lumped mass of each tied level. The resulting weighted rearrangement gives a strictly convex discrete energy and hence a unique finite element equilibrium. On meshes satisfying a discrete comparison principle, the discrete equilibrium is nonnegative and bounded above by a computable barrier. A Poisson-preconditioned rearrangement iteration converges on each fixed mesh and admits computable a posteriori energy and energy-norm bounds. As polygonal finite element domains converge to the physical cross-section, the discrete equilibria converge strongly in $H^1_0(\Omega)$ and uniformly on $\overline\Omega$ to the unique Grad–Mercier equilibrium. Reconstructed equilibrium sources converge weak-* in $L^{\infty}(\Omega)$ to the original nonlocal source.

We study the two-dimensional periodic resistive compressible magnetohydrodynamic equations with constant shear viscosity and magnetic diffusivity, bulk viscosity $\rho^\beta$, and pressure $\rho^\gamma$. For every $\beta>1$ and $\gamma>1$, we construct global finite-energy renormalized weak solutions from strictly positive bounded $W^{1,q}$ densities, $q>2$, and arbitrary $H^1$ velocity and magnetic data. The construction combines approximation-independent finite density moments, generalized effective-flux identities, and a joint logarithmic renormalization of the density and its entropy defect. An augmented relative energy containing the $\beta$-power density potential yields weak–strong uniqueness against positive $H^2$ strong solutions for the same full pressure range. We also establish global strong existence for a large-background family when $\gamma>\max\{5-2\beta,\beta+1\}$. Writing $m=\int\rho_0$, the density perturbation is bounded in the scaled norm $m^{(\gamma-3)/2}\|\rho_0-m\|_{H^2}$, while the fixed $H^2$ velocity and magnetic norms need not be small. For sufficiently large $m$, frequency-dependent acoustic dissipation, staged vorticity weights, and a separate initial-layer flux estimate close a simultaneous bootstrap. The admissible pressure exponents have infimum $7/3$, with the endpoint excluded. For $\beta>2$ the relevant damping rate is the minimum of the viscous and pressure-relaxation rates; keeping both rates extends the argument without an upper restriction on $\beta$. The strong theorem has no upper restriction on $\gamma$ under this scaled density hypothesis; for fixed additive density perturbations it applies in the stated lower range up to $\gamma=3$.

We study the realization problem for point schemes of skew polynomial algebras. We approach this problem combinatorially by translating it into the study of certain $3$-uniform hypergraphs, which we call point hypergraphs. We first give a homological criterion characterizing point hypergraphs. Using this criterion, we study hypergraphs obtained by deleting triangles from triangulations of connected orientable closed surfaces. Deleting exactly one triangle always gives a non-point hypergraph. When the triangulation has no separating nonfacial $3$-cycles, deleting either no triangles or at least two triangles gives a point hypergraph, and the one-triangle deletions are minimal non-point hypergraphs with respect to taking induced sub-hypergraphs. It follows that point hypergraphs cannot be characterized by finitely many forbidden induced sub-hypergraphs. Next, for each point hypergraph, we construct an affine moduli variety of skew polynomial algebras realizing it, and determine its dimension. For point hypergraphs arising from the above surface construction, we obtain an explicit dimension formula in terms of the number of vertices, the Euler characteristic, and the number of deleted triangles. Finally, we characterize point hypergraphs on six vertices in terms of a four-vertex local condition together with a single exceptional obstruction.

We show that, under some mild assumptions, every quasi-pointed category C carries a canonical near-torsion theory (T,F), where T consists of the 0-supported objects, that is, those admitting a morphism to the initial object 0, and F consists of the subterminal objects. The coreflection and the reflection of an object A are then given by the projection $A \times 0 \to A$ and by the strong-epimorphic part of $A \to 1$, respectively. Using the general construction of the "largest" non-pointed torsion theory associated with a near-torsion theory, we prove that (T,F) is a (non-pointed) torsion theory in the full subcategory of those objects A for which the canonical sequence $t(A) \to A \to f(A)$ is short exact, in the sense that it is both a kernel and a cokernel diagram. This subcategory is the whole of C whenever C is sequentiable, that is, a regular protomodular and quasi-pointed category. We describe the resulting torsion theories in the category of Lie-Rinehart algebras and in the categories of crossed modules over a fixed group. We also show that (T,F) is a torsion theory in the (non-protomodular) category of monoids over a group G all of whose elements have finite order.

Equi-isoclinic (EI) and equichordal (EC) tight fusion frames (TFFs) are types of minimally coherent (maximally orthogonal) subspace packings that arise in several applications, such as compressed sensing. They seem challenging to construct, and the problem of characterizing their existence remains mostly open. In this paper, we characterize those special complex EITFFs and ECTFFs that arise from isometries whose cross-Gram matrices commute, as is the case with filter banks. In particular, we show that every filter bank EITFF is equivalent to a direct sum of equiangular tight frames (ETFs). More generally, we show that every filter bank ECTFF is analogously equivalent to a novel generalization of ETFs that we dub stratified unit norm tight frames (SUNTFs). We further establish a general theory of SUNTFs, providing various explicit constructions of them, and some necessary conditions on their existence. All of our techniques are explicit, yielding direct constructions of incoherent filter banks.

Given a morphism of Abe-Bott-Samelson bimodules, we construct an algorithm for computing its double-leaves coefficients. The key ingredient is a modification of Bruhat upper-triangularity which allows us to use forward substitution to solve for the double-leaf coefficients. We provide a complete implementation of the algorithm in the Sage computer algebra system. We use this to compute the Zamolodchikov 3-color relation in type $H_3$ for balanced realizations. We further simplify the 3-color relation using thick Soergel calculus and express it as a difference of two thick Elias–Williamson diagrams.

We show that there exists $C\in\mathbb{N}$ such that $K_t$-minor free graphs are $Ct$-colorable. The proof was found by GPT-6 Astra, following the directions by the authors.

We consider the initial value problem of the semilinear heat equation in the Fujita supercritical range for which global-in-time solutions may exist. The main result shows the existence of a separating norm $\|\cdot\|_X$ in the sense that, for nonnegative initial data $u_0\in L^\infty(\mathbf{R}^n)$, the smallness of $\|u_0\|_X$ implies global-in-time existence, whereas the largeness of $\|u_0\|_X$ implies finite-time blow-up. The separating norm is unique up to equivalence on nonnegative data.

A separable metric space has strict measure zero if its packing measure induced by any gauge is zero. We provide a few characterizations of strict measure zero, look at cartesian products and prove that in every Polish locally compact group every set of strict measure zero is null-additive and that in the Cantor set, Euclidean spaces and tori the two notions are equivalent.

The quantum $n$-coloring problem, given graph $G$, asks whether there exists a perfect quantum strategy for the $n$-coloring game of $G$. Previously it was known that quantum $n$-coloring is undecidable only for $n=3$. We extend this to all $n\geq 3$ with one elementary reduction. Letting $G'=(G\square K_3)\lor K_{n-3}$, we show that $G$ is quantum $3$-colorable if and only if $G'$ is quantum $n$-colorable.

Let $X: M^n\to \mathbb S^{n+1}(1)$ be a closed minimal hypersurface with constant scalar curvature. Chern conjecture says that a closed minimal hypersurface with constant scalar curvature, $S\geq 2n$ if $S>n$, where $S$ is the squared norm of the second fundamental form. As a partial result, Peng-Terng \cite{pt1, pt2}, Yang-Cheng \cite{yc1, yc2, yc3} and Suh-Yang \cite{sy} obtained that if $S>n$, then $S>n+ \dfrac{3}7n$. Recently, Chen \cite{c} has proved $S>n+\frac{26}{59}n$ if $S>n$. Since resolving Chern conjecture is a hard problem, as a midway for resolving Chern conjecture, one wants to prove $S>n+\frac12 n$ if $S>n$. In this paper, we give a positive solution for this midway problem. We should remark that our topological condition is complete which is weaker than closed in Chen \cite{c}.

Projection-free methods replace projections by linear minimization oracles, making their convergence fundamentally sensitive to the geometry of the feasible region. We develop a geometric framework for this dependence in Riemannian Frank–Wolfe optimization. A power-type double-geodesic modulus quantifies the intrinsic geometry of the feasible set, and an exact transfer principle shows that this modulus induces a Frank–Wolfe scaling margin modulated by the differential of the Riemannian logarithm map. The resulting factorization separates the contributions of feasible-set geometry and ambient Riemannian distortion. Rauch comparison provides explicit curvature-dependent certificates for this distortion, while on the positive-definite cone equipped with the affine-invariant metric the logarithm differential admits an exact spectral characterization in terms of relative eigenvalue spread. Combined with a quadratic error bound, the power-type geometry yields the convergence rate $O(t^{-p/(p-1)})$ for the gap-feedback short-step policy. For every fixed $p>3$, a compact feasible set in a hyperbolic plane attains the matching rate $\Theta(t^{-p/(p-1)})$ under the same policy. The geometry-derived exponent $p/(p-1)$ is therefore sharp for the gap-feedback short-step policy over the geometry-controlled class considered here.

We investigate the global existence and energy decay properties of a weakly coupled evolution system consisting of two wave equations in a bounded domain. The system exhibits a unique structural asymmetry: a nonlinear logarithmic damping term acts exclusively as a friction-like feedback mechanism on the first wave equation, while the second equation remains completely undamped and relies solely on linear cross-coupling to transfer and dissipate its energy. We establish two primary objectives. First, by utilizing the Faedo-Galerkin approximation scheme in conjunction with the structural monotonicity properties of the non-Lipschitz logarithmic nonlinearity, we prove a rigorous global existence result for strong solutions. Second, we provide a rigorous mathematical proof demonstrating the stability and asymptotic energy decay of the solutions under suitable constraints on the cross-coupling parameter. These results provide new insights into the stabilization mechanisms of coupled hyperbolic systems under degenerate nonlinear damping.