Papers

We establish new recursions for Hurwitz class numbers with polynomial weights. In contrast to previous recursions, our results decouple class numbers of even and odd discriminants. Our main tool is the vector-valued holomorphic projection operator applied to mock modular forms. We invoke representation theory to connect the relevant spaces of vector-valued modular forms to spaces of classical new and old forms. We thereby leverage the vanishing of spaces of vector-valued cusp forms not available in the scalar case.

We model Rozansky-Witten theory of $T^*X$ using modules over $\operatorname{Perf}(X\times\mathbb{A}^1)$ viewed as sheaves of categories over $X\times\mathbb{A}^1$. This is in parallel to Tamarkin's approach to nonconic Lagrangians in $T^*X$ via the singular support of sheaves on $X\times\mathbb{R}$. More specifically, we construct the objects of Rozansky-Witten theory of $T^*X$ given by hybrid Lagrangians of which graphs and conormals are special cases. As a consistency check, we show that $\operatorname{Hom}$s between such objects are given by suitable matrix factorizations.

For compact Kähler manifolds $(M,g,J)$ with negative scalar curvature we study the existence problem for non-trivial Einstein deformations of $g$, that is small time curves $g_t$ of Einstein metrics with $g_0=g$. No asssumption on the complex structure $J$ is made; also we do not assume that the metrics $g_t$ are Kähler w.r.t. $J$. We determine explicitly the obstruction to third order Einstein deformation for $g$; that is we fully solve the equations $(\Ric^{g_t})^{(k)}(0)=0$ for $1 \leq k \le 3$ in terms of the Taylor expansion $g^{-1}g_t=\id+th_1+\tfrac{t^2}{2!}h_2+\tfrac{t^3}{3!}h_3+o(t^4)$ at $t=0$. Up to a suitable gauge transformation we show that third order integrability for the Einstein equation amounts to Maurer-Cartan type equations and polynomial identities relating the coefficients $h_3,h_2,h_1$. This result is interpreted in terms of the underlying complex geometry of $M$ by means of the Cayley transform of the metric $g$; the Cayley transform is also used for formulating conjectures for the higher order Einstein deformation problem.

We show that H-convergence for multiplication type operators as envisioned by Murat and Tartar in the 1970's always implies nonlocal H-convergence as introduced in 2018 in Calc.~Var.~PDE 57(6):159. In contrast to earlier findings, the results presented here work for arbitrary space dimensions, are not bound to a certain geometry of the underlying domain, and do not explicitly require an underlying Hilbert complex for the application of any particular version of the div-curl lemma. We extend classical theory and the main results to more general differential operators with different boundary conditions and orders. Furthermore, the present results confirm homogenisation formulas used in the literature of which we failed to find an explicit proof. As a consequence, H-convergence for multiplication operators in divergence form problems will always imply H-type convergence for a different variational problem for free.

We present a novel parameter-free discontinuous Galerkin (dG) finite element method (FEM) for the time-dependent Maxwell system formulated as a saddle point problem. We establish the stability of the proposed semi-discrete problem and derive optimal error estimates in energy and \( {\bf L}^{2} \) norms for the electric field variable, as well as in \( L^{2} \) norm for the potential function. To the best of our knowledge, this work provides the first optimal \( {\bf L}^{2} \)-norm error analysis for the second-order time-dependent saddle point Maxwell equations using any variants of FEMs. Additionally, we propose several complete discrete time-integrators and verify the optimal convergence results through examples in both 2D and 3D setups.

Myroshnychenko, Tatarko, and Yaskin constructed a body $K$ in $\mathbb{R}^n$, $n \geq 5$, with the property that there is exactly one hyperplane $H$ passing through $c(K)$, the centroid of $K$, such that the centroid of $K\cap H$ coincides with $c(K)$. This construction provided answers to questions of Grünbaum and Loewner for $n\geq 5$, which are still open in dimensions $3$ and $4$. We study analogues of these questions in the settings of hyperbolic space $\mathbb H^n$ and $s$-concave functions on $\mathbb R^n$.

In this paper, we develop a novel class of high-order uniformly accurate exponential integrators for charged-particle dynamics under strong magnetic field. The small parameter $0<\varepsilon\ll 1$ induces rapid temporal oscillations, rendering traditional numerical methods prohibitively expensive due to severe step-size restrictions. To address this issue, a linearization technology that introduces auxiliary polynomial variables is employed to recast the original charged-particle dynamics as a higher-dimensional system. Classical exponential integrators are subsequently applied to this augmented formulation, which inherently carries richer structural information, thereby yielding a family of uniformly accurate exponential integrators that can reach arbitrarily high order without requiring any order conditions. For strong magnetic fields under the maximal-ordering scaling, we rigorously establish through algebraic techniques that the schemes with auxiliary polynomial variables of degree $k\geq2$ achieve an improved position error bound of $\mathcal{O}(\min\{\varepsilon^{k+1}h,\varepsilon h^{k+1}\})$ and a uniform velocity error bound of $\mathcal{O}(\min\{\varepsilon^{k}h,h^{k+1}\})$, where $h$ denotes the time-step size. Under the additional condition that the magnetic field gradient vanishes at the origin, these bounds sharpen to $\mathcal{O}(\min\{\varepsilon^{k+2},\varepsilon h^{k+1}\})$ and $\mathcal{O}(\min\{\varepsilon^{k+1},h^{k+1}\})$ for the position and velocity, respectively, for $k\geq1$. The same sharper bounds hold for constant strong magnetic fields. Numerical experiments confirm the advantages of the proposed methods.

In this paper, we study the gradient descent dynamics for jointly training both layers of a one-hidden-layer ReLU network to fit a linear target function. Concretely, we consider a realizable setting where inputs are drawn i.i.d. from a Gaussian distribution and labels follow a planted linear model. This stylized framework captures salient features of end-to-end training in inverse problems and certain auto-encoder models. Despite its apparent simplicity, the dynamics remain poorly understood, in part because the loss landscape contains multiple non-strict saddle points, making it unclear why gradient descent from random initialization reliably escapes bad stationary regions. We provide a detailed characterization of the optimization landscape and prove that gradient descent from a moderately small random initialization-simultaneously training both layers-converges to a global minimizer at a linear rate with order-wise optimal sample complexity. Our analysis tracks the trajectory through three phases: an alignment phase in which hidden weights progressively align with the planted direction while the output weights maintain the correct sign pattern; a growth phase in which the norms of both layers increase while preserving alignment; and a local refinement phase in which the aligned neurons rapidly converge to the planted direction, yielding fast local convergence. To rigorously show that GD avoids non-strict saddles, we develop trajectory-level control arguments for the end-to-end dynamics. In addition, we establish novel uniform concentration results that hold along the entire trajectory, and are essential for obtaining order-wise optimal sample complexity. We corroborate our theory with extensive experiments across a range of configurations.

A mobile phone vibrating on a hard surface often sounds qualitatively unlike ordinary audiovisual recordings when reproduced through laptop loudspeakers. We propose that part of this perceptual distinctiveness can be described as a nested periodicity: a first-order cepstral structure reflecting the vibration period and its multiples, and a second-order cepstral structure reflecting repeated spacing within the first-order cepstrum. Treating the perceptual effect as real and using a deliberately transparent synthetic signal chain, we model six stages: mechanical generation, surface and air propagation, microphone capture, encoding and decoding, laptop-speaker playback, and re-recording or post-processing. The synthetic analysis shows that the first-order cepstral periodicity is preserved across the chain, whereas a cleaner bimodal or quasi-bimodal second-order cepstral signature is most evident at the mechanical source and at laptop-speaker playback. The result supports, but does not prove, the hypothesis that laptop reproduction can re-emphasize a latent contact-vibration periodicity that is less cleanly expressed in intermediate recorded and encoded forms. We frame second-order cepstral bimodality as an exploratory descriptor of contact-vibration playback rather than as a completed perceptual metric. Required validation includes recordings of real devices, controlled playback transfer functions, perceptual judgments, and comparisons against ordinary speech, music, and environmental recordings.

In recent years, researchers have actively contributed to the field of graphs associated with algebraic structures and ordered structures. It is a fundamental question to ask whether we infer algebraic or ordered structure from associated graphs and vice versa. In this paper, we gave characterizations about comparability graphs and associated lattices. In particular, we determined some properties of lattices that are preserved under the graph isomorphism. We have also provided a technique to construct non-isomorphic lattices with isomorphic comparability graphs. Also, we find two classes of lattices in which the graph isomorphism gives the lattice isomorphism.

We study the evaluation of Rényi multi-entropies $S^{(q)}_n$ in Random Tensor Network (RTN) states in the large bond-dimension limit. For the case of Rényi index $n=2$ and arbitrary number of parties $q$, we prove that that multi-entropies are determined by minimal multiway cuts through the network. When the minimal multiway cut is degenerate, we characterize the full minimizer set via compatible families of minimal cuts and give a criterion for all minimizers to come from ordinary cut partitions. For $n=2$, this gives a natural generalization of the minimal cut description of bipartite entanglement to multipartite systems with arbitrarily many parties. For the case of integer $n>2$, we show that the minimal multiway cut conjecture is in general not true by providing explicit counter examples for both the single random tensor and for the network built from isometric tilings. We discuss the implication for our results on the multipartite entanglement structures in RTN and holography.

Offline multi-objective optimization (Offline MOO) seeks Pareto-optimal designs from static datasets without additional environment interactions. Existing generative methods typically guide sampling with external surrogate or preference models, which adds training complexity and may provide unreliable guidance. We propose ParetoPilot, a plug-and-play method that guides designs to Pareto front at inference time using a pre-trained conditional diffusion model without any surrogate. ParetoPilot introduces an Infer-Perturb-Guide (IPG) engine within the reverse diffusion process. IPG first infers the individual conditional target for each sample in the batch by aligning its conditional and unconditional predictions. It then perturbs these targets collectively across the batch, balancing convergence toward the Pareto front and diversity among samples. Finally, the engine guides the generative trajectory toward the Pareto front by injecting these perturbed targets via standard Classifier-Free Guidance (CFG). Experiments on 51 tasks demonstrate that ParetoPilot achieves the best overall ranking among 16 methods and competitive hypervolume improvement.

We develop a novel iterative approach for solving the incompressible magnetohydrodynamics problem. The main idea is to split the velocity-momentum and magnetic induction equations with respect to the diffusive terms. As a result, we get a smaller system that is iteration-level-dependent, along with two Stokes systems that need to be assembled only once. We also extended the scheme to the Elsässer variables reformulation of the equations. For both schemes, we established boundedness and convergence. Several numerical experiments are presented to show the effectiveness of the schemes.

Lebesgue's universal cover problem asks for the infimum of the areas of planar sets that contain a congruent copy of every planar set of diameter at most one. We work in the convex Brass$-$Sharifi three-test-set framework, where the test sets are a closed disk, an equilateral triangle, and a regular pentagon of diameter one. For each normalized placement $v$, let $A(v)$ denote the area of the convex hull of these three test sets. We construct a finite certificate proving $A(v)\ge 0.833$ throughout the admissible normalized domain. The threshold $0.833$ yields a certified improvement over the Brass$-$Sharifi lower bound $0.832$ within the same convex three-test-set framework. The proof is a finite-cover argument. The admissible domain is covered by finitely many parameter domains, and each domain carries a local lower-bound certificate. The non-witness domains are certified by supporting local records. On the witness domains, the local bound is obtained from an inner-witness polygon construction. The witness points lie in the three test sets and determine an ordered polygonal region certified to be simple and positively oriented. Its area is bounded below by interval orientation and shoelace estimates. Since this certified polygonal region lies inside the corresponding convex hull, its area gives a lower bound for the hull area. Combining the local inequalities with the finite cover yields $\alpha_{cvx}\ge 0.833$, where $\alpha_{cvx}$ is the infimum of the areas of convex universal covers.

For any $N\geq 2$, we construct a global solution of the energy-critical focusing wave equation in dimension four which blows up in infinite time at $N$ prescribed points $z_1,\ldots,z_N\in \mathbb R^4$, provided that the points form one orbit under a finite group of orthogonal symmetries. We denote by $c:=2\sum_{j\ne k}|z_j-z_k|^{-2}>0$ the corresponding interaction coefficient, which is independent of $k$. The common concentration scale satisfies \[ \log\frac{1}{\lambda(t)} = \left(\frac{9c}{4}\right)^{1/3}t^{2/3}+O(t^{1/3}) \qquad \text{as } t\to+\infty . \] This concentration rate comes from a genuinely four-dimensional effect: the borderline decay of the ground state makes the interaction between different bubbles enter the leading order parameter dynamics.

Real estate valuation is a structured regression problem in which prices are governed by heterogeneous feature types, sparse regional effects, nonlinear interactions, and the practical logic of comparable properties. Standard multilayer perceptrons treat each row as an isolated vector and must learn locality, scale sensitivity, and categorical matching from supervision alone. Gradient-boosted decision trees provide strong tabular baselines, but their feature-centric splitting mechanism does not explicitly model the retrieval of similar historical observations. This paper presents RowNet, a retrieval-based neural architecture for real estate price-per-square-meter prediction. RowNet represents a query property through pairwise similarity features against a memory bank of labeled properties. A first retrieval layer estimates a coarse target from feature-only similarities. A second layer augments the memory comparison with target-consistency features and uses multiple learned attention heads to retrieve complementary comparable sets. A final mixture-of-experts module combines learned gating, residual correction, entropy regularization, and head-diversity regularization to produce the prediction.

This document provides in-depth details for the derivation of the univariate splitting algorithm developed in arXiv:2606.01530. The algorithm approximates the standard, 1-D Gaussian distribution with a mixture of uniformly spaced homoscedastic Gaussian components. The solution is found by minimising the squared $L^2$ norm of the mismatch between the approximation and the original Gaussian. This text presents asymptotic analyses of the proposed splitting in the limit of small step $h$ between the mixand means and in the limit of large number of mixands $M$.

For given graphs $G_{1}, G_{2}, \dots, G_{t}$ and $G$, let $G\rightarrow (G_{1}, G_{2}, \dots, G_{t})$ denote that each $t$-coloring of $E(G)$ yields a monochromatic copy of $G_{i}$ in color $i$ for some $i\in [t]$. The size Ramsey number, $\hat{r}(G_{1}, G_{2}, \dots, G_{t})$ is the minimum size of $G$ such that $G\rightarrow (G_{1}, G_{2}, \dots, G_{t})$. A graph $G$ is a size Ramsey minimal graph for $(G_{1}, G_{2}, \dots, G_{t})$ if $G\rightarrow (G_{1}, G_{2}, \dots, G_{t})$ and $e(G)= \hat{r}(G_{1}, G_{2}, \dots, G_{t})$. A star forest is a vertex-disjoint union of stars, and a uniform star forest is a star forest with the same size of each component. In 1978, Burr, Erdős, Faudree, Rousseau and Schelp, and in 2025, Davoodi, Javadi, Kamranian and Raeisi completely characterized the size Ramsey minimal graphs for uniform star forests. In this paper, we completely characterize the size Ramsey minimal graphs for uniform star forests in multicolors.

When assessing the causal effect of an exposure on two or more outcomes in an observational study, a linear combination of outcomes may lessen the sensitivity of a test of the global null hypothesis to potential unmeasured biases. While all linear combinations of scored outcomes can be considered using Scheffe projections or constrained variants thereof, finding the combination that minimizes sensitivity to unmeasured biases requires corrections for multiple testing, which can erode power, especially when many outcomes are of interest. To mitigate this issue, we propose splitting the sample into a planning sample to identify an optimal linear combination and an analysis sample to conduct inference. We provide a novel characterization of the set of linear combinations for which this approach is guaranteed to achieve the same asymptotic power as full-sample alternatives and conduct extensive simulation studies that demonstrate enhanced power in finite samples. Finally, we apply our method to investigate the effects of poverty on the emergence of cardiovascular disease risk factors in children and adolescents. We discover adverse consequences on outcomes related to body composition, physical activity, and tobacco exposure. Although the impact of poverty on elevated tobacco exposure shows some robustness to unmeasured confounding, the other findings remain sensitive to potential biases.

We extend the exponential stability theorem for the three-dimensional incompressible Navier-Stokes equations from hyperbolic 3-space $\HH^3$ (established in a companion paper) to complete simply connected Riemannian 3-manifolds $(M^3, g)$ with pinched negative sectional curvature $-b^2 \leq K \leq -a^2 < 0$ and bounded geometry (including a strictly positive injectivity radius). The deformation Laplacian $\Delta_\Def = \Delta_B + \Ric$ remains the viscous operator, selected by Lagrangian kinematics. We prove that the {exact} system admits a unique global mild solution for small $L^3$ data, with exponential decay at a rate determined by the spectral gap of the Stokes operator. The extension overcomes three obstacles absent on $\HH^3$: (i) the semigroup factorisation $e^{t\Delta_\Def} = e^{-2t}e^{t\Delta_B}$ fails because $\Ric$ is not a scalar multiple of the metric; (ii) the Leray projector no longer commutes with $\Delta_\Def$; (iii) the exact spectral gap is unknown. We resolve (i) unconditionally, without any curvature restriction, by observing that the Ricci perturbation $V = \Ric + 2a^2 g$ is negative semi-definite and applying a Trotter product bound with the diamagnetic inequality. We resolve (ii) by an algebraic reduction of the commutator $[\PP, \Delta_\Def]$ to the complementary projector $(I-\PP)$ applied to the shifted Ricci endomorphism, giving a clean zeroth-order bound proportional to the curvature variation $b^2 - a^2$. This is the sole source of a curvature pinching constraint. We resolve (iii) via McKean's theorem, the diamagnetic inequality, and the Weitzenböck identity. The Fujita-Kato temporal singularity exponent $1/2 - 3/(2p)$ is unchanged from the $\HH^3$ case, confirming that the ultraviolet scaling obstruction is local and geometry-independent, driven fundamentally by an unresolvable temporal scaling mismatch.

We establish the well-posedness of the 3D rotating incompressible Navier-Stokes equations with critical initial data $u_{0,\Omega}\in X_{0,q,p}^{\Omega}$ for $p<5$, where $X_{0,q,p}^{\Omega}$ is defined by the norm \begin{equation*} \begin{aligned} &\|u_{0,\Omega}\|_{X_{0,q,p}^{\Omega}}:= \Omega^{3- \frac{6}{q}}\|u_{0,\Omega}\|_{\dot{B}_{q,\infty}^{-7+\frac{15}{q}}}^{\ell_\Omega} +\|u_{0,\Omega}\|_{\dot{B}_{p,\infty}^{-1+\frac{3}{p}}}^{h_\Omega}. \end{aligned} \end{equation*} This extends the previous results by Chen, Miao, and Zhang (\cite{CMZ2013}). The main ingredients are a new global-in-time dissipative-dispersive estimate for the Stokes–Coriolis semigroup and corresponding bilinear estimates. Furthermore, we establish the vanishing rotational limit for the 3D rotating Navier-Stokes equations as $\Omega\rightarrow 0^{+}$.

Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Turán density $\pi_{\mathrm{co}}(F)$ is the supremum over all $\gamma \in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $\gamma n$ edges. In this paper, we prove that there is a linear $k$-graph $F$ with $0<\pi_{co}(F) < \varepsilon$ for any $\varepsilon>0$. The special case $k=3$ solve a question proposed by Ding, Lamaison, Liu, Wang and Yang (JLMS, 2025). The main method combines an affine-plane-type incidence structure over a finite field and elementary number-theoretic arguments.

The present paper continues Arveson's program on curvature, Euler characteristic, and Gauss-Bonnet-Chern type formulae for Hilbert modules. For regular unitarily invariant complete Nevanlinna–Pick spaces on the unit ball in \(\mathbb{C}^d\), we prove that Arveson's Gauss-Bonnet-Chern formula holds unconditionally over the full multiplier algebra. Together with the polynomial-ring case, this shows that the validity of the formula depends essentially on the coefficient algebra. We then solve Arveson's finite defect problem in the same framework by giving a sharp classification of finite defect submodules. This problem is natural in curvature theory, since both curvature and Euler characteristic are governed by the defect structure. Finally, using commutative-algebra methods, we apply this classification to unitary-equivalence rigidity for finite-codimensional submodules of vector-valued reproducing kernel Hilbert modules.

We find the asymptotic ratio between the storage capacities when enforcing real pre-activations in a complex hypothesis class as opposed to complex ones in the same class. We use weights drawn from the complex Gaussian, which converge asymptotically in norm to the square root of dimension almost surely. Our methods depend on Gardner volume-type comparisons at critical capacity. Our proof relies on an application of the Harish-Chandra-Itzykson-Zuber (HCIZ) formula, nonstandard in literature. With the HCIZ formula, we may obtain a more robust approximation for the final asymptotic ratio. This strategy is applicable to our work specifically since we integrate over the unitary and orthogonal compact manifolds, facilitated via the Weyl integration formula and the Haar measure.

In this paper, we study the Bakry-Émery estimates for GNS- and KMS-symmetric semigroups in terms of the Fourier multiplier of the gradient form and the iterated gradient form in the framework of quantum Fourier analysis. We also systematically investigate the intertwining properties for bimodule GNS- and KMS-symmetric quantum Markov semigroups and compare with the Bakry-Émery estimates. A number of examples of GNS- and KMS-symmetric semigroups satisfying these intertwining properties are presented.

We prove a comparison principle for a class of second-order Hamilton–Jacobi–Bellman equations on the Wasserstein space whose second-order term is generated by a general common-noise Hessian. The main difficulty is that the relevant second-order direction is induced by a state- and measure-dependent coefficient, so the associated perturbation of the measure is no longer a translation or a fixed state-dependent transformation. We introduce a nonlinear flow of measures and use it to transform the Wasserstein-space equation into an augmented equation on $[0,T]\times \mathcal P_2(\mathbb R)\times\mathbb R$, where the general Hessian becomes an ordinary second derivative in the auxiliary variable. The construction may be viewed as a measure-dependent Lamperti transform: it removes the common-noise direction at the level of the equation, but unlike the classical one-dimensional Lamperti transform it permits degeneracy of the coefficient and dependence on the conditional law. We establish the spatial, measure-derivative, and negative-Sobolev estimates for this flow that are needed in the viscosity argument. Under structural assumptions on the transformed Hamiltonian, these estimates yield a Crandall–Ishii type comparison theorem for semicontinuous viscosity sub- and supersolutions. This gives, to the best of our knowledge, the first viscosity comparison framework of this kind for the filtering-driven equations considered here, and opens a new class of second-order PDEs on spaces of measures with state- and law-dependent common-noise directions. As an application, we identify the value function of a controlled stochastic filtering problem with state- and law-dependent common noise as the unique viscosity solution of its dynamic programming equation. We also explain how the same change-of-variable viewpoint applies to Zakai-type Kolmogorov equations on spaces of finite positive measures.

We construct a weight-zero infinitesimal bialgebra structure on the $\bk$-module spanned by planar binary trees, using the under product $\backslash$ of Aguiar–Sottileand a root-recursive coproduct $\DeltaLR$. We prove that $\DeltaLR$ is coassociative and satisfies the infinitesimal derivation rule with respect to $\backslash$, hence gives a unitary infinitesimal bialgebra distinct from the usual Loday–Ronco Hopf structure. We also obtain an elementary vertex-cut formula, establish freeness properties for unitary $(\backslash,\vee)$-algebras and unitary infinitesimal $(\backslash,\vee)$-bialgebras, and identify the construction with the infinitesimal coproduct transported from planar rooted forests.

We prove a high-dimensional primitive inhomogeneous Diophantine approximation result for fixed non-singular simultaneous frequencies. The frequency class is explicit and has full Lebesgue measure.

Conventional patchified Transformers operate on uniform spatial partitions, distributing computational effort evenly across the domain irrespective of local features. This inflexible tokenization scheme is inherently limited in its ability to efficiently represent and process solutions to complex PDEs. To address this, we propose MeshTok, an adaptive mesh refinement (AMR)-inspired tokenization and sequence modeling framework. This method selectively refines spatial regions exhibiting sharp gradients, transient features, or multiscale structures, generating a heterogeneous set of multiscale tokens defined on a fixed simulation grid. These tokens are processed within a unified Transformer sequence, enabling the model to simultaneously capture coarse-grained global context and fine-grained local details without requiring specialized architectural components. Although adaptive refinement moderately increases token count, it promotes a more targeted allocation of computational resources to physically informative regions, which we view as a practical inductive bias rather than a formal optimality guarantee. Experimental evaluations across multiple PDE families and benchmark datasets demonstrate that MeshTok consistently improves the efficiency-accuracy trade-off compared to uniform-grid baselines. This suggests adaptive multiscale tokenization as a scalable and generalizable design principle for neural PDE modeling. Code is available at https://github.com/SCAILab-USTC/MeshTok.

Age of Information (AoI) has become a central metric for the design of wireless update systems, especially in applications where fresh measurements support tracking, estimation, and control. Despite its popularity, the use of mean AoI or peak AoI as a surrogate for closed-loop performance is often motivated by intuition rather than by a control-theoretic derivation. This paper examines whether minimizing the mean AoI is in fact optimal for networked control systems. For scalar linear time-invariant systems with delayed intermittent updates, we show that, under state-independent scheduling policies, the infinite-horizon LQR tracking problem reduces to an optimization over the distribution of inter-scheduling intervals. The resulting objective depends on higher-order statistical moments, and in unstable or correlated regimes on exponential moments, of the inter-scheduling process rather than only on its mean. Consequently, policies with identical mean AoI can induce substantially different tracking costs. We further extend the analysis to disturbances with exponentially decaying autocorrelation and derive equivalent cost formulations that expose the role of the full interval distribution. Finally, we evaluate the theory using real vehicle trajectories from the NGSIM US-101 dataset. The empirical results match the predicted performance trends, demonstrating that mean AoI alone is insufficient for control-oriented network design.