Papers

We study a fast reaction limit for a two-component reaction-diffusion system in which one component diffuses and the other does not. In the regime considered here, the nondiffusing component converges to zero as the reaction rate tends to infinity, while the reaction term has a nonzero limit. As a result, the limiting equation for the diffusing component retains a time-integral term. Our results cover both homogeneous Dirichlet and Neumann boundary conditions. We prove convergence of the diffusing component and determine the limiting behavior of the reaction term and the rescaled nondiffusing component. The proof combines monotonicity with respect to the reaction rate with barrier estimates adapted to the time-integral term. Under an additional separation condition on the initial data, we also prove uniqueness of the limiting initial-boundary value problem.

We construct a bounded Lipschitz domain in $\mathbb{R}^3$ that is starlike about the origin and whose first positive Neumann eigenvalue is simple. A corresponding eigenfunction, after a choice of sign, has its unique global maximum at the origin. The construction uses domains with increasingly fine angular structure and convergence to a weighted eigenvalue problem whose first positive eigenfunction is radial.

This paper introduces a new cumulative residual Rényi entropy (CRRE), which addresses limitations of the existing cumulative residual Rényi entropy. The conventional measure suffers from two primary drawbacks: its failure to converge to the cumulative residual entropy (CRE) as the order parameter approaches unity, and its lack of guaranteed positivity. Our proposed CRRE resolves these issues, offering a theoretically consistent and always nonnegative measure of uncertainty in residual lifetimes. We explore its mathematical properties, including its relationship with existing information measures. Furthermore, the concept is extended to its dynamic version, and its time-dependent characteristics are investigated. A non-parametric estimator of the proposed CRRE is developed, its almost sure convergence is established, and is validated through Monte Carlo simulation. The effectiveness of CRRE as a tool for quantifying uncertainty is confirmed through its application in chaotic theory and financial data.

Bilinear Compressive Security (BCS) is a lightweight encryption scheme that combines compressed sensing with a fresh random convolution filter to improve resistance against key-recovery attacks under key reuse. Previous work derived information-theoretic guarantees for BCS by relating key identifiability to phase retrieval under the assumption that an adversary has access to the full ciphertext distribution induced by the random filter. In practice, however, an adversary only observes finitely many encryptions, leaving the security of BCS in the finite-sample regime largely unexplored. In this paper, we study key recovery from only finitely many observations. We first develop a covariance-based attack that estimates the linear measurements from repeated encryptions of the same plaintext and subsequently recovers the secret measurement matrix. We analyze the estimation procedure and derive finite-sample guarantees together with corresponding sample-complexity bounds. We further investigate the complete attack experimentally and compare its observed recovery behavior with the previously established information-theoretic identifiability results. Our experiments show two qualitatively different recovery regimes on the structure of the measurement matrix. For dense matrices, including Gaussian matrices, the number of measurements required for successful recovery appears to grow linearly with the signal dimension. This differs from the worst-case behavior suggested by information theory. In contrast, for matrices with sparse rows, we observe a quadratic growth consistent with the theoretical bound. These results suggest that the sparsity structure of the measurement matrix plays an important role in the security of BCS and should be taken into account when selecting parameters and designing practical instances of the scheme.

We give a parabolic proof of the Fu-Wang-Wu theorem for semilinear elliptic equations associated with generalized Gauduchon metrics. Fixing a Hermitian metric $\omega$, we introduce a flow $\partial_t v-\Delta v=\psi(\vert\nabla v\vert^2)+\langle B,dv\rangle-c(t)-f(x)$ for the conformal factor, whose stationary points solve the semilinear elliptic equation of Fu-Wang-Wu. Under the growth condition $\liminf_{t\to+\infty}\psi(t)/t^\mu>0$ for some $\mu>\tfrac{1}{2}$, we prove that the flow exists uniquely for all time and converges; the limit solves the elliptic equation and is unique up to addition of a constant. Our convergence argument does not rely on a gradient-flow structure or a monotone energy functional. In particular, this recovers, by a flow method rather than the continuity method, the existence and uniqueness theorem of Fu-Wang-Wu on $k$-th generalized Gauduchon metrics, providing a new parabolic approach to their elliptic theory.

Second-order flows represent an emerging computational framework for approaching stationary points of variational energies, characterized by damped hyperbolic partial differential equations. In this paper, we develop a scalar auxiliary variable (SAV) method for this class of flows, referred to as SAV-E. The resulting scheme is linear and decoupled, and we prove unconditional modified-energy stability. We further establish subsequential convergence to stationary points for the Ginzburg-Landau and conserved Swift-Hohenberg models under suitable assumptions. Numerical experiments support the analysis and demonstrate robustness and efficiency compared with representative baselines and the corresponding gradient flows.

Let $\DD$ be $\R$, $\C$, or $\HH$, and let $H=\GL_p(\DD)\times\GL_q(\DD)$ be the block diagonal subgroup of $G=\GL_{p+q}(\DD)$. We prove that $\dim\Hom_H(\pi,\eta)\leq1$ for every irreducible Casselman–Wallach representation $\pi$ of $G$ and every smooth character $\eta$ of $H$.

Scattering of sound waves propagating in a compressible fluid is linked to the quantum dynamics generated by a class of non-Hermitian Hamiltonian operators. The scattering amplitude is expressed in terms of the corresponding evolution operator. This is used to provide a new proof of the reciprocity theorem in acoustic scattering, conduct analytic low-frequency acoustic scattering calculations, and study the scattering of sound waves by a finite planar array of delta-function point scatters embedded in an infinite acoustic slab. A detailed discussion of acoustic scattering properties of a homogeneous acoustic slab including a pair of delta-function point scatters on one of its faces is provided.

We study the optimisation landscape of the maximum mean discrepancy (MMD) over weighted atomic measures. For the one-dimensional Gaussian kernel, we show that, for any compactly supported target distribution, sufficiently many particles ensure that every local minimum has arbitrarily small MMD. Moreover, any stationary point whose MMD remains bounded away from zero is unstable under an unweighted gradient flow over the particle locations. These results hold both when the weights are constrained to the simplex, and when they are unconstrained but optimised at each step. When the target is itself finitely supported, we further strengthen this result: beyond an explicit particle threshold, there are no spurious local minima, and every spurious stationary point is unstable. To our knowledge, these are the first finite-particle overparameterisation guarantees of their kind for weighted MMD optimisation.

We study the data-driven approximation of control-affine systems, where the drift and each control vector field exhibit a low-dimensional structure. While the ANOVA (ANalysis Of VAriance) decomposition is tailored to low-dimensional structures, a straightforward implementation does not exploit control affinity, rendering the approximation computationally expensive. We propose a coupled least squares approach for given state-control samples, whose ansatz is affine linear in the control by construction and can be realized by a preprocessing of the ANOVA index sets. Based on a Marcinkiewicz–Zygmund inequality for the resulting ansatz space, we prove a probabilistic $L_2$-error bound with explicitly given constants. Separate error control on drift and input vector fields renders the approach flexible to, e.\,g., parameter changes. Finally, we numerically illustrate the identification of the coupling structure via global sensitivity indices on discretizations of a nonlinear advection-diffusion equation, the shallow water equations and the viscous Burgers equation.

In this paper, we consider the endpoint Sobolev regularity of Radon transforms associated with real-analytic functions in the plane. We establish all of the failure of endpoint regularity for real-analytic phases. On the one hand, the critical endpoint regularity $L^p(\mathbb{R}^2)\rightarrow L^p_s(\mathbb{R}^2)$, corresponding to extreme points of the reduced Newton polyhedron of phase function, is true if only and only if $p=2$. For generic extreme points, this failure of endpoint regularity generalizes a result of Christ \cite{christ}. On the other hand, we find that the endpoint Sobolev regularity also fails on a bounded segment of the reduced Newton polyhedron when the bisectrix $y=x$ intersects one of unbounded faces at a relative interior point. In particular, our results imply that the Radon transform is not bounded from $L^p$ into $L^p_{1/n}$ with $p=n/(n-1)$ or $p=n$, where the phase functions are real-valued homogeneous polynomials $S(x,y)=a_1x^{n-1}y+a_2x^{n-2}y^2+\cdots+a_{n-1}xy^{n-1}$ with $n\geq 3$ and $a_1a_{n-1}\neq0$. This answers a problem of Phong and Stein in 1994. Moreover, the main results of this paper also give examples which show failure of critical Sobolev regularity for degenerate Fourier integral operators satisfying finite type conditions formulated in \cite{Seeger93,seeger2}. Based on earlier results and our proof of several remaining cases, we conclude that all of other critical endpoint Sobolev estimates are true for Radon transforms associated with real-analytic phases in the plane. Therefore, all failures of endpoint regularity estimates are clearly determined.

Policy mirror descent (PMD) refers to a class of policy optimization methods associated with different mirror maps. For a broad class of decomposable Legendre mirror maps encompassing many standard choices, Li and Wei [2026] show that the PMD policy sequence converges to an optimal policy from any admissible initialization under any positive constant step size, even when the optimal policy is not unique. In this note, we provide a counterexample with assistance from OpenAI's GPT-5.6 model, showing that PMD with a non-decomposable Legendre mirror map need not converge in policy.

Shannon entropy is a standard measure of ciphertext randomness, evaluated either globally over the byte histogram of the entire ciphertext or locally as the average over non-overlapping blocks of ?xed size. While the global statistic captures only the overall byte distribution, the local statistic is also sensitive to non-randomness that is con?ned to parts of the ciphertext. Both measures are commonly assessed against reference values derived under the assumption of independent uniform bytes. However, STL encodes most numerical data as single-precision IEEE-754 ?oats, and format validity requires them to be ?nite, causing nonuniform byte probabilities and dependence between the two bytes of each exponent pair. We derive the exact mean and variance of entropy under the valid-STL (VSTL) distribution, which assigns equal probability to every admissible sequence of triangle records, and provide second-order expansions that keep the global reference values practical to compute at large ?le sizes. Using these expansions, we establish a two-sided hypothesis test for the global entropy statistic, whose limiting value is lowered from 8 to 7.99991479609 bits. The format constraints also change the local entropy reference: applying the capacity energy criterion to triangle records selects a block size of 1950 bytes instead of 1936 bytes, with an exact reference mean of 7.903106732975 rather than 7.902469316647 bits. Although small, these shifts are su?cient to place the VSTL reference means outside the conventional 95% acceptance intervals at practical ?le sizes, making unadjusted entropy tests prone to falsely rejecting ideal VSTL ciphertexts. Monte-Carlo simulations con?rm all reference values within sampling error. We therefore recommend using the format-speci?c reference values derived here to evaluate STL ciphers.

This paper studies resolvent expansions at all four spectral thresholds for the discrete Schrödinger operator $H=-\Delta+V $ on $\Z^3$, where $\Delta$ denotes the discrete Laplacian on $\mathbb{Z}^3$ and $V(n)$ is a real-valued decaying potential of $\mathbb{Z}^3$. At the thresholds 0, 4, 8, and 12, we first give a complete classification of the threshold types and derive the corresponding resolvent expansions in weighted $\ell^2$-spaces, including the non-generic cases of resonances and eigenvalues. As a result, we establish the Jensen-Kato type time-decay estimates of the evolution $e^{itH} P_{ac}(H)$ to discrete Schrödinger equations with thresholds singularities on $\Z^3$, with decay $O(t^{-3/2})$ when all thresholds are regular and $O(t^{-1/2})$ otherwise. These results provide a three-dimensional lattice analogue of the Jensen–Kato classical analysis at zero energy threshold on $\mathbb{R}^3$.

GPU-accelerated first-order methods are increasingly effective for solving large-scale linear programs (LPs), yet LP preprocessing remains predominantly CPU-based and can become costly. A key challenge is that dependencies among presolve reductions and frequent modifications to sparse LP data make efficient GPU parallelization difficult. To address this challenge, we develop GPU-Preprocessor, a GPU-native LP preprocessing framework that integrates folding and presolve. To efficiently implement folding on GPUs, we adapt a randomized partition refinement algorithm to LP folding, using sparse matrix-vector products and parallel sorting to construct stable row and column partitions. For presolve, we introduce a new piecewise-linear reformulation rule that exploits underlying convex piecewise-linear structure to obtain sparser LP representations. We also develop a framework that combines phase scheduling with a Scan-Plan-Apply design to coordinate presolve rules and reduce sparse-matrix reconstruction overhead through batched updates. Numerical experiments on three LP benchmark sets show that GPU-Preprocessor achieves greater average problem-size reductions than PSLP and Gurobi Presolve, with SGM10 preprocessing-time speedups of up to $43.9\times$ and $677.7\times$, respectively. Here, SGM10 denotes the shifted geometric mean with a 10-second shift. On the largest tested quadratic assignment problem LP relaxation, these speedups reach $126.3\times$ and $2{,}207.9\times$. When integrated with the GPU-accelerated LP solver HPR-LP, GPU-Preprocessor achieves SGM10 speedups of $2.5\times$ to $5.0\times$ in combined preprocessing and solving time over HPR-LP without preprocessing. The code is available at https://github.com/PolyU-IOR/GPU-Preprocessor.

Let $M$ be a rank-one symmetric space of compact or noncompact type and real dimension $m=kn$, where $k\in\{1,2,4,8\}$ is the real dimension of the underlying division algebra. Meng and Wang proved sharp isoperimetric inequalities for reciprocal sums of the first $p$ or $\ell$ nonzero Neumann eigenvalues, with indices strictly below $m$ in the nonconstant-curvature cases. We keep the whole $m$-dimensional transplanted first eigenspace of an equimeasurable geodesic ball and treat its mass and stiffness forms as a coupled generalized eigenvalue problem. For every bounded connected smooth domain in a noncompact rank-one symmetricspace we prove the full sharp estimate \[ \sum_{j=1}^{m}\frac{1}{\mu_j(\Omega)} \ge \frac{m}{\mu_1(B)}, \] where $B$ is a geodesic ball with $|B|=|\Omega|$; equality characterizes geodesic balls. The Hopf-direction contribution is encoded by a positive, unital, trace-preserving projection on symmetric matrices, and the finite-dimensional closure is supplied by a positive-projection Jensen inequality. In compact type, under the localization hypothesis of Meng–Wang, the same full estimate is obtained whenever $\cot^2 R\ge k-1$, where $R$ is the radius of the comparison ball, as well as in the isotropic endpoint cases and for Hopf-balanced angular second moment. Consequently the complete $m$-term inequality holds for all complex and quaternionic projective spaces under the original localization assumption, and for the Cayley plane when $R\le \arctan(1/\sqrt7)$. In the remaining admissible Cayley interval we exhibit a finite-dimensional obstruction showing that the affine tangent-matrix bounds alone cannot yield the full trace estimate. This obstruction concerns the proof mechanism and is not a counterexample to the unresolved spectral inequality.

This study develops and validates an adaptive parametric method for sizing safety stock when forecast errors are autocorrelated, heteroscedastic, asymmetric, and heavy-tailed. Using four years of weekly forecasts from a fast-moving consumer goods company, the analysis identifies substantial positive autocorrelation, departures from normality, systematic forecast optimism, and heterogeneous relationships between forecast volume and error variance. The proposed method integrates several corrections into a modular, sequential safety-stock architecture. It constructs cumulative forecast errors over the operational action period, selects an SKU-specific parametric distribution, and determines a baseline protection point using an expected-shortage condition consistent with the target beta fill rate. A forecast-scaled module adjusts dispersion when error variance depends on forecast volume, while a Cycle-End Exposure safeguard limits within-cycle shortage risk when large replenishment lots increase exposure. Performance is evaluated through discrete-event simulation against the company's current method. The proposed method reduces average inventory by 9.9% and maximum inventory by 19.5%, with no statistically significant deterioration in beta fill rate or average stockout duration. The proportion of weeks without stockouts declines slightly, without evidence of more persistent or operationally severe shortages. The heteroscedasticity module provides selective benefits, with greater service improvements as variance dependence intensifies. The resulting framework is interpretable, adapts to SKU-specific statistical profiles, links distributional features to safety-stock outcomes, and can be integrated into existing ERP environments.

We determine the minimal stretch factors among all orientation-reversing pseudo-Anosov mapping classes of the closed orientable surfaces of genus two, three and four.

Repeated high-fidelity state and gradient evaluations make large-scale PDE-constrained optimization computationally expensive. Reduced-order modeling provides a natural way to reduce the cost of these repeated high-dimensional computations. However, existing reduced-order modeling approaches often rely on converged high-fidelity states for model construction or updating, while their reduced spaces may lose accuracy as the design moves beyond the region represented by the available data. To address these limitations, we present Reduced-order Trajectory-based Optimization (RETO), an online framework that constructs local reduced-order models (ROMs) directly from unconverged primal solver trajectories. Each local ROM is built from snapshots collected during a finite number of high-fidelity primal iterations, without requiring full-order convergence, and supports multiple low-cost state evaluations, adjoint-based gradient evaluations, and design updates. New solver trajectories are generated at updated design points to reconstruct the local ROM and continue the optimization. Across three distinct PDE-constrained optimization problems, RETO achieves solution quality comparable to that of full-order optimization, with speedups of approximately 8 to 22 times in cases without discontinuities in the state fields. Even in the transonic inverse-design case involving moving shocks, RETO retains an approximately threefold speedup. The transonic results further show that the online local ROM progressively captures evolving shock-related flow features as new primal solver trajectories are incorporated. These results demonstrate the value of unconverged primal solver trajectories as an online information source for reduced-order optimization, allowing the reduced basis to evolve with the design rather than requiring a basis that covers the entire design space in advance.

Let $f(z,w)=(p(z),q(z,w))$ be a polynomial skew product of $\mathbb{C}^2$ of degree $d\geqslant 2$ and let $\mu$ and $\mu_p$ be the Green measures associated to $f$ and $p$ respectively. Let $\lambda_1 > \lambda_2$ and $\lambda_p$ be the Lyapunov exponents of $(f,\mu)$ and $(p,\mu_p)$ respectively. We show that if $\lambda_2 = \lambda_p$, then $\mu$ is exact dimensional and: \begin{equation*}\dim_H\mu =\frac{\log d}{\lambda_1}+\frac{\log d}{\lambda_2}.\end{equation*} The proof relies on the continuity of the conditional measures obtained by Jonsson, on the fact that $\mu$ has constant Jacobian $d^2$, and on a fibered version of a normal form theorem for the inverse branches of the iterates of $f$.

We prove that a bounded pseudoconvex domain with piecewise $C^2$-smooth boundary which has a finite volume quotient for the complete Kähler-Einstein metric is biholomorphic to a product of complex unit balls. More generally, the same conclusion holds for a bounded domain possessing a boundary point satisfying Kodama's conditions (C.1)-(C.5), provided the quotient has finite volume for a complete $\operatorname{Aut}(D)$-invariant Kähler metric. For piecewise Levi-flat boundary the product is a polydisk and in the smooth pseudoconvex case it is a single ball.

Among the various space interpolation operators used in high-order semi-Lagrangian schemes, the case of finite element interpolation presents a somewhat incomplete stability theory, partly due to the impossibility of applying in a standard form the framework of Von Neumann analysis, even in the constant coefficient case. In this work, the stability of the P_2 semi-Lagrangian scheme is stated in the case of the constant coefficient equation on a uniform mesh, using a suitable extension of the Von Neumann technique to treat block circulant matrices.

Judicious partition problems ask for vertex partitions of graphs or hypergraphs that optimize several quantities simultaneously. In this paper, we investigate judicious partitions of graphs and hypergraphs in which each class meets many edges and establish the following results. (1) Every hypergraph with nonnegative edge weights and all edges of size at most $2$ has a bipartition in which each vertex class meets edges of total weight at least $(w_1-\Delta_1)/2+2w_2/3$, where $w_i$ is the total weight of the edges of size $i$, and $\Delta_1$ is the maximum weight of an edge of size $1$. (2) Every hypergraph with all edges of size at most $3$ has a bipartition in which each vertex class meets at least $m_1/2+3m_2/4+7m_3/8+o(m_1+m_2+m_3)$ edges, where $m_i$ is the number of edges of size $i$. (3) For any integer $k\ge4$, every graph with $m\ge2(k-1)(k-2)$ edges has a partition into $k$ classes, each meeting at least $2m/(2k-1)$ edges. Both the coefficient of $m$ and the edge threshold are best possible. (4) For all sufficiently large $m$, every $4$-uniform hypergraph with $m$ edges has a partition into four classes, each meeting at least $4m/7$ edges. The first three statements are conjectures of Bollobás and Scott, and the last is a conjecture of Bollobás and Thomason on 4-uniform hypergraphs. All of the above results are best possible.

Non-periodic functions on closed intervals are extended into sufficiently smooth periodic functions through a continuous Hermite interpolation framework with necessary and sufficient endpoint conditions, using smooth blend-to-zero operators and short extrapolation recurrences. A matrix-free method is introduced for generating these smooth periodic extensions on the fly directly from function values at equidistant points. No precomputed continuation operator or auxiliary basis construction is required. The resulting extensions can then be used for FFT-based differentiation of non-periodic functions. Numerical tests are provided to show the effectiveness of the method, including for the highly accurate solution of a 1D linear advection equation.

We study operator systems arising from Peter-Weyl and Dirac spectral truncations of $\mathrm{SU}(2)$ in the sense of Connes-Van Suijlekom. We determine operator system invariants for these systems, including the $C^*$-envelope and uniform bounds on the propagation number. For Peter-Weyl truncations, we find $\mathrm{prop} \leq 2$ for all connected compact Lie groups, generalizing known results for the torus. We also carry out an analysis of the spectral localizer by Loring and Schulz-Baldes on $\mathrm{SU}(2)$. In this analysis, we find numerical evidence that an index map for operator systems previously proposed by Van Suijlekom only captures the right information in a suitable interval of the parameter $\kappa$, instead of the limit $\kappa \downarrow 0$.

We prove the sharp volume bound $\mathrm{vol}(\mathcal{F})\ge 1/6$ for complex foliated surfaces of general type with positive geometric genus. We also study the equality case and apply the bound to the algebraic integrability and finiteness of small volumes on irregular surfaces.

We obtain a uniform lower bound for the drift of random walks on hyperbolic spaces in terms of joint minimal displacement. Our main theorem applies to weakly acylindrical actions on Gromov-hyperbolic spaces and is quantitative in all relevant constants. For discrete isometry groups of Hadamard manifolds with pinched negative curvature, the required assumptions are automatically satisfied by the Margulis lemma. This yields uniform gaps at zero in drift spectra for pinched negatively curved Hadamard manifolds, hyperbolic groups, and mapping class groups acting on curve graphs.

We consider a discrete memoryless state-dependent channel with noiseless feedback, where the transmitter uses randomized encoding to jointly perform message transmission, identification, and state sensing. In this setting, the transmitter sends a conventional transmission message and an identification message while estimating the unknown channel state sequence under an average distortion constraint. We establish a single-letter characterization of the randomized identification-transmission-sensing capacity-distortion region. The result reveals a single-letter separation structure: the communication rates are determined by the averaged channel, subject to the sensing-induced constraint on admissible input distributions. Specifically, the transmission rate is characterized by mutual information, while the identification rate is characterized by output entropy under the same distortion constraint. The achievability proof combines constant-composition transmission coding, output partitioning, and combinatorial identification coding, while the converse follows from an averaged-channel reduction, distortion lower bounds, and feedback-identification converse arguments.

Computational compressive imaging, such as single-pixel imaging and lensless imaging, is a promising imaging technique, enabling the reconstruction of objects from under-sampled measurements through compressed sensing algorithms or deep learning methods. However, the demand for electronic computational resources grows rapidly when recovering high-resolution and multi-dimensional images and training neural networks with numerous parameters, which has become a bottleneck for the widespread adoption of this technique. Here, we propose a physics-driven reconstruction framework that formulates imaging inverse problems as Ising models and directly recovers target objects through the self-evolution of non-von Neumann Ising machines toward the ground state of the Hamiltonian. We employ a simulated-annealing-based Ising solver to demonstrate this proof-of-concept by reconstructing objects in both single-pixel imaging and a lensless diffuser camera at sampling rates ranging from 1% to 25%. By further incorporating a foreground-sparsity constraint and a smoothness constraint, the reconstruction quality of our method is comparable to or even better than that of mainstream algorithms. This approach has the potential to substantially reduce the electronic computational burden of inverse reconstruction, paving the way for real-time, low-power computational imaging.

We study free-material optimization for nonlinear bond-based nonlocal diffusion and its localization as the interaction horizon $\delta$ vanishes. Bond conductivities are subject to pointwise bounds and a global material budget constraint. After normalization, the asymptotic behavior of these problems is governed by the fraction $\theta_\delta$ of potentially active bonds that can be assigned the maximal conductivity. We identify three distinct limiting regimes. If $\theta_\delta\to0$, the normalized optimal values converge to those of the classical vanishing-material optimal design problem, equivalently formulated as a Lipschitz-constrained state problem or a measure-valued flux problem. For $\theta_\delta\to\theta\in(0,1)$, we obtain a directional local free-material problem governed by the upper tail of the directional $p$-strain over position-direction space. Thus the state variable localizes, while the limiting constitutive law retains an internal directional variable inherited from the bonds. If $\theta_\delta\to\theta\geq1$, the limit is the isotropic local $p$-Dirichlet problem. A convex minimax argument eliminates the normalized conductivity variable exactly, after which the analysis combines convex duality, nonlocal compactness, and $\Gamma$-convergence for convex integrands. We prove a unified $\Gamma$-convergence result for the reduced state energies whenever $\theta_\delta\to\theta\in[0,+\infty)$, yielding convergence of normalized optimal values and compactness and identification of optimal states throughout the full parameter range. At the vanishing-material endpoint, we additionally recover optimal measure-valued local fluxes from optimal nonlocal two-point fluxes.