Papers

In this paper, we propose two hierarchical algorithms for charge deposition and electric-field interpolation that apply to both the sparse-grid combination technique (SGCT-PIC) and hierarchical sparse-grid (HSG-PIC) particle-in-cell methods. The two algorithms are inspired by the fast multipole method (FMM) and exploit clusters of particles associated with a directed acyclic graph (DAG) of particle-populated boxes to reduce the number of particle–mesh interactions. The particle–mesh interactions are governed by piecewise-polynomial kernels, so that the associated multipole expansions are exact, requiring neither truncation nor approximation, and are valid in both near- and far-field regions, thereby eliminating the need for multipole-to-local translations. The arithmetic complexity of the charge deposition and field interpolation steps is reduced from $\O(p^d n^{d-1}N)$ to $\O(p^d(N+M))$, where $M=2^{dn}$ denotes the number of full-grid mesh nodes and is typically no larger than the particle population in the considered regime, $M\lesssim N$. Numerical experiments in two-dimensional configurations demonstrate charge-deposition speedups of $8.2\times$–$66.9\times$ for SGCT-PIC and $3.1\times$–$18.8\times$ for HSG-PIC, and field-interpolation speedups of $4.1\times$–$62.6\times$ and $4.2\times$–$13.7\times$, respectively, depending on the particle-per-cell ratio, while preserving the exact particle–mesh interactions. The speedups increase with the particle-per-cell ratio, reflecting the reduced dependence of the hierarchical algorithms on the number of particles and their increasing advantage for large particle populations.

We study the compactness question for certain special Lagrangians in semiflat SYZ fibrations (resp. associative submanifolds in Donaldson's proposal of collapsing coassociative K3 fibrations), and give some criterion for when gradient graphs emerge from the adiabatic limit. This gives a partial converse to the Donaldson-Scaduto proposal.

This paper investigates the open cover specification property and its role in the study of topological dynamical systems. We show that this property is a topological invariant and a natural extension of the classical topological specification property from uniform spaces to general topological spaces, and that it is preserved under infinite products with compositions. We further prove that on regular spaces, the strong open cover specification property implies Devaney chaos, and that on locally compact, first countable Hausdorff spaces, it guarantees both distributional chaos and Li Yorke chaos. These results demonstrate the significance of the open cover specification property in linking topological structure with dynamical complexity.

We develop a geometric framework for learning deterministic and stochastic forced Hamiltonian systems with neural networks. Motivated by the Lagrange-d'Alembert principle and the theory of variational integrators, we introduce the notion of a Lagrange-d'Alembert map and establish a $C^r$ convergence theorem for first-order one-step methods. Building on these results, we propose Generalized Forced Hamiltonian Neural Networks (GFHNNs), a class of structure-preserving neural networks obtained by concatenating Lagrange-d'Alembert-Euler maps, and prove a universal approximation theorem for this architecture. We further extend the framework to parameter-dependent systems, leading to Parametric Generalized Forced Hamiltonian Neural Networks (PGFHNNs). By interpreting the multiple Stratonovich integrals appearing in the Stratonovich-Taylor expansion as parameters, the same framework can be applied to stochastic forced Hamiltonian systems whenever information about the underlying Wiener process is available. Our numerical experiments demonstrate that the proposed geometric architectures provide significantly improved long-time stability and accuracy compared to non-geometric residual neural networks, while requiring substantially less training data to achieve a comparable level of performance.

For a $C_4$-free graph $G$ on $n$ vertices — one with no induced cycle on four vertices — we study the two-sided extremal problem for the triangle density $\tau$: How large and how small can $\tau$ be for given edge density $\varepsilon$ and clique-number density $\kappa = \omega(G)/n$? We give lower and upper bounds for $\tau$ in terms of $\kappa$ and $\varepsilon$. The two bounds sandwich $\tau$, and their compatibility forces a lower bound for $\kappa$ in terms of $\varepsilon$. When the clique complex of $G$ is $2$-Leray over a field $\Bbbk$, the resulting bound on the clique-number density lies between the previous best $C_4$-free bound and the sharp chordal bound. It improves on the former for every $\varepsilon \in (0,1)$. The lower bound is elementary. The upper bound is homological, obtained by passing to the Stanley–Reisner ring of the clique complex. When the complex is $2$-Leray, its Betti table has at most two linear strands. The two first entries in the first strand encode edge and triangle densities, and the strong structural form of a Boij–Söderberg decomposition constrains what these entries can be, yielding the upper bound. For $2$-Leray graphs with no holes in the range $[4,g]$ we give a conjecturally sharp bound. We further ask questions concerning the triangle bound for any $C_4$-free graph.

We establish upper and lower bounds for relative entropy contraction of generalized quantum depolarizing channels and semigroups. Our bound provides tight first order asymptotic of the contraction rate in terms of the dimension constant. One side estimate are based on sharp reverse ratio and convexity of relative entropy of two states, which can be derived from the recently introduced Hockey-Stick quantum $f$-divergence, and also independently, Bogoliubov–Kubo–Mori quantum Fisher information metric. The other side follows from the existence of index achieving pure state with respect to a general conditional expectation. Our results extend to the complete entropy contraction rate, tensor stable estimates for product dynamics. Examples include quantum depolarization, dephasing, and compact group symmetrization, with consequences for the decay rate of coherence and asymmetry.

When the control field of a qubit, a polarization state, or a spin-$\tfrac12$ system is swept through a level degeneracy, its direction traces an open curve on the Bloch sphere whose endpoints are antipodal, and the geodesic rule for the open-path geometric phase becomes ambiguous: infinitely many geodesics close the path, and different closures enclose different solid angles. We resolve this ambiguity in closed form. A coordinate-free monopole connection defines the open-path solid angle $\Omega[C]$ intrinsically, and displacing the degeneracy by $\epsilon\uhat$ closes the path with enclosed solid angle $\Omega(\epsilon\uhat)=\Omega[C]+2\alpha+O(\epsilon)$, where $\alpha$ is the azimuth of the transverse part of $\uhat$ measured from the principal normal of the control curve at the crossing. The identity between geometric phase and enclosed solid angle therefore holds for exactly one closing geodesic—the great circle in the osculating plane ($\alpha=0$)—supplied by the curvature at the degeneracy. Berry's $\pi$ invariant under reversal of the displacement and the values $\pm\pi/2$ under a reflection symmetry follow as corollaries, and the pure-state limit of the finite-temperature Uhlmann phase selects the osculating-plane closure automatically, turning the heuristic closing rules of the open-path literature into a computable prescription.

We investigate the existence problem of frame measures for uniform consecutive-digit Bernoulli convolutions under the separated scaling condition. Given parameters $N\geq 2$ and $0<\rho<1/N$, we prove that if $\rho^{-m}=B$ for some integers $m\geq 1$ and $B\geq 2$, with $N\nmid B$, then the associated $N$-Bernoulli convolution $\mu_{\rho,N}$ admits no frame measure. The proof introduces a new cyclic mask-quotient obstruction adapted to the multi-character structure of the consecutive-digit framework.

Reading an optical quantum reservoir costs repetitions growing super-exponentially with feature order: what it can afford is set by its detector, not its optics. For reservoirs encoding data in a parametric pump's phase, one conservation law fixes what is readable and what it costs. Pairwise photon exchange conserves an integer phase charge: across an ensemble of input masks, order-D readout reaches exactly the assemblies of at most D unit-charge kernels; degree-one homodyne readout is universal for fading-memory functionals along weak-squeezing families, at shot cost polynomial in accuracy; and at fixed squeezing no finite degree reaches every sector, at a computable distance. Nonlinearity sits in the optics, not the detector, whose per-shot variance is fixed at every order. In a hardware-faithful digital twin-no device was built-the rule is visible: on an open RF corpus a displacement-encoded control at identical photon number loses 17.7 accuracy points, as the charge algebra predicts.

In this paper, we study the existence of nontrivial solutions for the following Gross-Pitaevskii system involving mass-critical exponent: \[ \left\{ \begin{array}{ll} -\Delta u_{1}+V_1(x)u_{1}=a_{1}u_{1}^3+\beta u_{1}u_{2}^2+\mu u_{1}& \hbox{ in }\Omega,\\ -\Delta u_{2}+V_2(x)u_{2}=a_{2}u_{2}^3+\beta u_{2}u_{1}^2+\mu u_{2}&\hbox{ in }\Omega, u_{1},u_{2}\ge 0 &\hbox{ in }\Omega, u_1=u_2=0 &\hbox{ on }\partial\Omega, \end{array}\right. \] with the constraint \[ \int_\Omega (u_1^2+u_2^2)=1, \] where $\Omega$ is an unbounded smooth domain in $\mathbb{R}^2$, $a_1, a_2, \beta$ are positive parameters, $V_i$ are trapping potentials, and $\mu\in\mathbb{R}$ is an unknown Lagrange multiplier. We derive the existence of solutions by computing the Leray-Schauder degree for the parameters $a_1,a_2, \beta$, which are away from some critical values. The system may have semi-trivial solutions of the form $(u_1, 0)$ or $(0, u_2)$. Our novelty is that we provide mechanisms ensuring that the solutions we find are nontrivial, i.e., $u_1>0$ and $u_2>0$.

We study the generalized Hilbert operator \[ \mathcal{H}_b f(z)=\int_0^1 f(t)\,\frac{(1-t)^b}{(1-tz)^{b+1}}\,dt, \qquad b>0, \] acting on the Hardy spaces $H^p$ for $1\leq p\leq \infty$. We establish the precise operator norm \[ \|\mathcal{H}_b\|_{H^p\to H^p}=B\!\left(\frac1p,b+1-\frac1p\right) \] for every $1<p<\infty$ and, by continuous extension to $b=0$, recover the classical norm $\pi/\sin(\pi/p)$. We also prove that $\mathcal{H}_b$ is bounded on $H^1$ for every $b>0$, in contrast with the classical Hilbert operator, and we obtain the sharp restricted norm estimate \[ \|\mathcal{H}_b\|_{H^1_0\to H^1}=B(1,b). \] We also determine the exact norm \[ \|\mathcal{H}_b\|_{H^\infty\to \mathcal B}=\frac{1}{b+1}+2. \]

This study demonstrates a rationally enriched Chebyshev (REC) trunk for deep operator network (DeepONet) surrogate models of singularly perturbed and high-Péclet transport problems whose solution profiles are characterized by thin localized boundary or wall layers. The REC trunk combines Chebyshev polynomial dictionary elements with rational dictionary elements constructed using the adaptive Antoulas-Anderson (AAA) algorithm. Over five independent training runs, the resulting REC-trunk DeepONet is evaluated against a vanilla DeepONet and a Chebyshev-trunk DeepONet whose prescribed dictionary consists only of Chebyshev polynomials across three problems whose singular perturbation parameters are diffusion-to-advection ratios: a singularly perturbed scalar boundary-value problem (BVP), the thermal entrance problem with a prescribed wall temperature, and the concentration entrance problem with an absorbing wall. Across the held-out test profiles, the REC-trunk DeepONet improves over the vanilla DeepONet and remains comparable to the Chebyshev-trunk DeepONet in predicting the scalar profile, with its clearest advantage over the Chebyshev-trunk DeepONet appearing when the perturbation parameter lies between $1.00\times10^{-4}$ and $1.78\times10^{-4}$, where it reduces the profile-error metrics by up to $19.5\,\%$ relative to the Chebyshev-trunk DeepONet. In predicting the wall-normal temperature and concentration profiles, the REC-trunk DeepONet reduces the profile-error metrics by up to $60.2\,\%$ and $32.2\,\%$ relative to the vanilla and Chebyshev-trunk DeepONets, respectively, while suppressing artificial near-wall oscillations as the Péclet or mass-transfer Péclet number ranges from $10^{2}$ to $10^{4}$.

In this paper, we study a class of eigenvalue problems involving both local and nonlocal operators, namely the classical Laplacian and the fractional Laplacian, under mixed boundary conditions. More precisely, we consider the problem \begin{equation}\label{1} \left\{ \begin{aligned} \mathcal{L}u &= \lambda f(u), \quad u>0 &&\text{in }\Omega,\\ u&=0 &&\text{in }U^c,\\ \mathcal{N}_s(u)&=0 &&\text{in }\mathcal{N},\\ \frac{\partial u}{\partial\nu}&=0 &&\text{on }\partial\Omega\cap\overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_\lambda$} \end{equation} where \( U=\Omega\cup\mathcal{N}\cup \bigl(\partial\Omega\cap\overline{\mathcal{N}}\bigr), \) \(\Omega\subseteq\mathbb{R}^n\) is a bounded open set with smooth boundary, \(\lambda>0\) is a real parameter, \(f\) is continuous function with \(f(0)=0\), and \[ \mathcal{L}=-\Delta+(-\Delta)^s, \qquad s\in(0,1). \] We establish a characterization theorem for the existence of positive weak solutions to problem (P_{\lambda}). Motivated by the classical elliptic framework developed by Molica Bisci and Rădulescu \cite{MolicaBisciRadulescu2017}, we establish a corresponding characterization result for mixed local-nonlocal operators under mixed boundary conditions.

Lee and Marquis exhibited convex cocompact hyperbolic reflection groups in dimension 5 whose limit sets are homeomorphic to the 3-sphere, but none of whose finite-index subgroups can be realized as 4-dimensional real hyperbolic lattices. Using different methods, we furnish right-angled examples.

Quadratic moment-based quasi-posteriors depend on a weighting matrix governing both the contributions of moment conditions and the scale of posterior uncertainty. We propose curvature-calibrated quasi-Bayesian (CCQB) updating, which iteratively evaluates the inverse of the estimated moment covariance (or long-run covariance) at the center of the resulting fixed-weight quasi-posterior. A two-part stopping rule monitors changes in the center and Monte Carlo error in estimating it. Under local regularity conditions, the ideal calibration map is contractive, and its fixed point is root-N consistent. The calibrated quasi-posterior and the sampling distribution of its center have the inverse Godambe information matrix as their root-N-scale covariance. Simulations show substantial gains over identity weighting. Further iteration has little effect in linear designs but improves calibration for some parameters in nonlinear settings, although nominal coverage is not uniformly achieved. An application to longitudinal childhood wheeze data illustrates the method under within-subject dependence.

For every prescribed integer $N\ge2$, we construct uniformly rotating unit-vorticity vortex patches $\omega=\mathbf{1}_D$ for the planar Euler equation such that $D$ is connected and $\mathbb{R}^2\backslash D$ has exactly $N$ bounded connected components. These components are genuine zero-vorticity holes, rather than opposite-sign vortex patches or regions carrying a second nonzero vorticity level. The angular velocities lie in the rigidity-compatible interval $(0,1/2)$, and the domains converge in measure to the Rankine disk as the holes collapse. The construction starts from a fixed co-rotating polygonal configuration of $N$ unit-vorticity vortex patches and removes a shrinking spatial copy of that configuration from the Rankine disk. An exact complement identity solves all inner-boundary equations before the outer circle is perturbed. The remaining defect is generated by the $N$-th exterior multipole and has size $\varepsilon^{N+2}$. The resulting outer correction feeds back into the normalized inner problem at size $\varepsilon^{2N}$. Separate renormalization of the outer and inner equations produces a limiting affine system with a lower-triangular derivative. Its diagonal blocks are the nonresonant Rankine operator and the angular-velocity-augmented linearization of the fixed seed configuration. We also determine the first corrections to the outer boundary, the hole boundaries, and the angular velocity.

For every fixed $k\ge2$, we give a randomized one-pass insertion-only algorithm that colors an $n$-vertex $k$-uniform hypergraph of maximum degree $\Delta$ with $O(\Delta^{1/(k-1)})$ colors using $\widetilde O_k(n)$ bits of working memory. As a graph-theoretic result of independent interest, we also prove a tight palette-sparsification theorem for general uniform hypergraphs. Independently sampled lists of $\Theta(\sqrt{\log n})$ colors from a palette of size $O(\Delta^{1/(k-1)})$ preserve colorability with high probability; the list-size dependence is asymptotically optimal. These results extend to bounded-rank hypergraphs. We complement the algorithm with a deterministic lower bound: for every fixed polylogarithmic semi-streaming space bound, there are polylogarithmic values of $\Delta$ for which any deterministic one-pass algorithm requires $\exp(\Delta^{\Omega(1)})$ colors.

We revisit the Moore–Read construction of the $\nu=1/k$ Laughlin states on compact Riemann surfaces, deriving their conformal blocks from $U(1)_k$ Chern–Simons theory through the CS/WZW correspondence. We compute their explicit monodromies under quasi-hole transport and Aharonov–Bohm flux insertion, and conjecture that these coincide with the corresponding adiabatic holonomies. Under this identification, we recover classical results on Laughlin states including the flux-averaged Hall conductance $1/k$ via a vector bundle of Laughlin states over $Jac(\Sigma)$. We also relate our construction to the recently developed algebro-geometric approach to higher genus Laughlin states.

Let $X$ be a metrizable space, and let $\mathrm{Met}(X)$ denote the space of metrics compatible with the topology of $X$, regarded as a subspace of the space of continuous pseudometrics with the topology induced by the supremum metric. We first prove that if $Z$ is a discrete space whose cardinality is the first uncountable cardinal, then $\mathrm{Met}(Z)$ is not Borel. As a consequence, if $\mathrm{Met}(X)$ is Borel, then $X$ is separable. Combined with a theorem of Koshino, our result yields that the space of bounded compatible metrics on a metrizable space $X$ is completely metrizable if and only if $X$ is $\sigma$-compact. We also establish non-Archimedean analogues for spaces of ultrametrics. We further prove componentwise versions of these results, where two metrics belong to the same component if their distance is finite.

Rather than prescribing an evaluation interval a priori, the proposed framework characterizes each evolving unsteady flow using a sequence of time windows. A consecutive-window convergence criterion is introduced to automatically identify a representative time window within its fully developed stage. The objective evaluation and discrete adjoint analysis are then carried out consistently over the identified representative time window. The framework is implemented using a regularized lattice Boltzmann method-based large-eddy simulation (LBM-LES) solver together with a partial bounce-back fluid-solid model. The consecutive-window convergence criterion is first validated using the backward-facing step flow. Cylinder-flow applications are then employed to investigate the influence of different flow regimes on the proposed framework. The wake-flow recovery problem verifies its effectiveness for unsteady topology optimization, while U-bend optimization further demonstrates its capability to identify and reorganize complex vortical structures.

In this paper, we develop an evolutionary generalized transferable neural network (Evo-GTransNet) solver for parabolic partial differential equations, formulated as a retained-space fixed-feature Galerkin method of lines. A GTransNet provides the prescribed spatial dictionary, while only the retained output coefficients evolve, thereby avoiding nonlinear training during time integration. To address severe mass-matrix ill-conditioning, we apply a quadrature-weighted truncated singular value decomposition (SVD) to select the numerically resolved trial space, followed by a separate rescaling that makes its basis orthonormal with respect to the assembly-quadrature mass inner product. Rank truncation modifies the approximation space, whereas the subsequent orthonormalization changes only its coordinate representation and preserves the retained discrete functions in exact arithmetic. The resulting semidiscrete coefficient system has an identity mass matrix, and we establish a semidiscrete energy law for symmetric linear parabolic problems. With the implicit midpoint scheme for time discretization, we further prove the contractivity of the method and derive a conditional fully discrete error estimate in which the error is controlled by the retained-space approximation error and the consistency defects. Numerical experiments demonstrate second-order temporal convergence using repeated feature samples together with separate assembly and validation quadratures, and quantify the accuracy of the retained space in the presence of severe raw-mass ill-conditioning. For the high-frequency and multiscale benchmark problems considered here, GTransNet achieves the smallest mean validation errors among the tested fixed-feature dictionaries at the same nominal output dimension.

The notion of $p$-derivation as introduced by Buium has a rich history of applications in arithmetic geometry. Working over $\ZZ_p$, Buium-Ralph-Simanca showed that arithmetic differential operators built from these determine $p$-adic analytic functions and vice versa. Notably, over $\ZZ_p$, there is only one $p$-derivation. In this article, we consider the same question for ramified extensions $A_\pi$ with uniformizer $\pi$. This has the effect of passing from ordinary to partial differential operators, since such extensions can enjoy multiple $\pi$-derivations. We introduce a notion of analytic functions which are naturally determined by these operators in a way analogous to that in the unramified case, but with a significantly richer structure. We show that under mild conditions, analytic functions and partial differential operators determine each other in this setting.

For a full-dimensional rational polytope $P\subset\mathbb{R}^d$ and a real dilation parameter $t>0$, the integer point enumerator is defined by $L_{P}(t):= |tP\cap\mathbb{Z}^d|$. We determine exactly which translation vectors $\mathbf y=(y_1,\ldots,y_d)\in\mathbb{R}^d$ have the property that the single translated counting function $t\longmapsto L_{P+\mathbf y}(t)$, with $t\in\mathbb{Q}_{>0}$, uniquely determines $P$ among all full-dimensional rational polytopes in $\mathbb{R}^d$. The necessary and sufficient condition is that $1,y_1,\ldots,y_d$ be linearly independent over $\mathbb{Q}$. In particular, we may use the explicit algebraic vector $\mathbf y^* := (2^{1/(d+1)},2^{2/(d+1)},\ldots,2^{d/(d+1)})$ in every dimension $d$. The sufficiency proof recovers the primitive facet inequalities from isolated discontinuities of the counting function, while necessity follows from an affine-unimodular obstruction.

Kendall's tau and Spearman's rho, two widely used dependence measures in statistics and risk management, are often treated as interchangeable, yet can disagree sharply: Schreyer et al.~(2017) established the exact region of attainable $(\tau,\rho)$ pairs. We study the complementary question of equality, namely, identifying nontrivial families of copulas $C$ satisfying $\tau_C=\rho_C$. We prove that this equality holds for every factorizable copula of the form $C_{e,\alpha}\ast C_{\beta,e}$, where $\alpha$ and $\beta$ are piecewise linear monotonic surjections (PLMS). For these PLMS-generated copulas, we study further dependence measures, including Chatterjee's rank correlation coefficient and tail dependence coefficients, revealing some useful algebraic formulas and unexpected phenomena. In particular, Chatterjee's coefficient can exhibit extreme asymmetry.

We present a compact formula for the exact partition function of the $d$-dimensional arithmetic Ising model (AIM). For a $2\times2$ system, we express it analytically using the $q$-Hurwitz-Lerch zeta function and derive explicit forms for the free energy and entropy. Additionally, we find that the entropy increases at high temperatures, supporting the presence of entropic order.

We present a macroscopically conservative, rank-adaptive method for solving a hybrid Vlasov-Fokker-Planck (VFP) equation in which the ions are treated kinetically, and the electrons are treated as a fluid. Solving this system poses several coupled computational challenges: the curse of dimensionality, conservation of macroscopic quantities, stiffness arising from the multiscale nature of the model, and structure preservation. To address these challenges, we combine several established methods, each targeting a specific difficulty, into a unified framework for solving this nonlinear system. Specifically, we employ the recent Reduced Augmentation Implicit Low-rank (RAIL) method for rank adaptivity in velocity space to combat the curse of dimensionality, high-order implicit-explicit time stepping to handle the multiscale stiffness, and the Local Macroscopic Conservative (LoMaC) procedure for conservative truncation of the solution. The RAIL and LoMaC methods are extended from Cartesian to cylindrical coordinates. The resulting framework accommodates implicit rank-adaptive time integration while remaining conservative and leveraging structure-preserving discretizations. We verify the scheme on a suite of test problems and simulate a standing shock to demonstrate the importance of conserving the macroscopic quantities.

We prove uniform a priori estimates for solutions to degenerate complex Monge–Ampère equations in big cohomology classes, using both auxiliary-function technique developed by Guo, Phong and Tong [On $L^\infty$-estimates for complex Monge-Ampère equations, Ann. of Math. (2) 198 (2023), no.1, 393-418], and quasi-psh envelope approach developed by Guedj and Lu [Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, J. Eur. Math. Soc. (JEMS) 27 (2025), no. 3, 1185-1208.]. As an application, we apply our method to prove the Moser-Trudinger and Brezis-Merle-type inequalities for complex Monge-Ampère equations.

A rectangular data matrix is often modelled as noise plus a signal of low rank. When that rank is fixed the picture is classical: a signal direction must exceed a critical strength before it produces a singular value outside the noise bulk, and above the threshold the singular vectors of the observation retain a definite, computable fraction of the planted direction. We ask what survives when the number of signal directions grows with the dimension. Everything here rests on one hypothesis, which we call a subspace local law: seen from inside the signal subspace, the resolvent of the noise should look like a scalar. We show that this hypothesis alone locates the outliers and counts them, and, in a holomorphic form, determines the outlier singular subspaces as well. The conclusions are stated through spectral projectors rather than individual singular vectors, so they remain meaningful when spike strengths collide or come closer together than the error of the approximation, which at growing rank they must. We then verify the hypothesis in four genuinely different settings, from deterministic noise viewed through randomly oriented signal directions to independent entries with fixed deterministic ones, and for Marchenko–Pastur noise we compute every constant explicitly.

A task-effective proposal is not yet a justified physical action. In networked Physical AI, a proposal may be understood while valid, timely, proposal-bound evidence or the authority required to finalize an action remains unavailable. We call this mismatch the verification gap and introduce a Post-Semantic Communication Framework for the systems interface between proposal formation and physical execution. The framework begins with application-declared evidence requirements, represents qualifying observations as evidence records, validates supporting and conflicting records through one path, and separates evidence sufficiency from authorized finalization and a downstream runtime gate. It further distinguishes evidence transfer, which can enlarge the record set reachable by a finalizer, from evidence coordination, which can suppress transmission around records already held at the finalization endpoint. Finite-state framework checks verify that the evaluator implements the declared distinctions consistently. Under the declared model, the controlled communication study exposes a finalizer-dependent asymmetry: sender-finalized Feedback uses evidence transfer to expand evidence reachability throughout the feasible plotted region, whereas receiver-finalized Feedback uses coordination to suppress redundant payload until loss, latency, freshness, and deadline costs shift selection to One-way. Finally, an episode-level reporting schema defines common denominators for future measured Physical-AI studies.

Let $D\subset\mathbb{R}^n$, $n\ge2$, be a bounded convex domain, and let $u_D$ be the torsion function for the restricted half-Laplacian. We prove that $D^2u_D$ is negative definite at every point of $D$. The argument is based on the reflected harmonic extension in a slit domain. Quantitative Schauder estimates in slit domains yield parameter-uniform estimates for the first and second derivatives of the edge remainder; a Schur-complement calculation then determines the inertia of the extended Hessian near the slit edge. Superharmonicity of the logarithmic Hessian determinant and the Gleason–Wolff zero-set theorem exclude interior degeneracy. A method of continuity starting from the unit ball proves the result for smooth uniformly convex domains, and an exhaustion argument treats arbitrary bounded convex domains.