Papers

We construct an explicit Hermitian polynomial in six selfadjoint variables, $f=-1+\omega+\omega^*+M\sum_{j=1}^{15}(2-u_j-u_j^*)$, with integer coefficients, degree $72$, and exactly $33$ monomials. Here $\omega,u_1,\ldots,u_{15}$ are specified words, $*$ reverses words, and $M$ is a specified positive integer. Its normalized trace is at least $3/4$ on selfadjoint matrix contraction tuples of every dimension, but equals $-1$ at a specified tuple of selfadjoint unitaries in a group von Neumann algebra. Thus $f$ is a counterexample to the algebraic formulation of Connes' embedding conjecture. We also construct a Hermitian quartic in $37$ selfadjoint variables, with coefficients in $\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}$ and $387$ monomials, attaining the same trace bounds without norm restrictions on selfadjoint matrix inputs. Degree four is minimal in this unrestricted setting. Finally, an encoding in two selfadjoint variables yields a counterexample on the contraction domain with integer coefficients and degree at most $48$; two variables are minimal.

We establish a global uniqueness result at the critical Sobolev regularity for the isotropic Calderón problem in all dimensions $n\ge 4$. More precisely, for every $n\ge4$ and every bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$, any real-valued, uniformly elliptic scalar conductivity $\gamma\in W^{1,n}(\Omega)$ is uniquely determined by its full Dirichlet-to-Neumann map. This settles a long-standing open problem in the field. To prove uniqueness at this endpoint, we develop an averaged trace norm method, which combines complex geometrical optics with trace norm estimates averaged over wave orientations.

We prove that every holomorphic expanding map on a compact connected complex manifold is biholomorphically conjugate to an affine expanding endomorphism of a complex infra-nilmanifold.

Let $p$ be a large prime and $\chi$ a non-trivial Dirichlet character modulo $p$. We study the character sum \[ \sum_{a \sim A} \sum_{b \sim B} \sum_{c \sim C} \alpha(a,b) \beta(c)\chi(f(a) + bc), \] where $z \sim Z$ means $Z\le z < 2Z$, $f \in \mathbb{Z}[X]$ is of small degree $k$, and $\boldsymbol\alpha=(\alpha(a,b))_{a\sim A,b\sim B}$ and $\boldsymbol\beta=(\beta(c))_{c\sim C}$ are two complex coefficients. We prove non-trivial upper bounds for this sum in either of the two cases: (1) $\boldsymbol\beta\equiv1$, $k=2,3,4,5$ and $A,B,C>p^{\frac{1}{8}+\varepsilon}$, (2) $\boldsymbol\beta$ general, $k=2,3$ and $A,B,C>p^{\frac{1}{6}+\varepsilon}$, where $\varepsilon>0$ is fixed. This work was originally motivated by an intermediate result of Ganguly and Rajan (2023) on counting $2\times2$ matrices over $\mathbb{F}_p$ with irreducible characteristic polynomials, where the entries are in short segments. The new bounds here allow us to count such matrices in much shorter segments.

This survey provides an overview of the existence of stable manifolds for tangent to the identity biholomorphisms in $(\mathbb{C}^2,0$) and its connection with the existence of invariant curves for formal vector fields.

In the $q$-color online Ramsey game, Builder and Painter play on an infinite independent set of vertices. At each step, Builder draws an edge and Painter immediately assigns it one of $q$ colors. Builder aims to force a monochromatic copy of a fixed graph $H$. We prove that, for every $q \ge 2$ and $d \ge 1$, Builder can force a monochromatic copy of any $d$-degenerate graph $H$ while drawing a graph of degeneracy at most $d$. The bound $d$ is tight, and this resolves in the affirmative a problem of Conlon, Fox and Sudakov.

We design and analyse an unfitted hybrid high-order (HHO) method for the elastic wave equation in a medium made of two components separated by an imperfect interface of linear slip type, across which the traction is continuous and the displacement jump is proportional to the traction through a compliancy tensor $\bK=\alpha\bI+(\beta-\alpha)\bn\otimes\bn$. The mesh is not fitted to the interface: the discrete unknowns are doubled in the cut cells, the small cuts are cured by a cell agglomeration procedure, and no unknown is attached to the interface. The two specific ingredients of the method are a local symmetric strain reconstruction in each cut subcell, which incorporates the interface condition through the regularised interface stiffness $\bS_h=(h_T\delta^{-1}\bI+\bK)^{-1}$ in the spirit of Hansbo and Hansbo {\em{A finite element method for the simulation of strong and weak discontinuities in solid mechanics.}} {Comput. Methods Appl. Mech. Engrg.}, 193, 2004, and an interface stabilisation built from the same matrix. For the space semi-discrete problem we prove that the discrete bilinear form is coercive and continuous, and we derive an energy-error estimate of order $h^{k+1}$ and an $L^2$-error estimate of order $h^{k+2}$, with constants independent of the compliancy parameters and of how the interface cuts the mesh. The scheme is combined either with the Newmark scheme, which conserves a discrete energy exactly, or with singly diagonally implicit Runge–Kutta schemes of order up to four. Numerical experiments in two dimensions confirm the predicted convergence rates for $k\in\{1,2,3\}$, the robustness with respect to the compliancy over sixteen orders of magnitude, and illustrate the propagation of elastic waves across an unresolved slipping interface.

sGS-PADMM [21, 14, 6] is a powerful and versatile class of convergent multi-block ADMM solvers for implementations on moderate-sized linear semidefinite programming (SDP) problems. In this paper, we further enhance this class of algorithms for solving SDP problems by proposing a new multi-clique decomposition approach, allowing substantial improvements in applications on large-scale sparse SDPs (e.g., where $n > 1000$) with conducive aggregate sparsity patterns. Our SDP decomposition strategy mainly aims to reduce the estimated PSD projection cost after decomposition, in contrast to common decomposition algorithms that are encumbered with minimizing the overlaps between cliques. This feature is made possible by our novel linear-space projection approach that is capable of efficiently processing a large number of overlap constraints via simple averaging steps. For the numerical experiments, we demonstrate the performance of our solver – named McADMM for Multi-clique ADMM – on a number of large-scale SDP instances that arise from relaxations of some important quadratically constrained quadratic programming (QCQP) problems. The performance of McADMM is contrasted against other state-of-the-art decomposition-based solvers as well as the non-decomposed sGS-PADMM to highlight our key contributions. We additionally develop a GPU implementation of McADMM and demonstrate that it can substantially accelerate the decomposed solver.

Numerical investigations of dynamo action have remained active in fluid mechanics over the past decades, presenting numerous challenges and open problems. Meanwhile, the development of structure-preserving methods and finite element exterior calculus (FEEC) inspires a revisit of numerical dynamo studies and an exploration of existing open questions in computation. In this paper, we present a FEEC approach for dynamo problems. In particular, we investigate structure-preserving finite element schemes for computing the spectra and pseudospectra of advection-diffusion operators of differential forms. The schemes and their analysis are based on finite element de Rham complexes.

We study sparse Kahane–Salem–Zygmund constructions and weighted Hardy–Littlewood inequalities for homogeneous polynomials. For supports of cardinality $n^{d+o(1)}$, we determine the sharp power of $n$ governing the smallest norm of a unimodular $m$-linear form on $\ell_{p_1}^n\times\cdots\times\ell_{p_m}^n$; in the diagonal case, this yields the missing polynomial growth exponent in the coefficient-versus-supremum norm problem for $2\le p\le m$ and $2\le r\le\infty$. We then introduce a diagonal weighted Hardy–Littlewood functional which, on $s=p\ge m$, agrees exactly with the classical Hardy–Littlewood coefficient norm with the same optimal constant. We determine the optimal diagonal weight exponent for $2\le p\le m$ and $1\le q\le2$, on the full critical line $p=m$, and on a sharp part of the region $q>2$; at $q=\infty$ the optimal weight exponent is obtained for every $2\le p\le m$. The sparse coefficient estimates provide the matching dimensional obstructions.

We investigate the eigenvalues of Hermitian Toeplitz matrices generated by the symmetric Fisher-Hartwig symbol $\hat a(z) = (1-z)^{\alpha/2}(1-z^{-1})^{\alpha/2}$ for $\alpha \in (0,2)$. Using Dirichlet-Neumann bracketing of the discrete Laplacian, we establish explicit, non-asymptotic bounds for the individual eigenvalues. As a consequence, we prove that all eigenvalues are simple. We also obtain a two-term approximation for every eigenvalue, with explicit bounds on the remainder, as the size of the matrix tends to infinity. While known results are limited to $\alpha > 1$, we bridge this gap by covering the full range of $\alpha \in (0,2)$. Our approach uses a construction of approximate eigenvectors that has not previously been applied in this setting.

Iterative fine-tuning on synthetic data causes model collapse: output diversity narrows as rare patterns are progressively lost, a signature most visible as phrase-level repetition. Existing mitigations either require model log-probabilities, an external oracle, or continued access to real human data. Here we develop a new approach grounded in mathematical information theory: the non-parametric Kontoyiannis entropy rate estimator $h_k$, computed entirely from raw text via match-length statistics, with no model of any kind. We show that this is in fact a superior training-data filter on text-diversity metrics in a fully-synthetic, single-lineage fine-tuning setting. In a six-generation QLoRA collapse experiment on Llama-3.1-8B, logprob-based filtering (the most established model-access-requiring baseline) provides no significant text-diversity benefit on any metric ($p > 0.23$), whereas $h_k$-filtering yields $+42\%$ unique trigrams, $+30\%$ vocabulary, and $-19\%$ repetition (all $p < 0.001$). We validate $h_k$ as a cross-domain entropy proxy ($\beta = 0.924$, $R^2 = 0.746$) and collapse detector ($\rho = +0.454$, $p < 0.0001$) across 4~domains, 2~temperatures, 2~generator–scorer model pairs, and 1{,}520 generated documents. Our results demonstrate that information theoretic approaches to collapse mitigation are efficient, and suggest new approaches for maintaining multi-agent diversity.

The operation of residential energy hubs with multiple energy carriers (electricity, heat, mobility) poses a significant challenge due to the energy storage differences in time-constants, round-trip efficiencies and self-discharge rates. Usually, thermal storage exhibits flexibility in yearly planning optimizations or long-term scenarios. However, as optimization horizons shrink (1-48hs) so does their supplied value due to the lower round-trip efficiencies. To avoid this early depletion during operation this paper proposes a data-driven highway to steer the short-term daily control towards long-term optimality. The proposed methodology also presents how to optimally size the thermal storage and avoid yearly simulations and how all of this is related to nonlinearities in the daily operation. The presented framework links seasonal and daily optimizations through dynamic terminal sets and value functions. The seasonally-aware nonlinear economic model predictive controller achieves the most balanced performance, with the second best mean grid cost of all MPCs at -\texteuro 209. It also achieves better battery degradation control than its linear counterparts (between 26-34%) and the best thermal comfort of the nonlinear benchmarks. Nevertheless, the data-driven seasonal highway restrains the ability to control battery degradation and slightly increases computational time.

We study an incompressible visco-morphoelastic system in three dimensions in which the Eulerian effective strain is transported by the Zaremba-Jaumann rate and coupled to a Navier-Stokes momentum balance through a Kelvin-Voigt stress. The strain remains symmetric and splits into a scalar transport-relaxation equation for its trace and a deviatoric equation containing the nonlinear stretching and Jaumann terms. We regularize both equations by diffusion of order $\varepsilon$, which admits a mechanical interpretation as weakly nonlocal remodeling with a characteristic length scale. For fixed $\varepsilon>0$ we construct Leray-Hopf type weak solutions by a truncated Galerkin scheme and remove the truncation a posteriori using a maximum principle for the scalar variable. As $\varepsilon\to0$, the scalar variable converges strongly in $L^p((0,T)\times\Omega)$ for every $1\le p<\infty$, by a DiPerna-Lions commutator argument on bounded Lipschitz domains, allowing the stretching term to be identified in the limit. In contrast, the Jaumann commutator is a product of two weakly convergent sequences, and its limit cannot be identified from the available estimates. We therefore obtain a defect \[\boldsymbol{\mathcal{D}}\in L^2(0,T;(H^2(\Omega;\mathbb{S}_0))^\ast)\] in the limiting deviatoric equation. We give two sufficient conditions for $\boldsymbol{\mathcal{D}}=0$. The energy-level cancellation used in related corotational viscoelastic models does not close here because the coefficient of the stretching term is itself an unknown.

Let $(\mathcal{C},\mathbb{E},\mathfrak s)$ be an extriangulated category equipped with a proper class $\xi$ of $\mathbb E$-triangles.We establish stability under iteration for the quasi-$\xi$-Gorenstein projective and injective subcategories and extend the relative Ext-vanishing criteria for their finite homological dimensions to larger coefficient subcategories. An example in a bounded homotopy category shows that $\mathcal{P}(\xi)\subsetneq\mathcal{QGP}(\xi)\subsetneq\mathcal{GP}(\xi)$. For module categories, every quasi-Gorenstein projective module over a left perfect ring or a commutative Noetherian ring of finite Krull dimension is projective. When a commutative ring admits a finite chain of trace and nilpotent quotients ending in an Artinian ring and the endomorphism rings at the trace steps have finite right global dimension, we construct finite families of quasi-projective test modules that detect prescribed upper bounds on projective dimension for modules of bounded cardinality through Ext vanishing in finitely many consecutive degrees.

We prove that the tangent sheaf of $\SU_C(n,L)$ is slope stable for every smooth complex projective curve $C$ of genus $g\geq2$, every $n\geq2$, and every determinant $L$. For $g\geq3$, we prove a more general stability theorem on moduli spaces of principal bundles for connected reductive groups, using cohomology sheaves associated with representations in place of tangent sheaves. The adjoint representation gives tangent sheaf stability for the case of almost simple groups. The moduli spaces may be singular, and stability holds with respect to every ample Cartier polarization.

We consider the semiclassical magnetic Schrödinger operator in dimensions one and two. The first result is that an even one dimensional semiclassical potential is spectrally determined by the Schrödinger operator. The second result states that a two dimensional radial magnetic field is spectrally determined by a purely magnetic Schrödinger operator. This extends the earlier work of Guillemin-Wang and answers a conjecture of theirs therein.

Large language models (LLMs) are increasingly deployed as computational engines in autonomous decision-making and planning loops, yet their systems and control treatment remains hindered by architectural simplifications, index conflations, and informal descriptions of tool interactions. This paper presents a control-theoretic formulation of decoder-only language models as multi-index, multi-rate systems, and sets the stage for a stochastic hybrid systems framework to govern the multi-phase dynamics of agentic tool interaction. We formalize the architecture across three hierarchically coupled evolution indices: (i) an ultrafast feedforward cascade of transformer blocks across layer depth, where layer normalization is cast as a spherical projection and key–value caching is proven to be an exact internal state realization via causal prefix invariance; (ii) an uncontrolled stochastic difference recursion over token generation steps, where finite context truncation induces a time-homogeneous Markov chain; and (iii) an autonomous mode-switching mechanism governing transitions between token generation and tool execution regimes, where tool invocations are triggered upon trajectory arrival at switching manifolds, followed by exogenous state jump maps that augment the context string with external observations. By defining a prompt-dependent evaluator over successive evaluable claims, we obtain a task-level error process whose fault-free histories and expected error growth admit bounds under conditional fault-hazard and error-drift assumptions.

We consider two coupled linear heat equations on different spatial domains that interact through a lower dimensional interface. This models conjugate heat transfer. The problem is solved using Dirichlet–Neumann waveform relaxation. This allows the subproblems to be solved using separate codes, a so called partitioned approach. Our overall goal is to develop more efficient partitioned methods, and to this end, we want reliable error estimates. Here, we use an exponentially weighted Fourier technique to derive new error estimates in $l^2$ for finite time $T$ in the fully discrete setting. These describe both linear and superlinear behavior. We show that the fully discrete estimate is close to a previously obtained time discrete estimate and independent of $\Delta x_1$ and $\Delta x_2$ when the CFL number in the subsolvers is large. We also show that the convergence behaviour depends on the ratio $\Delta x_1/\Delta x_2$ when the CFL number is small. Our numerical experiments show that the fully discrete estimate is accurate across a wide range of grid sizes $\Delta x_1, \Delta x_2$, time step sizes $\Delta t$ and $T$.

Given a prime $p$, the Nottingham group $G(p)$ is the group of formal power series $x+a_2x^2+a_3x^3+\cdots$, $a_i\in{\bf Z}/p{\bf Z}$, under substitution. For $n\geq 1$, we write $G_n(p)$ for the quotient of $G(p)$ by its normal subgroup $K_n=\{x+a_{n+1}x^{n+1}+a_{n+2}x^{n+2}+\cdots\,|\, a_i\in {\bf Z}/p{\bf Z}\}$, namely the group of truncated polynomials $x+a_2x^2+\cdots+a_nx^n$, $a_i\in {\bf Z}/p{\bf Z}$, under substitution, which is a $p$-group of order $p^{n-1}$ generated by $x+x^2,\dots,x+x^n$. In this paper, we determine the power-commutator presentation of $G_{15}(p)$ relative to $x+x^2,\dots,x+x^{15}$. This, in turn, automatically gives the power-commutator presentation of $G_{n}(p)$ relative to $x+x^2,\dots,x+x^{n}$ for any $1\leq n<15$, as well as the initial segments of the power and commutator relations in $G_{n}(p)$ corresponding to $x+x^2,\dots,x+x^{15}$ for any $n>15$.

We study the genus-$19$ Albanese fibration of the Cartwright–Steger surface and its order-three symmetry. Passing to the orbifold quotient of the elliptic base gives a relation among the two handle monodromies and the three Dehn twists about the vanishing cycles. The two real vanishing paths give cycles whose real fixed points lie on different ovals of the invariant fiber. For a cyclic triple cover, we express the intersection of a curve with its image under the deck transformation as a signed crossing count on the quotient. We determine the two local branch values of a degree-$72$ bicanonical pencil near a node and study a symmetric bicanonical pencil. A general pencil also gives a finite degree-$72$ map from a blowup of the surface to the product of the elliptic curve and a projective line. The monodromy translates of the nodal vanishing classes span the first homology of a smooth fiber.

We show that free groups admit amenable actions on unital simple AF-algebras. This settles, for free groups, the last remaining case of the existence problem for amenable actions of non-amenable groups on classifiable simple C*-algebras. The proof relies on the Baire category theorem. This result was obtained with substantial assistance from GPT-6 Astra.

The Kannan–Lovász–Simonovits (KLS) conjecture asks whether every isotropic log-concave probability measure on $\mathbb R^n$ has a Cheeger constant bounded below by a universal positive constant. We prove that $\psi_n\le C$ for a universal constant $C>0$, resolving the KLS conjecture. We also prove that $C_P(\mu)\le C'$ for every isotropic log-concave probability measure $\mu$ on $\mathbb R^n$, with a universal constant $C'>0$.

We continue the study of $k$-limited domination in graphs, a domination variant in which each vertex of a dominating set may dominate at most $k$ vertices outside the set. This concept extends classical domination by incorporating capacity constraints on dominating vertices. We improve the general lower bound for the $k$-limited domination number $\gamma_k^L(G)$ by employing the concept of $k$-capacitated domination. As a consequence, we characterize graphs satisfying $\gamma_k^L(G)=\lceil \frac{n}{k+1} \rceil$. We also refine the known upper bound for graphs with $k<\delta(G)$ using the $k$-limited packing number. Under this condition, we describe graphs attaining $\gamma_k^L(G)=n-k$. These results extend and unify previous investigations for the case $k=1$ and provide a complete characterization of graphs attaining the extreme values of the $k$-limited domination number under the considered assumptions.

A strong odd $k$-coloring of a graph $G$ is a proper $k$-coloring such that every color appearing in the neighborhood of a non-isolated vertex appears an odd number of times. The minimum $k$ for which $G$ admits a strong odd $k$-coloring is the strong odd chromatic number, denoted by $\chi_{\text{so}}(G)$, of $G$. Given a graph $G$ and an integer $k$, \textsc{strong odd $k$-colorability} problem asks whether $G$ admits a strong odd $k$-coloring. It is known that STRONG ODD $k$-COLORABILITY is NP-complete in general graphs. In this paper, we prove that the problem is NP-complete on perfect elimination bipartite graphs for $k\geq3$, which is a subclass of bipartite graphs. Furthermore, we show that $\chi_{\text{so}}(G)$ is inapproximable within a factor of $O(n^{\frac{1}{2}-\varepsilon})$ for every $\varepsilon>0$. On the positive side, we obtain a linear time algorithm to compute an optimal strong odd coloring for block graphs. From a parameterized perspective, we present an FPT algorithm for STRONG ODD $k$-COLORABILITY when parameterized by treewidth. Moreover, we show that the problem cannot be solved in time $(k-\varepsilon)^{\texttt{tw}}n^{O(1)}$ for every $k\geq3$ and $\varepsilon>0$ when parameterized by treewidth under SETH. Furthermore, we show that STRONG ODD $k$-COLORABILITY does not admit a polynomial kernel when parameterized by feedback vertex set. Lastly, we prove that STRONG ODD $k$-COLORABILITY is W[1]-hard when parameterized by clique-width.

Wireless digital twins (WDTs) are promising enablers for developing and evaluating AI-native radio access networks, yet constructing a high-fidelity WDT typically requires substantial manual effort to integrate heterogeneous information and infer unknown propagation-related parameters. This paper proposes Agentic WDT (AWDT), an end-to-end agentic framework for autonomous WDT construction and calibration using readily available environmental information and measurements readily obtainable from commercial smartphones. AWDT comprises three agents: EnvAgent constructs the propagation environment, OpAgent infers BS and sector configurations, and MatAgent calibrates radio material properties. The agents iteratively refine the WDT using discrepancies between LTE/NR reference signal received power (RSRP) measurements and ray-tracing predictions. Real-world experiments show that AWDT reduces the RSRP prediction MAE from 11.46 to 5.25 dB, demonstrating a substantial improvement in ray-tracing fidelity. Evaluation with an independent measurement system further demonstrates cross-device transferability with lightweight device-specific bias adaptation, highlighting the potential of agentic AI for scalable WDT construction and calibration with limited prior knowledge.

Weak modal logics can be obtained by restricting the axioms and inference rules governing the modal operator. At the extreme, one may consider the least modal logic, in which no specifically modal axiom or inference rule is imposed. We introduce closure semantics, a new possible-world semantics for such weak modal logics, and establish soundness and strong completeness for a range of systems. We also give truth-preserving translations between closure models and other possible-world semantics, including neighborhood, FMT, and Kripke semantics.

Prescribing the speed of gradient flow on the risk itself, by the dynamics $\dot w=-u(E(w))\nabla E(w)/\abs{\nabla E(w)}^{2}$, makes the risk $e(t)=E(w(t))$ obey $\dot e=-u(e)$ exactly, whatever the landscape~$E$; the time needed to reach zero risk from $e_0$ is $\int_0^{e_0}\dd e/u(e)$. Minimizing this time alone is ill posed, and we study the regularized problem $\inf\{\int_0^{e_0}(\tfrac\lambda2\abs{u'}^{2}+1/u)\,\dd e:\ u\in H^{1}(0,e_0),\ u\ge0,\ u(0)=0\}$, $\lambda>0$. We prove that the minimizer exists, is unique, and is a linearly scaled cycloid, and we show that the optimal rate behaves like $u^{*}(e)\sim(9/(2\lambda))^{1/3}e^{2/3}$ near zero risk: the exponent $2/3$ is the one found in \cite{betti2026holder} by a power-law ansatz, and it lies in the Hölder window $(\tfrac12,1)$ where the arrival is in finite time with vanishing weight speed. The proof follows the classical route: existence by the direct method, uniqueness by strict convexity, positivity of the minimizer away from the origin, and the explicit integration of the Euler-Lagrange equation.

Let $a_t(n)$ denote the coefficients of the Nekrasov–Okounkov hook length generating function $F_{t^2}(x)$, and let $b_t(n)$ denote the coefficients of the $t$-core partition generating function $C_t(x)$. We prove a sharp prime-power dichotomy: for every integer $t\ge 2$, the congruence $$ a_t(n)\equiv b_t(n)\pmod t $$ holds for all $n\ge 0$ if and only if $t$ is a prime power. When $t=p$ is prime, the congruence strengthens to modulo $p^2$. For $t=6$, and more generally for every integer that is not a prime power, the congruence fails. This identifies prime powers as the exact moduli for which these two partition-theoretic coefficient sequences are arithmetically equivalent.

Weak perturbations of the same size can lead to terminal Wasserstein errors of different orders. For infinitely divisible laws smoothed by a Gaussian, we differentiate their distribution functions with respect to variance-weighted Lévy characteristics. This includes distributional directions caused by moving atoms. We obtain first- and second-order expansions that are uniform over Lipschitz test functions. When the first response is nonzero, Gaussian analyticity gives the signed second coefficient of the Wasserstein–1 distance. When the first response vanishes, a finite second displacement moment is enough to obtain a quadratic response. Its coefficient is strictly positive if the displacement dispersion is nonzero. For local balanced remeshing, the error is comparable to the grid-alignment variance. This gives sharp quadratic grid rates for compactly supported densities and nonaligned atomic sequences. We derive metric speed and length along admissible non-atomic curves, and use the response to build a linear program with a proved bound on how much its error exceeds the infimum over the allowed class and consistent quadrature. We also prove a sharp transition as smoothing vanishes and give a multivariate second-order expansion. The appendix gives state-dependent response formulas and further stability estimates.