Papers

We study higher-order global estimates for the heat equation on Riemannian manifolds, both for static metrics and for metrics evolving under the Ricci flow. Under minimal geometric assumptions, we derive first-order regularizing estimates for log-solutions of the heat equation, together with upper second-order bounds with explicit constants. Our quantitative approach is based on integral duality methods proposed by L.\ C.\ Evans, J.-M.\ Lasry and P.-L.\ Lions in different settings.

Let $\lambda=\{k_1,\ldots,k_q\}$ be a partition, and let $|\lambda|=k_1+\cdots+k_q$. A $|\lambda|$-list assignment $L$ of a graph $G$ is a $\lambda$-assignment if its color set can be partitioned into $q$ disjoint sets $X_1,\ldots,X_q$ such that $|L(v)\cap X_i|=k_i$ for every vertex $v$ and every $i\in[q]$. This notion, introduced by Zhu [J. Combin. Theory Ser. B, 2020], puts ordinary coloring and list coloring in the same framework. A theorem of Alon [Random Structures Algorithms, 2000] states that every graph with minimum degree $d$ has choice number at least $(1/2-o(1))\log_2d$. Saxton and Thomason [Invent. Math., 2015] later used the hypergraph container method to replace $1/2$ by the sharp constant $1$. It is natural to ask whether a similar phenomenon holds for every fixed partition $\lambda$. Minimum degree alone is not sufficient: balanced complete bipartite graphs have arbitrarily large minimum degree but are always $\{1,1\}$-choosable. We show that the appropriate replacement is the minimum $q$-codegree, defined for $|V(G)|\geq q$ by $\delta_q(G)=\min\{|N_G(S)|:S\subseteq V(G),\,|S|=q\}$. More precisely, for every partition $\lambda$ there exists an integer $d$ such that every graph $G$ with $\delta_q(G)\geq d$ is not $\lambda$-choosable. Let $f(\lambda)$ be the least such $d$. For every fixed $q$, we prove $f(\lambda)\leq2^{(2q+o(1))|\lambda|}$ as $|\lambda|\to\infty$, while $f(\lambda)\geq(q+1)^{-1}(1+1/q)^{|\lambda|}$ for every $\lambda$. For the partition $\{k,\ldots,k\}$ with $q$ equal parts, we determine the threshold asymptotically: $f(\{k,\ldots,k\})=\rho_q^{-(1+o(1))k}$ as $k\to\infty$, where $\rho_q$ is the unique $x\in(0,1)$ satisfying $x=(1-x)^q$. When $q=1$, our result implies $\operatorname{ch}(G)\geq(1-o(1))\log_2\delta(G)$.

This is the first paper in a series on the Gromov-Witten theory of the universal abelian variety over the moduli space of principally polarized abelian varieties of dimension $h$. We conjecture that the generating series of Gromov-Witten classes, when summed over the degree against the principal polarization, is a cycle-valued quasimodular form for $\mathrm{SL}_2(\mathbb{Z})$ and satisfy a holomorphic anomaly equation. These conjectures generalize the quasimodularity of the Gromov-Witten theory of elliptic curves to higher dimension and raise interesting questions regarding enumerative mirror symmetry for abelian varieties. In genus $1$ it specializes to a conjecture of Greer and Lian which was proven by Iribar Lopez after tautological projection. We also discuss a special family of abelian varieties with a conjectural relation to Siegel quasimodular forms of higher genus. The main result of the paper is a proof of the conjectures in genus $2$ after tautological projection. For that we introduce quotient Gromov-Witten invariants which are indexed by the characteristic polynomial of the curve class and are shown to determine all descendent Gromov-Witten invariants satisfying a degree conditions. We then give an explicit formula for all genus $2$ quotient invariants after tautological projection as the Shimura lift of the product of two Eisenstein series. The formula is based on a curious modular identity derived in a joint appendix with Brandon Williams.

We prove some conjectures of K. C. Au concerning the descents of cyclotomic levels in certain sums over multiple polylogarithms. Via answers to a question of Au in some particular cases, we also confirm Broadhurst's conjecture on honorary multiple zeta values at even weights.

We study properties of the calculus $\varphi\mapsto\varphi(A,B)$ for self-adjoint operators with commutator $[A,B]$ in the Schaten–von Neumann class $\boldsymbol{S}_p$. It turns out that this calculus defined on the Besov class $B_{\infty,1}^1({\Bbb R}^2)$ in the case $p\le2$ admits a commutator Lipschitz estimate and is multiplicative modulo $\boldsymbol{S}_p$. On the other hand in the case $p>2$ there is no commutator Lipschitz estimate. There is no commutator Lipschitz estimate in the operator norm as well.

We perform a large-$x$ asymptotic analysis of the Fredholm determinant of an integrable integral operator with generalized sine kernel, where $x$ controls the strength of the oscillations along the integration contour of the operator and will play the role of the distance variable in applications to the correlation functions of integrable quantum systems. Our generalized sine kernel involves a number of functional parameters that will allow us to adapt it to the analysis of the dynamical correlation functions of the Lieb–Liniger model at finite temperature and for all positive values of the coupling constant. It will also allow us to consider a class of equilibrium correlation functions that are governed by generalized Gibbs ensembles. Our work is based on the analysis of a matrix Riemann–Hilbert problem that is canonically connected with our integrable integral operator.

We answer a question of Li–Liu–Xu: every $n$-dimensional klt germ $x\in(X,\Delta)$, where $n\ge2$, satisfies the sharp inequality \[ \widehat{\operatorname{vol}}(x,X,\Delta)\le n^{n-1}\operatorname{mld}_x(X,\Delta), \] with equality if and only if $\Delta=0$ near $x$, and analytically, $(x\in X)\cong\frac{1}{r}(1,\ldots,1)$ for some $r\ge1$. We also prove that, in fixed dimension and with coefficients in a fixed finite set, $\widehat{\operatorname{vol}}/\operatorname{mld}$ is discrete away from zero. As applications of the sharp inequality, we obtain lower bounds for minimal log discrepancies of log Fano pairs.

We formulate the inverse problem in information geometry within a two-point tensorial framework and solve it for general metric-affine manifolds, without imposing any curvature or torsion constraints. The construction is explicit and starts directly from the given geometric data: the metric tensor is paired to the affine structure, realized through a local parallelism obtained from parallel transport. This yields a contrast bi-form inducing the original metric-affine manifold. The inverse problems for statistical manifolds admitting torsion and for statistical manifolds are then recovered by homotopical reduction. In this way, suitable pre-contrast and contrast functions are obtained, including several established constructions for statistical manifolds and SMATs. We apply the general theory to reductive homogeneous pseudo-Riemannian manifolds endowed with invariant affine connections. Particular attention is devoted to semisimple Lie groups with Cartan-Schouten connections and to odd-dimensional spheres equipped with Berger metrics.

The lamplighter group over the planar integer grid is proved to not be bi-Lipschitz equivalent to any metric space of negative type, so in particular it does not admit a bi-Lipschitz embedding into $L_1$. This shows the existence of finitely generated metabelian groups on which word metrics are never comparable up to constant factors to conditionally negative definite (CND) kernels, and that the property of admitting a word metric-comparable CND kernel is not preserved by wreath products.

We consider a degenerate chemotaxis system of the form \begin{align}\label{star}\tag{$\star$} \left\lbrace \begin{array}{r@{}l@{\quad}l} &u_t=\nabla\cdot\big(D(u)\nabla u-uS(u)\nabla v\big)+f(u,v),\\ &v_t=\Delta v+g(u,v),\end{array}\right. \end{align} in a bounded domain $\Omega\subset\mathbb{R}^{N}$ with smooth boundary subjected to no-flux and homogeneous Neumann boundary conditions. Herein, the diffusion coefficient $D\in C^0([0,\infty))\cap C^1((0,\infty))$ is assumed to satisfy $D(0)=0$ and $D'(s)\geq 0$ on $(0,\infty)$, and there are $s_0\in(0,1]$ and $d>0$ such that $D(s)\geq ds^{m-1}$ on $[0,s_0]$ and that \begin{align*} s D'(s)\leq C_D D(s)\quad\text{for }s\in[0,s_0]. \end{align*} The sensitivity function $S\in C^2([0,\infty))$ and the source term $f\in C^{1}([0,\infty)\times[0,\infty))$ in the first equation are supposed to be nonnegative. The source term $g\in C^{1}([0,\infty)\times[0,\infty))$ of the second equation can in fact be negative. Prototypical choices for $g$ are $g(u,v)=-uv$ and $g(u,v)=-v+u$. We show under suitable assumptions on weak solutions to \eqref{star} on $\Omega\times(0,T_0)$, that whenever the smoothly bounded domain $\omega\subset\mathbb{R}^N$ and $T\in(0,T_0)$ are such that \begin{align*} \overline{\omega}\subseteq \Omega,\qquad u_0>0\ \text{ in }\ \overline{\omega},\qquad\text{ and }\qquad u>0\ \text{ on }\ \partial\omega\times(0,T), \end{align*} then \begin{align*} u>0\quad\text{in }\ \overline{\omega}\times[0,T). \end{align*} In particular, any dead cores that appear during the evolution must have developed from regions that were already part of the initial zero set.

Componentwise weak convergence of signed synaptic kernels does not, by itself, determine the fast-synapse limit of a sparse threshold-reset network. Within a causal event protocol with clamped refractoriness and smooth positive-delay kernels, we construct two families whose excitatory and inhibitory measures converge weakly to $\delta_0$ while their microscopic arrival orders are reversed. A target fires in the excitatory-first family and not in the inhibitory-first family precisely when $x+a-b<\theta\le x+a$. Strict margins preserve this response under perturbations of the target state, aggregate E/I pulse masses, and bounded drift. The macroscopic effect persists on a moderately sparse Dale-compatible random block graph with $q_N\to\infty$ and $q_N/N\to0$. The two systems share their graph and initial data. Along every deterministic joint scale $\varepsilon_N\downarrow0$, their population-averaged firing counts differ by $1/2+o_{L^1}(1)$. A bounded-degree construction and a later probe show that the discrepancy is macroscopic and can persist through reset. Fixed positive-delay kernels with finitely many classes admit a stable regime. Before grazing, typewise-mixing sparse networks converge to a delayed class mean-field system. Directed Erdos-Renyi graphs yield the bound $O_P(\lambda_N^{-1/2}+\|\pi_N-\pi\|_1)$ when $\lambda_N\to\infty$ and $\lambda_N/N\to0$. This separates stable averaging at a fixed delay from singular collapse. In the latter, componentwise weak convergence discards signed arrival-order information needed by the threshold-reset response.

Lossy compression is conventionally driven by a task-agnostic distortion (e.g., MSE or MS-SSIM), yet in many emerging applications the receiver cares not about uniform fidelity but about a downstream task whose relevant content varies across the signal and evolves over time. We formulate task-aware compression as a weighted rate-distortion problem, in which a single codec is driven by a separable, per-component weighted distortion whose weights encode task importance and may depend on the source. We introduce task consistency, i.e., that minimizing the weighted distortion also minimizes the true task loss, and characterize when it holds: for linear tasks, the task loss admits a weighted-MSE form with signal-independent weights under suitable cross-term conditions, while for nonlinear tasks, an integrated-gradients analysis motivates separable task-aware weights. We show how task symmetry and irrelevance further constrain the admissible weights. Guided by this theory, we realize the weight-conditioned code in a single learned Vision Transformer (ViT) codec whose token-level attention natively consumes a per-component importance vector, so one fixed backbone is re-targeted at runtime, from universal (task-agnostic) to task-specialized operation, purely by swapping the injected weights, without retraining, while producing a single human-viewable reconstruction steered to the active task. On downstream face-analysis tasks, a single model reaches 91.4% accuracy at 0.034 bpp on a localized task, within 1.9% of a task-specific codec (93.3%) and well above a universal codec (76.9%). Such task-adaptive compression suits bandwidth-constrained perception systems, e.g., in Physical AI, where the active task drifts and per-task retraining is infeasible.

A two-parameter hierarchy of classes of finitely generated modules over an artin algebra is described by using the $\mho$-quiver. As an application, it is shown that the stable category of reflexive modules is equivalent to two other categories via the mutually quasi-inverse equivalences induced by $\Omega$ and $\mho$.

Atiyah's Minkowski-space version of the configuration-of-points construction assigns to an admissible marked configuration of $n$ worldlines a collection of $n$ binary forms of degree $n-1$, whose roots are the ordered retarded celestial directions. He conjectured that these forms are always linearly independent. We disprove this conjecture for every $n\ge3$. For $n=3$, an explicit planar one-parameter family yields a real coefficient determinant with exactly one simple zero in a specified interval. At this parameter, all six ordered celestial roots are distinct and the coefficient matrix has rank exactly two. A null-translation construction then multiplies the first three forms by a common factor and produces counterexamples for every $n>3$. Consequently, within the class of complete pairwise disjoint timelike affine lines, universal independence holds at $n=2$ and fails for every $n\ge3$; for each of the counterexamples, the normalized Atiyah–Sutcliffe determinant is defined and vanishes.

Let $\theta$ be a regular cardinal, $R$ a ring, and $M$ a left $R$-module. We prove that for every ordinal $\alpha$ there is a set $I_\alpha \subseteq M^{<\theta}$ of size at most $\beth_\alpha(|R| + \theta)$ such that every parameter-free $L_{\infty,\theta}$ formula of rank at most $\alpha$ is equivalent in $M$ to an infinitary Boolean combination of cosets $\overline{a} + \phi(M)$, where $\overline{a} \in I_\alpha$ and $\phi$ is an infinitary positive existential formula of rank at most $\alpha$. The main ingredient in the proof is a combinatorial lemma which says that given $\kappa$ subgroups of an abelian group, there is a set of at most $2^\kappa$ points which tests whether any family in which each member is either empty or a coset of the corresponding subgroup covers the whole group. The proof proceeds by applying this lemma fiberwise to show that the relevant complete Boolean algebras of positive-existentially definable cosets are closed under projections.

A small polygon is a planar polygon of diameter at most one; let $A_n$ be the largest area at order $n$. Using the global characterization of the even-order maximizers established in a companion paper, we determine their asymptotic geometry. After scaling the angular deficits near the unique pendant diameter, the exact critical equations converge to an autonomous second-order recurrence. Its marked boundary condition selects a unique positive half-line orbit, equivalently the unique minimizer of an explicit strictly convex action. The orbit approaches the regular state with stable multiplier $(-3+\sqrt5)/2$, producing an alternating, exponentially damped boundary layer. A uniform finite-cycle shadowing theorem transfers this profile to the true maximizers. For every fixed depth, an excursion-clipping argument proves that the positive variational finite section is the unique global minimizer on the limiting geometric domain of the Bingane–Mossinghoff construction; this conclusion is expressly distinct from minimization on a looser algebraic box. The sections converge sharply, with two-step error ratio $|(-3+\sqrt5)/2|^4$. The limiting constant $q_*$ has an exact variational definition and a certified rational enclosure. For even $n\to\infty$, $A_n=\frac\pi4-\frac{5\pi^3}{48n^2}-\frac{q_*\pi^3}{n^3}+O(n^{-4}).$ We also identify the leading gap from the Foster–Szabo upper bound and prove that $A_n$ is represented, up to an exponentially small error, by a real-analytic function of $1/n$.

The resonance graph of a hexagonal system is connected, which shows that a perfect matching can be transformed into any other perfect matchings by a series of flips along hexagons. However, the resonance graph of a coronoid system (with holes) is not necessarily connected. Saldanha et al. (Discrete Comput. Geom. 14 (1995) 207-233) used homology and cohomology theory to obtain three versions of criteria for two tilings of a quadriculated region in the plane to be in the same connected component of the flip graph. Inspiblack by the combinatorial version, in this paper we use a purely graph-theoretical approach to give a criterion in terms of simple invariant\textcolor{black}{—flow} across cuts between holes/exterior face for two perfect matchings of a coronoid system $G$ to be in the same connected component of its resonance graph. As a corollary we obtain a criterion for the resonance graph of a coronoid system to be connected. We also discuss whether such \textcolor{black}{criteria} are applicable to nanotubes, and construct a nanotube whose resonance graph is connected, which disproves a conjecture proposed by Tratnik et al. (MATCH Commun. Math. Comput. Chem. 74 (2015) 175-186).

We classify a broad family of weak first-order set theories, under ordinary parameter-free interpretability, by the first-order arithmetical theories with which they are mutually interpretable. This also determines their consistency strength. Set theories that correspond to the same arithmetical theory are often related by deductive extension. It is therefore enough to interpret a stronger set theory in the arithmetical theory and to recover the arithmetical theory in a weaker set theory; all intermediate cases then follow. The set theories under consideration fall roughly into three classes: theories with neither Power Set nor Infinity, theories with Power Set but without Infinity, and theories with Infinity but without Power Set. We conclude with a brief account of the higher levels that remain to be investigated.

In nonsmooth optimization, identifiable sets describe the local region eventually reached by sequences converging to a prescribed critical point. When such a set is a $C^2$ manifold on which the objective restricts to a $C^2$ function, it is called an identifiable manifold. Their appeal lies in what they enable: many first-order methods identify these manifolds in finitely many iterations, after which the iterates enter a region in which the problem is effectively smooth. Consequently, many powerful tools and guarantees from smooth optimization transplant naturally to the nonsmooth setting. Owing to these properties, much existing work has focused on characterizing conditions that guarantee their existence. In this work, we study a complementary question: under what conditions is a critical point devoid of any identifiable manifold? We answer this by developing a deterministic branching criterion for a broad class of finite-max composite optimization problems, characterizing when a critical point admits no identifiable manifold. This criterion is surprisingly mild in certain classes of problems: it holds with high probability for overparameterized robust low-rank recovery and almost surely at common interpolators of random minimax regression, suggesting that the absence of identifiable manifolds may be the rule rather than the exception in modern optimization.

We study the nonlinear stochastic Cahn-Hilliard equation per- turbed by additive colored noise. We show almost sure existence and regularity of solutions. We introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to 1. We also prove strong convergence without known rate.

Let $S_k(\alpha;K)$ denote the exponential sum over $k$-free integers in the short interval $(N-K,N]$. For $s>0$, we prove essentially tight bounds on the $s$-th moments of $S_k(\alpha;K)$ whenever $K \gg N^{\theta_{k,s}+\epsilon}$ for some $\theta_{k,s}<1/2$. As an immediate consequence, we obtain a lower bound for the $L^1$-mean of the Möbius-twisted exponential sum over short intervals of length at least $N^{0.49685}$. Moreover, we show that further improvements on all of these results would follow immediately from improvements to an $\ell^2$-estimate involving the Möbius function.

Let $G$ be a simple graph of order $n$ with adjacency matrix $A= (a_{ij})$. The determinant and the permanent of the matrix $A$ are defined as \[\mathrm{det}A= \sum_{\sigma \in S_n}\mathrm{sgn}(\sigma) \prod_{i=1}^n a_{i\sigma(i)}\quad\text{and}\quad\mathrm{per}A= \sum_{\sigma \in S_n} \prod_{i=1}^n a_{i\sigma(i)},\]respectively. The polynomials $\phi(G;x) =\mathrm{det}(xI-A(G))$ and $\pi(G;x) =\mathrm{per}(xI-A(G))$ are called the characteristic polynomial and the permanental polynomial of $G$, respectively. Two graphs are said to be nearly cospectral with respect to the determinant (resp. permanent) if the difference of their characteristic (resp. permanental) polynomials is a constant. Lv et al. introduced the nearly cospectral graphs problem with respect to the determinant, and provided partial results in the case modulo 4. In this paper, we mainly prove that the corresponding results also hold for the nearly cospectral graphs problem with respect to the permanent. The determinant and permanent are the immanants corresponding to the irreducible characters $(1^n)$ and $(n)$ of the symmetric group $ S_n $, respectively. Here, the immanant $d_\lambda(A)$ of $A$ is defined as \[d_\lambda(A) = \sum_{\sigma \in S_n} \chi_\lambda(\sigma) \prod_{i=1}^n a_{i\sigma(i)},\] where $\chi_\lambda$ is the irreducible character of $ S_n $ indexed by the partition $ \lambda $. The immanantal polynomial of $G$ associated with $ \chi_\lambda $ is given by $ \phi_\lambda(G;x)=d_\lambda(xI-A) $. In this paper, we also establish a similar result for nearly immanantal cospectral graphs in $\mathbb{F}_2[x]$ for all irreducible characters $\chi_\lambda$.

We prove that every isotropic Gaussian mixture with finitely many components has finitely many modes. In one dimension, classical theory of Chebyshev systems gives the sharp bound of at most $n$ modes for an $n$-component mixture. In several dimensions, however, it has remained open whether every such mixture has finitely many modes. Our main result is the stronger statement that the entire critical set has finite cardinality, which is proven by combining real analytic curve selection theorem and Ax's functional-transcendence theorem. As an application, we show that, for Gaussian location mixtures, every nonparametric maximum likelihood estimator (NPMLE) based on a finite dataset is finitely supported. More specifically, all NPMLEs share the same finite set of allowable atom locations.

The Bethe-Hessian is a symmetric matrix for which the negative spectrum has been observed to encode the informative structure of sparse stochastic block models. We prove that, in the stochastic block model where all vertices have expected degree $d>1$, the number of negative eigenvalues of the Bethe-Hessian is exactly the number predicted by the eigenvalues of the planted model lying outside the bulk spectrum. The condition $d>1$ is optimal, and matches a regime in which existing spectral approaches based on larger non-Hermitian matrices apply. Our result extends a theorem of Stephan and Zhu, who established the same conclusion under the assumption $d\geq 2$. Our improvement relies on two main ideas. First, we construct test vectors on the $2$-core, where degree fluctuations are substantially smaller, and then extend them to the entire graph while controlling the quadratic form. Second, we construct the test vectors using an isotropic basis of the underlying Markov random field, with coefficients adapted to each relevant planted eigenvalue. This allows us to control the fluctuations of the test vectors throughout the sparse regime.

Isogeny classes of abelian varieties over finite fields were described by Tate and Honda. Deligne proved that the category of ordinary abelian varieties over a finite field is equivalent to the category of ordinary Deligne modules. Centeleghe and Stix extended Deligne's results to the whole category of abelian varieties, but, if the base finite field is not prime, then the target category is not related to Deligne modules. In this paper we introduce generalized and twisted Deligne modules and give a more direct generalization of the Deligne theorem.

We study the Random Quadratic Form (RQF) on a sphere in the presence of random Brownian forcing. We show that the forcing does not effectively change the law of the process but affects the synchronization properties of the system. While the RQF without forcing exhibits partial synchronization due to the intrinsic symmetries, the introduction of an arbitrarily small forcing results in long-term symmetry breaking and leads to full synchronization. In this work we focus on the small forcing regime and recover the multiscale behavior of the two-point process. We show that in the first stage the model converges to an anti-polar configuration due to the symmetries of the RQF and in the second stage the two clusters meet due to the symmetry breaking phenomenon. The model is motivated by continuous-time machine learning models such as Neural ODEs and continuous-time formulations of transformers. In particular, the results of this work explain the role of the bias and the scale of its initialization.

We propose two-level decorrelated coding (TLDC), a novel coded modulation scheme for the $D_4$ lattice that combines Voronoi shaping with a two-stage decoding process to achieve lattice shaping and coding gains at low complexity. In TLDC, the decoded values of the first level allow the several random variables in the second level to become approximately uncorrelated. The resulting independence of the variables in level two permits decoding in parallel or consolidation into a larger codeword, enhancing performance. TLDC supports flexible choice of FEC within each level. Using bit-interleaved or multi-level polar codes at each level, the resulting coded modulation scheme exhibits a gain of up to 0.5 dB over analogous state-of-the-art coded modulation schemes on a 16-QAM under AWGN at block sizes of 64 and 1024 bits.

We establish quantitative mean-field convergence and propagation of chaos for repulsive Coulomb gradient flows at the bounded-density regularity of the limiting equation. The argument couples the dissipative modulated-energy identity with the normalized quadratic transport cost of the full $N$-particle law. The remaining negative mean-square force-error term is used through an exact completion of squares after mollification: the non-Lipschitz remainder is absorbed by this negative term, while a sharp first-order commutator estimate is applied to the mollified Lipschitz field. For the Coulomb equation, the sharp $L^\infty$ decay gives the density envelope $m(t)=\|\rho_0\|_{L^\infty}/(1+t\|\rho_0\|_{L^\infty})$. A density-adapted transport weight and mollification scale $m(t)^{-1/d}$ yield an Osgood comparison. Thus, for every $d\ge2$ and $\rho_0\in\mathcal P_2(\mathbb R^d)\cap L^\infty(\mathbb R^d)$, we obtain quantitative comparison with the global bounded-density Coulomb solution on every prescribed finite interval. For tensorized initial data, the normalized squared Wasserstein distance of the full $N$-particle law, the expected modulated energy, and the time-integrated mean-square force error are bounded by $N^{-2\gamma_{T,d}/d}$ for $d\ge3$ and $((1+\log N)/N)^{\gamma_{T,2}}$ for $d=2$, where $\gamma_{T,d}=(1+T\|\rho_0\|_{L^\infty})^{-c_d}$. For $d-2<s<d$, we also prove Riesz weak–strong stability for prescribed reference solutions in $L^\infty(0,T;B^{s-d+2}_{\infty,q})$, with Gronwall, Bihari, and Osgood comparisons according to $q$, together with uniqueness in the stated Besov class. Finally, an outlier construction separates modulated-energy convergence and Kac chaos from normalized Wasserstein convergence of the full $N$-particle law.

We study sign-changing viscosity solutions of the singularly perturbed fully nonlinear equation $$ F(D^2u_\varepsilon) = \frac{\alpha}{\varepsilon} \beta\left(\frac{u_\varepsilon}{\varepsilon}\right) \qquad\text{in }B_1\subset\mathbb R^n, $$ where $F$ is uniformly elliptic and $\beta\in C_c (-1,1)$ is nonnegative. We prove the scale-sharp estimate $$ \|\nabla u_\varepsilon\|_{L^\infty(B_{1/2})} \leq C\left( \|u_\varepsilon\|_{L^\infty(B_1)}+\sqrt\alpha \right), $$ with $C$ depending only on the dimension, the ellipticity constants, and $\beta$, and independent of $\varepsilon$ and $\alpha$. This removes a longstanding compactness obstruction in the analysis of fully nonlinear two-phase singular perturbations. The difficulty is structural: at positive $\varepsilon$ there is neither a free boundary nor a prescribed transmission law, while the general fully nonlinear setting provides no monotonicity formula capable of controlling the interaction of the two phases. The proof develops a diffuse counterpart of the De Silva–Savin decay-versus-Lipschitz alternative. Exact planar transitions furnish the local comparison geometry, and curved-test compactness carries this geometry across collapsing reaction layers. An intrinsic transition-region estimate reduces the problem to linear growth from buffered level boundaries. The resulting dyadic continuation is closed by a large-slope stopping argument: bounded accumulated slopes yield the desired growth directly, whereas unbounded slopes force the effective reaction strength to vanish after normalization and lead to a contradiction. The estimate is quantitatively optimal and supplies the scale-invariant compactness framework required for the subsequent sharp-interface analysis.

We develop a new convex-integration scheme, inspired by \cite{BCK26}, for the inviscid surface quasi-geostrophic equation on the two-dimensional torus. For the explicit, nonoptimized exponent $\bar p=\frac{4}{3}+10^{-5},$ we prove a flexibility theorem for weak solutions in the standard momentum formulation with active scalar \[ \theta\in C([0,1];L^{\bar p}(\mathbb T^2)). \] More precisely, any two prescribed mean-zero states in $L^{\bar p}(\mathbb T^2)$ can be approximated at the initial and final times by such a solution. The perturbations are constructed from localized, concentrated traveling SQG profiles whose centers move along rational directions and whose radii depend on the Reynolds stress. Time averages of auxiliary sources along these trajectories reconstruct the preceding-stage stress, while a two-dimensional bilinear null-form estimate compensates for the derivative loss caused by the nonlocal constitutive law. Exploiting the time-locality of the iteration, we also obtain a dense subset of the mean-zero space $L^{\bar p}(\mathbb T^2)$ such that every initial datum in this subset admits at least two distinct momentum weak solutions. Thus, the construction establishes both flexibility and nonuniqueness beyond the concentration-critical exponent $p=4/3$.