Papers

We study optimal control problems for a stationary reaction–diffusion model describing the spatial distribution of a tumor cell population with Gompertz growth. The control $m(x)$ represents a treatment term acting as a density-dependent removal rate and it is subject to $L^{1}-L^{\infty}$ constraints. When the intrinsic growth rate is constant, the uniform distribution of the treatment is shown to be the unique minimizer. For the maximization problem, we prove that every optimal control is of bang-bang type. In addition, we show that in the one dimensional case and for sufficiently large diffusion rates, the positivity set of optimal controls is an interval sticking to one of the extrema of the domain. Finally, numerical simulations complement the theoretical analysis and explore regimes that are not fully covered by the results proved in the paper. The computations confirm the bang-bang structure of maximizers, and illustrate how the shape of optimal controls and the associated states are affected by spatial heterogeneity in the growth rate, localized admissible treatment regions, and the diffusion coefficient. Moreover, they reveal a monotone dependence of the optimized total population on the diffusion coefficient: this is a new phenomenon with respect to the logistic setting.

Let $k$ be a perfect field and let $A$ be a representation-infinite finite-dimensional $k$-algebra. We prove that the Auslander–Reiten quiver of $A$ has infinitely many connected components. This establishes, for finite-dimensional algebras over perfect fields, a conjecture of Auslander, Reiten, and Smalø concerning Artin algebras. Over an algebraically closed field, the proof combines a localized polynomial representation embedding with semilinear twists induced by field automorphisms. The passage from a perfect field to its algebraic closure is obtained by separable base change: we prove that if the Auslander–Reiten quiver of $A$ has only finitely many components, then the same holds for the scalar extension to the algebraic closure.

A central problem in spectral Turán theory is to understand the relationship between the spectral extremal family ${\rm SPEX}(n,F)$ and the ordinary extremal family ${\rm EX}(n,F)$. For many forbidden graphs $F$, it is known that ${\rm SPEX}(n,F)\subseteq{\rm EX}(n,F)$ holds for infinitely many $n$, while only a few examples have been identified where the two families are disjoint. In this paper, we study this problem for suspensions of balanced trees. A tree is balanced if its two bipartition classes differ in size by at most one. Let $T$ be a balanced tree on $2k$ or $2k+1$ vertices and $\widehat T$ be its suspension which is obtained from $T$ by adding one new vertex adjacent to every vertex of $T$. Our first main result establishes a tight upper bound for the spectral Turán number of $\widehat T$ for sufficiently large $n$ provided that $T$ satisfies some mild assumptions. Our second result determines for which integers $k$ and which non-path balanced trees $T$ on $2k$ or $2k+1$ vertices there are infinitely many integers $n$ such that ${\rm EX}(n,\widehat{T})\cap {\rm SPEX}(n,\widehat{T})=\emptyset$.

We discuss the question of when a homology or cohomology class with twisted integer coefficients of a manifold $X$ is realised by a submanifold. While this question is classical in nature, providing an answer requires relatively modern techniques from parametrised homotopy theory. More specifically, we introduce cobordism classes twisted by a coefficient system and then define a twisted Thom space $\operatorname{M^\mathrm{tw}O}(n)$ over $\operatorname{BO}(1)$, which serves as the classifying object for this cobordism theory under a twisted Pontryagin-Thom construction. As a result, a twisted homology class is realisable if and only if its Poincaré dual is the image of the twisted Thom class in $\operatorname{M^\mathrm{tw}O}(n)$ under a parametrised map $X \to \operatorname{M^\mathrm{tw}O}(n)$ over $\operatorname{BO}(1)$. By showing that $\operatorname{M^\mathrm{tw}O}(n)$ is parametrised homotopy equivalent to the unbased homotopy orbits of $\operatorname{MSO}(n)$ under the orientation-reversing involution, we are able to construct its parametrised Postnikov tower to derive obstructions to realisability. We conclude by giving the first known examples of non-realisable integer homology classes in non-orientable manifolds.

We use deformation theory to show that a cover of smooth, projective, geometrically connected curves $\pi\colon C\to D$ over a $p$-adic field can be $p$-adically perturbed to obtain a nearby simply branched cover $\pi'\colon C'\to D$ and that, if $D=\mathbb{P}^1$ and $\pi^*\mathcal{O}_{\mathbb{P}^1}(1)$ is very ample, then we may take $C'=C$. As an application, we give a formula for the wild conductor exponent of a curve at a prime $p>d$ in terms of the ramification data of any degree $d$ cover $C\to\mathbb{P}^1$.

We develop Interval Algebraic Quantum Field Theory (IAQFT), a finite-precision operational formulation of algebraic quantum field theory (AQFT). Physical experiments provide only finitely many observations at finite resolution and therefore determine regions of compatible states rather than uniquely specified point states. IAQFT represents this information by quantum parcels—non-empty convex weak$^*$ open regions of state space—with observational parcels determined by finitely many finite-precision expectation constraints. Exact AQFT states remain indispensable mathematical objects and are recovered under appropriate parcel refinement. We establish reduction to exact states, measurement update and no-signalling, and formulate locality through compatible parcel nets. In finite dimensions an exact Jacobian formula for invertible Lüders updates gives a volume-contraction criterion. Spacelike vacuum correlations and strict CHSH violations persist throughout sufficiently small observational parcels. Reeh–Schlieder cyclicity becomes finite-precision local reachability, while parcel equivalence of vacuum refinements implies vacuum GNS equivalence and hence does not evade Haag-type obstructions. Tomita–Takesaki modular automorphisms act on parcels, measurement update is modularly covariant, and modular conjugation induces a duality between normal parcels of an algebra and its commutant. State-dependent modular correlations are continuous under predual-norm variation of faithful normal states; shrinking parcels with vanishing KMS defect recover a KMS state. Bisognano–Wichmann yields a parcel formulation of the Unruh effect, recovering the exact KMS relation and Unruh temperature under refinement. Finally, limiting projection faces and their conjugation actions recover the Murray–von Neumann factor classification.

We prove that the Bergman metric of a (possibly unbounded) pseudoconvex domain with an exponentially flat infinite type boundary point cannot be Einstein. Our result provides further evidence in support of the Cheng–Yau conjecture beyond the finite type setting.

We develop a combinatorial theory for finite arrangements of connected semialgebraic curves with ordinary multiple intersections, governed by a local node contribution $\psi$ that determines global geometric and topological properties. We prove exact region-count formulas, characterize maximal arrangements, and extend the deletion–restriction recurrence to general curve arrangements. In the algebraic setting, we prove that the absence of triple points ($k_x=3$) is a sufficient condition for the OS-type algebra to factor through cohomology; the converse, however, fails already for line arrangements, where the classical Orlik–Solomon relations ensure factorization even in the presence of triple points. For line arrangements we compute the discrepancy between the simplified OS-model and $H^2$, showing it is governed by nodes with $k_x\ge4$ and equals $\sum_{k_x\ge4}\binom{k_x-1}{2}$. The node contribution appears in the mixed Hodge structure via the Euler characteristic; for arrangements in normal crossing position we compute the full weight decomposition of $H^2$ and show it is Hodge–Tate exactly when every component has genus zero, recovering the line-arrangement case as $\dim\operatorname{Gr}^W_4H^2=\psi$. The defect complex for concurrent lines reveals that exactness obstructions require curve-wise incidence data. Finally, we introduce binomial node invariants $\{\Psi_k\}$, prove $\Psi_2$ is the universal linearly locally additive invariant, and show $\psi=\Psi_1-\Psi_0$.

The simply typed {\lambda}-calculus is a model of computation where typed terms correspond to proofs of intuitionistic propositional logic (IPL) via the Curry-Howard correspondence. Justification logic is an operational modal logic in which the standard box modality is replaced by an explicit proof term, allowing the logic itself to reason directly about proofs of its formulas. Standard justification logics reason about proofs of IPL by embedding Hilbert-style axiomatic proofs of IPL as proof terms of the logic. We instead introduce a justification logic of the {\lambda}-calculus, in which the proof terms of the modality are exactly the typed {\lambda}-terms themselves: a modal logic that reasons about computation and proof simultaneously, as both notions coincide under the Curry-Howard interpretation. First, we provide an axiomatisation of this logic. We then propose a natural deduction system and a Curry-Howard interpretation, where a formal connection between the term calculus and the proof terms of the logic is provided. To complete the picture, we give a Gentzen-style sequent calculus for which we prove cut-elimination, and consequently obtain a normalisation result by working in the negative fragment.

Liouville numbers form a classical class of transcendental real numbers characterized by exceptionally strong rational approximations. A theorem of Maillet shows that non-constant rational functions with rational coefficients preserve the Liouville property, motivating a question of Mahler on whether analogous phenomena hold for transcendental functions. In this paper, we address this problem for real functions of finite smoothness. For any $\varepsilon>0$, we construct an uncountable set of $C^k$-functions on $\mathbb{R}$, dense with respect to the topology of uniform convergence on compact sets, mapping $\mathbb{Q}$ into itself and satisfying $\operatorname{den}(f(p/q)) \le q^{2k+\varepsilon}$, and deduce that such functions preserve Liouville numbers. In contrast, we prove a rigidity result about a $C^{2k+1}$-function mapping $\mathbb{Q}$ into itself and satisfying $\operatorname{den}(f(p/q)) \ll q^k$.

Subset sums over finite abelian groups lie at the intersection of additive combinatorics, design theory, and coding theory. Let $G$ be a finite abelian group, and let $\cB_k^x$ be the family of $k$-subsets of $G$ whose elements sum to $x\in G$. This paper studies when the incidence structure $(G,\cB_k^x)$ is a block design. The elementary abelian $p$-group case was settled by Falcone and Pavone. Pavone (Des. Codes Cryptogr. 91 (2023), 2585–2603) further asked whether, for an arbitrary finite abelian group $G$, the zero-sum incidence structure $(G,\cB_k^0)$ can be a nontrivial $2$-design only when $G$ is an elementary abelian $p$-group. We settle this open question in the stronger form that, for every $x\in G$, $(G,\cB_k^x)$ can be a nontrivial $2$-design only if $G$ is an elementary abelian $p$-group. The proof develops a character-theoretic approach to subset-sum designs, using character sums over the blocks to constrain the structure of the character group $\widehat G$. The approach also yields a complete characterization of subset-sum $1$-designs and general arithmetic restrictions on subset-sum designs over arbitrary finite abelian groups, extending the corresponding results previously known for finite abelian $p$-groups.

Even though entropy methods have been used in combinatorics for at least five decades, only in recent years has their use really proliferated. There are now tens, if not hundreds, of combinatorial papers that crucially rely on the notion of entropy and exploit the various powerful identities and inequalities relating entropies. In this short survey article, we give a selective overview of these works and discuss several of them in more detail, outlining some of the key ideas.

We prove a multidimensional extension of a strong Hales-Jewett theorem that simultaneously and "directly" extends Ramsey's theorem and Hindman's theorem. The proofs show the effectiveness and simplicity of the techniques based on iterated nonstandard extensions that have been recently developed. Unlike existing ultrafilter proofs, our arguments to prove the strong Hales-Jewett theorem assume neither minimal nor idempotent ultrafilters. To demonstrate this, we translate our proof of the strong Hales-Jewett theorem into an ultrafilter proof that requires only non-principal ultrafilters.

We prove a local well-posedness criterion for the Vlasov–Poisson system on $\R^d_x\times\R^d_v$, $d\geq2$, under an anisotropic assumption on the initial distribution. The datum has finite mass, its weighted velocity supremum belongs to $L^p_x$ for some $p>d$, and it has an arbitrarily small positive Hölder regularity in the velocity variable, uniformly with respect to velocity and with the same spatial $L^p$ control. The main estimate is a nonlinear mixing bound \[ [\rho(t)]_{C^\alpha_x}\lesssim t^{-d/p-\epsilon}C(f_0), \qquad \epsilon>0~~\text{small}. \] Thus the density is integrable in time with values in a positive spatial Hölder class, and the corresponding electric field belongs to $L^1_tC^{1,\alpha}_x$. We construct a solution by a Schauder fixed point on the density and prove uniqueness by a Loeper-type stability estimate.

We introduce a new notion of positive dependence, namely multivariate total positivity of order two for conditional distributions, denoted cMTP$_2$, that explicitly incorporates the conditioning variables. This property is weaker than multivariate total positivity of order two for densities, but stronger than multivariate total positivity of order two for distribution functions, stochastic monotonicity, and tail monotonicity. It thus provides a natural intermediate concept linking these classical notions of positive dependence. We verify the cMTP$_2$ property for several distributions and copula families, including Archimedean copulas, and establish stability under Markov product transformations. As a consequence, we derive comparison results for measures of directed dependence arising from Markov structures, including Chatterjee's rank correlation and a related sensitivity measure.

Based on a 1981 work by G. Talenti, which established a comparison principle for smooth solutions of the Monge-Ampère equation alongside related two-dimensional a priori estimates, this paper explores the connection to the Blaschke-Santaló inequality and its reverse form by K. Mahler (1939). More specifically, in this work, a quantitative version of one of Talenti's a priori estimates is proved. We exploit the interplay between Talenti's analytical framework and Mahler/Blaschke-Santaló inequalities via convex geometry, demonstrating how this link can be utilized to gain sharp insights into the Alexandrov-Bakelman-Pucci (ABP) maximum principle for elliptic PDEs.

The layer number of a finite point set is the number of iterations needed to delete it by repeatedly removing the vertices of its convex hull. Ambrus, Hsu, Peng, and Yan conjectured that the layer number of the $d$-dimensional integer grid $\{1,\ldots,n\}^d$ is of order $n^{2d/(d+1)}$ for every fixed $d$. We prove this conjecture. Let $P_i$ be the convex hull of the point set remaining after $i$ steps, and let $Z_n$ be the convex hull of the lattice points in the Euclidean ball of radius $n$. For every step that leaves a nonempty point set, the Minkowski sum $P_{i+1}+Z_n$ contains no vertex of $P_i+Z_n$. Integrality of normalized lattice volume, together with the Bárány–Larman vertex estimate for $Z_n$, gives a lower bound, independent of $i$, on the resulting volume decrease. Summing over $i$ yields the matching upper bound, even when $P_i$ is lower-dimensional. For $d\ge2$, the same upper bound holds uniformly over all nonempty subsets of $\{1,\ldots,n\}^d$.

Let $(X,G)$ be a tame flow and let $K$ be an Ellis group of its enveloping semigroup $E(X,G)$. Although $K$ is a compact Hausdorff topological group in its $\tau$-topology, the inclusion $K\hookrightarrow E(X,G)$ need not be Borel. We show that normalized Haar measure on $K$ nevertheless determines, via the Riesz–Markov theorem, a canonical regular Borel probability measure $\mu_K$ on $E(X,G)$, called its Haar decompression. Haar decompression gives an exact ergodicity description. For an arbitrary flow $(X,G)$, amenability of its Ellis flow is equivalent to hereditary amenability of all finite powers $X^n$. In the tame setting, $(E(X,G),G)$ is amenable if and only if $(X,G)$ is hereditarily amenable. Moreover, for the minimal left ideal $\mathcal{M}$ in $E(X,G)$ containing $K$, $\mu_K$ is $G$-invariant if and only if $(\mathcal{M},G)$ is amenable; when this holds, $\mu_K$ is the unique ergodic measure on $\mathcal{M}$ and $\mathcal{M}=\overline{K}$. Then, we conclude that the ergodic measures of $E(X,G)$ are precisely the Haar decompressions associated with amenable minimal left ideals. We also prove that every minimal tame amenable flow is uniquely ergodic and that every ergodic invariant measure on a tame flow has minimal support. Consequently, every ergodic measure on a tame ambit arises by evaluating a suitable Haar decompression.

As an homage to the memory of Hans Willem Capel, I present some personal reminiscences and thoughts that come to my mind in this special occasion.

We give explicit formulas for the smallest non-trivial irreducible representation $V$ of quantum affine superalgebras in types D$_{2\vert 1;\alpha}$, $\dim V=18$ (in the all-fermionic parity), F$_{3\vert 1}$, $\dim V=41$ (in the distinguished parity), and G$_{2\vert 1}$, $\dim V=32$ (in the distinguished parity), both in the Drinfeld-Jimbo and in the new Drinfeld realizations. We use this information to obtain an explicit expression for the corresponding $R$-matrices.

We introduce a class of goodness-of-fit tests for the spherical Cauchy model on the unit hypersphere. The proposed procedures exploit the invariance of the spherical Cauchy family under Möbius transformations: after estimating the parameter by the sample Möbius mean, the observations are transformed to approximate spherical uniformity, and a projection-based uniformity statistic is applied to the resulting sample. We show that, under mild conditions, the resulting tests are exactly distribution-free under the null hypothesis, so that exact critical values can be arbitrarily well approximated by Monte Carlo simulation. We study the Möbius mean as a population functional, establish its existence and uniqueness under mild conditions, and prove equivariance and asymptotic linearity of its empirical counterpart, which coincides with the spherical Cauchy maximum likelihood estimator. We derive the asymptotic null distribution of the test statistic and show that it coincides with that of the underlying uniformity statistic after removing the degree-one spherical-harmonic component, which corresponds to the tangent space of the spherical Cauchy model. We establish consistency against fixed alternatives and characterize local powers through the spherical-harmonic decomposition of contiguous alternatives. Monte Carlo experiments demonstrate the finite-sample accuracy of the asymptotic approximations and the empirical power of the proposed tests. A real data example is treated.

In a recent breakthrough, Kalmynin proved a conjecture of Sárközy on additive irreducibility of the set of quadratic residues in a prime field. More recently, Kim, Yip, and Yoo initiated the study of a multiplicative analogue of the conjecture for shifted multiplicative subgroups. Specifically, they showed that for an odd prime $p$, a proper multiplicative subgroup $G$ of $\mathbb F_p^*$, and $\lambda\in G$, there do not exist sets $A,B\subseteq \mathbb F_p^*$ with $|A|,|B|\ge 2$ such that $AB=(G-\lambda)\setminus\{0\}$. In this paper, when $\lambda \in \mathbb F_p^* \setminus G$, we completely resolve this problem in the equality case from a Stepanov bound in a prime field.

Virtual Artin groups were recently introduced by Bellingeri-Paris-Thiel as a generalisation of virtual braid groups. In this article, we initiate a geometric study of these groups by constructing an analogue of the Deligne complex for virtual Artin groups, and we prove that it is CAT(0) for all locally reducible defining graphs (a class that contains in particular two-dimensional graphs and graphs without any label $3$, and which is generic in the sense of Goldsborough-Vaskou). As applications, we classify finite subgroups of locally reducible virtual Artin groups, showing that such groups are conjugated into an isomorphic copy of the corresponding Coxeter subgroup. We also prove an analogue of the $K(\pi, 1)$-conjecture for locally reducible virtual Artin groups: we show that these groups are virtually torsion-free and admit a cocompact model of classifying space for proper actions of minimal dimension, equal to the virtual cohomological dimension of the group.

We present a new proof of Wolpert's Magic Formula, stating that any Fenchel–Nielsen coordinates on the Teichmüller space of a closed surface are Darboux coordinates for the Weil–Petersson symplectic form. Our approach relies on a de Rham cohomology model for the tangent space to the character variety model of Teichmüller space, in which, by the seminal work of Goldman, the Weil–Petersson form is expressed by a natural symplectic form, called the Goldman form. We extend the work of Fillastre and Seppi, who have managed, using that approach and Stokes's Theorem, to provide a new proof of Wolpert's sum of cosines formula. Using the unique isometric symmetry on any hyperbolic pair of pants, we also introduce a way, given a pair of pants decomposition of a closed surface, to construct an associated linear involution on every tangent space of its Teichüller space. That allows us to derive a new proof of Wolpert's Magic Formula and deduce a self-contained proof of the closedness of the Goldman form on Teichmüller space.

Let $\rho(G)$ denote the adjacency spectral radius of a graph $G$ of order $n$. We determine the sharp constant in an open problem of Nikiforov (2008) on cycles of consecutive lengths. For every $\varepsilon>0$ and all sufficiently large $n$, if $G$ is an $n$-vertex graph with $\rho(G)>\sqrt{\lfloor{n^2/4}\rfloor},$ then $G$ contains a cycle $C_\ell$ for every integer length $3\le \ell\le (\frac{3-\sqrt5}{2}-\varepsilon)n.$ The constant $(3-\sqrt5)/2$ is best possible, as shown by the split graph $K_k\vee\overline K_{n-k}$ with $k\sim(3-\sqrt5)n/4$. Our result improves all previous results [LAA2008, CPC2020, JGT2023, JGT2023, GC2024]. The proof combines the degree form of Szemerédi's regularity lemma, a spectral matching theorem of Feng-Yu-Zhang, Weyl's inequality, a refinement of Łuczak's connected-matching embedding method, and other ideas.

This paper shows that interaction is unnecessary for order-optimal 1-bit mean estimation under finite central moments. For distributions satisfying $|\mathbb{E}X|\leq\lambda$ and $\mathbb{E}|X-\mathbb{E}X|^k\leq\sigma^k$ for a fixed $k>1$, we construct a fully non-adaptive public-coin protocol that fixes every measurable 1-bit query before communication. All localization and refinement queries are generated in a single batch; a subsequently decoded coarse center changes only how the stored refinement bits are interpreted. Two complementary constructions realize this decoder-side refinement: a finite dyadic scheme based on periodic residues and a continuous-scale scheme based on shifted random grids. Up to $k$-dependent constants, the refinement cost is $(\sigma/\epsilon)^2\log(1/\delta)$ for $k>2$, $(\sigma/\epsilon)^2[1+\log(\sigma/\epsilon)]\log(1/\delta)$ for $k=2$, and $(\sigma/\epsilon)^{k/(k-1)}\log(1/\delta)$ for $1<k<2$. Together with the additive localization cost $1+\log(\lambda/\sigma)$, these rates answer the Lau–Scarlett open problem for arbitrary measurable 1-bit queries in the affirmative. In the parameter range covered by existing small-error, high-confidence lower bounds, the resulting sample complexity is minimax optimal.

In this paper we study an equation driven by a non-local operator of order "near zero" with mixed Dirichlet-Neumann boundary conditions, modelling a biological system consisting of a population self-competing for resources in a closed environment and subject to an extremely heavy-tailed dispersal process. We establish several results concerning the existence of states in which the population can survive requiring different assumptions on the non-linear reaction term and on the various configurations of the sets where the Dirichlet or Neumann conditions hold.

A subspace of $\mathbb{F}_q^n$ is called cyclically covering if the union of $\sigma^i(U)$ can cover the whole space $\mathbb{F}_q^n$, where $\sigma$ is the cyclic shift, $0 \leqslant i \leqslant n-1$. Let $h_q(n)$ be the largest possible co-dimension of a cyclically covering subspace of $\mathbb{F}_q^n$. We show that $h_2(2p) = 2$ for every prime $p$ such that $2$ is a primitive root modulo $p$. By constacyclic codes, we show that $h_q((q-1)n) = 0$ when $h_q(n) = 0$ and $\gcd(n,q-1) = 1$. We also derive a lower bound on $h_q(n)$ by the concept of support weight distribution, which is important in coding theory. Finally, using irreducible cyclic codes, we present several families of $n$ such that $h_q(n) = 0$.

Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we refine the blow up analysis for sequences of solutions of a class of perturbed singular Liouville equations which share the phenomenon of “blow up and concentration without quantization". The problem is delicate because we are dealing with the exact threshold value above which one meets the well known “concentration without quantization" phenomenon, as recently pushed forward in [C.S. Lin, G. Tarantello, C. R. Math. Acad. Sci. Paris (2016)] and in [Y. Lee, C.S. Lin, G. Tarantello, W. Yang, Comm. PDE. (2017)]. We overcome these difficulties by developing different strategies for “fast" and “slow" blow up, combining refined Pohozaev identities, elliptic estimates and “Sup+CInf" inequalities in the same spirit of [C.C. Chen, C.S. Lin, Comm. An. Geom. (1998)]. In the process, we also derive a new sharp Harnack type inequality by using a moving-plane method, following the ideas in [D. Bartolucci, C.C. Chen, C.S. Lin, G. Tarantello, Comm. P.D.E. (2002)].

Neural spectral PDE solvers often learn an entire unresolved vector field even when an inexpensive approximate model can already capture most of the trajectory. Here we introduce Perturbative-NeuSA, a residual formulation that decomposes the target solution into a low-fidelity background and a high-resolution perturbation, so that only the unresolved dynamics is learned. Starting from the exact perturbation equation, the method combines a fixed spectral operator, a background-dependent correction, the background defect in the target PDE, and an optional neural closure. This construction makes the roles of physical structure and neural closure separately measurable. Across 2D Burgers, Klein-Gordon, and heterogeneous 2D wave equations, the deterministic structured solver outperforms the trained NeuSA baseline while requiring no neural-network training. The largest gains occur on Burgers, where the deterministic correction reduces training and extrapolation errors by factors of 24 and 44, respectively. In addition, a Klein-Gordon sweep over seven background resolutions shows that the effect of the closure is conditional: it improves a poor background by 3.6 times, becomes neutral at intermediate resolutions, and degrades a well-resolved background. For the wave equation, however, the closure provides an additional 18% reduction when the remaining residual is interface-localized. Multi-initial-condition diagnostics further show that the useful closure regime depends on the initial-condition spectrum and can disappear in extrapolation when structured correction already captures the dominant Burgers dynamics. Perturbative-NeuSA therefore reframes neural closure as a conditional, diagnosable correction governed by background fidelity, residual organization, and compatibility with the closure model.