Papers

Consider an adjoint triple of triangle functors between two nice enough triangulated categories. In this paper, we are looking for conditions under which the silting, respectively, cosilting property ascends or descends via at most left, respectively, at most right adjoint.

Accurately predicting the spatiotemporal evolution of amyloid-$\beta$ and tau proteins at the individual level is critical for improving the diagnosis and treatment of Alzheimer's disease. We consider the problem of constructing patient-specific digital twins that model the propagation of these biomarkers on the cortical surface using reaction–diffusion dynamics. A major challenge is that the underlying nonlinear aggregation mechanisms are unknown and must be inferred from sparse, noisy, and heterogeneous longitudinal PET imaging data. To address this, we develop a data-driven framework that learns biomarker dynamics directly from clinical observations. The approach combines operator learning with reduced-order representations to infer governing equations of disease progression from data. Using this framework, we achieve predictive accuracies of 87% for amyloid-$\beta$ and 81% for tau. Building on the learned dynamics, we further formulate a PDE-constrained optimal control problem to design personalized therapeutic strategies that regulate pathological protein propagation. By integrating data-driven dynamical modeling with treatment optimization, the proposed digital twin framework provides an interpretable and predictive platform for understanding disease progression and enabling precision interventions in neurodegenerative disorders.

Transfer systems as studied in equivariant algebra admit minimal generating sets, analogous to bases in linear algebra. It is natural to wonder if minimal generating sets form a matroid. We show that this happens only for lattices which are linear orders, or for cyclic groups of prime power order. In this case, we compute some invariants of the resulting matroid.

A Zoll magnetic system on an oriented closed surface $M$ is a Riemannian metric $g$ together with a function $\lambda\colon M\to \mathbb{R}$, such that every unit speed solution of the ODE $\ddot \gamma(t)=\lambda(\gamma(t))\gamma(t)^\perp$ is periodic and the minimal period depends continuously on $\gamma$. The trivial example is given by $g$ with constant curvature $K$ and $\lambda\equiv {\rm const.}$ such that $\lambda^2+K>0$. This article exhibits non-trivial Zoll magnetic systems for every genus-for genus $\ge 2$ these are the first such examples. The approach is twistor theoretic: To a general magnetic system $(g,\lambda)$ one associates its transport twistor space $Z(g,\lambda)$, which is the unit disk bundle $DM$, equipped with a degenerate complex structure that encodes the magnetic flow. For the trivial Zoll magnetic systems explicit holomorphic blow-down maps $\beta\colon Z(g,\lambda)\to W$ into certain ruled surfaces $W\to M$ are constructed, mapping $\partial Z(g,\lambda)$ onto a Lagrangian $P\subset W$. For small Lagrangian perturbations $P'\approx P$ the procedure can be reversed and this results in a large class of (non-trivial) nearby Zoll magnetic systems.

Sampling from an unnormalized target density by reversing an Ornstein-Uhlenbeck diffusion requires the score of each noise-perturbed marginal law. Two exact identities are available: Tweedie's identity and a target-score identity, each yielding unbiased finite-reference score estimators for the OU-marginal score. Score estimators induced by scalar blends of Tweedie and TSI score estimators can reduce variance, but they are too rigid for singular or strongly anisotropic targets. We formulate blended score estimation as a conditional risk-minimization problem over matrix valued blending coefficients, referred to as gates. Our central result is to show the optimal matrix valued gate for blended score estimation is given \[ G_\star(y,t) = \alpha_t^2 \left(\alpha_t^2 I_d + \gamma_t\, \mathbb{E}[H_0(X_0)\mid Y_t=y] \right)^{-1}, \qquad H_0=-\nabla^2\log p_0 .\] Here $\alpha_t = e^{-t}$ and $\gamma_t = 1-e^{-2t}$ are the OU coefficients, and the conditional expectation is under the OU posterior of $X_0$ given $Y_t=y$. We call this formula the Laplace-Fisher Gate Identity (\LFGI{}). Because the Tweedie-TSI disagreement has conditional mean zero, the gate changes the score-estimator variance but not its expected value. We derive the variance-optimal matrix gate, record the Gaussian special case, and establish finite-reference consistency and stability bounds for estimating the gate from weighted reference samples. We then use the finite-reference LFGI score estimator for normalized density evaluation in Bayesian inverse problems. In regimes where MCMC pilot samples and derivative information are already available, LFGI uses those byproducts to construct a normalized surrogate for the posterior density. The resulting surrogate supplies information that the MCMC samples alone do not provide: posterior-energy evaluation, model-evidence estimation, and downstream density-based diagnostics. On a PDE-constrained inverse-problem benchmark, the LFGI surrogate improves posterior-density calibration and sampling diagnostics relative to the other tested score-estimator classes. Experiments using LFGI with known model evidence check absolute evidence calibration in both Gaussian and non-Gaussian settings.

A star coloring of a graph $G$ is a proper coloring in which no path on four vertices is bicolored. The star chromatic number $\chi_{\star}(G)$ is the minimum number of colors in a star coloring of $G$. In this work we study star colorings from the perspective of forbidden induced subgraphs, focusing on three subclasses of chordal graphs. We provide both a structural characterization and a characterization in terms of forbidden induced subgraphs for star $3$-colorable chordal graphs; such characterizations yield a simple certifying recognition algorithm, running in time $O(|V|+|E|)$, for this class. We also characterize split graphs that are star $4$-colorable and star $5$-colorable in terms of (finitely many) forbidden induced subgraphs, again deriving linear-time certifying recognition algorithms. Finally, we study star colorings of $2$-trees and $2$-paths: we characterize the $2$-paths that are star $4$-colorable, prove that every $2$-path is star $5$-colorable, and exhibit a $2$-tree on $21$ vertices with star chromatic number $6$ such that any proper induced subgrahp has star chromatic number $5$.

We determine a geometric condition necessary and sufficient for an $m$-dimensional manifold in Euclidean space to support a probability measure $\mu$ satisfying the Fourier decay bound $|\widehat{\mu}(\zeta)| \lesssim_\varepsilon |\zeta|^{\varepsilon - m/2}$ for all $\varepsilon > 0$. As a result, for each $m > 0$, we explicitly determine the largest codimension of an $m$-dimensional smooth submanifold $M$ of Euclidean space which is a Salem set. This largest codimension is precisely expressible in terms of the Radon-Hurwitz numbers. In particular, we find that the only odd dimensional manifolds which can be Salem sets are hypersurfaces, and that the largest codimension of an $m$-dimensional manifold which is a Salem set is upper bounded by $2 \lg(m/2) + 3$, and equal to $2 \lg(m/2) + 3$ when $m$ is a power of 16. The proof strategy, which involves covering manifolds by certain stationary sets associated with the Fourier transform on that manifold, is robust, and we demonstrate its use by proving that all nondegenerate curves in $\mathbf{R}^n$ have Fourier dimension equal to $2/n$, and find an alternate proof of a result of Junjie Zhu on the Fourier dimension of hypersurfaces with a fixed number of vanishing principal curvatures.

We work towards completely classifying all hyperelliptic Atkin-Lehner quotients of Shimura curves $X_0(D,N)/W$ with level $N$ coprime to $D$ and $W \le W_0(D,N)$, extending, on the one hand, a result of Ogg that provided such a classification for the trivial quotients (the case $W = 1$), and on the other hand, results of Furumoto and Hasegawa that provided such a classification for modular curves (the case $D = 1$). As a byproduct of our methods, building on the works of Guo and Yang, we also obtain models for some quotients of genus at most two, answering some questions of Padurariu and Saia.

Alternative means have recently attracted considerable attention in matrix analysis and operator theory. In this paper, we investigate the linearization problem for alternative means, namely the question of determining when a mean can be expressed as an affine combination of the matrices under consideration. We first prove a conjecture of Choi, Kim, and Lim for the Wasserstein mean. More precisely, we show that the Wasserstein mean $\text{A} \diamond \text{B}$ is linearizable if and only if $\text{A}\text{B} = \text{B}\text{A}$ and $\left|\text{Spec}(\text{A}^{-1}\text{B})\right| \leq 2$. We further establish a general rigidity theorem for a large class of alternative means. Specifically, we prove that an analogous characterization holds whenever the representing function $f$ is of the form $f(x) = \sqrt{h(x)}$, where $h$ is a non-affine operator monotone function. As consequences, we obtain linearization criteria for several families of alternative means, including logarithmic, harmonic, and power-type means.

Every Cartesian three-dimensional PDE solver hides a structural secret that production CFD codes have used for half a century and that graduate-level textbooks rarely state plainly. The derivative matrices, the compact Padé line solves, the Galerkin mass inversions, the alternating-direction-implicit substeps, and even the fast Poisson and Helmholtz diagonalization transforms all factor along the coordinate axes and collapse into repeated one-dimensional banded kernels executed along the grid lines. The three-dimensional operator exists only on paper; it is never assembled, factored, or stored. This paper is the manual for that collapse. We derive the Kronecker-product algebra that makes it exact, carry it cleanly through central differences, compact schemes, tensor-product Galerkin, B-spline and isogeometric methods, collocation, ADI time stepping, and direct Poisson and Helmholtz solves, and bring into the open the three production tricks that turn the reduction into hardware-conscious floating-point throughput on real machines: the multi-right-hand-side reshape that exposes a sweep as one batched line kernel (a dense BLAS-3 GEMM when the line factor is dense or element-local, a banded or stencil kernel when it is not), the sum factorization that rescues high-order Galerkin from the $O(p^{2d})$ quadrature trap, and the pencil decomposition that keeps every direction contiguous across an MPI cluster. For fixed stencil width or fixed polynomial degree, the compute cost stays $O(N)$ in the total number of unknowns $N = N_x N_y N_z$; the operator storage drops to $O(N_x + N_y + N_z)$ up to bandwidth constants; direct separable Poisson and Helmholtz solvers add the expected transform cost; the line kernels are embarrassingly parallel. These facts are familiar to practitioners but rarely assembled in one place; this paper collects them and shows how to use them.

We investigate the geometry of classical trajectories generated by separable two-dimensional polynomial potentials of the form $V(x,y)=\tfrac{1}{2}\big(x^{2N}+A\,y^{2N}\big)$, where $N=1,2,\ldots,$ and $A>0$. Special emphasis is placed on the emergence of nonlinear Lissajous figures and on the distinction between global and particular superintegrability in the Liouville sense. In the harmonic case ($N=1$) closed periodic orbits are a consequence of an additional global integral of motion whenever the frequency ratio is rational, rendering the system maximally superintegrable. In contrast, for anharmonic oscillators, already in the quartic case ($N=2$), the oscillation frequencies depend on the partial energies, so periodic Lissajous-type trajectories occur only under nonlinear resonance conditions fixed by the initial data. Accordingly, the extra conserved quantities that characterize these closed orbits are not global invariants but particular (trajectory-dependent) integrals that emerge only on the resonant trajectories. For higher-degree potentials $N\geq3$, the resonant trajectories are naturally described by hyperelliptic phase constraints rather than by a universal polynomial orbit equation.

Let $M$ be a manifold and let $\mathcal{M}$ be a simplicial model category. Given an object $A$ in $\mathcal{M}$, Tsopméné and Stanley constructed a topological space $\hat{A}$ that classifies homogeneous functors of degree $k$ from the poset of open subsets of $M$ into $\mathcal{M}$. They showed that the set of weak equivalent classes of such functors that maps disjoint union of $k$ open balls to $A$ is in bijection with the set $[F_k(M), \hat{A}]$ of homotopy classes of maps out of $F_k(M)$, the unordered configuration space of $k$ points in $M$. In this paper, we begin a study of the space $\hat{A}$, and we prove that $\hat{A}$ is weakly equivalent to the classifying space $B\mathrm{haut}(A)$, where $\mathrm{haut}(A)$ is the simplicial monoid of self weak equivalences of $A$. This proves a conjecture of Tsopméné and Stanley. Our result enables us to generalize the classification of homogeneous functors of Weiss for $\mathcal{M}=\mathcal{T}\mathrm{op}$ to any simplicial model category.

The paper investigates the construction of Lie algebroids and $3$-Lie algebroids via connections generated by finite families of differential operators and dual sections. We first recall the description of Lie and $n$-Lie algebroid brackets in terms of connections, and introduce an $n$-curvature operator whose $n$-Bianchi identity characterizes the fundamental identity. We then provide sufficient conditions under which such generating families determine Lie algebroid and $3$-Lie algebroid structures. The construction extends the single-operator approach and covers natural examples such as the Jacobi Lie algebroid. As an application, we construct a concrete $3$-Lie algebroid structure arising from Poisson Lie algebroid data.

We show that a simple irreversible Finsler geometry can be recovered uniquely and Lipschitz-stably from its travel time data. We introduce and use a version of Gromov–Hausdorff distance adapted to irreversible metric spaces. In contrast to reversible (e.g. Riemannian) geometry, even the question of stability becomes ill-defined without simplicity.

We study processes that consist of deterministic evolution punctuated at random times by disturbances with random severity; we call such processes semistochastic. Under appropriate assumptions such a process admits a unique stationary distribution. We develop a technique for establishing bounds on the rate at which the distribution of the random process approaches the stationary distribution. An important example of such a process is the dynamics of the carbon content of a forest whose deterministic growth is interrupted by natural disasters (fires, droughts, insect outbreaks, etc.).

We establish optimal Lipschitz lower bounds for proper smooth functions on three-dimensional Riemannian manifolds with Ricci curvature bounded below by negative constants, yielding a new family of width estimates for Riemannian bands using Gromov's $\mu$-bubble method, together with rigidity statements characterizing the equality case. One of the novelties of our approach lies in its ability to handle higher-genus boundary components, revealing a precise interplay between the width, the area of boundary surfaces, and the underlying topology. Finally, for a complete noncompact three-manifold $M$ with bounded geometry and scalar curvature $R_g\ge -6$, whose $H_2(M,\mathbb{Z})$ contains no spherical or toroidal classes, we prove a sharp lower bound for the boundary area. In the equality case, the manifold is shown to be isometric to an infinite hyperbolic band.

It is well known that, in Boolean algebras, the notions of precontact relation, quasi-modal operator, and subordination relation are interdefinable. In contrast, within the setting of distributive lattices, this equivalence holds only between quasi-modal operators and subordination relations, but not with precontact relations. In this paper, we study the class of bounded distributive lattices equipped with a precontact relation, referred to as precontact lattices. We also examine how conditions imposed on the precontact relation correspond to first-order properties in the Priestley dual. In addition, we characterize precontact substructures and strong precontact sublattices of precontact lattices in terms of a suitable lattice preorder. Finally, we introduce a notion of precontact congruence and identify the corresponding closed sets in the dual space.

Building on the null-infinity-layer construction, we develop an H1-conforming finite-element formulation of hyperboloidal compactification for the exterior Helmholtz equation. A change of coordinates maps infinity to a finite outer boundary, and a rescaling removes the leading oscillatory decay. We derive the transformed equation and a global sesquilinear weak formulation with bounded coefficients. The compactified boundary contributes an explicit boundary mass term, and its trace gives the far-field pattern up to a known normalization. We compare the resulting method with finite-element discretizations using perfectly matched layers (PML) and report benchmark results in two and three dimensions. Numerical experiments include scattering by a unit disk, resonance in a trapping geometry, a manufactured benchmark in three dimensions, and a submarine benchmark.

We investigate how reward design shapes the internal attention patterns of reinforcement learning agents trained for autonomous driving. Using three Perceiver-based agents that share identical architectures and training data but differ only in their reward configurations$\unicode{x2014}$ranging from basic violation penalties to continuous proximity penalties$\unicode{x2014}$we analyze cross-attention allocation across 50 real-world scenarios from the Waymo Open Motion Dataset. A central methodological finding is that naïve pooling of timesteps across episodes substantially underestimates the attention$\unicode{x2013}$risk relationship; within-episode correlation with Fisher z-transform aggregation is the appropriate statistic and reveals a robustly positive link between collision risk and agent-directed attention. Building on this validated methodology, we demonstrate two reward-conditioned effects: agents trained with navigation rewards allocate up to $2.0\times$ more attention to GPS-path tokens than those trained with additional proximity penalties$\unicode{x2014}$and $4.7\times$ more than agents with no navigation incentive$\unicode{x2014}$revealing that reward content directly determines which scene elements the encoder prioritizes, and continuous time-to-collision penalties create a $\textit{learned vigilance prior}$$\unicode{x2014}$elevated resting agent surveillance maintained throughout collision-free phases. In several scenarios, the complete-reward and minimal-reward models exhibit opposite attention$\unicode{x2013}$risk correlation directions, demonstrating that reward design can qualitatively reverse attentional strategy rather than merely modulating its magnitude. These results suggest that attention analysis is a practical diagnostic for verifying that a reward function produces the intended representational behaviour in safety-critical RL systems.

In this paper, we analyze the bit complexity of deciding whether a given polynomial can be represented as a sum of squares of polynomials. We show that the weak membership problem for the sum-of-squares cone lies in $\mathrm{P}$. Furthermore, we give a polynomial-time algorithm which computes, for a given polynomial and positive parameter $\epsilon$, an $\epsilon$-relaxed closest sum-of-squares polynomial.

We investigate a two dimensional flux limited Keller Segel FLKS system using classical Lie symmetry analysis. The complete Lie point symmetry algebra is determined and shown to consist of temporal and spatial translations together with planar rotations. An optimal system of one-dimensional subalgebras is constructed, yielding stationary, translationally invariant, radially symmetric, travelling-wave, and rotatingwave reductions. The corresponding reduced equations are derived systematically. The results demonstrate that the flux limiting mechanism eliminates the scaling symmetries of the classical Keller Segel model and significantly restricts the class of admissible invariant solutions. Exact spatially homogeneous equilibrium solutions are determined, and integral representations for stationary radial and travelling-wave profiles are derived.

We study one-parameter degenerations of rank-two vector bundles on complex threefolds to a rank-two torsion-free sheaf with an isolated point singularity. We prove a rigidity identity: the algebraic bubbling multiplicity of the central fiber equals one half of the Ext-length of the singularity of its reflexive hull. Furthermore, bubbling is forced when a family develops such an isolated point singularity. We use this identity to obtain smoothability obstructions and construct sharp local smoothings. We realize the local example from earlier joint work with Sun as a global degeneration of smooth Hermitian-Yang-Mills connections. Rescaling this degeneration produces a smooth non-flat Hermitian-Yang-Mills connection on $\mathbb C^3$ with density one at infinity, whose tangent cone at infinity has flat connection part and a multiplicity-one line as the blow-up cycle. We also construct smoothings of elementary modifications of projective-cone singularities with explicit algebraic bubbles. These examples give local models for Hermitian-Yang-Mills point bubbling in complex dimension three and distinguish this phenomenon from bubbling along complex codimension-two loci.

The binary binomial sequences correspond to the diagonals of the Pascal's triangle modulo 2. They have interesting properties such as they form a basis of the linear space of all binary sequences with period a power of 2. Other properties of these sequences (period, linear complexity, construction rules or relations among different binomial sequences) have been deeply analysed in detail previously. In this work, we study the binomial $p$-ary sequences for a prime $p$, its intrinsic characteristic and formation rules. We also prove that the family of $p$-ary sequences with period a power of $p$ form a vector space over $\mathbb{F}_p$ and that the family of binomial $p$-ary sequences is a basis of this space.

In this paper we consider the problem of minimization of a convex function that can be expressed as a maximum of affine linear functions. A convergence theorem is established and the minimizing sequence is constructed. The computational time is linear with respect to the step number. The obtained results are applied to eigenvalue problems such as eigenvalue minimization, symmetric stabilization, and the Riccati matrix inequalities. A number of examples are provided.

In this work we consider an extension of the Berestycki–Monneau–Scheinkman (BMS) model for speculative financial bubbles, in which the investor disagreement process is allowed to have jumps. While the original BMS framework assumes that disagreement evolves along continuous paths, our model accounts for sudden shifts in market sentiment through an independent Lévy jump process. Using optimal stopping theory and the Itô–Lévy formula, we show that the speculative bubble premium satisfies a non-local partial integro-differential equation (PIDE) with a moving obstacle. We develop a viscosity solution theory for this non-local obstacle problem. We prove a comparison principle by a doubling-of-variables argument adapted to the non-local jump integral, and we prove existence and uniqueness of the bubble price by Perron's method, constructing explicit continuous sub- and supersolutions. We then introduce a monotone Implicit–Explicit finite difference scheme for the bubble premium. Following the Barles–Souganidis framework, we show that the discrete operator preserves the M-matrix property and that the scheme converges locally uniformly to the unique viscosity solution under a state-dependent Courant–Friedrichs–Lewy (CFL) condition. At the end of the paper we implement the scheme via a PSOR–Picard algorithm and present numerical tests for four finite- and infinite-activity Lévy models.

We consider wideband channel estimation for extremely large-scale multiple-input multiple-output (XL-MIMO) arrays under hybrid analog-digital compression, in which a uniform linear array (ULA) is observed through far fewer radio-frequency (RF) chains than antennas. At a carrier frequency of 28 GHz with bandwidths reaching several hundred MHz, the standard narrowband polar-domain channel model fails: the near-field Fresnel curvature becomes subcarrier-dependent, and the compressed observation destroys the per-subcarrier spatial covariance structure that narrowband methods exploit. We propose the Wideband Cross-subcarrier Kullback–Leibler (WB-CL-KL) estimator, which jointly estimates angle and range directly from the compressed sample covariance, without full-array reconstruction, by fitting a structured Fresnel covariance model across orthogonal frequency-division multiplexing (OFDM) subcarriers via a cross-subcarrier Kullback–Leibler (KL) divergence criterion. We also derive the wideband compressed-domain Cramér–Rao bound (CRB) – the performance lower bound for this hybrid architecture – from the Slepian–Bangs formula, and decompose its gain over the narrowband bound into a data-diversity component of +27.093 dB and a geometric-diversity component of +0.701 dB, totalling +27.793 dB at B = 400 MHz (Propositions 1 and 2). In the single-path line-of-sight regime, WB-CL-KL attains a range root-mean-square error of 19.8 mm against a 19.9 mm bound at signal-to-noise ratio (SNR) = 10 dB, a ratio of 0.996. Under the 3GPP Urban Micro (UMi) path-loss and shadow-fading SNR distribution, it achieves a bound ratio of 0.959 at the median deployment SNR of 9.6 dB, indicating near-CRB operation at the representative deployment point, where the compressed-domain bound is evaluated at the scene-median geometry.

Let $\chi$ be a primitive Dirichlet character modulo $q>1$. Assuming the Generalised Riemann Hypothesis for $L(s,\chi)$ and that the non-trivial zeros $\rho=\tfrac12+i\gamma$ of $L(s,\chi)$ are simple, we prove lower bounds for the discrete moments $\sum_{0<\gamma\le T}|L'(\rho,\chi)|^{-2}$ and $\sum_{0<\gamma\le T}|L(2\rho,\chi^2)/L'(\rho,\chi)|^2$, uniformly in the conductor. The bounds capture the proportion $\beta/(1+\beta)$ of the conjectured asymptotics, where $\beta=\log T/\log qT$: this is one half whenever $\log q=o(\log T)$, recovering for fixed $q$ the Dirichlet analogues of theorems of Milinovich and Ng and of Sinha, and degrades to $1/(2+A)$ when $q=T^{A}$. We conjecture the true leading order asymptotics and their analogues when we average over the family of primitive characters modulo $q$.

We survey recent variational methods for strongly indefinite Schrödinger equations with sign-changing nonlinearities. The main object is an energy functional of the form \[ J(u)=\frac12\|u^+\|^2-\frac12\|u^-\|^2 -\int_{\mathbb{R}^N}F(u)\,dx+\lambda\int_{\mathbb{R}^N}G(u)\,dx, \] where the splitting $X=X^+\oplus X^-$ is induced by a spectral gap of the linear Schrödinger operator, and where the nonlinear part \[ I(u)=\int_{\mathbb{R}^N}F(u)\,dx-\lambda\int_{\mathbb{R}^N}G(u)\,dx \] is allowed to change sign. We discuss the generalized linking theorem developed for such functionals, and the abstract multiplicity theory for critical orbits in dislocation spaces. In the final part, we prove a new ground state result for the competing pure-power case \[ f(u)=|u|^{p-2}u,\qquad g(u)=|u|^{q-2}u,\qquad 2<q<p<2^*. \] More precisely, for $\lambda>0$ sufficiently small, the corresponding strongly indefinite functional possesses a nontrivial critical point of least energy among all nontrivial critical points.

We prove the following results. 1. First order model checking is fixed-parameter tractable on the class of finite fields, as a corollary of results of Ax on the theory of (pseudo)finite fields. 2. Every hereditary graph class first order definable in the class of finite groups is monadically stable, and thus has fixed-parameter tractable first order model checking. 3. Monadic second order model checking is not slicewise polynomial on the class of cyclic groups of prime-power order, unless E = NE.

Computing the viscous Stokes flow around a shape requires solving a boundary-integral equation, and for a new shape the solve must begin from scratch. Learned operators promise to spread this cost across shapes, but it is not clear what such an operator retains of the solver it replaces, or what determines whether it transfers to shapes it was not trained on. We make both questions answerable by choosing a problem which is exactly solvable except for a single term: a second-kind boundary-integral problem is solved exactly using a kernel-independent fast summation, while the boundary correction, which has no closed form, is learned. The solver's guarantees carry over unchanged: exactness on the closed-form part, $O(N)$ scaling, $SO(3)$-equivariance to machine precision, and an $O(N)$ differentiable adjoint. We then make precise what the learning contributes. It does not contribute to accuracy, differentiability, or scaling $O(N)$, all of which are provided by the solver. Learning contributes a one-time cost in that the forward map is trained once and then evaluated on a new shape in a single pass rather than resolved. Measured against baselines, the learned map is $5$ to $16\times$ more data-efficient than a black-box DeepONet and maintains a $\sim\!2.5\times$ lower in-distribution error than a geometry-aware operator, although that operator is stronger in the low-data limit and the learned map is less reliable out of distribution. We trace this fragility to the global parameterization and reduce it with a local equivariant kernel. The exactly-solvable setting yields clarity about the mechanism: geometric generalization is governed by invariance and coverage, not by conditioning or by capacity.