Papers

We establish a quantitative Gidas–Ni–Nirenberg theorem for the pure fractional Dirichlet problem \[ (-\Delta)^s u=k(x)g(u)\quad\text{in }B_1, \qquad u=0\quad\text{in }\mathbb R^n\setminus B_1, \qquad 0<s<1. \] For positive bounded solutions under a two-sided $L^\infty$ normalization, we prove \[ \mathcal D(u)\le C\,\mathcal D(k)^\gamma,\qquad 0<\gamma<1, \] where $\mathcal D$ measures spherical oscillation and outward radial increase. To the best of our knowledge, this provides the first quantitative Gidas–Ni–Nirenberg estimate for the pure fractional Dirichlet problem. The key new ingredient is a shift-uniform, weak Harnack inequality for shifted antisymmetric supersolutions, which overcomes the exterior-tail obstruction in the quantitative moving-plane argument. This mechanism extends to nonseparable nonlinearities.

Concentrated matrix-exponential (CME) distributions are random clocks approximating a fixed time, with quality measured by the squared coefficient of variation (SCV); a well-known construction builds such clocks from products of cosine-squared terms under shared exponential damping, but the number of these terms, and hence the number of parameters to optimize, grows with the order. Since exhaustive search over the full parameter space becomes prohibitively expensive at high orders, previous work has relied on low-dimensional heuristic parametrizations rather than the true optimum. We introduce a strictly larger class of common-damping harmonic densities, built from an arbitrary nonnegative trigonometric polynomial rather than such a product. For each fixed pair of scalar parameters, coefficient optimization reduces to a single eigenvalue problem, leaving a two-dimensional nonlinear search independent of the order. The enlargement leaves the infimum unchanged from that of the classical cosine-squared construction. Applying this exact characterization at the orders where the three-parameter heuristic allows comparison gives smaller reported SCV values, with larger gains at higher orders. Separately, an explicit harmonic construction gives an $O(N^{-2})$ upper bound on the attainable SCV, where $N$ is the ME representation budget, compared with the Erlang distribution's linear rate $1/N$.

An automorphism of the polynomial ring $k[x_1,\ldots ,x_n]$ over a field $k$ is said to be $\mathit{tame}$ if it can be obtained by composing affine automorphisms and elementary automorphisms, and $\mathit{wild}$ otherwise. Jung and van der Kulk showed that every automorphism of $k[x_1,x_2]$ is tame. In 1972, Nagata conjectured that a certain automorphism of $k[x_1,x_2,x_3]$ is wild. In 2003, Shestakov and Umirbaev proved this conjecture for $\mathop{\mathrm{char}}\nolimits k=0$. The purpose of this paper is to prove the conjecture for $\mathop{\mathrm{char}}\nolimits k\ge 7$. This is the first time that the existence of a wild automorphism has been confirmed in positive characteristic.

Let $P_d$ be the elliptope of $d \times d$ correlation matrices, and, for a standardised law $F$, let $S_d^F$ be the set of correlation matrices of random vectors with all margins $F$. We give two short proofs. First, an eleven-term representation of $|x|^4$ as a sum of fourth powers of linear forms on $\mathbb{R}^4$ yields $S_{11} \neq P_{11}$ for uniform margins. Second, the six diagonals of the regular icosahedron yield $S_6^F \neq P_6$ for arcsine margins. Together with a result of Devroye and Letac, this gives $S_d^F = P_d$ if and only if $d \le 5$, settling a conjecture of theirs.

Heavy-tailed response variables are routinely modeled as a function of covariates, but the asymptotic theory available for the resulting maximum likelihood estimators sidesteps certain technical difficulties by restricting the parameters to a compact set, which is not the natural parameter space. We consider a Fréchet model whose scale parameter depends log-linearly on a vector of covariates, while the tail parameter is constant, and we study the conditional maximum likelihood estimator of the tail parameter and the regression coefficients jointly. We show that this estimator is consistent over the full, noncompact parameter space, and that it is asymptotically normal, with the inverse Fisher information as limiting covariance matrix. Consistency is obtained by a specific compactification of the parameter space relying on the cosmic closure of Euclidean space. The extension of Wald's consistency theorem needed for that step, from single observations to blocks of observations, is stated in an abstract setting and is possibly of independent interest.

We derive localized maximum-norm bounds for bending moments computed by the Hellan-Herrmann-Johnson method for the clamped Kirchhoff plate problem. The main difficulty is to localize the discrete stress without leaving the HHJ space or violating its kernel constraint. Using symmetric-curl potentials, we construct a kernel-preserving localization and connect local Green stresses with a global discrete Green stress. This argument separates the local interpolation error from a weaker global pollution term. Under explicit regularity assumptions on the auxiliary problems, the resulting estimates give optimal pointwise convergence for bending-moment values and elementwise first derivatives. No logarithmic loss occurs for positive polynomial degrees, whereas the lowest-order value estimate retains a logarithmic factor. Under additional reflection symmetry, symmetric recovery improves bending-moment values for even polynomial degrees and first derivatives for odd degrees. The proved gains are one third and one half of an order, respectively. Numerical experiments confirm this parity dependence and exhibit gains close to one full order, exceeding those established theoretically.

We establish an exponential turnpike property for optimal boundary control of multi-dimensional first-order linear hyperbolic systems in which both the control and observation operators are unbounded and the cost functions may be nonquadratic. The main assumptions are internal and boundary dissipation conditions. These conditions ensure that the associated input-to-state and input-to-output maps are bounded, with operator norms independent of the time horizon T. This uniform boundedness is essential to the proof, which combines weighted energy estimates for the forward and adjoint equations, input-output duality, and strong convexity. Finally, we apply the result to the linearized two-dimensional Saint-Venant equations with sluice-gate control and illustrate the turnpike behavior numerically.

Suppose that $M$ is a finite-volume cusped hyperbolic three-manifold, equipped with a veering triangulation $\mathcal{V}$. We prove that the action of the fundamental group of $M$ on the veering two-sphere is a geometrically finite convergence action. Applying a result of Yaman, we deduce that the veering two-sphere is equivariantly homeomorphic to the boundary of hyperbolic space. As an application, we obtain Cannon-Thurston maps associated to veering triangulations. If $\mathcal{V}$ is layered we recover the classical Cannon-Thurston map. If it is not we obtain Cannon-Thurston maps that do not come from surface subgroups. These are the first such examples in the cusped case. Finally, we implement an algorithm to draw approximations of these Cannon-Thurston maps. This improves upon previous approximations obtained by Thurston and others.

We study the evolution of the speed of life history in populations competing for a single growth-limiting resource subject to demographic stochasticity and mass mortality events. We focus on a quasi-neutral regime in which competing types have equal resource-use efficiency but differ in life-history speed. In the large-carrying-capacity scaling limit, we reduce the model to a one-dimensional jump-diffusion supported on the manifold of ecological equilibria. The drift and diffusion components capture the joint effects of demographic stochasticity and density regulation, while the jump component is driven by the catastrophic mortality events. Using this limiting process, we derive a first-order approximation for the fixation probability of an invading type that differs slightly in life history speed from the resident population. Calculations show that small demographic events tend to favor slower life histories, whereas catastrophic mortality events create transient periods of resource abundance that benefit faster types. The interaction of these two evolutionary forces allows for the existence of a nontrivial evolutionarily attractive life-history speed, which is an increasing function of the frequency and intensity of the catastrophic mortality events. These results provide a rigorous mathematical framework for some classical r/K-selection arguments, and furthermore demonstrate how mass mortality events can maintain selection for faster life histories even in resource-constrained populations.

We derive exact analytical expressions for higher-order correlations of Gaussian level-set interfaces, establishing a direct link between bulk field statistics and interface geometry. This framework enables efficient reconstruction of disordered media, detailed modeling of {Gaussian foams}—structured, double-thresholded interfaces with tunable morphology—and analysis of {memory-enhanced anisotropy} reflecting directional persistence in surface structure. These results open new possibilities for characterizing complex transport, guiding stochastic reconstructions, designing materials with desired properties, quantifying the information content of correlation functions and modelling directional processes on irregular boundaries.

Let $\mathcal A$ be a simple unital $C^*$-algebra with real rank zero, nonempty tracial state space $\mathcal T(\mathcal A)$, and strict comparison of projections. We prove that every element $T$ in $\mathcal A$ with $\tau(T)=0$ for all $\tau\in\mathcal T(\mathcal A)$ is a single commutator if one of the following conditions holds: $(1)$ $\mathcal A$ has unique trace; $(2)$ $\mathcal A$ has stable rank one. We also prove that every nonscalar element in a simple unital purely infinite $C^*$-algebra is a single commutator. Applications include UHF algebras, irrational rotation algebras, and Cuntz algebras.

This paper provides a solution to the matrix $A_p$ conjecture. For every \(1<p<\infty\), we construct a family of \(2\times2\) matrix \(A_p\) weights $W$ with arbitrarily large characteristics for which \[\|H\|_{L^p(W)\to L^p(W)}\ge c_p[W]_{A_p}^{1+1/[p(p-1)]},\] where \(H\) is the Hilbert transform and \(c_p>0\) depends only on \(p\).

This paper studies the strong convergence order of structure-preserving stochastic theta Milstein methods for a class of index-$1$ stochastic differential algebraic equations (SDAEs) with time-dependent singular matrices and non-globally Lipschitz coefficients. The singular matrix is allowed to vary in time while preserving a fixed differential algebraic splitting, and the drift and diffusion coefficients may exhibit superlinear growth. By exploiting the index-$1$ algebraic-differential decomposition of the exact solution, we identify the Milstein coefficient of the reduced stochastic differential equation directly in the original SDAE variables and establish the well-posedness and constraint preserving property of the proposed method for $\theta\in[1/2,1]$. Under a coupled monotonicity condition and suitable polynomial regularity assumptions, the method is proved to preserve the algebraic constraints at all time levels and to converge with strong order one in the root mean square norm. Numerical experiments confirm the structure-preserving property and the theoretical convergence order.

The Fisher information matrix (FIM) determines the accuracy attainable in an estimation problem. For a parametric process with memory, it has a closed form only for narrow classes of models, and it cannot be computed when the model is available only as a simulator. This paper estimates the Fisher information rate of such a process, the per-observation limit of its FIM, from simulator output alone and without knowledge of the memory length. The construction uses the identity between the FIM and the local curvature of the Kullback–Leibler (KL) divergence. Directional divergence rates are estimated by context-tree weighting (CTW), and the matrix is recovered from them by least squares. We establish strong consistency in an iterated limit and a finite-sample bound, uniform over every working depth at least as large as the true memory length, which gives a mean error of order $n^{-1/3}$. The analysis also delivers a finite-sample rate for CTW-based estimation of the KL divergence rate. Experiments confirm the predicted exponents, show that the accuracy does not degrade as the working depth grows, and use the estimate to plan a sampling budget that attains a prescribed precision. The estimator stays accurate on a hidden-state source that has no finite memory length.

We prove a general boundary gluing principle for families of Riemannian metrics with pointwise curvature restrictions on manifolds with possibly non-compact boundary. This includes, as special cases, the classical gluing theorems by Perelman for metrics with positive Ricci curvature and convex singularities and by Gromov-Lawson and Bär-Hanke for metrics with positive scalar curvature and mean convex singularities. Using the language of algebraic curvature cones, our theorem implies and unifies many more previous gluing results. We also prove a boundary deformation principle that enables us to compare spaces of metrics with interior curvature restrictions and different boundary conditions. Our construction is based on the local flexibility lemma for open partial differential relations, as well as explicit 1-jet deformations of metrics along the boundary.

This monograph develops an introduction to global analysis centered on the interaction between differential geometry, functional analysis, partial differential equations, and variational methods on Riemannian manifolds. Beginning with smooth and Riemannian geometry, it develops Sobolev spaces, distributions, interpolation and fractional regularity, differential and pseudodifferential operators on vector bundles, elliptic theory, heat methods, bounded geometry, and trace theorems. It then treats Fredholm and index theory, culminating in the Atiyah–Singer index theorem, followed by geometric evolution equations and Ricci flow, infinite-dimensional geometry on Banach and Hilbert manifolds, and variational methods including the direct method, Palais–Smale theory, deformation arguments, the mountain pass theorem, and the Nehari method. Particular emphasis is placed on explicit proofs, the passage from local Euclidean estimates to intrinsic global statements, and the precise geometric hypotheses required in compact, noncompact, and boundary settings. The text is intended for advanced undergraduate and graduate students, as well as readers approaching global analysis from geometry or differential equations.

In this paper we consider a free boundary problem for the spreading of an invasive species in the habitat of a native competitor. We assume that the invasive species benefits from a climate shift that turns the environment from unfavorable to favorable at a constant speed $c$. We show that if the invasive species is an inferior one, then it must vanish, while the native competitor always persists. However, if the invasive species is a superior one, then a dichotomy occurs: either the invasive species vanishes and the native one persists, or the invasive one spreads and the native one vanishes. We also show that in the latter case the asymptotic spreading speed equals $\min\{c, c_0\}$, where $c_0$ is the spreading speed in the corresponding homogeneous environment. Numerical simulations are provided to illustrate our theoretical results.

Constraining reaction rate coefficients is a central challenge in the development of explicit atmospheric chemical mechanisms, particularly for autoxidation systems where many reaction pathways are only indirectly observed through high-resolution mass spectrometry. In this study, we evaluate rate-coefficient optimisation methods for a toy-case autoxidation mechanism using synthetic data with known ground truth. Two complementary approaches are compared: ODE-constrained neural-network optimisation, which provides efficient point estimates of uncertain rate coefficients, and the Markov Chain Monte Carlo (MCMC) approach, which samples the posterior distribution of rate coefficients and quantifies parameter uncertainty. The methods are tested using direct concentration observations and mass-spectral observations under different noise levels. For unperturbed and low-noise synthetic observations, both methods converged towards the known rate coefficients, with the neural-network optimiser providing faster point estimates. Under high-noise conditions (with the signal-to-noise ratio approximately S / N = 1), however, MCMC was substantially more robust in recovering the rate coefficients. The posterior analysis shows that mass-spectral aggregation broadens credible intervals even at low noise, and that high-noise mass spectra can leave many individual reaction rates weakly identifiable. Posterior predictive validation nevertheless shows how broad parameter uncertainty constrained by MCMC remains consistent with accurate reproduction of the observable mass spectrum. These results demonstrate that point-estimation and Bayesian sampling methods provide complementary information: neural-network optimisation is effective for informative data, whereas MCMC is essential for diagnosing uncertainty, non-uniqueness, and identifiability in noisy or aggregated inverse problems.

We consider ground states of a two-component logarithmic fermionic system in $\mathbb{R}^d$, where $d\ge 1$ is arbitrary. We prove that up to translations and scalings, ground states of the logarithmic system are the $L^\infty $-limits of ground states for a two-component $2p-1$ power-law fermionic system as $p\searrow 1$. As a byproduct, we also establish a sharp logarithmic Sobolev inequality for orthonormal functions in \(\mathbb{R}^d\), whose optimizers are, up to scalings, the minimizers of a constraint variational problem associated with the logarithmic system.

This paper focuses on difference-of-convex (DC) composite optimization problems with set-inclusion and manifold constraints. We develop an inexact proximal-linearized nonsmooth exact penalty (NEP) method that penalizes the conic constraint while retaining the manifold constraint explicitly. At each outer iteration, the method approximately solves a strongly convex proximal-linearized subproblem over the current tangent space, using computable inexactness criteria, adaptive parameters, and a retraction-based acceptance mechanism. Under boundedness of the iterates and penalty parameters, we establish an $O(\epsilon^{-2})$ worst-case outer iteration complexity for finding an $\epsilon$-approximate KKT point of the original problem. We further develop a complexity-transfer framework relating uniform primal objective-gap and KKT-residual decay, or dual objective-gap decay, to total inner-iteration complexity bounds. This yields an $O(\epsilon^{-4})$ overall oracle complexity with either accelerated semi-proximal ADMM or FISTA applied to the dual subproblem as the inner solver. Using semi-proximal ADMM as the inner solver improves the bound to $O(\epsilon^{-2}\log(1/\epsilon))$ under an additional uniform error-bound condition. We further establish convergence of the entire iterate sequence to a KKT point under the same boundedness assumption and the Kurdyka-Lojasiewicz property of the associated potential function. Numerical experiments demonstrate the competitive performance of the proposed method on row-sparse PCA, clustering, robust SVMs, and nonconvex SOCPs.

We introduce an equivalence relation on exponential families, and prove that two exponential families are equivalent in this sense if and only if the corresponding induced Hessian manifolds are isomorphic. Moreover, we classify exponential families of order 1 with constant Hessian sectional curvature. To this end, we show that for an exponential family of order 1, it has constant Hessian sectional curvature if and only if the natural exponential family equivalent to the given family has a quadratic variance function (NEF-QVF). The classification coincides with that of NEF-QVF by Morris (1982) essentially.

We investigate local conserved quantities in non-Hermitian bosonic lattice systems whose underlying Yang–Baxter structure remains unclear. Focusing on a one-dimensional bosonic chain with quartic interactions of creation operators, we provide a new representation of the local conserved quantities previously constructed by Sanatani and Shiraishi. Using Fourier transformation and trigonometric identities, we systematically derive these conserved quantities and show directly that they form a mutually commuting family. We further extend the construction to interactions extending beyond a single site, cubic interactions, asymmetric hopping, and an on-site potential. Our results provide a unified framework for constructing and characterizing local conserved quantities in this class of non-Hermitian many-body systems.

For two-dimensional norms that generate rotationally-symmetric norms in all higher dimensions, necessary and sufficient conditions are established for them to be subspaces of $L_1$. \

Let $\widetilde{X}$ be the blow-up of a smooth projective variety $X$ along a smooth subvariety $Z \subset X$ satisfying a numerical positivity condition. We study virtual cycles of moduli spaces of stable maps to $\widetilde{X}$. We lift the numerical blow-up formula of Chen and Du to a pushforward identity for virtual cycles of moduli spaces of genus-zero stable maps to $\widetilde{X}$ and $X$. Using virtual dimension calculations for the fixed loci and a free-leaf contraction morphism, we show that all correction terms in the master space localization formula vanish. Variants of this method yields further blow-up formulas, including a generalization of Gathmann's formula. We also give a counterexample to the point-blow-up conjecture in higher genus (proposed by Hu), and establish an absolute–relative correspondence in all genera under positivity and dimension assumptions, comparing the Gromov–Witten theories of $(\widetilde{X},E)$ and $X$, where $E$ is the exceptional divisor.

For $a,b,c\in\mathbb R$, we consider the Forelli–Rudin operators $$ T_{a,b,c}f(z):=(1-|z|^2)^a\int_{\mathbb D}\frac{(1-|w|^2)^b}{(1-z\overline w)^c}f(w)\,dA(w) $$ and their positive counterparts $$ S_{a,b,c}f(z):=(1-|z|^2)^a\int_{\mathbb D}\frac{(1-|w|^2)^b}{|1-z\overline w|^c}f(w)\,dA(w). $$ We obtain a complete and sharp classification of their weak- and restricted weak-type mapping properties in the hypersingular regime $$ \Omega_{\mathcal H}:=\{(p,q):1\leq p,q\leq\infty,\ p>q\}, $$ thereby substantially extending the recent work of the first and fourth authors on hypersingular Bergman projections. One of the main discoveries of this work is an intrinsic cancellation phenomenon associated with the complex Forelli–Rudin kernel: at certain critical endpoints, cancellation creates a sharp separation between the two operators, with $T_{a,b,c}$ remaining bounded while its positive counterpart $S_{a,b,c}$ fails to be bounded. Perhaps surprisingly, this cancellation is invisible in the strong $L^p$–$L^q$ theory established by Zhao and Zhou in 2022, where the two operators have the same boundedness range, and emerges only at the weak- and restricted weak-type levels. Allowing the parameters $a,b,c$ to vary, we show that the collection of all weak-type Forelli–Rudin pairs in $\Omega_{\mathcal{H}}$ is precisely $$ \mathcal{FR}_w=\{(p,q)\in\Omega_{\mathcal H}:1<p\leq2\}, $$ whereas the collection of all restricted weak-type Forelli–Rudin pairs is $$ \mathcal{FR}_{rw}=\{(p,q)\in\Omega_{\mathcal H}:p\neq\infty\}. $$ These ranges, together with all corresponding endpoint estimates and failures, are sharp. Our approach combines dyadic decompositions, probabilistic constructions, and weak-type Hardy estimates.

We study objective-only optimization of possibly nonconvex and nonsmooth functionals over probability measures on a separable Hilbert space, allowing the optimizer to be intrinsically non-Dirac. We introduce SCMO (Stochastic Control Measure Optimizer), a gradient-free particle method derived from entropy regularized stochastic control. After finite-particle and Galerkin approximations, a Cole–Hopf transform represents the optimal feedback as a Gibbs-weighted terminal displacement. SCMO approximates this feedback by sampling context clouds, replacing one particle at a time with candidate draws, scoring the resulting empirical measures, and applying exponential reweighting. SCMO uses separate covariances for candidate proposals and particle updates, termed matched when equal and nonmatched otherwise; our analysis covers both settings. In the matched case, we establish PDE-free qualitative convergence and a projection-first quantitative bound separating particle, Galerkin-projection, and entropy-regularization errors. For the practical multi-context implementation with nonmatched covariance, we prove finite-candidate consistency and show that the signed, curvature-dependent effect of nonmatched covariance can reduce the resulting upper bound on the approximation error. Finally, experiments on function-space, trajectory-law, and contact-rich manipulation problems show that SCMO handles nonsmooth and nonconvex objectives, escapes suboptimal local basins, and recovers prescribed multimodal, non-Dirac law structure. Code for reproducing our experimental results is available at https://github.com/HenryCHEUNG7373/SCMO.

We study distributional chaos in over-saturated sets under the non-uniform specification property. We prove that for any dynamical system with the non-uniform specification property, every non-empty over-saturated set exhibits distributional chaos of type 1. We further provide sufficient conditions for certain differences of over-saturated sets to exhibit distributional chaos of type 1. We also construct examples showing that some of these conditions are optimal.

We study the conformal parametrisation of smooth star-shaped planar domains and its dependence on perturbations of the boundary. The boundary correspondence of the normalized Riemann map is described by the Theodorsen equation, which we formulate in weighted Wiener algebras. We derive an explicit representation of the inverse of the linearized Theodorsen operator through a canonical Wiener–Hopf factorisation and obtain quantitative a posteriori estimates that are stable under perturbations of the boundary function. For a collection of nonperturbative reference domains, rigorous computer-assisted estimates yield explicit neighborhoods in a weighted Wiener algebra such that every boundary function in these neighborhoods admits a uniquely determined conformal parametrisation in the validated ball, with quantitative control of the corresponding Riemann map. Thus the validation applies to open, infinite-dimensional families of star-shaped domains. As an application, we consider the semilinear Dirichlet problem \[ -\Delta v=v^3\quad\text{in }\Omega, \qquad v=0\quad\text{on }\partial\Omega. \] Pulling the equation back by the validated conformal maps yields a problem on the unit disk. Uniform a posteriori estimates then imply persistence of the elliptic solutions throughout explicit neighborhoods of the reference domains.

We develop and analyze a class of Hermite-type methods for linear advection-diffusion equations in the case of one-dimensional periodic domains. The methods evolve both nodal values and cell averages and are therefore referred to as hybrid-variable (HV) methods. Using Hermite interpolation theory, we construct the HV methods to arbitrary order of accuracy, and prove that the method is supraconvergent in the sense that the spatial order of accuracy can be up to two orders larger than the local truncation error. In the second part, we prove all central HV methods for linear diffusion equations and linear advection-diffusion equations are stable at the semi-discretized level. Next, we prove the convergence of all stable HV schemes and provide error estimate in the discrete $L^2$-norm of the semi-discretized solution, which extends the supraconvergence from a single mode to all sufficiently smooth initial data. The theoretical predictions on the accuracy and stability of the methods are verified by extensive numerical tests.

Let $p$ be an odd prime, and write $A\RS B=\{a+b:a\in A,\ b\in B,\ a\ne b\}$. We classify all pairs $A,B\subseteq\Fp$ satisfying $|A|+|B|=p$, $|A|>|B|=\ell\ge1$, and $|A\RS B|=p-2$. After normalizing the two missing sums to $\{0,1\}$, we obtain exactly $\ell+1$ explicit models. The classification forces $B\subseteq A$ and yields, for fixed $p,\ell$, exactly $\binom p2(\ell+1)$ ordered pairs and $1+\lfloor\ell/2\rfloor$ equivalence classes under simultaneous affine transformations. We determine when either set is an arithmetic progression and compute exact difference-set cardinalities. For $\ell\ge3$ and $p\ge4\ell-5$, the cardinality $|B-B|$ determines the equivalence class at fixed $p,\ell$. The proof uses punctured translates and cyclic component counting. We also exhibit a critical pair in $\mathbb F_{13}$, with size gap three and $|A|+|B|=p-2$, that contradicts a proposed inverse statement below the boundary.