Papers

In this paper, we obtain the dispersive estimates and global well-posedness of the nonlinear magnetic Schrödinger equation in $\mathbb{R}^{3}$ with nonlinearity $|u|^{p-1}u$ with exponent $\frac{5}{3}< p<5$ when the initial value stays in a suitable space $\Sigma_{s}$. By proving the resolvent estimates with weight functions and the \textquotedblleft almost equivalence\textquotedblright between $(-\Delta_{A})^{\frac{s}{2}}$ and $(-\Delta)^{\frac{s}{2}}$, we obtain the Strichartz estimates of $|J_{A}(t)|^{s}u:=e^{\frac{i|x|^{2}}{4t}}(-t^{2}\Delta_{A})^{\frac{s}{2}}e^{\frac{-i|x|^{2}}{4t}}u$, therefore the dispersive decay estimates are obtained.

We study moderate deviations for the largest eigenvalue of the randomly deformed Gaussian unitary ensemble introduced by Johansson (Probab. Theory Relat. Fields, 138: 75–112, 2007). In the fixed-coupling regime, the rescaled largest eigenvalue converges to the convolution of the Tracy–Widom law and a Gaussian law arising from the random displacement of the spectral edge. We derive sharp logarithmic asymptotics for the right and left tails on growing Airy scales and obtain the corresponding phase diagram. The two tails have different transition scales. At criticality, the rate functions are nontrivial infimal convolutions of the Tracy–Widom and Gaussian rate functions. We establish a trace-norm Airy approximation that is uniform over typical diagonal configurations on a logarithmic window. A conditional convolution argument then combines the resulting tail estimates with the Gaussian moderate deviations of the edge displacement.

Subdiffusion occurs when long trapping or residence times of particles slow down spatial transport and produce a mean squared displacement proportional to $t^\alpha$, with~$0<\alpha<1$. We develop a general renewal–jump framework that combines the internal trapping dynamics with the reinjection of particles with spatial jumps. The internal trapping dynamics enters the macroscopic limit through the small-frequency behavior of a resolvent which limiting solution is not integrable. A growth mass condition determines the parameter~$\alpha$. Then the spatial density converges on the time scale~$\eps^{-2/\alpha}$ to a time-fractional diffusion equation. This criterion provides a common derivation for models with very different internal mechanisms. We first establish the abstract limit by Laplace transform in a general setting. We then show that the classical age-structured renewal model is a direct instance of the framework and apply the same criterion to an internal pathway model governed by a degenerate elliptic operator. Both yield the subdiffusion equation and its correct initial condition. The approach clarifies which microscopic property produces subdiffusion, how the anomalous exponent determines the macroscopic scaling, and why the initial distribution enters the limit through the initial spatial mass under the stated preparation assumptions.

We study the two-dimensional stationary Navier-Stokes equations in an unbounded wedge with small rough perturbations of its angular boundaries. The Jeffery-Hamel flow in the corresponding straight wedge is taken as the effective background flow. Under a suitable small-flux condition, we prove the existence of weak solutions and establish an $H^1$-type energy error estimate of order $O(\epsilon^2)$. For sufficiently small wedge angles, we further derive weighted estimates and improve the squared $L^2$-error to $O(\epsilon^3)$.

This paper studies some fundamental problems in spectral geometry in the $p$-adic setting. By viewing the Vladimirov-Taibleson operator $D^{\alpha}$ as the $p$-adic counterpart of the fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ in the Archimedean setting, we prove Weyl's law for the Dirichlet operator and establish it for the Neumann operator outside an exceptional set with zero Lebesgue asymptotic density. For the Dirichlet operator, we derive the sharp estimate for the remainder and prove that the Weyl-Berry conjecture fails. Furthermore, we show that Pólya's conjecture fails in general, and we give geometric necessary and sufficient conditions for it to hold.

Let $M$ be an $m (\ge2)$-dimensional closed orientable submanifold in a Riemannian manifold of sectional curvature bounded above by $\delta$. We obtain the extrinsic upper bounds of the harmonic mean of the first $m$ nonzero eigenvalues of the Laplacian on $M$, which extend the previous related results for the first nonzero eigenvalues.

We prove that the generic part of the cohomology of Siegel modular varieties with torsion coefficients is concentrated above the middle degree, under a suitable notion of genericity that is optimal in the unramified case. Our method relies on the semi-perversity of the relative cohomology of the Igusa stack and on a trace formula computation that is made possible by a congruence technique introduced by Scholze and Fintzen–Shin. Compared with the work of Yang–Zhu, which handles Shimura varieties of abelian type via categorical local Langlands, our method is adapted to Siegel modular varieties but handles coefficients of arbitrarily small characteristic.

We determine the exact worst-case value, at every horizon $N\geq7$, of the smallest queried gradient norm generated by Nesterov's fast gradient method on smooth convex functions. Let $t_0=1$ and $t_{k+1}=(1+\sqrt{1+4t_k^2})/2$, and let $x_0,\ldots,x_N$ denote the points at which the method evaluates gradients. For every such $N$ and every dimension $d\geq N-4$, we prove \[ \sup_{\substack{f\in\F_{0,L}(\R^d),\ x_\star\in\arg\min f \norm{x_0-x_\star}\leq R}} \min_{0\leq k\leq N}\norm{\nabla f(x_k)}^2 =\frac{L^2R^2}{\sum_{k=0}^N t_k^2}. \] The relaxed-PEP upper bound is due to Kim and Fessler, who also reported tight numerical solutions of the exact-interpolation PEP at selected horizons. What remained missing was an analytic matching family valid uniformly over the horizon. For every $N\geq7$, we construct such a family using an FGM-specific spherical polytope $K_N$ and the standard projection-envelope function \[ f_N(x)=\max_{g\in K_N}\left\{\ip{x}{g}-\frac12\norm{g}^2\right\}, \qquad \nabla f_N(x)=\Proj_{K_N}(x). \] Every queried gradient has the same norm, and the vertices of $K_N$ are generated from a three-dimensional seed by a one-dimensional spherical cone lift. The lift preserves all projection inequalities and raises the adversary dimension by one at each horizon. The projection/Moreau-envelope template itself is classical; the new ingredients are the FGM-specific algebraic seed, the proof that it attains the relaxed bound, and the common-latitude lift that propagates this exactness to every $N\geq7$. We state precise hypotheses for that propagation and do not claim that every rank-one relaxed PEP admits such a seed.

Letzter, Methuku and Sudakov proved that sufficiently regular sublinear expanders contain nearly spanning cycles and paths. Montgomery subsequently conjectured that every regular sublinear expander of sufficiently large degree is hamiltonian. Recently, Chen, Liu, Wei and Yang disproved this conjecture by constructing $d$-regular sublinear expanders of order $n$ with $d=(1/2+o(1))\log^2 n$ and small circumference. They further posed the problem of determining whether, for every fixed $\varepsilon>0$, there exists a constant $C=C(\varepsilon)$ such that every sufficiently large $n$-vertex $d$-regular $(\varepsilon,d)$-expander with $d\ge C\log^2 n$ is hamiltonian. We answer this problem negatively by constructing, for every sufficiently small fixed $\varepsilon>0$, infinitely many nonhamiltonian $d$-regular $(\varepsilon,d)$-expanders with degrees far above the $\log^2 n$ scale.

Let ${\cal G}=\{G_1,\ldots,G_k\}$ be a collection of (not necessarily distinct) bipartite graphs on the same vertex bipartition $(X,Y)$, where $|X|=a$, $|Y|=b$ and $2\le k\le a\le b$. A rainbow matching of ${\cal G}$ is a set of pairwise disjoint edges that can be chosen from distinct members of ${\cal G}$. Denote by $q(G)$ the signless Laplacian spectral radius of a graph $G$. In this paper, we prove that if $q(G_i)\ge b+k-1$ for each $i\in\{1,2,\ldots,k\}$, then ${\cal G}$ admits a rainbow matching of size $k$ unless $G_1=\cdots=G_k\cong K_{k-1,b}\cup\overline{K_{a-k+1}}$, and show that the threshold is sharp and attained by the exceptional collection. The condition is also extended to larger collections for a prescribed level $t$. In addition, we obtain a lower bound for the rainbow matching number in terms of the ordered signless Laplacian spectral radii of the members, and provide a stability version of the extremal characterization. In the proofs, we use the shifting technique and a quotient matrix arising from an equitable partition of a signless Laplacian matrix.

Let $M_n$ be an $n\times n$ matrix with independent uniform sign entries. We prove that there exist absolute constants $C,c>0$ such that, for all sufficiently large $n$, \[ \mathbb{P}\!\left( \left|\operatorname{Per}(M_n)\right| \ge e^{-Cn}\sqrt{n!} \right) \ge 1-n^{-c}. \] This establishes the exponential scale lower bound suggested by Tao and Vu. The proof bounds the cumulative logarithmic loss of the sum of squared permanents of minors under row exposure.

The existing literature on model calibration focuses mainly on classification and probabilistic prediction. In this work, we address calibrated point predictions for the conditional mean. Although existing impossibility results preclude exact out-of-sample calibrated predictions, we develop calibrated confidence intervals that provide uncertainty quantification around such predictions. Our results come with distribution-free theoretical guarantees and are applicable in model-agnostic, finite-sample settings under exchangeability by leveraging conformal prediction. Calibrated confidence intervals rely on a general underlying binning scheme. We present two examples of such a binning scheme, one based on a data-independent partition and the other on isotonic regression. Under structural assumptions, we prove that calibrated confidence intervals based on isotonic regression come with strong asymptotic consistency properties and have an asymptotically vanishing width. We illustrate the empirical performance of our methods by applying them first in a simulated setting and then to a highly imbalanced insurance dataset.

We show that the Torelli group of a simply connected closed 5-manifold $M$, which is spin and has no 2-torsion elements in homology, is isomorphic to the bordism group ${\Omega}^{\mathrm{Spin}}_6(K(H_2(M), 2))$. For $H_2(M)$ with no 2- and 3-torsion we determine this bordism group, and give explicit constructions for the generators of the Torelli group. Furthermore, for the $g$-fold connected sum ${\#}^g(S^2 \times S^3)$ we completely determine its mapping class group. We apply our results to compute the stabilization and abelianization of the mapping class group of ${\#}^g(S^2 \times S^3)$, determine the group of isotopy classes of diffeomorphisms of $M$ that are homotopic to the identity, and study the embeddings of $S^3$ in $S^2 \times S^3$.

We study the GIT quotients of the Peterson variety $\mathrm{Pet}_n\subset \mathrm{GL}(n,\mathbb C)/B$ under a one-parameter subgroup $\lambda:\mathbb G_m \to T$ with respect to the linearization $\mathcal L(\chi)$ given by a regular dominant character $\chi$ in the root lattice. Using the Richardson stratification, we describe the semistable and stable loci explicitly in terms of subsets of simple roots. This determines the GIT chamber decomposition and the corresponding wall-crossing morphisms. In the deep chamber, the quotient is shown to be isomorphic to the weighted projective space $\mathbb P(1,2,\ldots,n-1)$. We obtain a complete chamber-theoretic characterization of normality and describe how normality varies with the choice of linearization. We also prove that the quotient is smooth if and only if $n\le3$, independently of the regular dominant linearization. These results describe how the singular geometry of the Peterson variety is reflected in the variation of its GIT quotients.

We study high-dimensional random-matrix limits of Gaussian Kullback–Leibler unbalanced optimal transport (KL-UOT). Under equal marginal penalties, the covariance action admits an exact log-determinant representation in terms of a nonlinear ridge product, together with a positive-semidefinite extension that remains finite at arbitrary aspect ratios. For independent real Wishart samples, strong asymptotic freeness gives the limiting free multiplicative convolution and almost-sure Hausdorff convergence of the ridge-product spectrum; independent Haar orientations yield the corresponding first-order limit for deformed populations. In the symmetric nonsingular identity-Wishart model, we derive an explicit $\eta$-transform and a low-degree algebraic equation that select the physical branch and determine the support interval, square-root edges, and extreme-eigenvalue limits. We further obtain all-aspect one-sample Marchenko–Pastur limits under finite fourth moments, real-Gaussian Bai–Silverstein fluctuations for $c<1$, and a joint random-matrix/penalty limit showing that sample-covariance noise produces the critical scale $\tau_p\asymp p$.

We prove bounds of order $n^{n/2}e^{O(n)}$ for the expected number of facets of high-dimensional random polytopes. First, let $\mu$ be a non-degenerate compactly supported even probability measure on $\R$ satisfying $\mu([x^\ast-s,x^\ast])\asymp s^\kappa$ near its right endpoint $x^\ast$. For every sufficiently small fixed $\alpha>0$, the convex hull of $N=\lfloor e^{\alpha n}\rfloor$ independent points with law $\mu^{\otimes n}$ has at least $n^{n/2}e^{-C_{\mu,\alpha}n}$ expected facets; this includes all symmetric finite-alphabet distributions. For every full-dimensional log-concave probability measure on $\R^n$, we prove that there exist $T\in[n,2n]$ and $N=\lceil e^Tn^{3/2}\rceil$ for which \[ n^{n/2}e^{-Cn} \leq \mathbb E f_{n-1}(P_N) \leq n^{n/2}e^{Cn}. \] Thus the scale $n^{n/2}$, up to exponential factors, is universal for log-concave measures in this high-dimensional exponential regime. Finally, we construct a symmetric isotropic full-support non-log-concave counterexample with only $(1+o(1))2^n$ expected facets.

We study the first-order oracle complexity of finding a queried point with small gradient in $\ell_p$ geometry, with particular attention to the information needed to adapt the unknown smoothness and distance scales. In the strict counted local value–gradient model, no finite complexity bound can depend only on $LR/\eps$ without a nondegenerate local scale observation: a one-dimensional construction keeps $LR/\eps=4$ while defeating every prescribed finite query budget. We resolve Diakonikolas's general-$\ell_p$ parameter-free extension question for every fixed $1<p<\infty$. Under a nondegenerate secant initialization, the method knows neither the smoothness constant $L$, the initial solution distance $R$, nor $f^*$, and returns a queried point $\widehat x$ with $\|\nabla f(\widehat x)\|_q\le\eps$. For fixed finite $p>2$, we first establish the dimension-free deterministic known-parameter upper exponent $p/(p+2)$ in $K=LR/\eps$, matching the published lower polynomial exponent under its horizon and dimension qualifications. The finite local routine fits the same observable scale–radius procedure, so this exponent is preserved without knowing $L$ or $R$. Writing $\Kbar=\max\{1,LR/\eps\}$, the post-initialization pair-oracle complexity is $O_p(\Kbar^{1/2})$ for $1<p<2$, $O(\Kbar^{1/2})$ for $p=2$, and $O_p(\Kbar^{p/(p+2)})$ for $p>2$, together with the additive calibration cost $O_p(\log(e+L/M_0))$ in every regime.

We develop a linear, unconditionally energy-dissipative, mass-lumped finite element method for the highly nonlinear Landau–Lifshitz–Gilbert (LLG) equation on quasi-uniform triangular meshes. The method is built on a projection strategy for enforcing the nonconvex pointwise constraint $|\mathbf{m}| = 1$, whose simultaneous preservation with unconditional energy stability remains challenging for standard finite element discretizations. The key innovation is a unified hybrid finite element-finite difference framework that underlies both the design and the analysis of the proposed method. In the scheme construction, we exploit the weak formulation and nodal structure of mass-lumped finite element method, while incorporating suitable interpolation operators and a node-wise length-preserving mechanism inspired by finite difference discretizations. This combination yields a linear scheme that preserves the node-wise unit-length constraint and satisfies a discrete energy dissipation law. The same hybrid framework also plays a central role in the error analysis, where the weak formulation and quasi-uniform mesh structure of finite elements are combined with interpolation-based and nodewise finite difference method to control the strongly nonlinear damping term and to establish an optimal-order error estimate. More importantly, the proposed method provides a unified framework that systematically integrates the geometric flexibility of finite element method with the pointwise constraint-preserving property of finite difference method, and thus offers a general strategy for designing and analyzing structure-preserving discretizations of constrained dissipative systems. Numerical experiments, including a classical blow-up simulation, confirm the predicted accuracy, energy dissipation, and robustness of the method.

We consider fractional Brownian motion with D-dimensional time in the domain G(T)=(-T<ti<T,tj>0 if j<d+1). Let P(T) be the probability that the process will not exceed a fixed level in G(T). We show that for large T, logP(T)=(D-dH)logT(1+o(1)) where H is the Hurst parameter of the process

Primal Dual Hybrid Gradient (PDHG) method has been verified to exhibit a two stage convergence behavior, in which a prolonged active set identification phase may be a major issue of slow convergence. In this paper, we propose Barrier PDHG (BPDHG), a nested algorithm which incorporates a logarithmic barrier function into the PDHG framework to alleviate this problem. We first establish convergence of the inner iterations, derive an error bound for the corresponding inner problem. Then we prove that the outer sequence generated by BPDHG approaches the optimal solution set of the LP problem we considered. Furthermore, we integrate the barrier technique into the {Primal Dual Linear Programming} (PDLP) framework to develop the corresponding Barrier PDLP (BPDLP) method. Numerical experiments show that the barrier modification can alleviate prolonged plateaus in the KKT residual on selected instances. We also investigate an empirical instance-dependent indicator for identifying LP problems on which BPDLP is more likely to outperform PDLP.

Under the assumption that the irreducible unitary representation $\pi$ is infinite dimensional, this paper improves the Helgason-Johnson bound in 1969 for $Sp(p, q)$.

We study Brownian motion killed upon exiting arbitrary open sets $\Omega\subset\mathbb R^n$, $n\in[3,\infty)\cap\mathbb N$. Let $\alpha\in(0,1]$. The infimal radius $d_\alpha(x)$ at which the absorber near $x$ carries an $\alpha$-fraction of the Newtonian capacity of the ball of the same radius provides a local measure of electrostatic trapping. We prove that this capacitary loading forces an absorption probability of at least $c_n\alpha$ within a dimension-dependent multiple of the diffusive time scale $d_\alpha(x)^2$. Independently, if $p\in (0,1)$ and $\tau_p(x)$ is the time by which Brownian motion started at $x$ has been absorbed with probability at least $p$, then \begin{align*} \int_\Omega\frac{|u\left(x\right)|^2}{\tau_p\left(x\right)}\,\mathrm d x \le\frac{2}{p^2}\int_\Omega|\nabla u\left(x\right)|^2\,\mathrm d x \end{align*} for every $u\in C_{\mathrm{c}}^\infty(\Omega)$. Together these statements give a capacitary Hardy inequality with constant $C_n\alpha^{-2}$ on every open set, answering a problem of Maz'ya, where $C_n$ is a positive constant and, moreover, the exponent of $\alpha^{-2}$ is sharp. As applications, we derive corresponding estimates for survival probabilities, a corresponding spectral lower bound (including a recovery of the Maz'ya–Shubin lower bound), heat dissipation, Schrödinger forms, and Fisher information.

Let $(M,g)$ be a smooth, closed, connected $d$-dimensional Riemannian manifold, and let $X_1,\ldots,X_n$ be i.i.d.\ with common law $f\,d\mathrm{vol}_g$, where $f\in C^4(M)$ is strictly positive. We derive a uniform intrinsic second-order expansion for symmetric three-vertex edge-indicator statistics supported on connected configurations, including the induced-path and triangle kernels. The second-order term separates density variation, normal-coordinate Jacobians, and the curvature-induced motion of the internal-chord boundary. Within this three-vertex connected symmetric class, a universal boundary–moment identity reduces the kernel-dependent contribution to a common intrinsic functional involving $\int_M f\|\grad f\|_g^2\,d\mathrm{vol}_g$ and $\int_M f^3\operatorname{Scal}_g\,d\mathrm{vol}_g$. For the normalized path–triangle contrast, the Euclidean leading term cancels. We construct a consistent estimator of this intrinsic functional and, in a denser bandwidth regime, we prove an exact-expectation-centered root-\(n\) central limit theorem via the first Hoeffding projection. For uniform sampling on a closed surface, the estimator consistently recovers the Euler characteristic. We also study the threshold radius at which the maximum degree of a binomial random geometric graph first reaches two. Using the active-triple intensity expansion and a dependency-graph Poisson approximation, we obtain the order-$n^{-3/d}$ correction to the log-survival law for $d>6$.

We show that for every sufficiently large genus $g$, there exists a closed hyperbolic surface $S_g$ with systole $\mathrm{sys}(S_g)\geq \log g-12\log\log g$. In particular, $$\liminf_{g\to \infty}\frac{\max\{\mathrm{sys}(S):S\in \mathcal{M}_g\}}{\log g}\geq 1,$$ improving the previously known bound $2/9$. This note is a continuation of our previous work on the diameter of finite covers arXiv:2608.12887, using the same framework of constant-twist pants decomposition to study systoles. The proof was developed by GPT-5.6 Sol through an extended discussion with the author.

Spectral Barron spaces, characterized by an \(L^1\)-based Fourier-Lebesgue norm, have earned significant attention in approximation theory due to their remarkable capacity to represent functions via shallow neural networks with controlled complexity. Meanwhile, recent theoretical advances have firmly established an intrinsic and profound connection between these function spaces and the regularity theory of elliptic partial differential equations. Building upon this foundational interplay, the present work undertakes a systematic and comprehensive investigation into the well-posedness of heat equations formulated within the spectral Barron spaces framework. Specifically, we rigorously establish the core aspects of well-posedness, including the existence, uniqueness, and stability of solutions, under suitable assumptions on the source terms and conductivity coefficients. We also investigate a typical parabolic inverse problem, namely the backward heat equation, for which we derive a logarithmic conditional stability estimate. To the best of our knowledge, this constitutes the first stability estimate for inverse problems within the spectral Barron space setting. Moreover, we extend our analytical results to address the more intricate setting of time-fractional heat equations, which govern anomalous diffusion phenomena and introduce nonlocal temporal memory effects. In this extended context, we provide a characterization of the corresponding heat kernels, deriving decay estimates, regularity properties, thereby enriching the theoretical landscape of evolutionary PDEs within the spectral Barron spaces setting.

Let $\mathcal C$ be an $(n+2)$-angulated category. Zhou proved that, when $n$ is odd, if the Auslander-Reiten $(n+2)$-angles generate the relations for the Grothendieck group of $\mathcal C$, then $\mathcal C$ is locally finite. Whether the corresponding statement remains valid for even $n$ is still open. In this paper, we give a partial affirmative answer to this problem by establishing a sufficient condition under which the same implication holds for even $n$. We further show that our sufficient condition is satisfied by a broad class of examples, thereby demonstrating that the result extends well beyond isolated cases.

We introduce a centroid-based block decomposition of the left-regular representation and use it to prove operator-valued Haagerup inequalities for finitely generated groups with centroid maps. The decomposition separates the three centroid growth conditions into distinct operator-norm contributions and, under a factorization hypothesis for the associated Schur multipliers, yields complementary lower bounds. For median groups of rank $\nu$, the centroid blocks are related to spherical operators, giving two-sided estimates with the explicit factor $\binom{\nu+r}{r}$; this applies in particular to right-angled Artin groups and to groups acting on finite-rank $\operatorname{CAT}(0)$ cube complexes. For finite-rank coarse median groups, we construct a coarse iterate median with error independent of the cardinality of the input set and obtain spherical-block estimates with factor $o(r^{1+\nu/2+\varepsilon})$ for every $\varepsilon>0$.

We study the Kac killing penalization for a one-dimensional recurrent Lévy process. We prove that the limit process possesses two equivalent characterizations by a Fredholm equation and by a Sturm–Liouville equation.

We study the emptiness formation probability (EFP) in the ground state of the repulsive one-dimensional Lieb-Liniger Bose gas. For a macroscopic empty interval of length $2R$, its leading asymptotic behavior is described by a rate function $f(\gamma_0)$, defined by $-\log P(R)\sim(\rho_0R)^2f(\gamma_0)$, where $\rho_0$ is the mean density and $\gamma_0$ is the dimensionless interaction strength. We propose a parameter-free integral equation for $f(\gamma_0)$. Starting from the exact dual-field Fredholm-determinant representation of the EFP, we show how the conjectured kernel arises formally and identify the uniform asymptotic statement that remains to be proven for a rigorous derivation. The conjecture reproduces the Tonks-Girardeau and weak-coupling limits, as well as the first correction obtained independently in both limits. It also agrees at the few-percent level with numerical minimization of the Lieb-Liniger hydrodynamic action over more than four orders of magnitude in coupling. The numerical calculation yields the corresponding emptiness instantons and their astroid-like vacuum regions.

We study the quadratic convexification of the compact nonconvex set ${G} := \left[1/2,2\right]^2 \cap \{(x_1,x_2)\in\mathbb R^2\mid x_1x_2\leq1\}, $ which is a square truncated by a hyperbolic arc. Although the ordinary convex hull of ${G}$ is a triangle, its lifted convex hull in the complete quadratic space retains the nontrivial geometry of the curved boundary. We characterize all extreme rays of the cone of quadratic polynomials nonnegative on $G$, including parameterized families of bounded tangent and bitangent rays. Using this classification and conic duality, we derive an exact finite semidefinite representation of the lifted convex hull of ${G}$. The analysis follows the general framework of our earlier work on an unbounded product-constrained region, but the bounded geometry creates new boundary-contact patterns and leads to a different finite organization of the nonnegative-quadratic cone.