Existence and Uniqueness of Nearest Stealthy Hyperuniform Configurations from Random Initial Conditions

Abstract

We prove that, for almost every uniformly random configuration of $N$ particles in a two-dimensional periodic square domain, there exists a unique nearest stealthy hyperuniform configuration, up to floppy-mode displacements, when a finite set of $m$ low-$k$ Fourier modes is constrained to vanish exactly. Under an explicit full-rank Jacobian assumption, the displacement component orthogonal to the tangent space of the finite-mode hyperuniform manifold is uniquely determined, while residual tangent-space degrees of freedom correspond to first-order floppy modes that preserve the constrained low-$k$ density fluctuations. We restrict attention to the regime of constrained-mode fraction $χ=m/(dN)$ in which such configurations are known to remain disordered rather than crystallize. To corroborate this framework, we implement a gradient-based generator network that iteratively displaces particles to minimize a combined loss functional of hyperuniformity, short-range repulsion, and smoothness, and whose dynamics is expected to approximate the theory's minimal orthogonal projection onto the hyperuniform manifold. The network drives generic random configurations toward disordered hyperuniform representatives, progressively suppressing long-wavelength density fluctuations as measured by the structure factor and number-variance exponent\textcolor{black}{, over a reciprocal-space window that extends well beyond the explicitly constrained modes and is bounded, as we show, by the structure-factor sum rule. Local bond-orientational analysis and the absence of Bragg peaks confirm that the resulting states remain disordered}. This integrated analytical and computational approach provides a rigorous geometric foundation for understanding stealthy hyperuniformity and for generating its disordered representatives from generic random configurations.