UED Reconstruction Programme — Paper 16: Emergence of the First Persistent SVE Configuration

Paper 16 of the UED Reconstruction Programme performs the second step of Phase II — Mathematical Closure by asking whether the Structured Vacuum Energy (SVE) ripple architecture can itself generate a finite, localized, dynamically persistent configuration rather than merely describe the conditions under which one could exist. Beginning from the Paper-2 balance–flux equations, the paper first proves that the linear source-free closure with positive relaxation and constant positive isotropic coupling admits no nonzero regular finite localized stationary solution under vanishing boundary response. This reproduces, directly within SVE dynamics, the finite-scale obstruction identified more generally in Paper 5. The local response is then extended to a measurable nonlinear odd expansion, while the spatial coupling is generalized to a wavenumber-dependent response. Finite-wavenumber selection requires a sign-changing effective spatial term regularized at higher order. From the measured spatial-response coefficients, characteristic length and time scales are derived rather than inserted as a particle radius or matter scale. After nondimensionalization, the candidate closure reduces to a cubic-quintic finite-wavenumber equation of Swift-Hohenberg type. The equation supports subcritical finite-amplitude states while the homogeneous background remains linearly stable, creating the mathematical conditions necessary for localized coexistence and hysteretic persistence. A declared three-dimensional numerical continuation experiment demonstrates a finite localized branch over a bounded control interval. Under the specified protocol, the lower localization boundary is bracketed by−0.35