This paper extends the Sc-Rubs formulation by exploring how the Laplacian operator governs the persistence and curvature of emergent scalar-field geometries within recursive domains.Building on the Law of Persistence introduced in earlier work, it formalizes the coupling between diffusion, curvature, and localized field potential τ(r), demonstrating how small variations in p (the geometric parameter) yield stable morphologies through balanced energy exchange. The study derives the persistence field equation φ = PF τ(r) and investigates its boundary effects under iterative Laplace transforms, showing that apparent “solid” geometries emerge as harmonics of a continuous equilibrium process. Numerical experiments and symbolic reconstructions illustrate the octahedral–spherical–cubic transition (p = 1 → ∞) and establish a parametric link between curvature stiffness β and truncation λ, giving rise to coherent polyhedral persistence domains. This paper forms part of the Sc-Rubs Modelling Series, a unified effort to describe how geometric form maintains existence through recursive scalar-field equilibrium.For figures and supporting data, visit https://sc-rubs.cloud.Related DOI: 10.5281/zenodo.17443937.