The Holographic Vacancy Law: A Surface Scaling Theorem for the Holosphere Lattice

We derive an explicit scaling law for topological vacancies induced by discrete angular packing on concentric Planck-scale shells in the Holosphere lattice. Let N_vac(R) be the cumulative vacancy count within radius R, where each shell is tiled with angular sites of area A_site. We prove a universal surface law: N_vac(R) = kappa_surf * [(R/a)^2 + theta(R/a)] + O(R/a), where kappa_surf = 4pi / (A_site / a^2), and theta ∈ {0,1} encodes a topological shift depending on whether the outermost shell is edge-anchored or center-aligned. The derivation incorporates annular packing bounds, spherical defect tilings, anisotropy corrections, and continuum limits. Assumptions A1–A4 define the lattice regularity and yield falsifiable diagnostics. The result provides a geometric foundation for angular entropy, defect transport, and surface thermodynamics in discrete spherical systems.