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

We present a rigorous geometric theorem for the scaling of topological vacancies in the Holosphere lattice, modeled as concentric Planck-scale shells tiled by nearly uniform angular nets. Let Nvac(R) denote the cumulative vacancy count within radius R. We prove a universal surface vacancy law Nvac(R) = kappa_surf * (R/a)^2 + O(R/a),where kappa_surf = 4pi / (Asite / a^2), valid under four explicit lattice assumptions (A1–A4) that constrain shell regularity, net density, reconnection budget, and defect conservation. The proof combines discrete annulus packing bounds, Gauss-Bonnet topological charge, harmonic averaging for patchy nets, and martingale concentration for randomized rounding. Robustness is demonstrated under curvature perturbations and anisotropy, with falsifiable diagnostics provided for simulation and experiment. Physically, vacancies act as localized sources of angular misalignment and coherence tension, generating bounded strain energy and entropy flux confined to surfaces. This surface law establishes the geometric foundation for angular entropy, defect transport, and holographic thermodynamics in discrete spherical systems, linking microscopic lattice frustration to macroscopic information scaling.