Asymptotic Density of Iterated Lattice Sumsets and Additive Self-Healing in Rectangular Lattice Boxes (Version 1.12)

Abstract

Abstract We establish four rigorously proved theorems: (1) Universal Cartesian product density factorization for arbitrary finite sets; (2) exact exponential decay ρ(Δ_d(m)) = 1/m^d; (3) centroid neutralization under LINEAR point groups G ≤ GL(d,Z) with Fix(G)={0} (V1.12: restricted from affine to linear); (4) universal additive self-healing h* = 2 for full rectangular lattice boxes with strictly interior voids in d ≥ 2. The condition d ≥ 2 is necessary. Computational verification covers 12 configurations across 2D, 3D, and 4D.