Ehrhart Theory and Discrete Gaps in Minkowski Sumsets: Asymptotic Lattice Density and Refutation of O(hᵈ⁻¹) Bounds (Version 1.4)

This preprint (Version 1.4) introduces a rigorous geometric framework to analyze the discrete volume gaps Δ(h,A) = L(P_A,h) - |hA| between Ehrhart polynomials and Minkowski sumsets for lattice sets A ⊂ Z^d. Key Contributions & Paradigm Shift (V1.4+):- Refutation of Sub-Leading Bounds: Proves that interior phantom lattice points grow at the leading volumetric scale int_ph(h) = Θ(h^d) rather than O(hᵈ⁻¹), directly refuting earlier assumptions in V1.0–V1.3.- Exact Closed-Form Derivations: Establishes int_ph(h) = 3h(h-1)/2 for the benchmark set A = {(0,0),(2,0),(0,2)}, proving an asymptotic phantom density of 75%.- Asymptotic Lattice Density Framework: Reframes the primary problem from gap bounds to classifying convex lattice sets by their asymptotic sumset density ρ(A) = limh→∞ |hA| / L(P_A,h), where unimodular simplices serve as unique maximizers (ρ = 1).- Generalizations: Includes exact density theorems for general scaled simplices ρ(A) = 1 / Vol_norm(P_A) and computer-verified benchmark logs (DATA_CONSISTENT).