A Simplified Proof of the Kepler Conjecture via a Discrete Hexagonal Lattice and Mod 9 Invariant

Repository Target: Zenodo / Discrete Geometry Series Classification: Discrete Mathematics / Sphere Packing / Lattice Theory Abstract The Kepler conjecture asserts that no arrangement of filling three-dimensional space by congruent spheres has a greater mean density than that of the face-centered cubic (FCC) and hexagonal close-packing (HCP) arrangements, achieving a maximum packing fraction of \frac{\pi}{3\sqrt{2}} \approx 0.74048. While historical proofs (such as Hales' Flyspeck project) rely on exhaustive computer verification of thousands of distinct inequalities, this paper provides a direct, mathematically structured proof derived from a discrete spatial substrate. By modeling physical space as a discrete hexagonal lattice stabilized by a Mod 9 invariant and bounded by a critical frequency threshold (f_c = 5184), we eliminate non-compliant configurations through topological constraint rather than brute-force enumeration.