Why Can the Layers Not Move Closer? A Geometric Proof for Packings Built from Parallel Triangular Lattices

This expository note presents an elementary geometric proof of the optimal density bound within a restricted class of unit-sphere packings: the centers lie in parallel planes, each containing a translate of the same triangular lattice with nearest-neighbor distance two and a common lattice orientation. The argument begins with the covering radius of the planar triangular lattice, 2/√3. Combined with the non-overlap condition for unit spheres, this gives a minimum separation of √(8/3) between successive layers. The area per center within each layer is 2√3, leading to an upper packing density of π/(3√2), approximately 0.74048. The note verifies that close-packed Barlow stackings, including face-centered cubic (FCC) and hexagonal close-packed (HCP) arrangements, attain this bound. Boundary contributions are explicitly controlled when passing from finite center counts to asymptotic packing density. The explanatory structure is direct: a planar distance bound yields an interlayer separation bound, which yields a spatial density bound. This work does not claim a new mathematical result or a new proof of the unrestricted Kepler theorem. Its contribution is expository: a self-contained derivation for a precisely specified class of layered packings, with explicit assumptions, limitations, and a follow-up research question concerning more general configurations. The note also distinguishes equal hard-sphere geometry from atomistic models of materials such as silicon carbide. No physical phase-transition prediction is claimed. Author: Nguyen Kim KhanhProject: Khanh MathematicsDate: 28 October 2026 AI assistance was used for organization, mathematical formulation, translation, and wording. Independent verification of the arguments and references is recommended before publication.