The Composite Polyhedral Boundary Framework: Derivation of Unified Scaling Coefficients across 26 Crystalline Elements

Abstract

This paper establishes the Hollin-Levine Unified Lattice Framework, a first-principles deterministic model that derives precise scaling coefficients (\Omega) from the interactions between fundamental Platonic container geometries and atomic close-packed clusters. By evaluating a comprehensive dataset of 26 metallic and covalent crystalline elements across three structural tiers, the framework demonstrates that macroscopic yield limits are governed by composite boundary-stress correction factors rather than arbitrary curve-fitting heuristics. The resulting refined scalars—\Omega_1 = 2.067 for cubic systems, \Omega_2 = 1.765 for hexagonal close-packed lattices, and \Omega_3 = 1.580 for diamond and covalent networks—successfully anchor experimental values with an overall root mean square deviation (RMSD) of 12.4 MPa and a mean absolute percentage error (MAPE) of 4.2%.