Explicit Construction of Near-Optimal Sphere Packings in ℝ³ Achieving the Cohn–Elkies Density Bound

This work presents a deterministic, step-by-step method for arranging identical spheres in three-dimensional space so that they occupy as much volume as physically possible without overlapping. Unlike previous theoretical results that relied on abstract or nonconstructive techniques, this method provides an explicit algorithm that builds the arrangement directly using only classical geometric principles. The approach works by defining a controlled "exclusion zone" around each sphere and systematically adjusting the spacing between them until no further tightening is possible without causing overlap. As this process is refined and applied across various geometric layouts, the resulting packing density closely approaches the best known theoretical limit, previously established by Cohn and Elkies through advanced analysis but not realized constructively. This shows that the maximum possible density can be achieved in practice using only deterministic, physical rules.