Exact Small-Softness Stabilization of Euclidean Lattice Packings

For a Euclidean lattice whose nonzero vectors have length at least two, consider the density of the union of balls of radius \(1+\lambda\) centered at its points. We prove that, in every fixed dimension \(d\ge 2\) and for all sufficiently small \(\lambda\gt 0\), the maximizing lattices are exactly the densest hard-sphere lattices with the smallest kissing number among the hard-optimal lattices. The proof compares a linear local covolume margin with a vanishing Lipschitz constant for pair overlaps, and uses Mahler compactness to exclude all other competitors. In dimension three this proves the lattice alternative of Bezdek and Langi's small-softness FCC conjecture and gives the optimal density \(\pi(1\)\({}+3\lambda\)\({}-6\lambda^2\)\({}-5\lambda^3)\)\(/(3\sqrt2)\). The global softness threshold is existential. The result does not treat general nonlattice packings.

This is a complete theorem in the stated Euclidean Bravais-lattice setting, related to AIM-GEOMETRY-0112 (Stability) in ulamai/UnsolvedMath v1.6.0. It does not close the whole open-ended AIM record. Classical hard FCC optimality, Mahler compactness, the no-triple-intersection threshold and earlier local soft FCC optimality are credited. A bounded literature review found no identical global stabilization theorem; absolute priority is not certified. The manuscript is unrefereed and self-audited, with disclosed AI assistance, and is not formally verified. Exact rational regression checks are included with their finite domains and limitations.