A certified refinement and asymptotic analysis of the Kuznetsov-Sahinidis diameter bound for Lennard-Jones clusters

Abstract

Kuznetsov and Sahinidis (J. Glob. Optim., 2025) prove distance bounds that confine optimal Lennard-Jones clusters and shrink the search region of deterministic solvers; their diameter bound charges each unit-width layer the loosest internal energy $-\binom{n}{2}$. We replace this by a certified estimate built from their own subset inequality and the proven minima $V_5^*$, $V_6^*$, and minimize the resulting layer bound over population profiles. Centred decreasing profiles supply candidate minima; an arrangement-free relaxation, whose only classical ingredient is the rearrangement inequality for sequences, closes every certificate over all profiles; and all comparisons are re-verified in directed-rounding arithmetic. For $5 \le N \le 200$ this certifies a one-layer improvement of the published bound at 92 sizes. The improvement resolves no open global-optimization case: it is a rigorous tightening of a published a priori bound, with a precise account of the mechanism. A direct downstream test on the only tractable sizes ($N \le 6$) finds the diameter box is not the binding resource for a deterministic solver there, and no solver runs at the sizes the refinement affects ($N \ge 38$); we therefore present the result as a theoretical note. We also derive the asymptotic form of the bound, $ρ_{KS} = N - Θ(\sqrt{N})$, resolving a point left open by Kuznetsov and Sahinidis; the gain of the refinement itself grows like $Θ(\sqrt{N})$ layers, with an exact asymptotic constant.