{"ID":6497610,"CreatedAt":"2026-07-13T01:19:40.13847098Z","UpdatedAt":"2026-07-14T01:36:59.12045529Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.09555","arxiv_id":"2607.09555","title":"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.","short_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 thei...","url_abs":"https://arxiv.org/abs/2607.09555","url_pdf":"https://arxiv.org/pdf/2607.09555v1","authors":"[\"Guillaume Lecomte\"]","published":"2026-07-10T16:02:02Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math-ph\"]","methods":"[]","has_code":false}