AI 中文总结
研究对 Lennard-Jones 团簇的 Kuznetsov-Sahinidis 直径界进行认证细化与渐近分析。核心方法是构建认证估计、用中心递减分布找候选最小值等。主要贡献是在 92 个大小上改进界，推导渐近形式，增益增长如$\Theta(\sqrt{N})$层。
AI 中文摘要
Kuznetsov 和 Sahinidis（《全球优化杂志》，2025 年）证明了限制最优 Lennard-Jones 团簇并缩小确定性求解器搜索区域的距离界；他们的直径界为每个单位宽度层赋予最宽松的内能$-\binom{n}{2}$。我们用基于他们自己的子集不等式和已证明的最小值$V_5^*$、$V_6^*$构建的认证估计来取代它，并在总体分布上最小化由此产生的层界。中心递减分布提供候选最小值；一种无排列松弛，其唯一的经典成分是序列的重排不等式，在所有分布上封闭每个证书；并且所有比较都在有向舍入算法中重新验证。对于$5 \le N \le 200$，这在 92 个大小上认证了已发表界的一层改进。该改进并未解决任何开放的全局优化情况：它是对已发表的先验界的严格收紧，并精确说明了机制。对唯一可处理大小（$N \le 6$）的直接下游测试发现直径框不是那里确定性求解器的约束资源，并且没有求解器在细化影响的大小（$N \ge 38$）上运行；因此我们将结果作为理论说明呈现。我们还推导了界的渐近形式，$\rho_{KS} = N - \Theta(\sqrt{N})$，解决了 Kuznetsov 和 Sahinidis 留下的一个问题；细化本身的增益增长类似于$\Theta(\sqrt{N})$层，具有精确的渐近常数。
英文摘要
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.
