The 2026 disproof of Erdős's unit-distance conjecture and Sawin's subsequent explicit quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes finite parameters whose choice is not fully optimized. This report formulates the finite parameter-selection task as a variant of a nonlinear integer programming problem and proposes an open-source Python verification pipeline, first validated by reproducing Sawin's published parameter choice and then applied to computationally improved certificates. The main computational contribution is an integer optimization and checking procedure for the sets of primes $T$ and $S_Q$, the integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. The optimization pipelines are intentionally lightweight and replicable on standard hardware: we propose a deterministic greedy construction heuristic, a Tailored Integer Evolution Strategy with repair operators for number-theoretic feasibility, and a two-parent discrete-recombination variant. Four certificate levels are compared: Sawin's published example with $δ=0.0141144286784982\ldots$, a greedy optimization certificate with $δ=0.0151718056372133\ldots$, a Tailored Integer Evolution Strategy certificate with rational $R=6672416/100000$ and $δ=0.0152616610684193\ldots$, and a Tailored Integer Evolution Strategy with discrete recombination, again with $R=6672416/100000$, giving $δ=0.0152628688170072\ldots$. Consequently, subject to Sawin's explicit criterion being applied exactly as cited, the best current certificate supports the cautious clean statement $u(n)>n^{1.0152}$ for arbitrarily large $n$.

翻译：2026年对埃尔德什单位距离猜想的证伪以及Sawin随后提出的显式定量改进表明，对于平面上的$n$个点，其最大单位距离数$u(n)$可在固定正数$\varepsilon$下超过$n^{1+\varepsilon}$。Sawin的显式界对任意大的$n$给出了超过$n^{1.014}$的单位距离数，并揭示了参数选择尚未完全优化的有限参数集。本报告将该有限参数选择任务形式化为非线性整数规划问题的变体，并提出一个开源Python验证流程——首先通过复现Sawin已发表的参数选择进行验证，随后应用于计算改进后的证书。主要计算贡献在于对素数集$T$与$S_Q$、整数重数$k(p)$以及有理数编码的实参数$R$的整数优化与校验程序。优化流程刻意设计为轻量且可在标准硬件上复现：我们提出一种确定性贪心构造启发式算法、一种结合数论可行性修复算子的定制整数进化策略，以及一种双亲离散重组变体。对四个证书级别进行了比较：Sawin已发表示例（$\delta=0.0141144286784982\ldots$）、贪心优化证书（$\delta=0.0151718056372133\ldots$）、采用有理数$R=6672416/100000$的定制整数进化策略证书（$\delta=0.0152616610684193\ldots$），以及同样采用$R=6672416/100000$但结合离散重组的定制整数进化策略证书（$\delta=0.0152628688170072\ldots$）。因此，在严格按引文精确应用Sawin显式准则的前提下，当前最佳证书支持如下谨慎的简洁结论：对任意大的$n$，有$u(n)>n^{1.0152}$。