The 2026 disproof of Erdős's unit-distance conjecture and Sawin's 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 integer parameters whose choice is not fully optimized. This report treats Sawin's parameter selection as a nonlinear integer optimization problem and develops an open-source Python optimization and verification pipeline for certificates involving prime sets $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. After reproducing Sawin's certificate with $δ=0.014114\ldots$, the pipeline yields improved certificates with the same $T$. We develop a tailored integer evolution strategy achieving a certificate with $δ=0.015263\ldots$ and supporting the cautious statement $u(n)>n^{1.0152}$ for arbitrarily large $n$. For extended ramified prime ranges, the Emmerich--Cordella certificate obtained with the same framework reports $u(n)>n^{1.031}$ for $\#T=67$, illustrating the importance of enlarging $T$. Very recent MathOverflow discussions, brought to the author's attention as of version~4, report further improvements, including certificates above $δ>0.035$ and beyond $δ>0.036$. Some of these improvements may rely not only on larger prime ranges but also on modified constraint systems and additional degrees of freedom that deviate from Sawin's original formulation. Beyond this application, the work illustrates how randomized optimization heuristics can improve, verify, and refine explicit certificates for combinatorial geometry through nonlinear integer optimization.

翻译：2026年对Erdős单位距离猜想及其量化的Sawin改进的反证表明，对于固定正数$\varepsilon$，$n$个平面点中单位距离的最大数量$u(n)$可超过$n^{1+\varepsilon}$。Sawin的显式界给出任意大$n$下超过$n^{1.014}$的单位距离，其整数参数选择尚未完全优化。本报告将Sawin参数选择视为非线性整数优化问题，开发了一个开源Python优化与验证管道，用于处理涉及素数集合$T$和$S_Q$、整数重数$k(p)$以及有理编码实参数$R$的证书。在复现Sawin的$\delta=0.014114\ldots$证书后，该管道在相同$T$下生成了改进证书。我们开发了定制整数进化策略，获得$\delta=0.015263\ldots$的证书，支持任意大$n$下$u(n)>n^{1.0152}$的谨慎陈述。对于扩展的分歧素数范围，使用同一框架获得的Emmerich-Cordella证书在$\#T=67$时报告$u(n)>n^{1.031}$，说明扩大$T$的重要性。截至第4版，作者注意到的近期MathOverflow讨论报告了进一步改进，包括$\delta>0.035$及超过$\delta>0.036$的证书。其中部分改进可能不仅依赖更大素数范围，还依赖偏离Sawin原始形式的修正约束系统与额外自由度。除该应用外，本研究展示了随机优化启发式方法如何通过非线性整数优化改进、验证并精化组合几何显式证书。