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Sawin定量改进的反证表明，平面点集中单位距离的最大数目$u(n)$可对固定正数$\varepsilon$超过$n^{1+\varepsilon}$。Sawin的显式界给出了对任意大$n$超过$n^{1.014}$个单位距离，并暴露了参数选择未完全优化的问题。本报告将Sawin的参数选择视为非线性整数优化问题，开发了一个开源Python优化与验证管线，用于处理涉及素数集$T$和$S_Q$、整数重数$k(p)$以及有理编码实参数$R$的证明。在复现Sawin的$\delta=0.014114\ldots$证明后，该管线在相同$T$下得到了改进的证明。我们定制了整数进化策略，获得了$\delta=0.015263\ldots$的证明，支持对任意大$n$谨慎断言$u(n)>n^{1.0152}$。在扩展的分歧素数范围内，使用相同框架得到的Emmerich–Cordella证明报告了$\#T=67$时$u(n)>n^{1.031}$，说明了扩大$T$的重要性。截至第4版，作者得知的最新MathOverflow讨论报告了进一步改进，包括$\delta>0.035$及$\delta>0.036$以上的证明。其中一些改进不仅依赖于更大的素数范围，还依赖于对Sawin原始公式的约束系统修改与额外自由度。除该应用外，本研究展示了随机优化启发式如何通过非线性整数优化改进、验证并精炼组合几何中的显式证明。