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 integer parameters whose choice is not fully optimized. This report starts from Sawin's nonlinear integer optimization problem and develops an open-source Python verification pipeline. The pipeline is first validated by reproducing Sawin's published parameter choice and is then applied to computationally improved certificates. We optimize and verify certificates involving sets of primes $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. The implementation is deliberately lean, so that all results can be replicated on standard hardware and the procedures can be extended. We compare a deterministic greedy heuristic, a tailored integer evolution strategy with two-sided geometric, or discrete-Laplace, integer mutation and repair operators for number-theoretic feasibility, and a two-parent discrete-recombination variant. Four certificate levels are reported: Sawin's published example with $δ=0.0141144286784982\ldots$, a greedy certificate with $δ=0.0151718056372133\ldots$, a tailored integer evolution strategy certificate with $R=6672416/100000$ and $δ=0.0152616610684193\ldots$, and a recombination variant with the same $R$ and $δ=0.0152628688170072\ldots$. Consequently, the best current certificate supports the cautious statement $u(n)>n^{1.0152}$ for arbitrarily large $n$. Beyond this unit-distance application, the work illustrates how randomized optimization heuristics can improve explicit certificates in pure mathematics and combinatorial geometry.

翻译：针对埃尔德什单位距离猜想在2026年被证伪，以及Sawin后续的显式量化改进表明，平面点集单位距离最大数目$u(n)$可在固定正数$\varepsilon$下超过$n^{1+\varepsilon}$。Sawin的显式界给出$u(n)>n^{1.014}$（对任意大的$n$成立），但其中整数参数的选择未完全优化。本报告从Sawin的非线性整数优化问题出发，开发了一套开源Python验证流水线。该流水线首先通过复现Sawin已发表的参数选择进行验证，随后应用于计算改进的证书。我们优化并验证了涉及素数集合$T$和$S_Q$、整数重数$k(p)$以及有理编码实参数$R$的证书。实现刻意保持精简，使得所有结果可在标准硬件上复现，且程序具有可扩展性。我们比较了确定性贪心启发式算法、定制化整数进化策略（含双几何分布/离散拉普拉斯整数变异与数论可行性修复算子）以及双亲离散重组变体。报告四个证书级别：Sawin已发表示例（$\delta=0.0141144286784982\ldots$）、贪心证书（$\delta=0.0151718056372133\ldots$）、定制化整数进化策略证书（$R=6672416/100000$，$\delta=0.0152616610684193\ldots$）及重组变体（相同$R$，$\delta=0.0152628688170072\ldots$）。因此，当前最佳证书支持谨慎结论：对任意大的$n$有$u(n)>n^{1.0152}$。除单位距离应用外，本工作展示了随机优化启发式方法如何改进纯数学与组合几何中的显式证书。