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 starts from Sawin's nonlinear integer optimization problem and develops an open-source Python optimization and verification pipeline, first validating it by reproducing Sawin's parameters and then applying it to improved certificates. We optimize and verify certificates involving prime sets $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. The implementation is lean and lightweight, so all results can be replicated on standard hardware and the procedures extended. We propose a deterministic greedy construction heuristic, a tailored integer evolution strategy with geometric mutation and repair operators to maintain number-theoretic feasibility, and an optional two-parent recombination variant. Four certificate levels are compared: Sawin's example with $δ=0.014114\ldots$, a greedy certificate with $δ=0.015172\ldots$, an evolution-strategy certificate with $R=6672416/100000$ and $δ=0.015262\ldots$, and a recombination variant, again with this $R$, with $δ=0.015263\ldots$. Consequently, the best reported certificate supports the cautious clean statement $u(n)>n^{1.0152}$ for arbitrarily large $n$ using the same set $T$ as in Sawin 2026, and a further improvement found with this framework hints at $u(n)>n^{1.031}$ for extended ramified prime ranges. Beyond this application, the work illustrates how randomized optimization heuristics can explore and improve explicit certificates in pure mathematics and combinatorial geometry.

翻译：2026年对Erdős单位距离猜想的证伪以及Sawin的量化改进表明，对于固定正数ε，平面点集的最大单位距离数u(n)可超过n^{1+ε}。Sawin的显式界显示，对于任意大的n，单位距离数超过n^{1.014}，其中整数参数的选择尚未完全优化。本报告从Sawin的非线性整数优化问题出发，开发了一个开源的Python优化与验证流程：首先通过复现Sawin的参数验证该流程，随后将其应用于改进验证集。我们优化并验证了涉及素数集T和S_Q、整数重数k(p)以及有理编码实参数R的验证集。该实现简洁轻量，所有结果均可在标准硬件上复现，且流程可扩展。我们提出了一种确定性贪心构造启发式算法、一种定制化整数进化策略（包含几何变异与修复算子以维护数论可行性），以及一种可选的双亲重组变体。比较了四个验证集级别：Sawin的示例（δ=0.014114…）、贪心验证集（δ=0.015172…）、进化策略验证集（R=6672416/100000，δ=0.015262…）以及重组变体（同样采用该R值，δ=0.015263…）。因此，最佳报告的验证集支持谨慎的明确结论：采用与Sawin 2026相同的集合T，对于任意大的n有u(n)>n^{1.0152}；而在此框架中发现的进一步改进暗示，对于扩展的分歧素数范围，u(n)>n^{1.031}。除该应用外，本工作展示了随机优化启发式算法如何探索并改进纯数学与组合几何中的显式验证集。