优化几何距离证明中的参数，让点间等距数量突破此前上限。
Optimizing Explicit Unit-Distance Lower-Bound Certificates
- 将参数选择建模为非线性整数优化问题，用程序自动搜索最优解。
- 改进后证书使等距数超过 n^1.0152，优于原论文的 n^1.0141。
- 适用于组合几何、数学证明自动化及优化算法研究者。
2026年对埃尔德什单位距离猜想的反证及Sawin的量化改进表明，平面上n个点之间的最大单位距离数u(n)可超过n^{1+ε}（ε>0）。Sawin的显式边界给出n^{1.014}以上的单位距离，但其参数未完全优化。本文将参数选择视为非线性整数优化问题，构建开源Python优化与验证流水线，处理素数集T、S_Q、整数重数k(p)及有理数编码的实参数R。复现Sawin证书δ=0.014114…后，获得相同T下的改进证书。采用定制整数进化策略，得到δ=0.015263…的证书，支持结论u(n)>n^{1.0152}对任意大n成立。扩展素数范围后，基于该框架的Emmerich–Cordella证书达到u(n)>n^{1.031}（#T=67），凸显扩大T的重要性。近期MathOverflow讨论指出进一步改进，部分证书δ>0.035，甚至超0.036，可能依赖更大素数范围或修改约束系统。本工作展示随机优化启发式在组合几何显式证明证书的优化、验证与精炼中的应用。
原文摘要 · Abstract (English)
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.
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