Optimizing Explicit Unit-Distance Lower-Bound Certificates

arXiv:2606.03419v3 Announce Type: replace-cross Abstract: The 2026 disproof of Erd\H{o}s'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 $\delta=0.014114\ldots$, a greedy certificate with $\delta=0.015172\ldots$, an evolution-strategy certificate with $R=6672416/100000$ and $\delta=0.015262\ldots$, and a recombination variant, again with this $R$, with $\delta=0.015263\ldots$. 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.