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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 determi","title":"Optimizing Explicit Unit-Distance Lower-Bound Certificates","url":"https://arxiv.org/abs/2606.03419","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.03419v3 Announce Type: replace-cross \nAbstract: 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. 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