Optimizing Explicit Unit-Distance Lower-Bound Certificates

arXiv:2606.03419v1 Announce Type: cross Abstract: The 2026 disproof of Erd\H{o}s'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 finite parameters whose choice is not fully optimized. This report formulates the finit