Optimizing Explicit Unit-Distance Lower-Bound Certificates 文章

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 finite parameter-selection task as a variant of a nonlinear integer programming problem and proposes an open-source Python verification pipeline, first validated by reproducing Sawin's published parameter choice and then applied to computationally improved certificates. The main computational contribution is an integer optimization and checking procedure for the sets of primes $T$ and $S_Q$, the integer multiplicities $k(p)$, and a rationally encoded real parameter $R$.