Title: Enhancing Lower-Bound Certificates for Explicit Unit Distances

Abstract: Following the 2026 refutation of Erdős’s unit-distance conjecture, Sawin provided an explicit quantitative refinement demonstrating that the maximum count of unit distances, denoted as $u(n)$, among $n$ points in a plane can surpass $n^{1+\varepsilon}$ for any fixed positive $\varepsilon$. Specifically, Sawin’s explicit bound establishes that there are more than $n^{1.014}$ unit distances for sufficiently large $n$, highlighting that certain finite parameters within the construction are not yet fully optimized. This study addresses the selection of these finite parameters by framing the issue as a variant of a nonlinear integer programming problem. To this end, we introduce an open-source Python verification pipeline. The pipeline was initially validated by replicating Sawin’s published parameter choices and was subsequently employed to generate computationally enhanced certificates. The core computational innovation involves an integer optimization and verification process targeting the sets of primes $T$ and $S_Q$, integer multiplicities $k(p)$, and a real parameter $R$ encoded rationally. Designed to be lightweight and replicable on standard hardware, our optimization pipelines utilize three specific approaches: a deterministic greedy construction heuristic, a Tailored Integer Evolution Strategy equipped with repair operators to ensure number-theoretic feasibility, and a two-parent discrete-recombination variant. We evaluate four distinct certificate levels: Sawin’s original published example featuring $\delta=0.0141144286784982\ldots$; a greedy optimization certificate achieving $\delta=0.0151718056372133\ldots$; a Tailored Integer Evolution Strategy certificate using $R=6672416/100000$ and $\delta=0.0152616610684193\ldots$; and a Tailored Integer Evolution Strategy with discrete recombination, also using $R=6672416/100000$, which yields $\delta=0.0152628688170072\ldots$. Therefore, assuming Sawin’s explicit criterion is applied precisely as referenced, the most robust certificate available today substantiates the conservative assertion that $u(n)>n^{1.0152}$ for arbitrarily large $n$.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC