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This study transforms the search for explicit unit-distance lower-bound certificates into a nonlinear integer optimization problem, utilizing an open-source Python pipeline to refine parameters and improve the exponent $\varepsilon$ beyond Sawin's original bound. By applying tailored integer evolution strategies and expanding prime set ranges, the research demonstrates how randomized optimization heuristics can systematically verify and enhance combinatorial geometry proofs. The work provides a reproducible framework for generating stronger mathematical certificates, highlighting the practical utility of algorithmic optimization in advancing discrete geometry.
Optimizing Explicit Unit-Distance Lower-Bound Certificates
arXiv:2606.03419v5 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 treats Sawin's parameter selection as a nonlinear integer optimization problem and develops an open-source Python optimization and verification pipeline for certificates involving prime sets $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. After reproducing Sawin's certificate with $\delta=0.014114\ldots$, the pipeline yields improved certificates with the same $T$. We develop a tailored integer evolution strategy achieving a certificate with $\delta=0.015263\ldots$ and supporting the cautious statement $u(n)>n^{1.0152}$ for arbitrarily large $n$. For extended ramified prime ranges, the Emmerich--Cordella certificate obtained with the same framework reports $u(n)>n^{1.031}$ for $\#T=67$, illustrating the importance of enlarging $T$. Very recent MathOverflow discussions, brought to the author's attention as of version~4, report further improvements, including certificates above $\delta>0.035$ and beyond $\delta>0.036$. Some of these improvements may rely not only on larger prime ranges but also on modified constraint systems and additional degrees of freedom that deviate from Sawin's original formulation. Beyond this application, the work illustrates how randomized optimization heuristics can improve, verify, and refine explicit certificates for combinatorial geometry through nonlinear integer optimization.
Abstract:The 2026 disproof of Erdős'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 treats Sawin's parameter selection as a nonlinear integer optimization problem and develops an open-source Python optimization and verification pipeline for certificates involving prime sets $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. After reproducing Sawin's certificate with $\delta=0.014114\ldots$, the pipeline yields improved certificates with the same $T$. We develop a tailored integer evolution strategy achieving a certificate with $\delta=0.015263\ldots$ and supporting the cautious statement $u(n)>n^{1.0152}$ for arbitrarily large $n$. For extended ramified prime ranges, the Emmerich--Cordella certificate obtained with the same framework reports $u(n)>n^{1.031}$ for $\#T=67$, illustrating the importance of enlarging $T$. Very recent MathOverflow discussions, brought to the author's attention as of version~4, report further improvements, including certificates above $\delta>0.035$ and beyond $\delta>0.036$. Some of these improvements may rely not only on larger prime ranges but also on modified constraint systems and additional degrees of freedom that deviate from Sawin's original formulation. Beyond this application, the work illustrates how randomized optimization heuristics can improve, verify, and refine explicit certificates for combinatorial geometry through nonlinear integer optimization.
17 pages, 9 figures. Added a declaration on the use of AI. Added references to further contributions discussed on MathOverflow, including a reference to the independently developed verification pipeline and certificate package by Tseng (2026), published on Zenodo
Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Computational Geometry (cs.CG); Neural and Evolutionary Computing (cs.NE); Combinatorics (math.CO)
G.1.6; G.2.1; F.2.2; I.2.8; G.4
arXiv:2606.03419 [math.OC]
(or arXiv:2606.03419v5 [math.OC] for this version)
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