aiqbench analysis
This study leverages open-source Python pipelines and tailored integer evolution strategies to improve explicit lower-bound certificates for the unit-distance problem, pushing the known exponent from 1.014 to 1.0152. The work demonstrates how randomized optimization heuristics, including discrete-recombination variants, can enhance pure mathematical proofs in combinatorial geometry. For a tech audience, the key takeaway is the availability of a lean, replicable implementation that validates advanced optimization techniques on standard hardware.
Optimizing Explicit Unit-Distance Lower-Bound Certificates
arXiv:2606.03419v2 Announce Type: replace-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 integer parameters whose choice is not fully optimized. This report starts from Sawin's nonlinear integer optimization problem and develops an open-source Python verification pipeline. The pipeline is first validated by reproducing Sawin's published parameter choice and is then applied to computationally improved certificates. We optimize and verify certificates involving sets of primes $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. The implementation is deliberately lean, so that all results can be replicated on standard hardware and the procedures can be extended. We compare a deterministic greedy heuristic, a tailored integer evolution strategy with two-sided geometric, or discrete-Laplace, integer mutation and repair operators for number-theoretic feasibility, and a two-parent discrete-recombination variant. Four certificate levels are reported: Sawin's published example with $\delta=0.0141144286784982\ldots$, a greedy certificate with $\delta=0.0151718056372133\ldots$, a tailored integer evolution strategy certificate with $R=6672416/100000$ and $\delta=0.0152616610684193\ldots$, and a recombination variant with the same $R$ and $\delta=0.0152628688170072\ldots$. Consequently, the best current certificate supports the cautious statement $u(n)>n^{1.0152}$ for arbitrarily large $n$. Beyond this unit-distance application, the work illustrates how randomized optimization heuristics can improve explicit certificates in pure mathematics and combinatorial geometry.
Abstract:The 2026 disproof of Erdős'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 integer parameters whose choice is not fully optimized. This report starts from Sawin's nonlinear integer optimization problem and develops an open-source Python verification pipeline. The pipeline is first validated by reproducing Sawin's published parameter choice and is then applied to computationally improved certificates. We optimize and verify certificates involving sets of primes $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. The implementation is deliberately lean, so that all results can be replicated on standard hardware and the procedures can be extended. We compare a deterministic greedy heuristic, a tailored integer evolution strategy with two-sided geometric, or discrete-Laplace, integer mutation and repair operators for number-theoretic feasibility, and a two-parent discrete-recombination variant. Four certificate levels are reported: Sawin's published example with $\delta=0.0141144286784982\ldots$, a greedy certificate with $\delta=0.0151718056372133\ldots$, a tailored integer evolution strategy certificate with $R=6672416/100000$ and $\delta=0.0152616610684193\ldots$, and a recombination variant with the same $R$ and $\delta=0.0152628688170072\ldots$. Consequently, the best current certificate supports the cautious statement $u(n)>n^{1.0152}$ for arbitrarily large $n$. Beyond this unit-distance application, the work illustrates how randomized optimization heuristics can improve explicit certificates in pure mathematics and combinatorial geometry.
Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Computational Geometry (cs.CG); Neural and Evolutionary Computing (cs.NE); Combinatorics (math.CO)
arXiv:2606.03419 [math.OC]
(or arXiv:2606.03419v2 [math.OC] for this version)
Full text under the publisher's open licence · 259 words