New optimization pushes unit-distance bound past n^1.0152

A new paper by Michael T.M. Emmerich (Leiden University) optimizes the explicit lower-bound certificates for the Erdős unit-distance conjecture, which was disproved in 2026. The conjecture asked for the maximum number u(n) of unit distances among n points in the plane. Sawin's refinement showed u(n) > n^{1+δ} with δ ≈ 0.01411, but left several integer parameters and a real parameter R unoptimized. Emmerich formulates this as a nonlinear integer programming problem and builds an open-source Python pipeline to search for better certificates.

The pipeline employs three optimization methods: a deterministic greedy construction, a Tailored Integer Evolution Strategy (TIES) with repair operators, and a two-parent discrete-recombination variant. Running on standard hardware, the best result raises δ to 0.0152628688…, corresponding to u(n) > n^{1.0152}. The paper compares four certificate levels, showing steady improvement. The work is important because it demonstrates that computationally guided number-theoretic searches can push the exponents in extremal combinatorics, and the open-source tools make the results fully verifiable. Subjects include Optimization, Computational Geometry, and Evolutionary Computing.