Optimizing Explicit Unit-Distance Lower-Bound Certificates

The 2026 disproof of ErdHos'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 n1+varepsilon for a fixed positive varepsilon. Sawin's explicit bound gives more than n1.014 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 SQ, integer multiplicities k(p), and a rationally encoded real parameter R. After reproducing Sawin's certificate with δ=0.014114ldots, the pipeline yields improved certificates with the same T. We develop a tailored integer evolution strategy achieving a certificate with δ=0.015263ldots and supporting the cautious statement u(n)>n1.0152 for arbitrarily large n. For extended ramified prime ranges, the Emmerich--Cordella certificate obtained with the same framework reports u(n)>n1.031 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 δ>0.035 and beyond δ>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.