{"id":"openai-math/Triangular-minimality-for-planar-Coulomb-renormalized-energy","pdf":"https://github.com/openai/math/blob/main/preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026","tags":["discrete-geometry","coulomb-energy","lattice-minimization"],"score":0,"title":"Triangular minimality for planar Coulomb renormalized energy","domain":"Convex and metric geometry","family":"090","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.MG","summary":"We prove the Sandier–Serfaty conjecture: the triangular lattice of covolume one minimizes planar Coulomb renormalized energy over all admissible curl-free fields with a unit uniform background. Combined with Bétermin and Sandier's asymptotic formula, this also proves the Brauchart–Hardin–Saff conjecture for the linear term of optimal ordered-pair logarithmic energy on the unit two-sphere. The proof uses a direct Voronoi-cell comparison with rigorous interval arithmetic.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":false,"family_title":"Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy"}