OpenAI · 090-04 · 2026-09-23 · Convex and metric geometry
Triangular minimality for planar Coulomb renormalized energy
preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026
Family 090. Universal optimality of the triangular lattice.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one. It also minimizes unit-background renormalized Riesz energies for and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.
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.
Preprint source and Lean scope
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A Lean scope note is stored for family 090, but it does not name this preprint. It names 090-01, 090-03. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.