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Triangular minimality for planar Coulomb renormalized energy

OpenAI · September 23, 2026

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Abstract

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.

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Possibly relevant literature (found by the agent, not checked)

  • Stefan Steinerberger, A set of points on the sphere with small Riesz energy (arXiv:2607.08049, July 2026). Constructs explicit point sets on the 2-sphere from pieces of the hexagonal lattice whose Riesz s = 2 energy constant matches the Brauchart–Hardin–Saff conjectured value; this is the Riesz analogue of Corollary 1.3, which by its own statement gives an asymptotic value and does not construct point sets.

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Cite

@misc{OAI:Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026,
  author = {{OpenAI}},
  title = {{Triangular minimality for planar Coulomb renormalized energy}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/paper.pdf}{OAI:Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026}},
  year = {2026}
}

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Family 090: Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy

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 $0\lt s\lt 2$ and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.

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