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

What OpenAI claims

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 < s < 2 and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy. (Lean)

Papers (4)

1. An atomic certificate for triangular-lattice universal optimality

   26 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove that the density-one triangular lattice minimizes the lower energy per particle for every nonnegative completely monotone function of squared distance, among all locally finite planar configurations of centered disk density one. The comparison includes infinite energies. The proof constructs sharp Gaussian Fourier minorants using an atomic interpolation certificate.

2. Universal optimality of the triangular lattice

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove universal energy minimality for the triangular lattice in the plane. Among locally finite configurations of centered density one, it minimizes the lower limit of centered-ball energy averages for every nonnegative completely monotone function of squared distance, including when the energy is infinite. We also prove triangular minimality for planar logarithmic and Riesz renormalized energies, with 0 < s < 2 in the Riesz case, and the corresponding jellium minima. The proof uses sharp Gaussian Fourier bounds, positive mixtures, and heat-kernel comparison.

3. A sharp Fourier certificate for planar circle packing

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We resolve the planar Cohn–Elkies sharpness conjecture: the two-point Fourier bound attains the optimal circle-packing density π/(2√3). We construct a radial Schwartz certificate and prove its global sign conditions using rigorous interval arithmetic and analytic estimates. The same certificate recovers the classical uniqueness of the triangular packing among periodic equality cases.

4. Triangular minimality for planar Coulomb renormalized energy

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.