Result 090, Convex and metric geometry

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<s<20\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.

Lean formalization Proof

The bigger picture

Why it matters

Why do points so often favor a grid of equilateral triangles? These manuscripts claim that this triangular lattice has the lowest limiting energy for a broad class of planar interactions, even against irregular arrangements.

What changes?

The manuscripts report minimality of the density-one triangular lattice among all planar configurations with finitely many points in each bounded region and point count divided by area tending to one in expanding centered disks. The objective is the lower limit of energy per particle averaged over those disks, including infinite energies. Allowed pair potentials are nonnegative completely monotone functions of squared distance: their successive derivatives alternate signs, starting with a nonpositive first derivative. Competitors need not be lattices.

What does that help mathematicians do?

They also report triangular minimality for renormalized, or background-adjusted, Riesz energies with inverse-distance exponent strictly between zero and two, and Coulomb energy, with uniform background density one. The Coulomb comparison covers all admissible curl-free fields, not just periodic arrangements. Combined with Bétermin and Sandier's asymptotic formula, that claimed result determines the linear term of optimal ordered-pair logarithmic energy on the unit two-sphere. Thus a planar minimization result fixes a precise correction to spherical energy.

Are there practical applications?

The immediate value is foundational: the claimed comparison would rule out every disordered planar competitor with lower limiting energy under the stated hypotheses. It supplies a sharp benchmark for mathematical models of interacting particles. This is an infinite-system conclusion, not an efficient algorithm for finding optimal finite configurations or evidence of a technological deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

4 manuscripts

An atomic certificate for triangular-lattice universal optimality

September 26, 2026 31 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026,
  author = {{OpenAI}},
  title = {{An atomic certificate for triangular-lattice universal optimality}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf}{OAI:An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026}},
  year = {2026}
}

Universal optimality of the triangular lattice

September 23, 2026 70 pages

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<20\lt s\lt 2 in the Riesz case, and the corresponding jellium minima. The proof uses sharp Gaussian Fourier bounds, positive mixtures, and heat-kernel comparison.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Universal-optimality-of-the-triangular-lattice-September-23-2026,
  author = {{OpenAI}},
  title = {{Universal optimality of the triangular lattice}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf}{OAI:Universal-optimality-of-the-triangular-lattice-September-23-2026}},
  year = {2026}
}

A sharp Fourier certificate for planar circle packing

September 23, 2026 49 pages

We resolve the planar Cohn–Elkies sharpness conjecture: the two-point Fourier bound attains the optimal circle-packing density π/(23)\pi/(2\sqrt3). 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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026,
  author = {{OpenAI}},
  title = {{A sharp Fourier certificate for planar circle packing}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf}{OAI:A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026}},
  year = {2026}
}

Triangular minimality for planar Coulomb renormalized energy

September 23, 2026 49 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/090.md.

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

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves that the density-one triangular lattice minimizes lower energy per particle among all locally finite planar configurations of centered-disk density one, for every nonnegative completely monotone function of squared distance. Infinite energies are allowed in the comparison.

It also constructs sharp radial Schwartz minorants for every Gaussian potential: each minorant lies below the Gaussian, has nonnegative real Fourier transform, agrees with the Gaussian at nonzero triangular-lattice points, and vanishes on nonzero dual-lattice points after Fourier transformation. For the stated parameter range, the formalization includes the explicit atomic interpolation construction.

The planar Cohn–Elkies sharpness conjecture asks whether the two-point Fourier method attains the optimal circle-packing density. The formalization constructs a radial Schwartz function ff on R2\mathbb R^2 with f^(0)=1\widehat f(0)=1, f(0)=2/3f(0)=2/\sqrt3, f^\widehat f real and nonnegative everywhere, and f(x)≤0f(x)\le0 whenever ∥x∥≥1\|x\|\ge1. These are the sharp Fourier certificate conditions yielding density π/(23)\pi/(2\sqrt3). The separate uniqueness statement for periodic equality cases is outside this selected theorem.

Comparator links

Result Comparator statement
Sharp Gaussian minorants from an atomic certificate AtomicGaussian.lean
Universal energy minimality of the triangular lattice TriangularEnergy.lean
Gaussian Fourier minorants with their construction TriangularGaussian.lean
Sharp Fourier certificate for planar circle packing PlanarPacking.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 11 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.