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Browse › Convex and metric geometry › 090 Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy

Universal optimality of the triangular lattice

OpenAI · September 23, 2026

Lean

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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 $0\lt s\lt 2$ in the Riesz case, and the corresponding jellium minima. The proof uses sharp Gaussian Fourier bounds, positive mixtures, and heat-kernel comparison.

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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI math release, Lean scope note for family 090 (lean/docs/090.md, release of 2026-10-06). The note attaches the Lean development to the atomic companion and the packing paper, not to this manuscript; its formalized universal energy statement matches this paper's Theorem 1.1, while the renormalized Riesz/logarithmic and jellium results here are not listed in its scope. [family-level]

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Cite

@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}
}

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Verification

The two finite checkers require Python 3 and python-flint==0.9.0:

paper_verify=$(mktemp -d)
python3 -m venv "$paper_verify/venv"
"$paper_verify/venv/bin/python" -m pip install -r verification/requirements.txt
"$paper_verify/venv/bin/python" -B verification/numeric_balls.py --output-dir "$paper_verify/results"
"$paper_verify/venv/bin/python" -B verification/arithmetic_bounds.py --output-dir "$paper_verify/results"

Run without -O or -OO. The manifest binds the finalized manuscript inputs in build/ and the checker sources. A zero exit status and complete success JSON are required for each run. See verification/README.md for the exact scope, source-version caveats, and output interpretation.

Lean formalization

OpenAI provides Lean files for this family: lean/docs/090.md. The Lean statements have not been checked by people against the paper's claims.

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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