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

A sharp Fourier certificate for planar circle packing

OpenAI · September 23, 2026

Lean

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Abstract

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

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Notes

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work in progress

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.

Follow-up works

Formalization

  • Ben Green, 100 open problems (problem list, 2024). formal-conjectures cites Problem 42 of this list as the source of the planar Cohn–Elkies sharpness question that this paper answers; the list is not in the paper's bibliography.

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

Digests and expositions

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Elsewhere

Independent sites; their content is not reviewed here.

Cite

@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}
}

From OpenAI's README for this paper:

Verification

The numerical certificate requires Python 3 and python-flint==0.9.0. The following runs it in a temporary copy so generated results stay separate from the supplied sources:

paper_root=$(pwd -P)
paper_verify=$(mktemp -d)
cp -R "$paper_root/build" "$paper_root/verification" "$paper_verify/"
python3 -m venv "$paper_verify/venv"
"$paper_verify/venv/bin/python" -m pip install -r "$paper_verify/verification/requirements.txt"
"$paper_verify/venv/bin/python" -B "$paper_verify/verification/verify_all.py"

The optional exact-rational six-by-six check uses only the Python standard library:

"$paper_verify/venv/bin/python" -B "$paper_verify/verification/check_sjj_rational.py"

Run without -O or -OO. See verification/README.md for the precise coverage, output interpretation, and limitations.

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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