OpenAI claim summary for Math-552
mathematicsTop Open Problemscontributionliterature_reportClaim: literature report
Summary of a claim published in openai/math; not a claim by this platform.
The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 090; 2026-09-23).
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
Cohn-Elkies Scheme for Circle Packings
OpenAI's claim
OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 090; 2026-09-23), the asserted result is that the planar two-point Fourier packing method attains density $\pi/(2\sqrt3)$ through a radial Schwartz certificate.Mathematical statement of the claimed result
Manuscript 1.
With Fourier convention $\widehat f(\xi)=\int_{\mathbb R^2}f(x)e^{-2\pi i x\cdot\xi}\,dx$, the claim supplies a real radial Schwartz function on $\mathbb R^2$ with \[ \widehat f(0)=1,\quad f(0)=2/\sqrt3,\quad \widehat f(\xi)\ge0\ (\xi\in\mathbb R^2),\quad f(x)\le0\ (|x|\ge1). \] These signs and values form the sharp linear-programming certificate for the triangular-lattice circle-packing density $\pi/\sqrt{12}$.Outline of the argument
For manuscript 1 (family 090; 2026-09-23), coupled cardinal interpolation produces a candidate function and its transform. Exact quadrature, interval residual bounds, and Bernstein coefficients verify the sign inequalities globally, including tails beyond the finite certificate region.Formal verification
For manuscript 1 (family 090; 2026-09-23), OpenAI lists Lean statements for this assertion: the planar two-point Fourier packing method attains density $\pi/(2\sqrt3)$ through a radial Schwartz certificate.The selected statement includes the clause used for this problem.
The linked family Comparator checks are AtomicGaussian.lean, PlanarPacking.lean, TriangularEnergy.lean, TriangularGaussian.lean.
Their named declarations include OAI.AtomicTriangular.sharp_gaussian_minorants_atomic, OAI.AtomicTriangular.sharp_gaussian_minorants_with_construction, OAI.AtomicTriangular.universal_energy_minimum, OAI.sharp_fourier_certificate.
This site has not executed the Lean code. Described statement coverage is not an independent proof check.
Status caveat
The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.Sources
- OpenAI, A sharp Fourier certificate for planar circle packing, 2026-09-23; citation key
OAI:A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: intro:main, intro:conditions. - Family 090: Lean scope.
- Comparator AtomicGaussian.json.
- Comparator AtomicGaussian.lean.
- Comparator PlanarPacking.json.
- Comparator PlanarPacking.lean.
- Comparator TriangularEnergy.json.
- Comparator TriangularEnergy.lean.
- Comparator TriangularGaussian.json.
- Comparator TriangularGaussian.lean.
- OpenAI catalogue entry.