Universal optimality of the triangular lattice in the plane (Cohn-Kumar)

For a locally finite configuration $\mathcal C\subset\mathbb R^2$ of density one and a potential $g$ , the energy per point is the lower limit of $N_R^{-1}\sum_{x\ne y\in\mathcal C\cap B_R}g(|x-y|^2)$ . Among lattices of covolume one the triangular lattice minimizes inverse-power sums (Rankin, Cassels, Ennola, Diananda) and Gaussian theta sums (Montgomery 1988), but these say nothing about non-lattice competitors. Cohn and Kumar (2007, Conjecture 9.4) conjectured that the triangular lattice, like $E_8$ and the Leech lattice, is universally optimal, with sharp Fourier auxiliary functions certifying it; Cohn, Kumar, Miller, Radchenko and Viazovska proved the analogue in dimensions 8 and 24. Does the triangular lattice of covolume one minimize energy per point among all density-one planar configurations, for every completely monotone function $g$ of squared distance?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Discrete geometry; energy minimization and universal optimality
Posed by: Henry Cohn and Abhinav Kumar (Conjecture 9.4, J. Amer. Math. Soc. 2007)
Year posed: 2007
Years open: 19y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 55 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: the density-one triangular lattice $A$ minimizes the lower energy per point $E_g$ among all locally finite planar configurations of centered-disk density one, for every nonnegative smooth completely monotone $g$ of squared distance, including Gaussians and all inverse powers, with infinite energies allowed and no separation assumption. The proof builds sharp radial Schwartz minorants $f_\alpha\le e^{-\pi\alpha|x|^2}$ with $\widehat f_\alpha\ge0$ touching at all nonzero lattice and dual-lattice points, and integrates over Gaussians. Section 8 transfers this, following Petrache-Serfaty, to triangular minimality of the logarithmic and Riesz $0<s<2$ renormalized (jellium) energies. It does not give a sharp auxiliary function for every mixed potential, does not classify minimizers, and does not treat other dimensions.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has a second manuscript, 'An atomic certificate for triangular-lattice universal optimality' (September 26, 2026), giving a separate construction of the sharp Gaussian minorants; the Lean formalization in the release is attached to that companion. Both manuscripts ship finite checkers (Python, python-flint) for their numerical inputs.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the Cohn-Kumar question. It proves $E_g(\mathcal C)\ge\sum_{a\in A\setminus 0}g(|a|^2)=E_g(A)$ for every smooth completely monotone $g\ge0$ and every locally finite configuration of centered-disk density one, infinite energies included. The manuscript states plainly that it does not construct a sharp auxiliary function for each mixed potential, the stronger per-potential quantifier in Cohn-Kumar's Conjecture 9.4; it identifies the minimum value, not all minimizers. Lean: lean/ComparatorChallenges/TriangularEnergy.json exists with solution_module OAI.Analysis.Triangular.Energy.Universal, whose file exists at the pinned commit; this challenge is not in lean/formalization.yaml. The statement TriangularEnergy.lean was read here: for every smooth nonnegative completely monotone $g$ on $(0,\infty)$ and every locally finite $\mathcal C$ with centered disk count over $\pi R^2$ tending to 1, the lattice energy (an extended-real series) is at most the liminf of disk energies of $\mathcal C$ and equals that of the lattice. This states the headline energy comparison. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.

Universal optimality of the triangular lattice in the plane (Cohn-Kumar)

What was actually shown

What the AI did

Verification

Sources

Discussion