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

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.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Universal optimality of the triangular lattice in the plane (Cohn-Kumar) · Universal optimality of the triangular lattice

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

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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.