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.

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.