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The matching claim for the triangular lattice in two dimensions is unproved.","posed_since":"2007","precise":"Prove that for every density $\\rho$ the triangular lattice minimizes the energy per point $\\sum_{y != x} f(\\mid x - y \\mid^{2})$ among all periodic configurations in $R^{2}$ of density $\\rho$, for every completely monotone f (including $\\exp(-a r^{2})$ and $r^{-s}$). Answer: proof or a periodic configuration beating the triangular lattice for some such f.","problem_ref":"b6f573745e9f262ee521363d74e123f93801358665647cc65a62a310dc8817b1","references":"","settled_by":"A proof by an interpolation or linear-programming argument in the style of the 8D and 24D case, or an explicit counterexample.","status_note":"Proved for $E8$ and Leech (Cohn, Kumar, Miller, Radchenko, Viazovska, Annals 2022); in 2D known only among lattices; Petrache and Serfaty (2020) showed it would imply crystallization at next order for 2D Coulomb and Riesz gases.","title":"Universal optimality of the triangular lattice in the 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