{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"d934d52a0111dfd2e710d40d713e8be71da00d0fdc5a20da9fad5714ca22c957","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["b6f573745e9f262ee521363d74e123f93801358665647cc65a62a310dc8817b1","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"e1ce2dbc7bc992e6fc14a1ba262d563830381c29aa7592981c4858b74db35bc0","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.crystallization.cohn-kumar-2d","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"In 2022 the $E8$ and Leech lattices were proved to be the best arrangements in 8 and 24 dimensions for every repulsive force from a broad class. 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 plane","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"57edffc99a32689752f3752b89c4a0507729281f8694517459cc747fce9fe397","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"7ecfeb88454bf0b371539895dd3ec207b843ca7729bac52bff5fe814c3cb9a98","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"tWzoYXP1mGTH3dds39o-iAUSYt3lsjAHORxARnYv2-r9azmf-xf1V3yNSIS0qT-jfPZowQdqrg_yXxNJklDVCQ"},"schema":"pubphys.envelope/1"},"record_hash":"57edffc99a32689752f3752b89c4a0507729281f8694517459cc747fce9fe397","leaf_index":3408}