{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"0e79128da313f39730e405d51c19783b4011e7cfa01a4960e47040eee1f282c5","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"86f519ea0de3fdd7c3d482d4cadf597d856de7806cdd0506e1fc8b508affed7c","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":"1","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 $\\operatorname{sum}_{y != x} f(\\mid x - y \\mid^{2})$ among all periodic configurations in $R^{2}$ of density $\\rho$, for every completely monotone f (including $\\operatorname{exp}(-a r^{2})$ and $r^{-s}$). Answer: proof or a periodic configuration beating the triangular lattice for some such f.","problem_ref":null,"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":"00bca35d0adb193ca8c103df758e6ebf929de8f1f684bf0377fdce9923d82670"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"b6f573745e9f262ee521363d74e123f93801358665647cc65a62a310dc8817b1","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"b137740a881230a4b4d1696b7d339dccad1d177a2e91a684bb4808c3f81a6d2e","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"vsPYg83B2T1qPu0AQEaZXf-c18HsrIyXRRRbFyOfr-XUTfeiaI7uFqTOv3drfG8cOnOFElwtmwmJ46HUlSzPAg"},"schema":"pubphys.envelope/1"},"record_hash":"b6f573745e9f262ee521363d74e123f93801358665647cc65a62a310dc8817b1","leaf_index":1607}