{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"4d7001079712bb65956885225e76ff999b9e46bafb685db50a7a743a5bd2c007","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011","e1fabcc9d99cc1fbb81083f240267eb6d9ab9d915051d745f3a047049e5172fd"],"salt":"1fcff4fc700cc153d7f9211e97ab23f7a51237c79faf4f8d3a37f96b2ccbaf3e","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":[],"external_id":"mathph.crystal-order-temperature.hard-sphere-solid","kind":"well-posed","literature_status":"open","n":"1","parents":[{"note":"","parent_revision":"e1fabcc9d99cc1fbb81083f240267eb6d9ab9d915051d745f3a047049e5172fd","relation":"special_case"}],"plain":"Hard balls with no attraction freeze into a crystal when packed densely enough, as computer simulations show above about 55 percent filling. Prove that such a crystalline equilibrium state exists.","posed_since":"1957","precise":"For hard spheres of diameter $a$ in $R^3$ at packing fraction $\\eta$ close to the close-packing value $\\pi/\\sqrt{18}$, about 0.7405, prove existence of a non-translation-invariant Gibbs measure, and more sharply a first-order freezing transition; simulations place fluid-solid coexistence at $\\eta$ about 0.494 and 0.545.","problem_ref":null,"references":"","settled_by":"A rigorous proof of crystalline long-range order for hard spheres at some $\\eta$ below close packing.","status_note":"","title":"Crystalline phase of hard spheres at high packing fraction","topic_ref":"ea8bde04090399eb2ce07a2138c4a5deafce80c26031655b989bc3381e0879d5"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"52ab7f80904bddf09b0a90e49f82cb596814f4ac89bc09596a485532dec88181","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"06cbb37664906d1485a65752a940737c282c55b54f601e7688bb2065d3f682ce","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"LIQs_iaDx85Io-JEjLsyQsMnN6KOF1arBzudJ-kXUa6kz5NYixvqrvbGzFy7XfJ7fekBhHva36fw_kY4xWnkBQ"},"schema":"pubphys.envelope/1"},"record_hash":"52ab7f80904bddf09b0a90e49f82cb596814f4ac89bc09596a485532dec88181","leaf_index":1603}