{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.topic/1","content_sha256":"0fc24a0d92d6d402e98c374343bbf4e8df66b991def41f437ca739b686e4aff1","created":"2026-10-03T07:17:52Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":[],"salt":"32ad6024d6ab6830d7e987ad03136d8b4b4857033a0164f5cf3ab341e103b6d8","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"topic"},"content":{"external_id":"mathph.crystal-order-temperature","field":"mathph","n":"1","review_cite":"Thomas Richthammer, Translation-invariance of two-dimensional Gibbsian point processes, Communications in Mathematical Physics, 2007","review_link":"https://doi.org/10.1007/s00220-007-0274-7","review_verified":"true","summary":"At low but nonzero temperature the atoms of a crystal vibrate around lattice sites, yet the lattice as a whole should stay in register over infinite distances. No one has proved this long-range order for any realistic system of particles moving in continuous 3D space.","title":"Crystalline order at positive temperature","topic_ref":null,"why":"It is the missing proof that solids exist as an equilibrium phase distinct from liquids in classical statistical mechanics."},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"ea8bde04090399eb2ce07a2138c4a5deafce80c26031655b989bc3381e0879d5","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"f53aa97bf29bc50c9cf713aa8719a7bf78ddb618b8c0bf714a825988cb4ff04c","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"NtJ5aFvpWlUaK8v6L69-d3OOtqA1I0mfpwpIo5TrRC8L-QtfCf7RdWLnT2UD2e_JffGwqPJ8FqQFGPhACOIkAw"},"schema":"pubphys.envelope/1"},"record_hash":"ea8bde04090399eb2ce07a2138c4a5deafce80c26031655b989bc3381e0879d5","leaf_index":255}