{"id":"openai-math/Ground-state-condensation-in-the-dilute-hard-sphere-gas","pdf":"https://github.com/openai/math/blob/main/preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026","tags":["mathematical-physics","bose-einstein-condensation","many-body","statistical-mechanics"],"score":0,"title":"Ground-state condensation in the dilute hard-sphere gas","domain":"Mathematical physics","family":"267","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.MP","summary":"We prove Bose–Einstein condensation in every ground state of a dilute three-dimensional hard-sphere Bose gas. At every sufficiently small fixed gas parameter, the constant orbital contains a positive fraction of the particles in the thermodynamic limit. The fraction can be chosen independently of the gas parameter, and the conclusion holds for arbitrary complex ground states.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":false,"family_title":"Positive-temperature Bose–Einstein condensation and exact quantum depletion"}