Bose-Einstein condensation of the dilute hard-sphere Bose gas in the thermodynamic limit, at zero and small positive temperature

Consider $N$ bosons on the torus $\Lambda_L=(\mathbb{R}/L\mathbb{Z})^3$ interacting through a repulsive short-range potential, for instance hard spheres of diameter $a$ . Condensation in the sense of Penrose and Onsager means that the constant orbital carries a macroscopic fraction of the particles: $\liminf N^{-1}\langle u_0,\gamma^{(1)}u_0\rangle>0$ as $L\to\infty$ with $N/L^3\to\rho$ fixed, for the ground state or the Gibbs state at a fixed temperature. This had been proved only in scaling limits where the box stays comparable to the healing length (Gross-Pitaevskii) or the interaction vanishes, and for a lattice hard-core gas at half filling. Does the dilute three-dimensional interacting Bose gas exhibit Bose-Einstein condensation in the thermodynamic limit at fixed density, in its ground state and at positive temperature?

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Argument
Field: Quantum many-body theory; dilute Bose gases
Posed by: Long-standing, after Penrose and Onsager's 1956 definition of condensation; Solovej's 2025 survey calls condensation in the thermodynamic limit a major open problem
Year posed: —
Years open: —
Solved: 2026-10-05
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 58 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Positive temperature (October 5): for every $a>0$ there is $\rho_*(a)>0$ such that for each fixed $0<\rho<\rho_*$ some fixed $T(a,\rho)>0$ gives $\liminf N^{-1}\langle u_{0,L},\gamma^{(1)}_{N,L,T}u_{0,L}\rangle>0$ for the exact canonical hard-sphere Gibbs state as $L\to\infty$ , $N/L^3\to\rho$ . Ground state (September 24 companion): absolute $\varepsilon_0,c_0>0$ with condensate fraction at least $c_0$ in every ground state when $\rho a^3<\varepsilon_0$ . A further companion gives a density-uniform bound for bounded nonnegative finite-range potentials at temperatures up to $\rho^2$ . Not shown: the transition temperature, condensation at moderate density, two dimensions, or potentials with attractive parts.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has five manuscripts (September 24 to October 5, 2026). The positive-temperature paper says it builds on, and reproduces the needed arguments of, two earlier manuscripts in the same family, so later results rest on earlier model output.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the October 5 positive-temperature manuscript and Theorem 1.1 of the September 24 ground-state companion were read against the problem as stated in the companion and in Solovej's survey section 5. The proofs were not refereed. The headline positive-temperature theorem has no Lean formalization. The ground-state companion does: lean/ComparatorChallenges/HardSphere.json exists with solution module OAI.Analysis.HardSphere.Main present at the pinned commit (not in the formalization catalogue). Its statement OAI.HardSphere.condensation_with_mixed was read: absolute eps0, c0 > 0 such that for rho a^3 < eps0 the constant-orbital occupation has liminf at least c0 along every thermodynamic sequence, for all ground vectors and ground-supported density operators. Not rebuilt. The temperature obtained is small and not tied to the transition.

Bose-Einstein condensation of the dilute hard-sphere Bose gas in the thermodynamic limit, at zero and small positive temperature

What was actually shown

What the AI did

Verification

Sources

Discussion