OpenAI claims quantum ferromagnet order and interacting Bose condensation

Your AI and this story

Key facts

• Magnetization (abstract, verbatim): ‘For every dimension d ≥ 3 and every spin S ∈ {1/2, 1, 3/2, …}, we prove that the nearest-neighbor isotropic quantum Heisenberg ferromagnet has a translation-invariant, spontaneously magnetized equilibrium state at every sufficiently low positive temperature. The same state satisfies the KMS condition for the zero-field dynamics and has magnetization at least S/4.’
• Why it was open (paper’s history): Mermin–Wagner rules out such order in d = 1, 2; Fröhlich–Simon–Spencer proved ordering for classical spins and Dyson–Lieb–Simon for the quantum antiferromagnet in d ≥ 3, but ‘The corresponding ferromagnetic problem resisted this approach: the reflection-positivity argument does not provide the needed ferromagnetic Duhamel infrared bound’
• Bloch’s law (Oct 5, verbatim): ‘We prove Bloch’s T^{3/2} law for the spontaneous magnetization of three-dimensional quantum Heisenberg ferromagnets at every fixed positive quantum spin… The leading magnetization deficit has the exact coefficient determined by the determinant of the quadratic one-magnon dispersion.’
• Bose gas (abstracts, verbatim): ‘We prove Bose–Einstein condensation in every ground state of a dilute three-dimensional hard-sphere Bose gas’ (24 Sept) and ‘there is a strictly positive temperature, independent of the volume, at which the exact canonical Gibbs state has a positive condensate fraction in the thermodynamic limit’ (5 Oct). The 24 Sept paper cites Solovej’s 2025 survey, which ‘identifies fixed-density ground-state condensation as a major open problem’
• Prior results (as the papers recount them) proved condensation only in scaling limits where the interaction weakens as the volume grows, e.g. Lieb–Seiringer’s trapped Gross–Pitaevskii theorem and Deuchert–Seiringer’s homogeneous positive-temperature theorem
• Lean (scope notes 271 and 267): spontaneous magnetization for all d ≥ 3 and all S, with a KMS state of magnetization ≥ S/4; ground-state condensation for ρa³ < ε₀, ‘without a positive-temperature assertion’. Both challenges accepted by oai-math-revised’s Comparator run of 25–26 Oct (standard axioms only)
• OpenAI released an abridged reasoning summary for family 271; its prompt asks the model to ‘Prove or refute’ the existence of such a magnetized state. Scientific American (8 Oct, ‘Einstein’s Baby’): ‘OpenAI’s model extended the prediction to a case with interactions. It used similar techniques to prove a major conjecture about the math of ferromagnetism—one that the famous mathematical physicist Freeman Dyson erroneously claimed with two collaborators in 1976.’