2D O(n) mass gap (Polyakov) and O(3) continuum limit claimed (OpenAI)

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Key facts

• Decay theorem (23 Sept, 20 pages; abstract, verbatim): ‘We prove exponential decay of two-point correlations for the classical nearest-neighbor O(n) model on the square lattice, for every n ≥ 3 and every finite positive inverse temperature. The estimate is uniform over finite free-boundary subgraphs and bounded nonnegative edge strengths. This resolves positively the all-temperature exponential spin-decay conjecture for these models.’ The paper cites Polyakov’s 1975 renormalization argument predicting mass generation at every positive temperature, earlier proofs at high temperature (Aizenman–Simon) and for large n (Kupiainen), and Aru, Garban and Sepúlveda’s 2025 statement of the conjecture as open; for n = 2 the claim fails (Berezinskii–Kosterlitz–Thouless phase)
• Continuum O(3) (4 Oct, 122 pages; abstract): ‘We construct a canonical interacting massive continuum limit of the two-dimensional nearest-neighbor O(3) model … The limiting fields satisfy the Osterwalder–Schrader axioms and have a nonzero connected four-point correlation on separated time supports. Their reconstructed theory has a unique vacuum, a nonzero vacuum complement, and a positive Hamiltonian gap on that entire complement.’ Companion (4 Oct, 26 pages): the spin field’s spectral measure has an isolated lowest-mass atom
• O(4) (23 Sept, 102 pages; 5 Oct, 109 pages): the full transfer gap is bounded by positive multiples of √β·e^(−πβ), with a positive gap at every finite β, and exactly m_lat(β) ∼ 32·exp(π/4 − 1/2)·√β·exp(−πβ) as β → ∞
• Lean (OpenAI scope note lean/docs/215.md): exponential decay for the 2D classical O(n) model, n ≥ 3, at every temperature, uniformly over finite square-lattice subgraphs with free boundary; ‘The infinite-volume limit and the separate O(4) spectral-gap claim are not included.’ Challenge ClassicalON: ‘accepted’ in the oai-math-revised comparator run of 25–26 Oct
• Scientific American (8 Oct, ‘Baby Yang-Mills’): Michael Douglas (Harvard): ‘This goes back 50 years, much like Yang-Mills, and is widely considered the ‘warm-up’ for Yang-Mills.’ Roland Bauerschmidt (NYU) on the first of the papers: ‘I haven’t digested it or understood all the details, but it seems very reasonable’; on the others, as the magazine reports it: ‘The second paper is horrible … If these results had been sent to me by a nobody, I would have deleted the e-mail’
• Physicists on X: Lautaro Vergara (7 Oct): ‘Polyakov’s conjecture on the O(N) mode has been demonstrated by OpenAI. Good. But, the 1/N expansion made it unavoidable.’ Zohar Komargodski, replying: ‘And it was known to be true from numerics for decades’. Rudy Pei (7 Oct): the twist estimate ‘never uses the sign of the couplings. So it extends to vector spin glasses: the 2D Heisenberg spin glass has no spin-glass order (finite χ_SG) at any T > 0’
• Overlap noted by readers: Marcus Bintz (X, 12 Oct) matched Lemma 2.1 and Proposition 2.2 of the O(4) mass-bound paper with Lemma 4.1 and Proposition 4.3 of the release’s spin-one Haldane-gap paper (see OpenAI proves the spin-one Haldane gap (periodic chains))