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Browse › Probability and statistical mechanics › 215 Canonical $O(3)$ continuum limit and exact $O(4)$ mass asymptotics

Exponential decay in two-dimensional classical O(n) models

OpenAI · September 23, 2026

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Abstract

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.

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Notes

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-23) and not checked by a person. May be wrong or incomplete. Report an error.

Follow-up works

Comments and discussion

  • Peter Hall and Joseph Howlett, The most exciting claims from OpenAI's heap of new proofs (Scientific American, 2026-10-08). Reports several release papers claiming to resolve the sigma-model analogue of the Yang-Mills mass-gap problem, quoting Michael Douglas and Roland Bauerschmidt (the first sigma-model paper "seems very reasonable", the second "horrible"); it does not identify the papers by title, and it is unclear whether this paper is among them. [family-level]
  • Rudy Pei (@PeaBrane10), post on X (2026-10-07). States that the twist estimate in the proof of Polyakov's conjecture "never uses the sign of the couplings", so it would extend to show the 2D Heisenberg spin glass has finite χ_SG at every T > 0.
  • Zohar Komargodski (@ZoharKo), post on X (2026-10-07). Replies that the O(N) result "was known to be true from numerics for decades".

Formalization

Possibly relevant literature (found by the agent, not checked)

Digests and expositions

  • AI Proved What?, Do spins forget? (Episode 215, October 2026). Explains the five papers’ claims, proof strategies and differing formalization scopes. [AI-generated per source] [family-level]

Elsewhere

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Cite

@misc{OAI:Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026,
  author = {{OpenAI}},
  title = {{Exponential decay in two-dimensional classical $O(n)$ models}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026/paper.pdf}{OAI:Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/215.md. The Lean statements have not been checked by people against the paper's claims.

Family 215: Canonical $O(3)$ continuum limit and exact $O(4)$ mass asymptotics

Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $O(3)$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice $O(4)$ model, determines the exact leading asymptotic of the full transfer gap, $m_{\mathrm{lat}}(\beta)\sim32e^{\pi/4-1/2}\sqrt\beta\,e^{-\pi\beta}$. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor $O(n)$ models with n ≥ 3 at every positive temperature.

Other papers in this family: