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The canonical massive continuum limit of the two-dimensional O(3) model

OpenAI · October 4, 2026

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Abstract

We construct a canonical interacting massive continuum limit of the two-dimensional nearest-neighbor $O(3)$ model with unit-length spins, no external field, and no topological term. Normalized by susceptibility and second-moment correlation length, the limit exists as the bare coupling tends to infinity through all positive real values, without selecting subsequences. 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.

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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") without identifying the papers by title. [family-level]
  • Marcus Bintz (@spurious_bows), post on X (2026-10-11). Notes that the "squaring probabilities" trick of another release paper's Lemma 4.1 also appears as Lemma 8.2/Prop 8.3 here and as Lemma 2.1/Prop 2.2 in the O(4) mass-bound paper.

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:The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026,
  author = {{OpenAI}},
  title = {{The canonical massive continuum limit of the two-dimensional O(3) model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf}{OAI:The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026}},
  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: