Claimed AI-led

Massive continuum limit of the two-dimensional O(3) model

OpenAI manuscripts claim to construct the continuum limit of the two-dimensional O(3) sigma model as an interacting relativistic quantum field theory with a unique vacuum and a positive mass gap, the long-standing simpler analogue of the Yang-Mills mass gap problem, and to prove exponential decay of correlations in the classical O(n) models for n at least 3 at every temperature.

Lab: OpenAI
Model: Unreleased internal OpenAI model, not named in the release
Field: Mathematics
Date: 2026-10-06

Sources

What was found

The main manuscript takes the nearest-neighbor lattice $O(3)$ model, normalizes by susceptibility and second-moment correlation length, and proves the limit exists as the coupling tends to infinity through all real values, without passing to subsequences; the limiting fields satisfy the Osterwalder-Schrader axioms, have a nonzero connected four-point function, and reconstruct to a theory with a unique vacuum and a positive Hamiltonian gap. A companion proves an isolated one-particle pole. Two further manuscripts treat $O(4)$ , ending with the exact asymptotic $m_{\mathrm{lat}}(\beta) \sim 32 e^{\pi/4 - 1/2} \sqrt{\beta}\, e^{-\pi\beta}$ for the lattice mass gap. A fifth proves exponential decay of two-point correlations for the classical $O(n)$ model with $n \ge 3$ at every finite inverse temperature, uniformly over finite subgraphs, the absence of a low-temperature ordered phase that physicists have expected since Polyakov's work of the 1970s.

Novelty check

Read CONTENTS.md and the family's Lean scope document at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb on 2026-10-12; the TeX introduction of the main manuscript could not be fetched from the repository's storage, so its account of prior work was not read. No earlier construction of the massive $O(3)$ continuum limit, and no proof of all-temperature decay for $n \ge 3$ , was found in the registry; both are standing open problems of constructive field theory and statistical mechanics in the literature the release cites.

Caveats and known objections

The Lean formalization covers only the classical $O(n)$ decay theorem on finite square-lattice subgraphs; its scope document says the infinite-volume limit and the $O(4)$ spectral-gap claims are not included, and nothing about the continuum limit, the Osterwalder-Schrader axioms or the mass gap is formalized. The catalogue records its review status as unchecked. Roland Bauerschmidt told Scientific American that one of the papers seemed very reasonable on a first read, while saying the details were not yet digested, and called another 'horrible', saying the rest were too hard for even an expert to verify; that is the only named outside assessment found. Michael Douglas described the problem to the same magazine as the widely accepted warm-up for Yang-Mills. Autonomy is graded ai-led for the same reasons as the other families split from this release.

Also recorded at

vibemathedo3-sigma-model-continuum-limit-mass-gap

Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.

Nobody outside the lab has checked this yet.

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History of this entry

2026-10-16CorrectedRecorded the vibemathed record for this result as a registration. It cites the same manuscript. No grade moved.
2026-10-12AddedEntered the registry graded Claimed and AI-led.

Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.

Community discussion

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Cite this entry

Plain text

whataifound.org. (2026). Massive continuum limit of the two-dimensional O(3) model. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-10-06-o3-model-mass-gap

BibTeX

@misc{whataifound-openai-2026-gap,
  title        = {Massive continuum limit of the two-dimensional O(3) model},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by OpenAI. Verification: Claimed. Autonomy: AI-led.},
  url          = {https://whataifound.org/finding/2026-10-06-o3-model-mass-gap}
}

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