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Sharp mass bounds for the two-dimensional O(4) model

OpenAI · September 23, 2026

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Abstract

We prove sharp mass bounds for the two-dimensional nearest-neighbor $O(4)$ model. For all sufficiently large inverse couplings β, the full transfer gap, including rotation-invariant local-observable sectors, is bounded above and below by positive multiples of $\sqrt\beta e^{-\pi\beta}$. We also prove that the model has a unique periodic local limit and a positive full gap at every finite β > 0, resolving the all-temperature lattice mass-generation conjecture for this periodic state. Under the stated cutoff scaling, the mass bounds are uniform in independently fixed physical units. A continuum spectral interpretation requires separate convergence hypotheses for the transfer semigroup and the block observables.

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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

  • Anton Ragin, The massive continuum limit of the two-dimensional O(4) sigma model (Zenodo, doi:10.5281/zenodo.23349319, v7, 2026-10-13). Constructs an O(4) continuum limit "conditionally on results of two recent unrefereed, AI-generated manuscripts released by OpenAI", one being the renormalization-group analysis of the lattice O(4) model; this manuscript is in its citation list. [AI-generated per source]

Comments and discussion

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

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026,
  author = {{OpenAI}},
  title = {{Sharp mass bounds for the two-dimensional O(4) model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026/paper.pdf}{OAI:Sharp-mass-bounds-for-the-two-dimensional-O4-model-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: