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

An Isolated Particle Pole for the Two-Dimensional O(3) Spin Field

OpenAI · October 4, 2026

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Abstract

We consider the continuum spin field constructed from the nearest-neighbor two-dimensional $O(3)$ model by the fixed prescription of the companion paper. We prove that its vector two-point spectral measure has a positive atom corresponding to the lowest mass, separated by a positive gap from all remaining mass support.

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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]

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Cite

@misc{OAI:An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-October-4-2026,
  author = {{OpenAI}},
  title = {{An Isolated Particle Pole for the Two-Dimensional O(3) Spin Field}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-October-4-2026/o3-particle-pole.pdf}{OAI:An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-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: