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Exact mass asymptotics for the two-dimensional O(4) lattice model

OpenAI · October 5, 2026

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Abstract

For the nearest-neighbor $O(4)$ model on the square lattice at inverse temperature β, we prove the exact low-temperature asymptotic $\displaystyle m_{\mathrm{lat}}(\beta)\sim 32\exp(\pi/4-1/2)\sqrt\beta\,\exp(-\pi\beta) \qquad (\beta\to\infty).$ Here the mass is the gap of the full Osterwalder–Schrader transfer operator, including rotation-invariant local-observable sectors, in units of one original lattice time step.

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  • 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:Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026,
  author = {{OpenAI}},
  title = {{Exact mass asymptotics for the two-dimensional O(4) lattice model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026/exact-mass-o4.pdf}{OAI:Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026}},
  year = {2026}
}

Lean formalization

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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: