The massive continuum limit of the two-dimensional O(4) sigma model

Conditionally on results of two recent unrefereed, AI-generated manuscripts released by OpenAI, on the renormalization group of the lattice O(4) model and on the continuum limit of the O(3) model, we construct the continuum limit of the nearest-neighbor O(4) spin model on the square lattice. Normalized by the susceptibility and the second-moment correlation length, the spin field converges, as the inverse temperature β tends to infinity through all real values, to a non-Gaussian Euclidean field theory that satisfies the Osterwalder–Schrader axioms. The reconstructed relativistic theory in 1+1 dimensions has a unique vacuum and a positive mass gap on the whole orthogonal complement of the vacuum. The second-moment correlation length satisfies ξ_β ∼ A β^(−1/2) e^(πβ) with A > 0, as predicted by two-loop asymptotic freedom. We also prove that compactly smeared fields have Gaussian-bounded exponential moments; that the infimum m of the spectrum on the vacuum complement equals the infimum of the spectrum on the one-field subspace; that the even part of the vacuum complement has no spectrum below 2m; and that H ≥ m(|Q₁| + |Q₂|) for the two commuting internal charges. The precise inputs from the two manuscripts, and the parts of their proofs that we adapt, are listed in Section 2.7. The construction, its proofs and the text were produced by a loop of AI agents designed by the author (GPT-6.1 Sol in Codex and Claude Opus 5.5); Section 12 describes the loop.