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. (Lean)
Manuscripts (5)
The canonical massive continuum limit of the two-dimensional O(3) model
October 4, 2026 · source · cite as \cite{OAI:The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026}
We construct a canonical interacting massive continuum limit of the two-dimensional nearest-neighbor O(3) model with unit-length spins, no external field, and no topological term. Normalized by susceptibility and second-moment correlation length, the limit exists as the bare coupling tends to infinity through all positive real values, without selecting subsequences. The limiting fields satisfy the Osterwalder–Schrader axioms and have a nonzero connected four-point correlation on separated time supports. Their reconstructed theory has a unique vacuum, a nonzero vacuum complement, and a positive Hamiltonian gap on that entire complement.
An Isolated Particle Pole for the Two-Dimensional O(3) Spin Field
October 4, 2026 · source · cite as \cite{OAI:An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-October-4-2026}
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.
Exact mass asymptotics for the two-dimensional O(4) lattice model
October 5, 2026 · source · cite as \cite{OAI:Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026}
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.
Sharp mass bounds for the two-dimensional O(4) model
September 23, 2026 · source · cite as \cite{OAI:Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026}
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.
Exponential decay in two-dimensional classical O(n) models
September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026}
We prove exponential decay of two-point correlations for the classical nearest-neighbor O(n) model on the square lattice, for every n ≥ 3 and every finite positive inverse temperature. The estimate is uniform over finite free-boundary subgraphs and bounded nonnegative edge strengths. This resolves positively the all-temperature exponential spin-decay conjecture for these models.
Result 215 · Probability and statistical mechanics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md