Canonical O(3) continuum limit and exact O(4) mass asymptotics

What OpenAI claims

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_lat(β) ∼ 32e^π/4-1/2√β e^-πβ. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor O(n) models with n ≥ 3 at every positive temperature. (Lean)

Papers (5)

1. The canonical massive continuum limit of the two-dimensional O(3) model

   4 October 2026 · Preprint, not peer reviewed · PDF

   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.

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

   4 October 2026 · Preprint, not peer reviewed · PDF

   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.

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

   5 October 2026 · Preprint, not peer reviewed · PDF

   For the nearest-neighbor O(4) model on the square lattice at inverse temperature β, we prove the exact low-temperature asymptotic

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

   23 September 2026 · Preprint, not peer reviewed · PDF

   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 √β e^-πβ. 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.

5. Exponential decay in two-dimensional classical O(n) models

   23 September 2026 · Preprint, not peer reviewed · PDF

   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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.