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.

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.

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.

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.

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.