From Lattice Mass Gap to Continuum SU(2) Yang–Mills Theory: O(4) Restoration and Osterwalder–Schrader Reconstruction

This paper develops the continuum-limit step in the Yang–Mills mass gap program for four-dimensional SU(2) gauge theory. Starting from a lattice Yang–Mills theory with a positive mass gap and exponential clustering, we construct a scaling family of lattice theories indexed by lattice spacing a → 0 and prove that the associated Schwinger functions admit subsequential limits. Using Dobrushin–Shlosman complete analyticity, we show that the lattice two-point function is analytic in a complex momentum strip, which allows the restoration of Euclidean rotational symmetry to be reduced to the suppression of finitely many anisotropy coefficients. The central technical mechanism proves that the self-energy anisotropy is O(a²). This is obtained through a per-polymer estimate in the cluster expansion, showing that anisotropy contributions from lattice artifacts remain O(a²) and do not accumulate in the polymer sum. This yields restoration of O(4) rotational symmetry in the continuum limit. Using these properties together with reflection positivity and exponential clustering, we apply the Osterwalder–Schrader reconstruction theorem to obtain a relativistic quantum field theory with a positive mass gap. All results are conditional on weak-coupling hypotheses (WC1–WC3) governing the lattice Yang–Mills measure. Under these hypotheses, the paper proves that the lattice mass-gap framework produces a continuum SU(2) Yang–Mills quantum field theory satisfying the Osterwalder–Schrader axioms with a positive mass gap.