Vanishing of Hypercubic Anisotropy in the Scaling Limit for SU(2) Yang–Mills Schwinger Functions

This paper completes the restoration of continuous rotational symmetry in the continuum scaling limit of SU(2) lattice Yang–Mills theory. A companion work constructed subsequential continuum limits of smeared plaquette Schwinger functions and verified all Osterwalder–Schrader axioms except continuous rotation invariance. The limiting Schwinger functions were shown to be invariant under translations and the discrete hypercubic symmetry group H4, leaving the restoration of the full Euclidean rotation group O(4) as the remaining step. The present paper proves that the hypercubic anisotropy of the lattice theory vanishes in the continuum limit. Two complementary regimes are analysed. In the regime where the renormalised physical mass diverges along the scaling sequence, connected Schwinger functions vanish and O(4) invariance holds trivially. In the physically expected regime where the renormalised mass remains finite, we prove that lattice anisotropy is suppressed by O(a^2). The argument combines three ingredients: (i) a hypercubic harmonic decomposition that isolates the leading ℓ = 4 anisotropy channel, (ii) a momentum–space dispersion analysis of the Wilson lattice action forced by hypercubic symmetry, and (iii) uniform complex-momentum analyticity derived from exponential clustering via Paley–Wiener arguments. These results imply that any subsequential continuum limit satisfying the conditions established in the companion continuum-limit paper is invariant under the full Euclidean group O(4). Together with reflection positivity, clustering, and the other Osterwalder–Schrader axioms already verified, this completes the symmetry requirements needed for Osterwalder–Schrader reconstruction of a relativistic quantum field theory. The remaining open condition for obtaining a non-trivial Yang–Mills theory with finite mass gap is a matching upper bound on the lattice mass gap ensuring that the renormalised physical mass remains finite along the scaling sequence.