Constructive Renormalization and Continuum Limit for Yang-Mills on S³

We adapt Balaban's constructive renormalization group program (1984–89) for pure Yang-Mills theory from flat tori to the three-sphere S³R, and prove that on this compact background all six structural difficulties of the flat-space analysis are resolved. The proper-time covariance decomposition uses the exactly known coexact spectrum λₖ = (k+1) ²/R²; the 600-cell polytope provides a natural blocking hierarchy with icosahedral symmetry; and the first perturbative RG step yields the correct one-loop coefficient b₀ = 22/3 for SU (2) from exact spectral sums. The key new result is that the Gribov diameter on S³ satisfies d·R = 9√3/ (2g), implying that the large-field region is empty within the Gribov horizon at physical coupling. Balaban's large-field analysis (Papers 11–12), which is essential on flat tori where field configurations can be arbitrarily large, becomes a one-line bound on S³. The one-step RG contraction is proven and iterated through all scales via the Bauerschmidt–Brydges–Slade mechanism. The sequence of lattice measures converges as a → 0 to a unique continuum probability measure satisfying all Osterwalder–Schrader axioms, with the mass gap established by the companion paper A. 1, 077 tests verify the RG infrastructure at 100% pass rate.