Preparatory Renormalization Estimates 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 — including the large-field problem that consumes approximately 100 pages of Balaban's Papers 11–12. The proper-time covariance decomposition is constructed using 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 in the 9-DOF truncation satisfies d·R = 9√3/ (2g), which implies that the large-field region is empty within the Gribov region at physical coupling. Consequently, Balaban's large-field analysis reduces to a one-line bound on S³, and the one-step RG contraction is proven. 1077 tests verify the RG infrastructure at 100% pass rate.