Universal-Cover Uniformization of Renormalized Tail Hierarchies from Graph-Gauge Atlases: Minimal Cyclic Descent, Orbifold Residues, and Deck-Rigid Recovery

The preceding globalization theorem in this series identified the exact obstruction to a global single-valued graph-gauge meromorphic realization of a compatible renormalized tail hierarchy: a flat residue line bundle \ _ _,: ₁ (_) ^, \ on the support manifold determined by the orbit support potential. The present paper converts that obstruction theory into an exact uniformization and classification theorem. Our first main result proves that every locally graph-gauge simple-pole hierarchy becomes globally single-valued on the universal cover. The pullback of _ is canonically trivial, the local Gaussian residue densities glue to a global analytic density, and all higher orbit transport fields are universally forced by (p, ). Our second main result is a sharp finite-monodromy criterion: the universal-cover uniformization descends to a connected finite cover if and only if is finite. In that case there is a canonical minimal connected cover \ p_: __ \ of degree ||, and because finite subgroups of ^ are cyclic, this minimal cover is automatically cyclic. Our third main result gives an exact orbifold descent theorem. Gauge classes of finite-monodromy hierarchies with fixed support potential and residue character are in natural bijection with projective classes of nonvanishing deck-equivariant Gaussian residue densities on the minimal cyclic cover. Equivalently, finite-monodromy hierarchies are exactly the descents of global graph-gauge meromorphic models on that cover. This yields a deck-rigid Torelli principle: within the finite-monodromy class, the hierarchy is completely determined by, the character, and the projective lifted density. Our fourth main result is quantitative. From finitely many overlap residuals on a good cone cover and finite-horizon orbit data of accuracy (N^-1), one can detect rational monodromy phases up to a prescribed maximal order, recover the exact minimal cyclic degree under a separation hypothesis, and build the lifted gauge on the recovered cover with the same asymptotic accuracy. The scope is deliberately precise. The paper concerns hierarchies whose local restrictions belong to the graph-gauge simple-pole image. It is not a theorem for arbitrary smooth hypersurface singularities, nor a globally conditioned numerical algorithm under arbitrary noise. Within this substantial class, however, the paper upgrades global obstruction theory into exact uniformization, cyclic descent, and complete finite-monodromy classification.