Coordinate-Free Global Geometrization of Renormalized Tail Hierarchies: Smooth Hypersurface Atlases, Branch-Permutation Monodromy, and Finite-Ray Interference Recovery

The preceding paper in this series solved the local-to-global problem for graph-gauge simple-pole atlases. Graph gauges, however, are coordinate artifacts, and they hide the first genuinely new global phenomenon beyond the single-branch regime: several smooth critical branches can contribute on the same exponential scale and therefore interfere. This paper gives a coordinate-free extension together with a clean multi-branch theory. We work with compatible renormalized-tail hierarchies on a connected positive ray manifold. First we prove an intrinsic local image theorem: once a hierarchy is transported into any local smooth normal chart, the graph-gauge obstruction tensors of the previous paper vanish if and only if the hierarchy comes from a coordinate-free smooth simple-pole hypersurface patch. Thus local realizability is independent of the chosen graph representation. Second, in the single-branch case the support potential determines a canonical coordinate-free smooth-critical atlas and conormal line bundle. The only global obstruction remains the residue cocycle; equivalently, a global single-valued realization exists exactly when a flat residue line bundle is trivial. The main new result concerns clean finite families of equal-support branches. Away from the phase-collision set, the whole renormalized tail orbit satisfies a full-\ (n\) interference law \ Tₙ^ (w) ==₁ᵐ _ () ⁿ (G_ (, w) +ₑ=₁^M-1n^-rG, ₑ (, w) ) + (n^-M), |_ () |=1, \ uniformly on compact ray bundles. On overlaps the ordered branch data glue only up to a locally constant permutation and constant residue multipliers, producing a nonabelian Cech cocycle with values in \ (Sₘ (^) ᵐ\). An ordered global meromorphic realization exists if and only if this class vanishes; otherwise one obtains a canonical twisted branch local system. Finally, under a uniform phase-gap hypothesis on a compact ray bundle, \ (2m\) consecutive scalar probes recover the local phase set and amplitudes with error \ ( (N+N^-1) \). Patching these local reconstructions over a finite good cover yields a quantitative finite-ray / finite-cover detector for branch-permutation monodromy and residue holonomy. The scope is explicit. We treat coordinate-free smooth simple-pole patches and clean equal-support branch families, not arbitrary singular varieties, caustic collisions, or globally conditioned algorithms under unrestricted noise. Within that precise regime, the paper replaces graph-gauge atlases by intrinsic smooth hypersurface geometry and identifies the exact new global invariant: a semidirect product of permutation monodromy and residue local systems.