The preceding paper in this series identified the local graph-gauge meromorphic image inside the intrinsic compatible hierarchy of renormalized tail data. The natural next question is global: if every patch of a connected ray domain is locally graph-gauge geometric, when do those local realizations glue to a genuine single-valued meromorphic singularity, and what remains when they do not? This paper gives an exact answer. We work with a compatible hierarchy on a connected positive ray manifold whose restriction to each member of a good cone cover belongs to the local graph-gauge simple-pole image. The support potential then determines canonical inverse-Legendre charts and hence a global analytic smooth-critical manifold _; in other words, denominator geometry always glues. The only global obstruction lies in the residue amplitude. From the order-zero local transport we build Gaussian residue densities \ dᵢ (x) =e^ᵢ (x) \, (² uᵢ (x) ) ^-1/2dx \ on each local graph chart. Our first main theorem proves that on every connected overlap these densities differ by a constant multiplier, \ dⱼ=e^c₈₉ (₈₉) _*dᵢ, c₈₉, \ where ₈₉ is the canonical overlap diffeomorphism induced by. The constants c₈₉ form an additive Cech 1-cocycle, equivalently a flat multiplicative cocycle g₈₉=e^c₈₉^. Our second main theorem states that a global single-valued meromorphic realization exists if and only if the class c H¹ (_;) vanishes. When c0, there is still a canonical twisted realization: the local residue amplitudes glue as a section of a flat line bundle c _, and all higher transport fields are universally forced by (, c, ). The third main theorem identifies the associated holonomy representation \ c: ₁ (_) ^, c ([) =₄ gₑ, \] as an exact orbit invariant recoverable from renormalized tail data alone. Thus the orbit theory detects not only local singular geometry but also global residue monodromy. The fourth main theorem gives a quantitative finite-cover detector. From finitely many local probes, finitely many overlap residuals, and a finite cycle basis of the nerve graph one reconstructs the obstruction cocycle and its monodromy with (N^-1) local finite-horizon accuracy, together with a constructive gauge-correction scheme whenever the obstruction vanishes to that precision. The scope is deliberately precise: the globalization theorem concerns atlases of local graph-gauge simple-pole patches; it is not a theorem for arbitrary smooth hypersurface singularities. Within this substantial class, however, the paper closes the gap between local geometric image theory and global meromorphic realization.
Mohammad Abu-Ghuwaleh (Tue,) studied this question.