Universal Morin Catastrophe Uniformization of Coordinate-Free Renormalized Tail Orbits: \ (Aₖ\) -Packets, Stokes Groupoids, and Finite-Window Classification

The preceding paper in this series resolved generic fold and cusp collisions of coordinate-free renormalized tail orbits by Airy and Pearcey packets. That closed the first caustic gap, but only at the first two Arnold levels. The structural next question is unavoidable: what happens at higher generic \ (A\) -type collisions, and can one still keep the theory global, coordinate-free, and diagnostically finite? This paper answers that question for generic Morin towers of bounded order. Fix \ (K 3\). We study compatible renormalized-tail hierarchies on compact ray bundles that admit local smooth simple-pole branch realizations and whose collision strata are of Morin type \ (Aₖ\) for \ (2 k K\). For each \ (k\) we prove a full-\ (n\) universal packet theorem: after subtracting spectator branches and renormalizing by \ (n^1/ (k+1) \), the collision packet is governed on the natural control scales \ xⱼ n^- (k+1-j) / (k+1), 1 j k-1, \ by a finite jet of the generalized Airy canonical integral \ ₖ (u) =䃐\! (^k+1k+1+₉=₁^k-1uⱼʲ) \, d. \ The law is uniform on bounded control boxes, holds for every \ (n\), and contains the fold/Airy and cusp/Pearcey theories as the cases \ (k=2\) and \ (k=3\). We then show that the chamberwise branch expansions of the previous papers re-emerge as sectorial asymptotics of these \ (Aₖ\) -packets. The continuation across anti-Stokes walls is controlled by universal \ (Aₖ\) -Stokes matrices acting on Lefschetz-thimble bases. On an adapted cover this yields a nonabelian Cech class with values in a finitely generated \ (A ₊\) -ₒₓ₎₊₄ₒ ₆ₑ₎ₔ₏₎₈₃, obtained by adjoining the new catastrophe blocks to the permutation--residue gluing data of the collision-free theory. We prove that this class is the exact obstruction to the existence of a global Morin-resolved smooth simple-pole atlas. Finally we derive a quantitative finite-window classification theorem. Using finitely many scalar probes on finitely many selected rays and a bounded number of consecutive \ (n\) -samples, one asymptotically identifies the local Morin order \ (k\0, 2, , K\\), separates it from all lower orders, and recovers the control parameters and leading amplitudes with error \ \! (N^-1/ (k+1) +N+N) \ under the natural local injectivity and noise-floor assumptions. The theorem is stated deliberately as a local finite-horizon asymptotic recovery result, not as a globally conditioned algorithm under arbitrary noise. The scope is explicit. We treat coordinate-free smooth simple-pole branch families with a fixed finite upper Morin order \ (K\). We do not claim a theory for \ (D\) - or \ (E\) -type catastrophes, higher-order poles, degenerate normal forms, or unrestricted global noise. Within that regime, however, the paper upgrades the caustic theory from the fold--cusp truncation to a complete \ (A ₊\) -tower: universal packets, Stokes continuation, exact global obstruction, and finite-window order classification.