Universal Morin Catastrophe Uniformization of Coordinate-Free Renormalized Tail Orbits: \(A_k\)-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 \(A_k\) for \(2≤ k≤ K\). For each \(k\) we prove a full-\(n\) universal packet theorem: after subtracting spectator branches and renormalizing by \(n1/(k+1)\), the collision packet is governed on the natural control scales \[ x_j~ n-(k+1-j)/(k+1), 1≤ j≤ k-1, \] by a finite jet of the generalized Airy canonical integral \[ _k(u)=∫Γ_0exp\!({τᵏ⁺¹}{k+1}+∑ⱼ₌₁ᵏ⁻¹u_jτ^j)\,dτ. \] 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 \(A_k\)-packets. The continuation across anti-Stokes walls is controlled by universal \(A_k\)-Stokes matrices acting on Lefschetz-thimble bases. On an adapted cover this yields a nonabelian {C}ech class with values in a finitely generated {\(A≤ K\)-Stokes groupoid}, 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 \[ \!(N-1/(k+1)+δ_N+η_N) \] 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 \(A≤ K\)-tower: universal packets, Stokes continuation, exact global obstruction, and finite-window order classification.