ADE Simple-Singularity Uniformization of Coordinate-Free Renormalized Tail Orbits:\ Vanishing-Cycle Packets, Picard-Lefschetz Stokes Groupoids, and Finite-Window Weyl Recovery

The preceding paper in this series completed the corank-one picture for coordinate-free renormalized tail orbits by resolving every bounded Morin tower of type \(A_k\). That left the next structural gap completely exposed: the orbit category still had no theorem for genuinely non-Morin simple collisions, hence no mechanism for corank-two or exceptional simple catastrophes. The present paper closes that gap inside the largest modality-zero class one can treat without leaving the simple-pole framework. We fix a finite family \( S⊂\{A_k,D_k,E_6,E_7,E_8\}\) of simple Thom-Arnold types and study compatible renormalized-tail hierarchies on compact ray bundles that admit local coordinate-free smooth simple-pole branch realizations with collision strata modeled by \( S\). For each \(Σ∈ S\) we choose a weighted-homogeneous core \(P_Σ\), a reduced Jacobi-monomial basis \(m_a\), and a distinguished Lefschetz-thimble basis. The first main theorem proves a full-\(n\) universal packet law: after removing spectator branches and renormalizing by the quasi-homogeneous factor \(n-γ_Σ\), the collision packet is governed on the natural control scales by a finite jet of the vector-valued canonical integral \[ I_Σ(u)=(∫_{ΓΣ,} eP_Σ(t)+∑_a u_a m_a(t)\,dt)₌₁μ_Σ. \] The law is uniform on bounded control boxes and contains the earlier \(A_k\)-packet theorem as the \(A\)-subtower. The second theorem upgrades local continuation to a global invariant. Sectorial asymptotics of the packet recover the chamberwise branch expansions, while wall-crossing acts by universal Picard-Lefschetz matrices. On an adapted cover these local changes of basis assemble into an ADE Stokes groupoid generated by branch permutations, residue multipliers, and type-\(Σ\) Picard-Lefschetz blocks. Its nonabelian {C}ech class is shown to be the exact obstruction to a global simple-singularity-resolved atlas on the chosen ray bundle. The third theorem is an inverse rigidity statement of Weyl type. From the packet germ together with its Picard-Lefschetz transport one recovers the Coxeter-Dynkin type, the tangent discriminant cone, the quasi-homogeneous scaling spectrum, and the reduced control jet up to the natural finite reflection symmetry and flat gauge. In particular, the local orbit germ distinguishes \(A\)-, \(D\)-, and \(E\)-regimes intrinsically. The fourth theorem is quantitative. For a fixed finite candidate family \( S\), finitely many scalar probes sampled on finitely many selected rays and a bounded window \(n=N,,N+J-1\) asymptotically classify the true ADE type and recover the leading control parameters and amplitudes with error \[ \!(N-1/h_Σ+δ_N+η_N) \] under the natural local injectivity, separation, and noise-floor hypotheses. The result is stated deliberately as a local finite-horizon asymptotic recovery theorem, not as a globally conditioned numerical algorithm under arbitrary noise. The scope is explicit. We treat coordinate-free smooth simple-pole branch families, a finite bounded simple-type list \( S\), and modality-zero collisions only. We do not claim modal singularities, higher-order poles, arbitrary singular varieties, or unrestricted global noise. Within that regime, however, the paper upgrades the caustic theory from the Morin tower to the full finite ADE simple-singularity level: universal packets, Picard-Lefschetz transport, exact global obstruction, Weyl-rigid inverse recovery, and finite-window diagnosis.