Caustic Uniformization of Coordinate-Free Renormalized Tail Orbits: Fold--Cusp Collision Normal Forms, Stokes-Filtered Monodromy, and Finite-Window Recovery

The preceding paper in this series gave a coordinate-free globalization theorem for smooth simple-pole branch families away from the phase-collision set. The remaining gap was structural, not cosmetic: when equal-support branches collide, the chamberwise oscillatory decomposition ceases to be uniform, branch amplitudes become singular, and the correct invariant is no longer a bare permutation--residue local system. One needs a caustic theory. This paper develops that theory for generic fold--cusp stratifications. We work with coordinate-free compatible renormalized-tail hierarchies that admit a finite clean family of smooth simple-pole branches with common support potential. Near a codimension-one fold wall we prove a full-\ (n\) Airy uniformization theorem: after subtracting spectator branches and renormalizing by \ (n^1/3\), the collision packet converges on the natural \ (n^-2/3\) scale to a universal Airy profile with holomorphic amplitude fields. Near a codimension-two cusp point we prove the corresponding Pearcey law on the \ ( (n^-1/2, n^-3/4) \) control scales. In both regimes the resolved profile is all-\ (n\), not merely subsequential, and the previous chamberwise branch laws are recovered as sectorial asymptotics of the catastrophe packet. Globalization changes accordingly. On an adapted good cover the local Airy, Pearcey, and regular branch bases glue by a matrix cocycle generated by branch permutations, constant residue multipliers, and universal catastrophe connection blocks. This defines a Stokes-filtered monodromy class in nonabelian Cech cohomology. We prove that this class is the exact obstruction to the existence of a global caustic-resolved smooth simple-pole atlas. Finally we derive a quantitative finite-window detector theorem. Using finitely many scalar probes on selected transverse rays and a bounded number of consecutive \ (n\) -samples, one asymptotically distinguishes the regular, fold, and cusp regimes and recovers the local control parameters with explicit \ (N^-1/3\) and \ (N^-1/4\) accuracy up to the measurement floor. The theorem is stated deliberately as a local finite-horizon asymptotic reconstruction result; it is not claimed as a globally conditioned arbitrary-noise algorithm. The scope is explicit. We treat coordinate-free smooth simple-pole branch families with generic \ (A₂\) and \ (A₃\) phase collisions. Higher Arnold types, degenerate Hessians, and unrestricted global noise conditioning remain outside the present paper. Within that regime, the main gap left by the previous coordinate-free geometrization is closed: phase-collision sets are no longer excluded, but are replaced by universal catastrophe packets, Stokes-filtered gluing data, and finite-window transition diagnostics.