Caustic Spectroscopy: Identifiability and Acquisition Limits of Topological Event Timing

Caustics carry the shape of a transparent surface, but their intensities are unstable under exposure and pose. We study a sparser, analytically tractable observable: sweep one knob of the illumination and record only the control values at which the caustic changes topology, a catastrophe spectrum. Isolated nondegenerate events are Morse critical points of the fold function, so their thresholds move with a closed-form differential, and the rank of the resulting event Jacobian certifies local identifiability: outright where it is full, which needs screen positions, and up to the sweep's symmetry group for thresholds alone, whose Jacobian that symmetry keeps below full rank. Paraxially the thresholds are the critical values of the surface's curvature fields. On simulated glass reliefs the differential matches finite differences to $9\times 10^{-11}$, and the singular values order a trained network's per-direction error across twelve nominal reliefs after one held-out calibration. The analysis also marks its own limits, and we measure each. Umbilic events are degenerate and leave the signature unchanged. Nine events per relief let a trained inverse recover half the shape variance up to rotation, measured against the best data-free predictor and carried mostly by their counts and types. Adaptive bisection localizes matched events eighteen times more precisely than a uniform sweep, but only on the idealized signature: injecting spurious transitions into it reproduces the end-to-end collapse, which is complete by a precision of $0.67$. All evidence is simulated.