Second-Order Geometry of Generative Paths: Fragility and Targeted Refinement

Generative probability paths can agree at their endpoints and in first-order transport summaries yet react differently to coarse sampling or small perturbations. We characterize the missing information at second order. For a dynamic path $({\mu }_t,v_t)$, a metric-compatible derivative on the moving $L^2({\mu }_t)$ bundle yields an exact Frenet decomposition of material acceleration into schedule-dependent speed variation and reparameterization-invariant bending. A first-order blindness theorem shows that endpoints, path length, and kinetic action may all remain fixed while bending energy becomes arbitrarily large, excluding any uniform upper certificate for bending energy based only on those summaries. A discrete-to-continuum error analysis then yields a fixed-budget evaluation density, while a spectral law identifies locally most-expanding perturbation directions. We also establish consistency of the material finite-difference estimator used at scale. Controlled constant-speed flows isolate bending; in CIFAR-10 Rectified Flow++, acceleration risk adds information beyond velocity and path length, and an independent 20,480-sample confirmation shows that it places the same refinement budget more effectively than random, uniform, or velocity allocation. In image editing, amplifying leading expansion modes increases non-target DINOv2, LPIPS, and RMSE drift relative to an orthogonal placebo. The results support two operational second-order risk channels associated with realized generative fragility, rather than a universal quality score or a faster sampler.