Beyond Uniform Lipschitzness: Pathwise Jacobian Stability for Flow Matching Samplers

Flow matching samplers integrate a learned velocity field. Existing error bounds often use its largest Jacobian norm over all states, even when generated paths visit only a small part of the space. We develop a framework that weights each velocity error and solver defect by the Jacobian growth that follows it between the compared paths. An exact error identity yields signed segment and tube bounds, with classical uniform Lipschitz estimates as a special case. In an explicit conditional flow matching example with nonzero velocity error, the signed bound equals the terminal Wasserstein error, while the classical bound is more than 500 times larger. We also determine what the population loss guarantees without a uniform Jacobian bound on the learned field. For a fixed population flow and a fixed linear growth class, the sharp worst case rate in state dimension $d$ is $\sqrt{\eta }[1+\log (1/\eta ){]}^{(d+4)/4}$, where $\eta$ is the population velocity loss. Polynomial state weights reduce the logarithmic factor. Tests on trained samplers connect path growth to terminal sensitivity and use later growth to select paths for numerical refinement. The same flow derivative gives a weighted Poincar\'e inequality for the output law.