Endpoint Singularity and Temporal Heterogeneity in Flow-Matching Transport

Flow matching learns a time-dependent velocity field transporting a source distribution toward the data distribution. We study the endpoint geometry of this transport. A lower-dimensional target forces the endpoint flow map to lose rank. In a quadratic Hamilton–Jacobi formulation, this appears through a Riccati equation for the velocity Jacobian whose critical eigenvalue corresponds to endpoint rank loss. We further confirm this criticality empirically: the divergence of a single trained model grows sharply as $t\to 1$. Fine-tuning modifies the learned velocity field and can be viewed as a perturbation of it. We measure the temporal profile of this perturbation and find that it is non-uniform in flow time. We use this structure to define a peak-based query distribution for dynamics-level watermark detection, in which the query budget is concentrated on flow times where the watermark signal has larger energy. On watermarked flow-matching models, the peak-based distribution is more robust to fine-tuning than uniform and fixed-mixture sampling, retaining higher detection accuracy at intermediate fine-tuning steps.