Straightness and Intrinsic Geometry in OT Flow Matching

Optimal transport (OT) flow matching promotes straight ambient trajectories, but exact ambient fitting need not recover intrinsic transport velocities on curved supports. We quantify this mismatch and examine whether tangent-projected straightness can detect it. For short circle arcs and full-dimensional product spherical bands, we prove shared ambient and intrinsic optimality and derive exact midpoint velocity-gap laws. The gap determines the leading excess kinetic action of the projected path, despite matching endpoint laws. A structured product-torus extension accommodates correlated source densities. Exact calibrations show that the diagnostic can vanish despite a positive intrinsic gap, while nonlinear representations can produce a positive score on a flat path. Separate CIFAR-10 experiments evaluate training behavior and sample selection without intrinsic ground truth. Across two scoring representations and three quality metrics, lower-score retention performs worse on average than matched random retention; reversing DINOv2 rankings yields metric-dependent gains. Projector ablations preserve this selection pattern. An exact finite-pool MMD identity explains why favorable bucket association need not improve the retained distribution.