Curvature and Sample-Level Path Dependence in Locally Optimal Conditional Transport

Conditional generators are often used to follow the same generated sample as its controls change, with its latent seed held fixed. Two edits can nevertheless produce the same final distribution in either order while sending that sample to different endpoints, a discrepancy that marginal metrics cannot see. We model each infinitesimal edit by its least-energy optimal transport (OT) velocity and define curvature as the failure of the two local OT motions to commute. Under our assumptions, zero curvature is equivalent to the existence of a single path-independent particle motion realizing both local OT fields. When curvature is nonzero, any shared-particle motion matching all marginals incurs positive excess action relative to the direction-wise OT benchmark, determined exactly by its integrated squared velocity mismatch from those fields. For a square loop of side $\varepsilon$, the paired-sample root-mean-square endpoint discrepancy satisfies $D_2(\varepsilon)=\varepsilon^2\|\Omega\|_{L^2}+O(\varepsilon^3)$, making curvature measurable from two opposite edit orders. A closed-form torus example gives the predicted scaling, finite-sample Fourier experiments recover the curvature signal, and a household-load study applies the audit to constrained profiles.