Validity-Aware Flow Matching on Lie Groups: What Transfers to Target-Free Generators?

Flow matching learns a marginal vector field from conditional training paths, yet many notions of validity concern the trajectory itself. Recent methods improve those conditional paths through geometry-aware or cost-aware interpolation, or impose constraints during fitting and inference. None of them settles which path properties survive marginalisation into a target-free generator. We study that gap on compact Lie groups, accounting for population marginalisation, finite-capacity regression and discrete sampling. At the population optimum, conditional flow matching preserves every one-time marginal, including terminal validity under state-only criteria, and expected additive state-time costs. The path law is another matter. Conditional paths and the population flow they induce can agree on every one time marginal and still differ in region hitting, path length and endpoint coupling. Integrated cost alone cannot control endpoint violations. Control becomes available once the deployed path satisfies a temporal modulus, of the kind a speed cap or a pathwise kinetic-energy bound supplies. We trace these effects on tori, rotation groups and molecular conformations using bounded, endpoint-pinned Lie-group paths with exact left-trivialised velocities. The experiments separate endpoint coupling, path design, regression and numerical sampling, and show where a conditional-path improvement survives deployment and where a validity claim needs evidence from the deployed generator itself.