When Is a Conditional Path Already Optimal? A Criterion for Flow Matching on Lie Groups

Under kinetic optimality, flow matching does not leave the conditional path free: that path is the geodesic of the metric the loss induces. On $\mathrm{SE}(3)$, where models now generate protein backbones, molecular poses and robot actions, that geodesic generally requires a boundary-value solve at a cost comparable to a training step, so implementations substitute one of three closed-form constructions. Yet how far the substitute sits from the curve it replaces has not been reported. We give that deviation in closed form for group interpolation: second order in the endpoint amplitude, with leading term $\frac{\tau (1-\tau )}{2}\Lambda (u)$. A deviation alone does not identify which constructions need changing, so we also prove a criterion for when it vanishes. A factorized construction solves the geodesic equation of a block metric $\mathrm{diag}(J,\beta I)$ exactly when the endpoint rotation axis is an eigenvector of $J$. The condition is readable off the loss, and every body-frame-invariant loss with a block-diagonal Gram operator satisfies it by Schur's lemma. For a given axis, what decides the matter is alignment rather than anisotropy: rotating a metric's eigenbasis at fixed eigenvalues takes the construction from solving the geodesic equation exactly to departing from it. Reading six released loss configurations at source level, in the coordinates where each loss is computed, we find four that already satisfy the criterion, without having been built for it. Where it holds, the departure stays at the solver floor and does not grow with amplitude. Where it fails, the construction sits at second order from the geodesic, and each level of a closed-form correction removes one further order. Whether the departure is zero is therefore decidable before any training. On a robot action head the criterion accepts, training instead on the geodesic of a frozen approximation to its metric changes the error on training episodes by less than $0.3$ percent, although the two paths differ by $3.7$ percent of that error in translation and $5.9$ percent in rotation.