Equivariance Begins at the Source: Characterizing Compatible Sources for Generative Robot Policies

Equivariant transport alone does not guarantee an equivariant manipulation policy. A compatible source distribution–one that transforms consistently with the observation–is required. We show that, under an invertible equivariant flow, the policy inherits the source's equivariance error exactly and therefore cannot correct an incompatible source. In particular, the commonly used isotropic Gaussian centered at the world-frame origin is not translation-equivariant—the KL divergence between the source and its translate grows quadratically with translation distance. We further characterize all compatible conditional sources through a unique frame-relative normal form. For robot actions, this characterization shows that translation requires observation dependence, whereas normalized Haar measure is the unique observation-independent rotational source. Building on these results, we propose GARLIC, an $SE(3)$-equivariant flow-matching policy that anchors its translational source at the end-effector, samples rotations from the normalized Haar measure, and uses an equivariant geometric-algebra transformer for transport. On RLBench, GARLIC improves task success and out-of-distribution generalization over existing baselines, demonstrates greater parameter efficiency, and transfers to real-robot manipulation.