Corollary: No general compact gauge action on the free shell

The prescription

does not define a tangent action on the unchanged massive free solution space for all compact smooth $\xi$. This conclusion does not depend on a proposed metric transformation.

Proof. The tangent space of the linear solution space is itself. Tangency at every solution is condition (i) above. A compactly supported Killing field vanishes on an open set and is therefore zero.