Proposition: Compact relative ghosts do not solve this problem

A smooth ghost coefficient compactly supported in spacetime and satisfying $\square c=0$ on the completion is zero. Consequently restricting ghosts to ${C_c}^{\infty }(U)$ does not produce a nontrivial interacting action on the unrestricted free ghost carrier.

Proof. A Cauchy surface earlier than the compact support has zero position and velocity data. Wave uniqueness gives the zero solution.