Proposition: Compact relative ghosts are not free ghost solutions

A smooth spacetime-compact ghost supported strictly above $\Sigma$ and satisfying $\square c=0$ is zero. Thus replacing the relative ghosts of (eq:relative-s) by nonzero homogeneous free ghosts is not an identification of domains.

Proof. The compact future support gives zero Cauchy data on $\Sigma$. Homogeneous Cauchy uniqueness gives $c=0$.