Lemma: Compact homogeneous ghosts vanish

A smooth solution of $\square c=0$, compactly supported inside an open time slab, is zero. The statement holds coefficientwise for a finite Grassmann algebra.

Proof. A Cauchy surface sufficiently near the lower end of the slab lies before the compact support. Both Cauchy data vanish there. Homogeneous uniqueness gives zero throughout the slab BGP.