Corollary: No retarded replacement of the original rods

The original rod inverse has instantaneous off-diagonal spatial support and temporal conormal singularities. It is not one of the causal inverses in Theorem thm:full-green.

Proof. Equation (eq:spkernel) is nonzero for arbitrary distinct spatial endpoints on the same time slice, whereas distinct same-time spatial points are spacelike for $\bar{g}$. A retarded or advanced bulk kernel vanishes there.