Proposition: Bulk and multiplet wavefront bounds

For $P=\mathcal{L},\mathcal{B},Q$,

Every component of the full Green matrices has the same corresponding upper bound.

Proof. Away from coincidence, the inverse equation is homogeneous. The normally hyperbolic principal symbol gives null propagation. The diagonal inverse singularity is conormal to the diagonal; the retarded or advanced support chooses the corresponding directed null relation. This is the bundle-valued distinguished-parametrix construction IS.

In the causal difference the inhomogeneous diagonal source cancels. Elliptic regularity excludes non-null covectors. Multiplication by ${\mathcal{A}}^{-1}$, and application of the differential operators in the full matrices, do not enlarge the wavefront set.