Proposition: Smoothness of the retained defect

The defect $\mathcal{D}$ is a smooth mixed bisolution:

Its vanishing is not implied by the canonical intertwining identities.

Proof. Using (eq:realpart), its imaginary part is

by (eq:intertwine). Hence $\mathcal{D}$ is real in a real trivialization.

Differential application to the contractions gives

A real distribution has sign-symmetric wavefront set. Consequently its wavefront relation is also contained in ${\mathcal{N}}_{-}$. The two nonzero frequency cones are disjoint. Thus ${\mathrm{WF}}^{\prime }(\mathcal{D})=\varnothing$, proving smoothness.

The bisolution equations follow from $\mathcal{B}K=-r{\mathcal{F}}^{†}Q$, ${\mathcal{F}}^{†}Q=-r^{-1}\mathcal{B}K$, and the field equations for the contractions. The causal identities remove the imaginary part, not the remaining smooth real part.