A pure-gauge background in the ramp

The homogeneous solutions can be replaced by free solutions with compact Cauchy data. Choose their initial data equal to those of $u,v$ on a ball containing the initial causal shadow of $\mathrm{supp}w$, and multiply by a larger compact cutoff. Finite propagation preserves equality on a neighbourhood of $\mathrm{supp}w$. Since $Q_w$ is local and supported there, the value (eq:Dnonzero) is unchanged.

Together with $b=C[u]/2$ or $C[v]/2$ and zero scalar and ghost fields, these are configurations of the full free multiplet.