Proposition: Admitted flat-gauge residual

The resulting expression satisfies

where $F^{(0)}=-\frac{1}{4}{\mathsf{P}}_{\mathrm{tr}}KC$. The residual is admitted. Its class modulo $\mathcal{J}$ is not decided here.

Proof. By (eq:cov2) and (eq:P),

The first term gives the same cancellation as in Theorem thm:main, with its same compact equation remainder. The second gives (eq:flat).

Its local differential source operator has coefficients supported in $\mathrm{supp}\xi$. The kernel is obtained by applying its formal transpose to $N_0$ in the second argument. It therefore has compact support in $C\times \mathrm{supp}\xi$ and wavefront in $Z_2$.