Lemma: Pairing through the inverse

For every smooth symmetric contravariant density $w$ on the slab,

In particular ${\mathcal{C}}_{3,\mathrm{D}}(f,g)[\gamma ,\varphi ]=\int {\beta }_{f,g}[\varphi ]\cdot {\mathcal{J}}_{\mathrm{D}}[\gamma ]$.

Proof. By (eq:alphabeta) and ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{E}_0^{\mathrm{R}}$, ${\mathcal{A}}_1({G_{\mathrm{gr}}}^{\mathrm{R}}w)=-\int \alpha \cdot \mathcal{P}{E}_0^{\mathrm{R}}w=-\int (\mathcal{P}\alpha )\cdot {E_0}^{\mathrm{R}}w$, since $\mathcal{P}$ is self-adjoint for the flat pairing. The compact smooth test $\mathcal{P}\alpha$ transposes through the zero-data inverse with no boundary term, because ${E_0}^{\mathrm{R}}w$ has zero data at $t=0$ and the advanced function vanishes near $t=T$ XA (Lemma on transposition). This gives $-\int {E_0}^{\mathrm{A}}(\mathcal{P}\alpha )\cdot w$. The last assertion is (eq:CD) with $w={\mathcal{J}}_{\mathrm{D}}[\gamma ]$; it is XII-D's derived root identity before the collar is inserted.