Theorem: Plateau decision for the named coefficient

Let $\xi \in {\mathfrak{g}}_U$ and $P_m\varphi =0$.

1. Always: ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}=-{\mathcal{A}}_1(K{\rho }_{\xi }[\gamma ];\varphi )+\int ({\mathcal{L}}_{\xi }{\beta }_{f,g}[\varphi ])\cdot A\gamma$, with the companion given by (eq:companion).

2. Under (CP): ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}=\int ({\mathcal{L}}_{\xi }{\beta }_{f,g}[\varphi ])\cdot A\gamma$.

3. Under (CP) and $A\gamma =0$ on $U$:

   $$\boxed{{\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}=0.}$$

4. Without (CP) the companion is the explicit remainder (eq:companion), supported as in (eq:companionsupport) inside $J^{+}(\overline{U})$, with no ramp-layer contribution. Without $A\gamma =0$ on $U$ the Einstein row is the explicit remainder (eq:einsteinrow).

Proof. (a) is Theorem thm:assembly with Propositions prop:companion and prop:einstein. (b) and (c) apply the two vanishing statements. (d) restates the displays.