Proposition: Cancellation on the named configuration subclass

Suppose $\Phi \in {\mathcal{C}}_{\mathrm{sol}}$, (CP) holds, and

Then ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}=0$. For a free-shell configuration, (eq:local-closed) is equivalent to $A\gamma =0$ on that neighbourhood. In particular, constant $b$ gives zero auxiliary pairing.

Proof. Under (CP), the companion vanishes. By compact integration by parts,

The compact tensor ${\mathcal{L}}_{\xi }\beta$ is supported in $S_{\xi }$. The result follows from (eq:local-closed) and Lemma lem:shell. A constant vector field has $D^{\ast }b=0$.