Proposition: Nonemptiness and inclusion

One has

Proof. Take $\gamma =b=0$, with arbitrary free scalar and ghost solutions. These configurations satisfy the shell relation and belong to ${\mathcal{C}}_W$. For $u\in {\mathcal{C}}_W$ and $\xi \in {\mathfrak{g}}_W$, the compact tensor ${\mathcal{L}}_{\xi }{\beta }_{f,g}[\varphi ]$ is supported inside $W$. Equation (eq:auxiliary) therefore gives ${\mathfrak{B}}_{\xi }(u)=0$.