Lemma: Reduction to one kernel

If $F\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ satisfies $F{|}_{{\mathcal{C}}_{\mathrm{sol}}}={\mathcal{C}}_3(f,g){|}_{{\mathcal{C}}_{\mathrm{sol}}}$, then $[u{]}_{\mathrm{sol}}$ meets ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$. Conversely, if $v\in [u{]}_{\mathrm{sol}}$ satisfies (eq:mc), then $F=⟨v,\gamma \otimes \gamma \otimes \varphi ⟩\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ and $F{|}_{{\mathcal{C}}_{\mathrm{sol}}}={\mathcal{C}}_3(f,g){|}_{{\mathcal{C}}_{\mathrm{sol}}}$.

Proof. Restrict $F$ to configurations in ${\mathcal{C}}_{\mathrm{sol}}$ with $c=\bar{c}=0$, which is allowed since zero is a free ghost; ${\mathcal{C}}_3$ has no ghost dependence, so the ghost-free part of $F$ already reproduces it. On ${\mathcal{C}}_{\mathrm{sol}}$, substitute $b=D\gamma$ in every $b$-slot: each kernel becomes $(\cdots \otimes D^{\mathsf{T}}\otimes \cdots )u_n$, compact, with wavefront set not enlarged by a differential operator Hormander ((8.1.11)), hence still satisfying (eq:mc). The shell is a linear space closed under the independent scalings $\gamma \mapsto s\gamma$, $\varphi \mapsto r\varphi$; comparing the coefficient of $s^{2}r$ isolates a single symmetric kernel $v$ of slot type $(\gamma ,\gamma ,\varphi )$ with $⟨v,\gamma \otimes \gamma \otimes \varphi ⟩={\mathcal{C}}_3(f,g)[\gamma ,\varphi ]$ on the shell. Polarization gives $v\in [u{]}_{\mathrm{sol}}$. The converse is Definition def:domain.