Proposition: Quantum compatibility

The operator $s_0$ preserves ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ and ${\mathcal{N}}_{\mathrm{sol}}$. It is a derivation of ${\star }_H$ precisely when

The condition holds for the frozen matrix. Thus $s_0$ descends to a nilpotent derivation of ${\mathcal{E}}_H$.

Proof. Differential operators do not enlarge wavefront sets. On the shell, $\mathcal{R}c$ is a wave solution and $D\mathcal{R}c=\frac{1}{2}\square c=0$; also $\square b=0$. Hence the odd linear variation is tangent to the shell and preserves its vanishing ideal.

Commuting the linear symbol variation through a contraction gives $\delta H$. The quantum Leibniz defect vanishes if $\delta H=0$. Conversely, applying the defect to two linear smearings detects every component of $\delta H$ as a scalar, proving necessity.

For the frozen data, both nontrivial ghost orderings cancel:

The pairs $(b,\bar{c})$ and $(\bar{c},b)$ vanish because $H_{bb}=0$. Every remaining pair has only zero contraction blocks.