Proposition: Gauge-fixing normalization

The gauge-fixing expression is

Eliminating $b$ gives

The reduced metric Hessian is exactly ${\mathsf{P}}_{\mathrm{gr}}$.

Proof. The odd Leibniz rule produces the minus sign in the ghost term. Equation (eq:operatoridentities) gives $s_0(\mathsf{C}r)=\square c$. The algebraic $b$-equation gives the stated value. The Hessian of $\frac{1}{4}\int (\mathsf{C}r{)}^2$ is $\frac{1}{2}{\mathsf{C}}^{\ast }\mathsf{C}$, which added to $A$ gives ${\mathsf{P}}_{\mathrm{gr}}$.