Lemma: Free off-shell identities

The differential $s_0$ is nilpotent off shell. It preserves $I_{\mathrm{free}}$ modulo integrated divergences. Algebraic elimination of $b$ gives

and hence the metric operator ${\mathsf{P}}_{\mathrm{gr}}$.

Proof. The square vanishes on each independent generator in (eq:s0); the square of an odd derivation is an even derivation. The metric variation vanishes by $AK=0$, the scalar is inert, and ${s_0}^2{\Psi }_0=0$.

Completing the square gives the auxiliary formula. Equation (eq:A) identifies the reduced metric operator.