Proposition: Unchanged free handoff

The quadratic gravitational and gauge-fixing coefficients of $S_{\mathrm{D},\mathrm{gf}}$ are the XI-A/XI-B coefficients. Their metric operator is

The auxiliary and ghost normalization of XI-B is unchanged.

Proof. The added functional in (eq:selected-action) is cubic in $\gamma$. Its first and second derivatives vanish at $\gamma =0$. It therefore changes neither the free metric Hessian nor the independent quadratic auxiliary and ghost terms.