Three classical objects

A broken master identity is not a BV complex. The three objects kept separate are: the unprepared invariant action $S_{\mathrm{inv}}$ with nilpotent $Q_{\mathrm{inv}}$; the nilpotent diffeomorphism BRST operator $s=s_0+\kappa s_1$; and the prepared action $S_D=S_{\mathrm{inv}}+V_{\sigma }+V_g$, whose square is broken. The convention follows XIV-A: one may write $(S,S)=\kappa \int c^{\nu }{B}_{\nu }+\cdots$, and

Following that convention does not repair the square.