The three-term variation formula

Off shell, the plateau difference is precisely $F_{P_m(X_{\xi }{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f)}$, not zero in general. General pullback of the scalar is not an automorphism of the free algebra: a non-isometric diffeomorphism need not preserve $P_m\varphi =0$. The theorem is instead background covariance at fixed free field, corresponding to the first completed coefficient of X-A's on-shell identity.