Compatibility with the retained three-tree identity

The X-F-ST direct identity acts on the classical background coefficients while holding the free fields fixed. For $\xi ,{\zeta }_p\in {\mathfrak{g}}_p$,

It is a different operation from $s$, which acts on the field variables.

For the mixed response,

The proof uses (eq:typed), ${\sigma }_{\epsilon }=1$ on $\mathrm{supp}\xi$, and the retarded left inverse on the compact tensor ${{\mathcal{L}}^{\mathrm{cov}}}_{\xi }r$. After compact smearing, the last term is a graviton-slot equation image.

This remains the cancellation supplied by the mixed tree. The matter and self trees are inert under that background transformation XF,XFST. No additional copy of the scalar second-order pair, matter tree, or mixed cancellation is added.

The relative BV construction does not establish that the relational observable is $\mathcal{Q}$-closed. Action symmetry, observable closure, and the fixed-free-input background identity are different claims. The supplied tree identity is not reinterpreted as a variational Møller theorem or an interacting quantum Ward identity.