Why the direct tree identity survives

For a compact metric test $a$, the second term has the equation representative

The preimage is smooth and compactly supported in $\mathrm{supp}\xi$. For the mixed observable, the corresponding distributional preimage is obtained by applying $-{{\mathcal{L}}^{\mathrm{cov}}}_{\xi ,y}$ to its transposed insertion–Green kernel. Compact generator coefficients cut off the source argument. Differentiation preserves the inherited sum-zero wavefront bound, so this is the graviton-slot equation image of the frozen tree carrier XF,XFST.

For clarity, the completed scalar identity used in that carrier is

One verifies it by setting $e=(1+L{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}{)}^{-1}c$, as in (eq:tests). The coordinate intertwiner on the plateau gives $c^{\prime }=F^{♯}c$. Naturality of the switched scalar Euler operator there gives $s^{\prime }=F^{\ast }s$ for its advanced solution $s={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}e$. Applying $P_m$ and expanding proves (eq:scalaridentity). The generator coefficients make the displayed preimages compact.