The completed scalar and three-tree interface

We record the insertion target to make the counting unambiguous XC,STRecords,XFST. For $q=\eta +\kappa l+{\kappa }^{2}j$, define $a_1[l]$, $A_2[l,j]$, $b_1[l]$, and $B_2[l,j]$ by

In particular $a_1[l]=-{\mathsf{P}}_{\mathrm{tr}}l$ and $b_1[l]=\frac{1}{2}{\mathrm{tr}}_{\eta }l$.

The coordinate coefficients $w[l],v[l,j]$ have zero position and velocity on $\Sigma$ and solve

Put

The affine decomposition is

With ${\hat{A}}_1[l]=\mathbf{\varphi }(d_1[l])$, the frozen three tree coefficients are

Here $G_2$ is the polarized quadratic coefficient of $J_q{G}^{\mu \nu }[q]$. These symbols mean the compact distributional kernels of the fixed-width tree sheet, not pointwise multiplication of singular fields.

The inherited expression is

The matter response equals $-2{\mathsf{P}}_{\mathrm{tr}}{{\mathsf{E}}_0}^{\mathrm{R}}(g{T}_0[\mathbf{\varphi }])$. The scalar second-order pair in (eq:tests) and the three coefficients in (eq:trees) are retained once.

The inherited support distinctions remain: the scalar slot lies in $C_f$; the matter cubic has support in ${C_f}^3$; graviton slots of the other trees can reach $D_f$. Full kernels are distributions. Suitable smooth scalar-slot contractions are elements of ${\mathcal{D}}_{C_f}$. No kernel-admission theorem is inferred from a scalar-test topology.