Proposition: Selected contraction outputs belong to

The following already-defined coefficient functionals belong to ${\mathcal{P}}_{\Sigma }$:

1. the coefficients of the products of $B_f$ with a smooth bulk linear field, in either order;

2. the pointwise term $B_f{B}_{f^{\prime }}$ and the scalar-contracted self-product coefficient

   $$T_{f,f^{\prime }}[h]=\int f(x)f^{\prime }(x^{\prime })w^{\mu }[h](x)w^{\nu }[h](x^{\prime }){{\partial }_{\mu }}^x{{\partial }_{\nu }}^{x^{\prime }}{\Delta }_m^{+}(x-x^{\prime }){\mathrm{d}}^{4}x{\mathrm{d}}^4{x}^{\prime };$$

3. the coefficients of $B_f$ with $W_g=\int g{h_{00}}^2$, in either order, when only this two-factor product is taken.

The assertion does not include the subsequent multiplication by another sharp insertion.

Proof. For a smooth linear metric test $a$, $H_{hh}a$ is smooth. The single contraction with $B_f$ is a regular scalar linear functional with coefficient obtained from $L(H_{hh}a)$. For a scalar linear test $u$, the contracted scalar factor is smooth and the output is a coordinate functional $\mathsf{J}(a_u)$ with smooth compact source. Reverse ordering has the same admission properties.

For (eq:selfT), the differentiated scalar covariance is paired against a smooth compact function for each smooth $h$. Polarization and the kernel theorem give a compact quadratic metric kernel.

Its first metric derivative is obtained by varying one of the two $w$ factors. In each term, the other $w[h]$, multiplied by its compact test, is smoothly smeared against the differentiated scalar covariance. That convolution is smooth. The varied factor therefore has the form $\mathsf{J}(a[h])$ with a smooth compact source. At every fixed configuration and after every directional derivative, the first metric derivative is a finite sum of $q_a$'s. Thus $T_{f,f^{\prime }}\in {\mathcal{P}}_{\Sigma }$. The possible double contraction vanishes because $H_{ww}=0$.

For $W_g$, the single graviton contraction leaves a functional of the form

up to its fixed combinatorial coefficient and sign. For fixed $\varphi$, smearing the $x$-slot gives a smooth function of $y$. One justification is to write it as a bulk covariance smeared with a kernel $q_a$: the latter has only timelike wavefront covectors, whereas the nonzero covariance covectors are null. The resulting composition is smooth. Equivalently, the Fourier expression (eq:K), smeared in $x$, has rapid high-frequency decay, including its reflected branch. After multiplication by $g$, the first metric derivative is a smooth compact kernel. Further directional derivatives have the same property. The output therefore belongs to ${\mathcal{P}}_{\Sigma }$.