Proposition: Mixed quadratic outputs

The finite contraction formulas for $B_f$ and $Q_q$, in either order, define compact polynomial coefficients belonging to ${\mathcal{P}}_{\Sigma }$.

This does not assert closure for arbitrary local quadratic polynomials or arbitrary later multiplications.

Proof. A graviton-only contraction leaves a scalar quadratic functional with a compact distribution kernel. Such a functional belongs to ${\mathcal{P}}_{\Sigma }$ because it has no metric dependence.

A scalar-only contraction leaves a quadratic metric functional of the form

with tensor indices understood and $D$ a differentiated scalar Hadamard distribution. Varying $w[h]$ gives $q_a$ with a smooth compact source: the $y$-factor is a smooth compact smearing at each fixed configuration. Varying $h(y)$ gives a smooth compact metric kernel obtained by smearing $D$ in $x$. Further directional derivatives have the same two forms. The output is therefore admitted.

For the double contraction, the numerical product is

In both direct and reflected branches of the first factor, the nonzero $y$-slot covector has the same temporal orientation as the $y$-slot covector of the scalar factor. It is nonzero and null. The two factors consequently cannot have opposite total wavefront covectors. The distributional multiplication criterion applies. Compact smearing gives a constant, which belongs to ${\mathcal{P}}_{\Sigma }$. Reverse ordering reverses both orientations and gives the same conclusion.