Proposition: A specified collection of admitted coefficient outputs

The following already-defined local contraction outputs are actual elements of ${\mathcal{P}}_{\Sigma }$:

1. coefficients of $B_f$ with a smooth compact linear metric or scalar functional, in either order;

2. the scalar-contracted metric quadratic functional

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

3. the single-contraction coefficient with $W_g=\int g{h_{00}}^2$, $g\in \mathcal{D}(V)$, in either order.

These are selected coefficient-membership statements. They do not define a quantum multiplication on ${\mathcal{P}}_{\Sigma }$.

Proof. A smooth linear metric smearing makes the contracted metric covariance smooth. Its image under the local $L$ is smooth on $V$. The remaining output is therefore a regular scalar linear functional. A smooth scalar linear smearing gives a smooth scalar function $v$, and the output is, up to the fixed contraction sign, $J(f\mathrm{d}v)$.

For (eq:selfcoefficient), pair the differentiated scalar covariance against the smooth compact two-variable coefficient at each fixed $h$. This defines a continuous quadratic functional with compact metric kernel. Varying one $w$-factor leaves the other factor smoothly smeared against the scalar covariance. The resulting source for the varied $w$ is smooth and supported in $\mathrm{supp}f$ or $\mathrm{supp}{f}^{\prime }$. Its first metric derivative is therefore a finite sum of $q_a$'s. Every further directional descendant has the same property.

For (iii), define the mixed kernel $H_{wh}=(L\otimes I)H_{hh}$ on the supported observation variables. It is a defined distribution: the retarded-step composition has the same transversality used in Proposition prop:generators. The left-ordered coefficient is

At fixed smooth $\varphi$, smearing the $x$-slot is precisely a metric covariance smeared with $q_{f\mathrm{d}\varphi }$. Lemma lem:smoothing makes the result smooth in $y$. Multiplication by $g$ gives a regular first metric derivative. Scalar differentiation replaces $\varphi$ by a fixed smooth direction and preserves the argument. Reverse ordering changes the covariance orientation, not this smoothness conclusion.