Corollary: Unnormalized root-attached coefficients

For each extension tuple $\{{\bar{p}}_{\mu \nu }\}$, each smooth $h$, and each real smooth scalar $\varphi$, the number

is well defined by compact support. It is linear in $h$ and in $\varphi$. Equivalently,

where compact cutoffs on the fixed effective support are understood.

Proof. Theorem thm:attach gives a compact output distribution, which acts on arbitrary smooth functions. The Hessian coefficients are linear in $h$, and the root is linear in $\varphi$.

To obtain (eq:smearfirst), multiply the root kernel by the smooth scalar, push forward its $x$-slot, and use the inherited root-kernel identity. The resulting $a_{f,g}[\varphi ](y)$ is smooth. The wavefront test just proved, together with proper support, permits the corresponding distributional composition and interchange of these pushforwards. Pairing against the reduced distribution gives (eq:smearfirst).