Proposition: Typing of the complete tested source

The map ${\mathsf{Q}}_G$ is a matrix-valued differential operator of order at most two:

Its coefficients are smooth and supported in $\mathrm{supp}G$.

In the Fourier convention where ${\partial }^{\nu }$ has symbol $(i\xi {)}^{\nu }$, its degree-two differential symbol is

This is a fiber-valued polynomial on $T^{\ast }M$. It is allowed to vanish and is not asserted to be elliptic.

Its Schwartz kernel is supported on the source diagonal $s=a$ and is a finite sum

These statements retain every coefficient of the actual source, including its ramp derivatives.

Proof. The copied quadratic source is a finite sum of local monomials

with constant fiber contractions and the fixed external ramp coefficients. Its directional derivative at $G$ is the sum of the two variations of every monomial.

Collecting the derivatives of $h$ gives (eq:QG). Each coefficient contains a derivative of $G$ of order at most two, multiplied by fixed smooth external data. It is therefore smooth and supported in $\mathrm{supp}G$.

The symbol (eq:Qsymbol) is the algebraic highest-degree differential polynomial of that operator. Lower-degree coefficients remain in (eq:QG); none is omitted.

The kernel identity follows from the distributional formula for differentiating $h(s)$ through the diagonal delta. It has exactly the stated support. No propagator or characteristic multiplication is used.