Corollary: Replacement in finite supported expressions

Assume Assumption ass:transfer-contract. A finite scalar expression built from compact smooth sources, smooth local differential operators, and the retained retarded or advanced scalar Green operations has the same retained test actions after replacing every flat scalar Green operation by its intrinsic Friedrichs counterpart. The assertion holds in fixed-support smooth seminorms when the window holds on the corresponding neighbourhoods.

Smooth supported Cauchy evolution is also replaced: the relevant propagators are $W_B(t)$ and ${\dot{W}}_B(t)$, compared with their flat counterparts under the same initial-to-observation window.

Proof. Apply Theorem thm:transfer to the innermost Green operation. Kernel equality on an open pair region persists under finite differentiation. Smooth differential operators with the declared compact coefficient supports produce the same next compact source. Induction proves the statement for the finite expression, with the retained-shadow hypothesis ensuring that no undeclared intermediate value is used.

For fixed-support seminorms, equality on the stipulated neighbourhoods gives equality of all derivatives used in the estimate. For Cauchy evolution, differentiate the positive-time equality for $W_B$, and use its common Cauchy normalization at zero. The data are smooth and compactly supported away from the apex, so these differentiations are legitimate.