The direct all-tree theorem

At $h=0$, $T_{\mathrm{mix}}$ vanishes, but its first background variation generally does not. Dropping a function because its value vanishes at one background would lose that variation.

An all-order route based on simultaneous pullback, retarded uniqueness, and a converse free-shell separation theorem is not proved. In the source-switched setting its variational interpretation would also require an additional construction. Only the direct finite-order proof above is released.