Corollary: Equation-vanishing changes cannot repair this witness

Suppose a corrected field derivation has

commutes with spacetime derivatives, and satisfies $(C_1\varphi ){|}_{\mathrm{free}}=0$ as a smooth function on every complete free configuration. Then it still fails unchanged free-shell descent on the witness of Theorem thm:nondescent.

Proof. On a free configuration, the identically zero smooth function $C_1\varphi$ and all of its spacetime derivatives vanish. Hence $P_m(C_1\varphi )=0$ there. The order-$\kappa$ value remains $2\kappa \vartheta$.