The plateau witness retains the switching terms

For the witness of Theorem thm:plateau-obstruction, $gK\xi =0$, so the entire forced response vanishes. Nevertheless,

Near $\mathrm{supp}f$, $g=1$, so the pointwise term $(g-1)X_{\xi }{\varphi }_0$ in (eq:switch-defect) vanishes there. The retarded switching contribution must therefore equal $-X_{\xi }{\varphi }_0$ on the observation region.

This is not a paradox. Constancy of the switch on the observation region does not make its derivatives vanish throughout the causal region sampled by the retarded inverse.

The nested support construction is shown in Figure fig:VI-plateau-witness.