Theorem: Contacts cannot repair an off-contact null defect

Suppose every permitted correction to a preliminary lift is supported in a closed contact locus $D\subset W$. If some $A\in \ker \mathsf{Rec}$ and some test $\psi$ supported in $W\setminus D$ satisfy $F_0(A)(\psi )\ne 0$, no permitted corrected lift descends.

Proof. Every allowed correction vanishes on $\psi$. Thus the corrected candidate still fails to kill $A$. Apply Theorem thm:descent.