Theorem: Separated null defects cannot be removed by contacts

Suppose every $a\in \mathcal{A}$ takes values in $V_C$. If there exist $A\in H$ and an open set $G\subset W\setminus C$ such that

then no member of $F_0+\mathcal{A}$ descends. Equivalently, ${\mathrm{Ob}}_{\mathcal{A}}(F_0)\ne 0$.

Proof. Every permitted correction restricts to zero on $G$. Therefore

for every $a\in \mathcal{A}$. No candidate annihilates $H$. Theorem thm:lifting proves the obstruction statement.