Theorem: Relative lifting obstruction

Let $O\subset W$ be an open reference domain, let $r$ be restriction to $O$, and $Y=\ker r$. Suppose a preliminary continuous full lift $F_0:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{D}}^{\prime }(W)$ already exists. For a linear permitted correction space $\mathcal{A}\subset \mathrm{Hom}({\mathcal{P}}_{\mathrm{int}},Y)$, let $H_0=\ker \mathsf{Rec}$, and define

assuming the reference restriction kills null presentations. A descending correction exists if and only if this obstruction is zero. When it exists, the descending candidates form an affine space over

All candidates have the same reference restriction.

Proof. The reference assumption makes $F_0(H_0)\subseteq Y$. Descent is precisely $F_0{|}_{H_0}+a{|}_{H_0}=0$. This is vanishing of the displayed quotient class. Subtract two solutions and use $r(a)=0$.