Theorem: Quotient descent versus non-feedback

Let $R_U\subseteq U$, let $T:V_{\mathrm{allowed}}\to W_{\mathrm{full}}$ be the allowed counterterm response, and let $F:U\to W_{\mathrm{full}}$ be the complete review response. For a complete left kernel $L_T$ of $T$,

where $\pi :W_{\mathrm{full}}\to W_{\mathrm{full}}/\mathrm{im}T$. The descended obstruction vanishes on every supplied class iff

Proof. Descent requires annihilation of relations after quotienting. Theorem thm:rank gives $\mathrm{im}T=\ker L_T$. Vanishing on all classes requires containment of $F(U)$, not only $F(R_U)$, in that image.