Corollary: Semiclassically normalized quantum descent is not automatic

Suppose a proposed differential on an unchanged free quantum carrier is required to have the displayed nonlinear scalar rule as its classical normalization, including the compact equation functional $F_f$. If its classical reduction preserves the unchanged free equations, such a proposal is incompatible with (eq:witness).

Proof. A well-defined differential sends the zero equation class to zero. Its assumed classical reduction would therefore make $s{F}_f$ vanish on every free solution. Equation (eq:witness) contradicts that requirement.