Proposition: Descent criterion

There is a unique map $M:\mathcal{K}\to {\mathcal{D}}^{\prime }(W)$ with $\tilde{M}=M\mathrm{Rec}$ if and only if

Annihilation after restriction to an open set proves only restricted descent.

Proof. A factorization annihilates every reconstruction-null presentation. Conversely, equal reconstructions have difference in $\ker \mathrm{Rec}$, so the condition makes their later images equal. This defines the unique factorization.

Restriction to an open set does not detect distributions supported in its complement. Restricted annihilation need not be global annihilation.