Proposition: Reconstruction descent

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

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

Proof. Factorization annihilates reconstruction-null presentations. Conversely, equal reconstructions have difference in $\ker \mathrm{Rec}$, so the condition makes their candidate images equal. This gives the unique factorization.

Restriction to an open set cannot detect a distribution supported in its complement, proving the qualification.