Proposition: Reconstruction descent

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

Agreement after restriction to an open $U\subset W$ proves this criterion only after that restriction.

Proof. A factorization annihilates every null presentation. Conversely, if two presentations have the same reconstruction, their difference is in $\ker \mathrm{Rec}$; the criterion makes their images equal. This defines the unique induced map.

Restriction to $U$ does not detect distributions supported in $W\setminus U$. Thus restricted annihilation need not imply global annihilation.