Proposition: Descent through reconstruction

Let

be a surjective linear reconstruction map, and let $\tilde{M}:\mathcal{P}\to V$ be linear. There exists a unique linear $M:\mathcal{K}\to V$ with

if and only if

Proof. A factorization annihilates every reconstruction-null presentation. Conversely, if two presentations reconstruct to the same kernel, their difference belongs to $\ker \mathsf{Rec}$. The proposed value $M(\mathsf{Rec}p)=\tilde{M}(p)$ is therefore independent of the presentation. Surjectivity gives uniqueness.