Proposition: Descent criterion for a coefficient-level construction

A linear candidate

defines a map on reconstructed kernels if and only if

When this holds, the induced map $\mathsf{M}(\mathsf{Rec}A)=\tilde{\mathsf{M}}(A)$ is unique.

Agreement of $\tilde{\mathsf{M}}$ with canonical contraction on $U_{\mathrm{can}}$ proves (eq:descent) there only. It does not prove that the same identity holds across $N$.

Proof. If $\tilde{\mathsf{M}}=\mathsf{M}\mathsf{Rec}$, a presentation in $\ker \mathsf{Rec}$ has image zero.

Conversely, if two presentations have the same reconstruction, their difference belongs to $\ker \mathsf{Rec}$. Condition (eq:descent) therefore makes their images equal, so the induced map is well defined and unique.

For a null presentation, canonical contraction of its zero reconstruction is zero on $U_{\mathrm{can}}$. That statement only determines the restriction of its proposed global image. A distribution supported outside $U_{\mathrm{can}}$ is not detected by this restriction.