Proposition: Reconstruction descent

Let

be the common reconstruction, and let

be linear. Then $\tilde{\mathsf{M}}$ defines a map on reconstructed kernels if and only if

When this holds, the induced map is uniquely given by

Agreement with canonical contraction on $O=U_{\mathrm{can}}$ gives only

where $r_O$ is restriction. It does not prove (eq:descent) across the complement of $O$.

Proof. If $\tilde{\mathsf{M}}=\mathsf{M}\mathsf{Rec}$, a null presentation has image zero. Conversely, two presentations with equal reconstruction differ by a null presentation; condition (eq:descent) makes their proposed images equal. The induced map is therefore well defined and unique.

On $O$, canonical contraction respects equality of reconstructed kernels. A null presentation thus has proposed image restricting to zero there whenever canonical agreement holds. That proves (eq:restricteddescent). A distribution supported in $W\setminus O$ is not detected by this restriction, so the global kernel inclusion does not follow.