Proposition: Canonical-domain agreement is insufficient

Agreement with canonical contraction on $U_{\mathrm{can}}$ can show

without proving $\tilde{\mathsf{M}}(A)=0$ on $W$.

Proof. An explicit model suffices. Let $W=\mathbb{R}$, $U_{\mathrm{can}}=\mathbb{R}\setminus \{0\}$, and $\mathcal{P}={\mathbb{R}}^2$. Fix a nonzero smooth distribution $k$, and set

Define

Its restriction to $U_{\mathrm{can}}$ is the zero canonical map. However, $(0,1)\in \ker \mathsf{Rec}$ and

Thus global descent fails despite canonical-domain agreement. All maps in this example are continuous on the finite-dimensional presentation space.