Corollary: Restriction does not prove global descent

Suppose a proposed presentation map agrees with canonical contraction on an open set $U_{\mathrm{can}}$. For $A\in \ker \mathsf{Rec}$, this proves only

It does not prove $\tilde{\mathsf{M}}(A)=0$ on $W$.

Proof. Restriction can fail to be injective. If $p\in W\setminus U_{\mathrm{can}}$, a nonzero delta distribution at $p$ restricts to zero on $U_{\mathrm{can}}$. Thus a possible distribution supported outside the canonical domain is not detected by canonical-domain agreement. This demonstrates the missing implication; it does not assert that such a term satisfies every other condition of the frozen map class.