Proposition: Exact finite schedule support

The Heisenberg pullback satisfies

If every block has diameter at most $\Delta$ in a declared metric $d$, then

The output restriction to $\mathcal{A}(Y)$ is determined by the input restriction to $\mathcal{A}(N_s(Y))$.

Proof. A block disjoint from the current support sees identity, and its channel adjoint preserves identity. One layer can therefore enlarge support only to blocks meeting that support. Backward induction proves the first inclusion.

If $x$ enters through a block containing a previously admitted site $z$, then $d(x,Y)\le \Delta +d(z,Y)$. Induction proves the metric envelope. Equal input restrictions give equal expectations of all pulled-back output observables, proving the final assertion.