Lemma: One-layer support bound

For every admissible realization,

and

Equal input reductions on $N_0(Y)$ give equal output reductions on $Y$. With an untouched reference $R$, equality of the joint $N_0(Y)$–$R$ input reductions gives equality of the joint $Y$–$R$ output reductions.

Proof. At one backwards tick, a block disjoint from the current support acts on an identity factor. Its adjoint is unital and therefore preserves that factor. All other blocks can enlarge the support only to their union. Induction proves (eq:adjoint-support).

Every point added at a backwards step is within distance $a_j$ of a point in the preceding backwards set. Following these steps and applying the triangle inequality proves (eq:metric-support).

The adjoint inclusion makes every output expectation depend only on the indicated input reduction. Output expectations determine the reduced output state. Tensoring the inclusion with the identity on $R$ proves the referenced assertion.