Corollary: Exact locality lower bound

If ${\Vert b\Vert }_2>n{a}_j$, the source-to-$b$ logical channel is a replacement channel. In particular, no admissible protocol can transmit payload influence, or implement ${\mathrm{id}}_D$, in time strictly less than

Proof. For $Y=\{b\}$, the source does not belong to $N_0(Y)$. Every encoded payload state has the same reduction on that ancestor set, namely the fixed auxiliary reduction. Lemma lem:support therefore makes the destination state independent of the payload.

Since $a_j=c{\delta }_j$, the condition $n{\delta }_j<{\Vert b\Vert }_2/c$ is exactly $n{a}_j<{\Vert b\Vert }_2$. A replacement channel cannot equal ${\mathrm{id}}_D$ when $D\ge 2$.