Proposition: Fixed architecture does not imply arbitrary precision

For fixed $n,m,A$, varying the Boolean table yields only finitely many logical channels. There is a unitary channel whose distance from all of them is positive. Consequently that fixed architecture and address length cannot approximate every unitary to every $\epsilon >0$.

Proof. There are $2^{2^m}$ Boolean tables. Their image is a finite set of channels. The unitary-channel family is infinite, so choose a unitary channel outside that image. Its minimum distance from the finite image is positive.