Proposition: Exactly consistent subdivisions with different limits

One finite qubit channel admits two exactly coarse-consistent subdivision families with different continuous-time limits. Each family is complete and reference stable.

Proof. Take the common coarse channel to be $\mathrm{id}$, and let

Both are continuous channel groups with ${{\mathcal{T}}_1}^a=\mathrm{id}$. At tick duration $1/n$, put

Then $({{\Phi }_n}^a{)}^n=\mathrm{id}$ exactly.

The bound

and expansion of the difference of two unitary conjugations imply

It holds with arbitrary references and uniformly in $t$.

At $t=1/2$, the first limit fixes $|+⟩$, whereas the second sends it to $|-⟩$. The corresponding output states are orthogonal. The limiting evolutions differ.