Proposition: No universal quantum return or finite CCR

The channel ${\mathcal{M}}_C$ is not invertible on all clock inputs. Finite matrices cannot satisfy $[T_C,H_C]=i\hslash I$ when $\hslash \ne 0$. Consuming $C$ does not remove $T$, $M$, or $g$.

Proof. The clock states $(|{\tau }_0⟩\pm |{\tau }_1⟩)/\sqrt{2}$, with the same payload, have identical outputs. A left inverse is impossible. Taking the trace proves the commutator assertion. The final statement follows because these geometric objects are prescribed parameters, not output factors of ${\mathcal{M}}_C$.