Proposition: No return, no residual inference, no finite CCR

For $d\ge 2$:

1. ${\mathcal{M}}_C$ has no universal left inverse on all clock–system states;

2. equality of record effects does not imply equality of recorded residual instruments;

3. finite matrices cannot satisfy $[T_C,H_C]=i{I}_C$.

Proof. The two clock states $(|{\tau }_0⟩\pm |{\tau }_1⟩)/\sqrt{2}$, tensored with the same residual, have the same image under ${\mathcal{M}}_C$. A left inverse would have to recover both distinct inputs from that image.

For the second assertion, on a residual qubit compare

Both instruments have effects $I/2$, but on $|0⟩$ their conditional residuals are $|0⟩$ and $|1⟩$. Thus their records agree on every input and their residuals need not.

Finally, $\mathrm{Tr}[T_C,H_C]=0$, whereas $\mathrm{Tr}(i{I}_C)=id\ne 0$.