Proposition: No return, and record is not residual

The clock instrument has no universal left inverse on its full input space. Its classical marginal does not determine the surviving state.

Proof. The clock states

with the same fixed $X$ input have identical outputs under ${\mathcal{M}}_C$, although their input density operators differ. A left inverse cannot recover both.

For the second assertion, take $P_0\otimes {\rho }_X$ and $P_0\otimes {{\rho }^{\prime }}_X$ with ${\rho }_X\ne {{\rho }^{\prime }}_X$. Both return the same deterministic classical record but different residual states.