Time dephasing preserves the code

The measurement environment must be included in a complete postmeasurement ledger. Nevertheless, measuring time alone does not make the quantum code unrecoverable.

For $p>0$, applying the original coherent-mode decoder after time dephasing instead gives

Indeed ${J_N}^{†}{J}_j={\overline{v}}_j{I}_C$. Its error is $(1-a)c_d$. This is decoder-mode failure, not an optimal-recovery obstruction. At $p=0$, use a constant decoder without defining $v$ or $a$.