An entropy-history control with opposite recovery

For completeness, append a degenerate inaccessible register $L\simeq C$ to a $b_{\ast }=1$ proxy. The returning model scrambles and releases the variable input while fixing $L$. The retaining model stores the variable input in $L$ and runs the same proxy on a fixed dummy input $|0{⟩}_C$.

For $|0{⟩}_C$, the accessible histories agree exactly for the same unitary. For every fixed pure input, both ensembles have the same Page and record means. At the coherent endpoint, recovery errors are respectively $0$ and $a_d$. The retaining complement carries the code isometrically. This proves that the entropy means and the supplied clock do not determine endpoint recovery CheungHawk01B,Hawk05B. It does not compare two models under an identical requirement of complete outgoing access and a unique empty residual.