Theorem: Matched complete entropy histories, different recovery

There are two energy-conserving sequential complete isometries with the same accessible spaces and complete residual spaces such that:

1. every pure logical input has the same radiation entropy at every stage in both models;

2. for one fixed logical input, their accessible density operators agree at every stage;

3. at complete transfer, one model has recovery error zero and complete leakage zero, while the other has recovery error $a_d$ and complete leakage $a_d$.

Proof. Use the schedule of Section sec:schedule, append a degenerate register $L\simeq C$, and keep $L$ inaccessible.

In model $G$, run the schedule on the variable input and leave $L$ fixed in $|0⟩$. In model $L_{\mathrm{ret}}$, retain the variable input in $L$ and run the schedule on a dummy input fixed in $|0⟩$. Both are isometries into the same output spaces. The transfer unitaries conserve (eq:energy); the additional logical labels in $L$ are degenerate and do not alter that ledger.

For pure inputs, the radiation entropy in $G$ is $h_2(q_n)$. The dummy schedule has the same entropy. For input $|0⟩$, the radiation states agree exactly at every stage.

At $q_N=0$, model $G$ returns $J_N|\psi ⟩$ and has fixed complete residual. Model $L_{\mathrm{ret}}$ returns the fixed state $J_N|0⟩$, while its complete residual retains $|\psi {⟩}_L$. Replacement recovery has error $a_d$. The residual channel is an isometric copy of the code with fixed spectators, whose distance to replacement is $a_d$ by Lemma lem:replacement and reversible compression.