Theorem: An extensive Page history without code release

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

1. for one fixed pure code input, their accessible states agree at every prefix;

2. the two ensembles have the same exact Page mean (eq:page) for every fixed pure code input;

3. the endpoint recovery errors are respectively $0$ and $a_d$, and endpoint complete leakages are respectively $0$ and $a_d$.

Thus even the extensive Page mean does not determine endpoint whole-code recovery.

Proof. Append a degenerate inaccessible register $L\simeq C$. In the returning model, run Section sec:model on the variable input and fix $L$ in $|0⟩$. In the retaining model, store the variable input in $L$ and run the same radiation schedule on the fixed dummy input $|0{⟩}_C$.

All transfer operations conserve (eq:H0); $L$'s logical labels are degenerate. Both are complete isometries into the same spaces. For the input $|0{⟩}_C$, the radiation histories agree exactly for the same $U$. Haar invariance proves that the returning model's mean entropy for any fixed pure input equals the dummy model's mean, namely (eq:page).

At the endpoint the returning radiation is $UJ|\psi ⟩$, with fixed complete residual. The retaining radiation is fixed at $UJ|0⟩$, while the residual contains $|\psi {⟩}_L$. The endpoint recovery errors follow from exact correction and the replacement benchmark. The retained logical copy has leakage $a_d$: a CPTP left inverse gives the identity-channel lower bound, and an embedded maximally mixed replacement gives the upper bound.