Entropy-history separation with residual memory disclosed

We now construct the counterexample rather than inferring it from similarity of spectra.

Let the preserving evolution above be denoted $G$. Fix a pure code preparation $|0⟩$. Construct a second evolution $L$ as follows:

1. retain the variable input $|\psi {⟩}_A$ in a separate inaccessible degenerate memory $F$;

2. run the entire $G$ sequence on an internal dummy code fixed initially to $|0⟩$;

3. emit the dummy sequence's residual through the same endpoint radiation embedding;

4. retain the variable memory $F$ at the endpoint.

The dummy code and all other ancillaries are fixed pure preparations, not additional variable inputs. Their common energies are included in the initial mass ledger.

Choose the endpoint memory's energy sector to have energy $m_{\ast }$ and degeneracy at least $d$. The preserving process occupies one fixed vector of this sector; the retaining process occupies the logical input in it. Thus the two processes can have the same macroscopic energy ledger while having different complementary channels.