Proposition: Energy drain, zero selected entropy, and termination are insufficient

There is a complete finite energy-conserving process that:

1. preserves the entire logical code globally;

2. transfers all of a declared core-energy excitation to an outgoing factor;

3. ends with zero core energy and zero accessible entropy;

4. terminates after one unitary;

5. has optimal outgoing recovery error $q_d$.

Proof. Let $F$ and $G$ be qubits and let $L\simeq C$ be a degenerate logical memory. Give the complete system Hamiltonian

Prepare

and apply the swap of $F$ and $G$. It commutes with $H_{\mathrm{tot}}$. Declare the outgoing access $R=G$ and complete residual $B=F\otimes L$. The resulting isometry is

The process is globally isometric, all core excitation has moved from $F$ to $G$, and the core energy is zero. The accessible state is the fixed pure state $|1⟩⟨1|$, so its entropy is zero and its channel is constant. The program has one unitary step. Lemma lem:constant-error gives outgoing recovery error $q_d$. The unknown logical code remains in the disclosed residual memory $L$.