Proposition: Global preservation, conserved energy, and termination do not imply outgoing recovery

There are finite complete preparations and unitary programmes with the same code, same access factorization, and same conserved complete Hamiltonian, which terminate after finitely many instructions and have complete isometric endpoints, but have opposite exact outgoing-recovery verdicts.

Proof. Prepare the code in $B\cong C$ and a blank $|0⟩$ in $R\cong C$. Fix a common self-adjoint matrix $h$ on both factors and the complete Hamiltonian

One programme applies $I_{RB}$ and halts. The other applies ${\mathrm{SWAP}}_{RB}$ and halts. Both commute with $H_0$ and preserve its complete probability distribution. Their complete endpoints are $V_{\mathrm{keep}}$ and $V_{\mathrm{send}}$ from Proposition prop:entropy.

Both preserve the code and every reference on the complete output. The first has a constant accessible output and recovery error $1-d^{-2}$. The second has constant complete complement and recovery error zero. The access factor $R$ has not changed.