Proposition: Equal entropy histories without recovery

A second bare-energy-conserving sequential model has the same radiation entropy $h_2(q_N)$ for every pure code input and every prefix, but constant radiation and recovery error $c_d$.

Proof. Store the variable code in a zero-energy degenerate spectator $V$. Prepare another source memory $D$ in the fixed excited state $|1{⟩}_D$, of energy $\omega$, and apply the same exchange pulses to $D$ and radiation. All Hamiltonians act as identity on $V$.

The radiation channel is

The unnormalized expression is valid even when $q_N=1$. Its two nonzero block eigenvalues are $q_N$ and $1-q_N$, so its entropy is $h_2(q_N)$. Any decoder produces a fixed state; Lemma lem:replacement gives optimal error $c_d$. The complete residual retains the original code in $V$.