Proposition: Recurrence of a fixed finite proxy

At fixed finite occupation cutoff and frequency mesh, the autonomous proxy has arbitrarily late times at which its prepared channel returns arbitrarily closely to its initial channel. The initially vacuum radiation channel is constant. Consequently,

Proof. A finite-dimensional Hamiltonian has finitely many eigenvalues. Simultaneous recurrence of their phases gives arbitrarily large times at which the unitary is arbitrarily close to the identity. Lemma lem:isometry then gives channel recurrence.

Optimal recovery error is one-Lipschitz in channel diamond distance: insert any decoder and use contraction, then take infima in both directions. The initial error is $a_d$. For every channel, the constant decoder has error $a_d$, giving the matching upper bound.