Proposition: Recurrence obstruction

At fixed finite occupation cutoff and frequency mesh, the autonomous proxy returns arbitrarily closely to its initial prepared channel at arbitrarily late times. For initially vacuum accessible radiation,

Proof. A finite-dimensional Hamiltonian has finitely many eigenvalues. Its phase flow lies in a compact torus. Simultaneous recurrence gives a sequence $t_j\to \infty$ for which the unitary approaches the identity. If the phase orbit is periodic, use multiples of its period; otherwise compactness and the group property give arbitrarily late approximate returns.

Lemma lem:bounds yields prepared-channel recurrence. Initially the accessible bath is vacuum, so its logical channel is constant and has recovery error $a_d$. Recovery stability gives errors tending to $a_d$ on the recurrence sequence. Every channel has error at most $a_d$ by Lemma lem:unitary-mixture, proving the limsup identity.