Proposition: Recurrence of a fixed finite proxy

At fixed occupation cutoff and fixed finite frequency mesh, the approximate total system is finite dimensional. Its prepared channel returns arbitrarily closely to the initial channel at arbitrarily large times. Consequently the finite proxy cannot, by itself, establish permanent information release.

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

Initially the reservoir is vacuum and its accessible channel is constant. These recurrences are incompatible with permanent high-accuracy recovery from that closed proxy.