Proposition: A fixed finite autonomous proxy does not prove permanence

For a finite-dimensional autonomous Hamiltonian, its complete prepared channel returns arbitrarily close to its initial channel at arbitrarily late times. If accessible radiation is initially fixed vacuum, then

Proof. The finitely many spectral phases have simultaneous approximate returns to one. Equivalently, compactness of the phase torus and the group property produce arbitrarily late returns; a periodic orbit uses multiples of its period. The propagator approaches the identity along such a sequence. Lemma lem:stability gives channel recurrence and recovery stability. The initial channel is a replacement with error $a_d$, and every channel has recovery error at most $a_d$ using the constant decoder.