Proposition: Recurrence is an arbitrarily late obstruction

For a fixed finite-dimensional autonomous Hamiltonian, there are arbitrarily late times at which its prepared complete channel is arbitrarily close to its initial channel.

If initially accessible radiation is fixed vacuum and the complete prepared map is an isometry, then

These statements do not assert positive leakage at every time.

Proof. The finitely many spectral phases evolve in a compact torus. Simultaneous recurrence gives arbitrarily late unitary returns to the identity; periodic cases use multiples of a period. Lemma lem:distances gives prepared-channel returns.

At time zero, radiation is constant, so its recovery error is $a_d$. The complete isometry with pure fixed radiation has an isometric residual encoding, whose leakage is $a_d$ by Lemma lem:constant and isometric invariance. Distance stability gives both values along the recurrence sequence. The universal upper bounds in Lemma lem:constant give the limsup equalities.