Theorem: Finite-chain recurrence obstruction

Fix finite $L$, a code of dimension $d\ge 2$, and access only to the finite reservoir. Then

The result does not prohibit useful finite-time recovery. It prohibits interpreting this closed finite system as permanently releasing the code into its reservoir.

Proof. The finite fermionic Fock space has finite dimension. Write its Hamiltonian spectral decomposition as

Simultaneous recurrence of the finitely many phases gives arbitrarily large times $t_j$ for which

For example, this follows from simultaneous Diophantine approximation; an exactly periodic orbit gives the same conclusion by its repeated periods.

At time zero the reservoir is fixed vacuum, so its code channel is constant and has optimal recovery error $1-d^{-2}$ by Lemma lem:replacement. Lemma lem:isometry and Lemma lem:optimal-stability give convergence of the recovery error to that value along $t_j$.

For every channel, discarding its output and preparing $I_C/d$ is an allowed decoder. Its error is $1-d^{-2}$. This supplies the matching global upper bound.