Bath dimension and initial entropy for closed repeated use of a quantum channel
We characterize the bath resources needed to supply repeated uses of a fixed finite-dimensional quantum channel in a closed device. For each horizon $T$, one bath, one initial state and one repeated unitary are fixed before the user. Each output is returned before the next input arrives; no reset, discard, fresh ancilla or uncounted controller is available. Approximation error must vanish against arbitrary adaptive users with quantum memory and references. Writing $r=\lim \log_2(R_T)/T$ for the bath dimension rate and $s=\lim S(ω_T)/T$ for the actual initial entropy rate, we prove that the achievable region is exactly $s\ge 0$, $r+s\ge h$ and $r-s\geκ$. Here $h$ is maximum entropy exchange and $κ$ is a smoothed independent-reference extension cost, with the zero-error limit taken before the supremum over full-rank inputs. Its exact fixed-input form is an affine transform of the zero-leakage quantum privacy funnel. The minimum dimension rate is $(h+κ)/2$. The proof combines entropy converses, a bath-dimension-independent support repair, and a closed adaptive implementation of encoder-only fully quantum Slepian--Wolf recycling. All seeds, clocks, workspace and retained residues are counted. Worked examples include dephasing, pure replacement and a qubit channel with $0<κ<h$. No computability of $κ$ or efficient circuit synthesis is claimed.