Thermodynamic Stability Constraints on Persistent Open Quantum Subsystems: Control Algebra, Physical Actuation and Energy Cost

Persistent control has to be realized by an interaction, with an error and a cost. We construct one such actuator: a degenerate bit coupled to a mode of a free chiral field through $U=\exp[-i\lambda X(b+b^\dagger)]$ and a smooth switch $g$. For independent diagonal bit inputs, the same unitary fixes the flip error, added stress $(g')^2/4$, switching work $\lambda^2\bar\omega$ and a positive quadratic cost on switch profiles. A flip with failure $\epsilon$ needs mean drive occupation at least $\theta_\epsilon(1-2\epsilon)/[4\sqrt{\epsilon(1-\epsilon)}]-\tfrac12$, where $\theta_\epsilon=\arccos(2\epsilon-1)$; the leading cost is $\pi/(4\sqrt\epsilon)$. Displaced squeezed states nearly attain it, whereas classical drives need order $\epsilon^{-1}$. A calibrated feedback task with scheduled fresh independent drives has an exact stationary loss, coupling rate and stability proof, with one acceptance condition joining calibration records to drive energy. A second bound prices response readout: if two branches differ only by probe delays bounded in difference by $\Delta\tau$, binary error $\epsilon$ requires $E\Delta\tau\ge\hbar[1-2\sqrt{\epsilon(1-\epsilon)}]/\zeta$, with $\zeta=\max_{x>0}(1-\cos x)/x$. Retained ancillas and joint decoding do not remove this bound, and a three-level construction saturates it across the allowed signed-delay families. A retained oscillator also supports a thermal branch: a coherent echo protects held quantum content, a mixed qubit port reads the coupling through one binary outcome, and a finite Gibbs bank restores the oscillator while retaining every spent output. These energy and memory accounts do not identify exported energy with removed entropy. The descent metric, physical work form and Lorentzian propagation form remain distinct.