Maintenance Is Not Restoration: Endpoint No-Go Theorems, Exact SSH Resource Horizons, and Quantum Rapid-Replacement Equality

Lower bounds based on the instantaneous free-energy loss of a target state are often interpreted as lower bounds on the power required to maintain that state. We show that this interpretation depends decisively on the control task. If maintenance only requires exact restoration at periodically sampled endpoints and the intervention period is unrestricted, the infimal average power can vanish even when the target has strictly positive instantaneous entropy production. A full-rank dephasing qubit gives an exact energy-conserving SWAP counterexample. We repair the formulation by separating endpoint restoration, bounded-latency maintenance, and continuous holding.For continuous-time finite Markov chains we adopt the established trajectory-relative-entropy holding cost and review the known reversible optimizer and Dirichlet-form representation. Our first composition result is then an exact geometric additivity theorem for suppressible and persistent generators sharing a detailed-balance reference. A two-state counterexample shows that this hypothesis is essential: channels with incompatible equilibria can cancel at a finite membrane width. A dual-rail pair of odd fermionic SSH chains supplies the microscopic layer: exact edge modes, a uniform bulk gap, and an explicitly filtered number-conserving Davies coupling produce two-sided holding-cost bounds without an assumed rate-inheritance bridge. The logical basis has fixed particle number and parity, so arbitrary logical coherence is physical under fermionic superselection. Finally, under a fully axiomatized resource-cell ledger, fresh target-state cells and energy-conserving SWAPs saturate the fixed-period free-energy bound; the correctly ordered rapid-control limit closes the SSH theorem for coherent logical targets. The fresh-copy construction is related to earlier collision-model stabilization work and is not claimed as a work-only controller. The remaining frontier is autonomous work-only control without preloaded target copies.