Operational Coherence Maintenance and the Quantum-Classical Boundary: Formal Definitions, Falsifiable Protocols, and an Outlook for Cognitive Systems

Maintaining quantum coherence against uncontrolled open-system dynamics is a control task with unavoidable thermodynamic cost. In a finite-dimensional setting with battery-assisted thermal operations at bath temperature T, we present the corrected operational hierarchy for maintenance power: an unconditional lower bound P_min(rho) >= k_B T sigma(rho) with sigma(rho) the entropy production rate of the target (proved in the appendix); the coherence-power bound P_min >= k_B T Cdot_loss(rho) under diagonal contraction; and incremental (extra) power bounds relative to thermodynamically efficient population-maintenance baselines, which become assumption-free exactly when the pinched target is stationary -- e.g. under pure dephasing, the very dissipator used in this paper's numerical protocol. Here C(rho) = S(rho||Delta[rho]) is relative-entropy coherence to energy pinching and Cdot_loss(rho) := -d/dt C(rho_t)|_{t=0}. These statements are operational, observer-independent, and geometry-free. We then present the Rate Inheritance Principle (RIP) as the falsifiable dynamical bridge between static clustering and decoherence rates, with its status: weak form a lemma under explicit hypotheses; strong form derived, with a frequency-resolved squared-amplitude exponent, in an exactly solvable quasi-free local-sink class; failure through near-zero Bohr-frequency channels realized within the secular Davies class. We provide falsifiable protocols distinguishing one-shot work from sustained maintenance power, including a numerical stress test (distance-independent influence floor without an interface vs collar-induced suppression) in a transverse-field Ising chain with remote dephasing. Finally, an explicitly speculative Outlook connects the resource boundary to the Free-Energy Principle for resource-limited agents, at a methodological (non-phenomenological) level.