The Conditional Maintenance Work Theorem: Operational Power Lower Bounds from Energy Pinching and a Split-Inclusion Blueprint for Type III AQFT

We derive operational lower bounds on the minimal thermodynamic power required to maintain quantum coherence against uncontrolled open-system dynamics. Work is defined as the consumption of non-equilibrium free energy stored in an explicit battery at bath temperature T, namely W := F_W(sigma_W) - F_W(sigma'_W) with F_W(sigma) = k_B T S(sigma||gamma_W), and allowed controls are battery-assisted thermal operations implemented by energy-conserving global unitaries. Coherence is quantified as a relative entropy to a conditional expectation, C_E(rho) := S(rho||E[rho]). Our main proved results are: (i) an unconditional maintenance bound P_min(rho) >= k_B T sigma(rho), where sigma(rho) >= 0 is the entropy production rate of the target under the uncontrolled semigroup (Spohn), which splits via the pinching Pythagorean identity as sigma = F_diag + C_loss; (ii) the coherence-power bound P_min >= k_B T C_loss(rho) under a diagonal-contraction hypothesis satisfied by Davies generators; and (iii) extra-power bounds for coherence stabilization relative to thermodynamically efficient population-maintenance baselines — version 1's "assumption-free" paired-infimum formulation is shown to be vacuous (the unlinked-pairs infimum is minus infinity), and the corrected statement is genuinely assumption-free precisely when the pinched target is stationary (e.g. pure dephasing), where the baseline is free. We further provide a conditional extension to general conditional expectations and a Type III split blueprint, with external geometric inputs encoded as explicit two-sided interface assumptions. All finite-dimensional claims are verified by an exact numerical suite distributed with the paper. These results provide an operational resource criterion for quantum-to-classical behavior, not a collapse theory.Version 2 corrects a sign error in the work definition and in the final step of v1's Appendix A, replaces v1's vacuous extra-power definition, and strengthens the main bound to an unconditional entropy-production form.