Thermodynamic limits on disturbance-rejection authority in constrained Markov networks: instantaneous contraction and finite-reservoir exhaustion

Thermodynamic limits on disturbance-rejection authority in constrained Markov networks: instantaneous contraction and finite-reservoir exhaustion. We distinguish two physically separate mechanisms that limit disturbance rejection in a constrained Markov system maintained near active constraint boundaries by bounded locally detailed-balanced support channels. First, even with fixed channel architecture, finite instantaneous thermodynamic supply makes part of the raw kinetic authority unaffordable. Second, when support channels draw on finite reservoirs, using the channels depletes those reservoirs and changes the kinetic authority available later. We formulate both effects at a fixed constrained state, then expose what happens when the same support currents deplete finite inventories that determine channel affinities. A direction-resolved authority margin measures maximum inward support beyond current holding requirement. A local calculation gives the sign of cross-directional erosion: load in one direction can accelerate loss of bounded authority in another direction through shared-reservoir depletion. In an exactly solvable four-state realization with three instantaneously decoupled focal support directions and two shared reservoirs, we derive the complete exact-maintenance horizon as a closed-form function of load. The focal state and support currents remain fixed while required activities rise toward saturation. Along approach to kinetic exhaustion, resolved dissipation and instantaneous thermodynamic price both decrease—even though bounded authority vanishes. This separation demonstrates that present affordability and future authority change via depletion are mechanistically and informationally distinct, and that scalar dissipation or cost alone cannot alert to approaching loss of support capability.