arXiv2026
At equilibrium, the ratio of two steady-state probabilities is a Boltzmann factor, set by a free-energy difference. Such ratios are the natural, normalization-independent readouts of both thermodynamics and information processing, and are often used as a measure of fidelity in biophysical systems. What becomes of these ratios once a system is driven from equilibrium, where the Boltzmann factor no longer holds? Representing Markov processes as graphs and their steady states as averages over a distribution on spanning trees, the \emph{arboreal distribution}, we track the ratio $π_i/π_j$ under driving along \emph{energetic edges}, where detailed balance is broken, relative to its equilibrium value. Our central finding is that a chosen ratio can stay exactly locked to its equilibrium value arbitrarily far from equilibrium, a phenomenon we call \emph{energetic protection}, whenever an algebraic equality between spanning-tree weights holds. Just as detailed balance constrains rates around cycles, energetic protection constrains weights across trees, providing a new mechanism for robustness against fluctuations in the driving force, such as variations in ATP concentration. Away from this equality, single-edge driving collapses the response onto two arboreal coefficients and forces it to be monotonic, so nonmonotonic single-edge control is impossible at any strength. With two energetic edges, protection and monotonicity combine into a \emph{thermodynamic switch} that holds a function at its equilibrium value for as long as desired and releases it sharply. Equilibrium is known for the limits it places on information processing. We show that new constraints, both no-go principles and exact invariances, survive far from equilibrium. These results reveal how the localization of energy expenditure governs the functional logic of nonequilibrium systems in physics and biology.