Thermodynamic, Informational, and Control Principles Shape Systemic Response Functions Across ScalesScaling laws

Biological, cognitive, and socioeconomic systems often display saturating responses to external stimuli. We propose that a unified class of rational response functions emerges naturally when three fundamental constraints—thermodynamic resource limits, informational encoding cost, and feedback-control stability—are jointly imposed. By minimizing marginal energetic, informational, and control cost, we derive a universal, dimensionless function: R(τ) = kτ/1 + kτ This saturating, concave, and monotonic curve appears across diverse domains, including enzyme kinetics, neural gain control, behavioral discounting, and agricultural labor efficiency. We advance this rational form as a null model of bounded optimality. Deviations from it—such as oscillations, bifurcations, or memory effects—may signal additional system complexity. This work provides a minimal mathematical grammar for systemic adaptation, analogous to Newton’s laws for motion.