Thermodynamics Constrains Life Systems to Rational Response Functions

This work proposes a universal mathematical constraint on the response functions of living systems. By starting from three core physical principles—finite energy input, inevitable information delay, and monotonic adaptive dynamics—we rigorously prove that all non-chaotic biological response functions must asymptotically converge to a low-order rational function of the form:S(\tau) = \frac{k \tau}{\tau_0 + \tau}This minimal rational structure simultaneously satisfies saturation, optimization, and stability conditions. It unifies a wide spectrum of empirical response curves across molecular, neural, and behavioral systems. Our theoretical results align with classical biological laws such as Michaelis-Menten kinetics and Herrnstein’s matching law, and are further supported by modern findings from neuroscience, sensory adaptation, and information thermodynamics.