Thermodynamic, Informational, and Control Constraints Select a Universal Rational Response Function

Biological responses across molecular, neural, and behavioural scales obey common energetic and informational limits. By casting energy , mutual information and closed-loop error into a dimensionless balance , and imposing saturation, monotonicity and concavity, we show that the unique closed-form solution is the rational function . The result unifies Michaelis–Menten kinetics, cortical gain control and hyperbolic discounting without additional parameters, and predicts a single dimensionless stiffness encoding each system’s free-energy cost per bit of information