We Unified the Response Laws of Life and Non-Life, and Discovered the Physical Meaning of τ₀

All biological and physical systems respond to stimuli. But until now, no unified law has described how these responses arise under fundamental constraints. In this work, we identify a universal response function that governs both living and non-living systems, emerging naturally under three general conditions: (1) thermodynamic resource limitation, (2) information-processing delay, and (3) control-theoretic stability. We prove that these constraints uniquely lead to a family of low-order rational functions, with the simplest form given by:\[\hat{S}(\hat{\tau}) = \frac{\hat{\tau}}{\hat{\tau}_0 + \hat{\tau}}\]This function captures the saturating, stable, and efficient dynamics found in enzymatic reactions, neural adaptation, behavioral responses, and even physical processes such as RC circuits and radioactive decay. Unlike exponential models, our formulation encodes explicit delay scales and bounded initial responsiveness, making it biologically and physically realizable. Through theoretical derivation and empirical validation, we demonstrate that this rational family constitutes the minimal mathematical grammar shared by all responsive systems in nature. We conclude that a universal response law exists—and it is rational.