A Unified Law of Biological and Physical Response Functions Reveals the Physical Meaning of τ₀

Across both living and non-living systems, a wide range of phenomena—from enzymatic reactions and neural adaptation to thermal relaxation and radioactive decay—exhibit saturating response behaviors. In this work, we demonstrate that a single rational response function of the form Ŝ(τ̂) = τ̂ / (τ̂₀ + τ̂) emerges as a universal solution when three fundamental constraints are simultaneously imposed: (1) thermodynamic resource limitation, (2) finite information-processing delay, and (3) control-theoretic stability. This function not only captures biological adaptation but also approximates classical physical systems under idealized assumptions. Most critically, we identify the parameter τ₀ as encoding an irreducible latency inherent to any responsive system—whether molecular, neural, behavioral, or physical—marking a foundational quantity that bridges life and physics. Our results suggest that the rational saturating response is not merely empirical but arises from first principles, revealing a deep structural constraint governing all systems capable of adaptation.