The Ukachi Scaling Laws: Fundamental Limits in Complex Systems

The Ukachi Scaling Laws: Fundamental Limits in Complex Systems We present the Ukachi Scaling Laws, a family of scaling relations describing system responses to exponentially growing inputs. From three complementary first-principles derivations (random-walk state exploration, renormalization-group scaling, and information-theoretic processing limits) we derive the canonical Ukachi form R(I) = k[logI]α, explain its domain of validity, and introduce a hybrid extension R(I) = k1[logI]α +k2Iβ, which couples information-limited amplification with classical allometric (power-law) behavior. We test the forms against simulated real-world datasets motivated by biology, physics, economics and neuroscience, present best-fit constants and goodness-of-fit metrics, and discuss the conceptual novelty and problems solved by this universality class. The manuscript includes reproducible fitting code snippets and tables with the simulated-data comparisons