Ukachi Degrees-of-Freedom Scaling Law: A Universal Framework for Complex Systems

We derive a universal scaling law that governs how systems respond as their effective degrees of freedom increase. The law reveals a critical invariant threshold, Π 1 Π=1, which predicts whether added complexity amplifies or suppresses system behavior. Applications in diffusion processes, machine learning, and entropy production demonstrate that this single framework captures phenomena as diverse as superdiffusion in porous media, overparameterization collapse in neural networks, and entropy runaway in small systems. The work offers a first-principles, cross-domain foundation for understanding complexity-driven phase transitions.