What question did this study set out to answer?

The research aims to establish a universal scaling law that describes system responses to increasing complexity.

May 3, 20260 citationsOpen Access

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

Key Points

The research aims to establish a universal scaling law that describes system responses to increasing complexity.
Derived a universal scaling law governing system responses to effective degrees of freedom.
Identified a critical invariant threshold, Π 1 Π=1, to predict system behavior changes.
Demonstrated applications in diffusion processes, machine learning, and entropy production.
The scaling law successfully describes superdiffusion in porous media, showing enhanced behavior.
Indicates overparameterization collapse in neural networks, leading to suppressed performance.
Captures entropy runaway in small systems, illustrating increased instability under complexity.

Abstract

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.

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