The Axiom of Topological Degrees of Freedom Formalization and Proof

In this paper, we formally establish and rigorously prove the Axiom of Topological Degrees ofFreedom. The axiom asserts that intrinsic structural complexity in a topological configurationadmits a stable and well-defined weighted aggregation of local scaling exponents.We construct a precise functional framework in which this principle is formulated explicitlyas an axiom. Within this setting, we demonstrate that the weighted effective dimension ismathematically well-defined, bounded, continuous, and stable under perturbations.This establishes the internal consistency and functional realizability of the axiom, providinga rigorous foundation for the concept of topological degrees of freedom as an intrinsic invariant.