The Axiom of Topological Degrees of Freedom Formalization and Proof

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