Weighted Effective Dimension and Structural Describability: A Functional Realization of Dimensional-Structural Describability

This paper develops a functional realization of Dimensional-Structural Describability through the weighted effective dimension. The conceptual foundation of the framework treats topological degrees of freedom as admissible structural conditions under which describability, boundary, continuity, connectivity, separation, and accessibility become meaningful within a descriptive regime. The present work does not attempt to prove the physical necessity of that axiom as a theorem. Instead, it establishes that the structural aggregation associated with the axiom admits a stable mathematical realization under explicit metric-measure assumptions. An admissible structural configuration is represented by a metric-measure space \ ( (X, d, ) \). Local structural contributions are encoded by a local scaling exponent \ ( (x) \), while admissible structural emphasis is represented by a normalized nonnegative weight \ (w (x) \). The resulting weighted effective dimension \ Dw=X (x) w (x) \, d \ is interpreted as a regime-indexed structural descriptor rather than as an ordinary spatial dimension, a new physical degree of freedom, or the totality of topological degrees of freedom itself. Under the standing assumptions that the local scaling exponent exists almost everywhere and belongs to \ (L^ (X, ) \), the weighted effective dimension is shown to be well defined, finite, bounded by the essential range of \ (\), continuous with respect to admissible structural weights, stable under bounded perturbations of local scaling exponents, and invariant under scaling-structure preserving equivalences with consistent transport of measures and weights. These results provide a static functional foundation for later time-dependent, dynamical, and physical extensions, without deriving those later extensions within the present paper.