Scaling Laws and Multifractal Spectrum Transformation Under Fractional Calculus: Theory and AI-Based Identification

Fractional calculus modifies the geometric roughness of functions and alters global fractal dimensions in a monotonic manner. However, a comprehensive description of how fractional operators transform full scaling structures and multifractal spectra remains incomplete. In this paper, we establish a unified scaling-law theory describing the action of Riemann- Liouville and Weyl-Marchaud fractional operators on local regularity, structure functions, and multifractal spectra. We prove that fractional differentiation of order [Formula: see text] induces a translation of local Hölder exponents by [Formula: see text], yielding an exact shift law for the multifractal spectrum. Furthermore, we derive an affine transformation rule for structure-function scaling exponents, showing that fractional operators generate linear deformations of scaling laws across statistical moments. An AI-assisted identification framework is introduced to recover fractional order directly from observed fractal signals. Numerical experiments on synthetic multifractal processes confirm the theoretical predictions and demonstrate the robustness of the proposed method. These results provide a geometric and scaling interpretation of fractional calculus and establish a unified framework connecting fractal geometry, scaling laws, and artificial intelligence.