The Fractal Stokes’ Theorem: Rigorously Definition, Proof, and Applications

This paper rigorously defines and proves the Fractal Stokes’ Theorem, a core tool in fractal geometry and topology. By introducing fractal manifolds, fractal chain complexes, and a new fractal boundary operator, we ensure the mathematical consistency of all terms. We provide a detailed proof of the theorem, focusing on measure convergence and commutativity of boundary mappings. Applications include stress tensor calculations in holographic duality and the study of fractal topological insulators in condensed matter physics. Through numerical simulations and algorithmic verification, the theorem’s correctness is demonstrated. References from Falconer, Connes, and others are cited for theoretical reliability.