Fractal AdS Solution Existence Proof and Global Properties: Fractional Connections and AdS/CFT Correspondence

By modifying the metric structure of fractal AdS spacetime, this paper rigorously proves the existence of solutions that satisfy Einstein’s field equations. A fractional connection system is constructed based on Caputo fractional derivatives, establishing a covariant fractional Bianchi identity. Through the analysis of null geodesics and conformal compactification structures, the global hyperbolicity of the spacetime is verified. Furthermore, a microscopic model for fractal conformal field theory (CFT) is proposed, predicting the dimensional dependence of the quasi-normal mode spectrum of fractal AdS black holes. This study provides a rigorous mathematical framework for the integration of fractal geometry with the AdS/CFT correspondence. Keywords: Fractal AdS spacetime; fractional differential geometry; Bianchi identity; conformal field theory; quasi-normal modes