Mathematical Framework of (∞,2)-category for Describing AdS and DS Duality

This paper proposes a mathematical framework for AdS/DS duality based on (∞,2)-category theory. By integrating the Langlands program, noncommutative geometry, and topological quantum field theory, it establishes a higher homotopy correspondence between fractal spacetime and quantum gravity. In this framework, AdS spacetime and fractal DS spacetime form a duality through a holographic embedding functor. Fractional diffeomorphism mappings regulate the fractal dimensions, and a curvature-entanglement entropy feedback mechanism realizes the covariant evolution of spacetime geometry and boundary quantum states. The theory predicts the logarithmic periodicity of fractal entanglement entropy, the Chern class generation mechanism of dark energy, and the observability of discrete graviton spectra, providing a rigorous mathematical foundation and experimental verification paths for quantum gravity.