Connectivity Engineering: Topological Enhancement of Quantum Entanglement Structure and Generalized Area Law by Fractal Geometry

Traditional holographic duality usually assumes a smooth and continuous spacetime background, where boundary entanglement strictly follows the area law. However, the real microscopic quantum gravity background may possess a non-trivial fractal structure. Within the framework of connectivity engineering, this paper investigates the regulatory mechanism of fractal networks (Sierpinski gasket) on quantum entanglement structures. Numerical results show that, unlike a one-dimensional gapped chain restricted to an extremely low area law constant, the entanglement capacity of the fractal network is significantly enhanced due to the presence of multiple branches and geometric shortcuts, approaching the upper bound of the Page curve at the bipartite symmetry point for this finite size. This topological resonance brought by self-similar hierarchical levels of the fractal demonstrates a generalized area law dominated by fractal dimension. This result proves that the fractal dimension of the underlying spacetime is the key geometric origin determining the long-range distribution of quantum information and the upper bound of entanglement capacity.