Quantum Gravity from Fractal Entanglement Geometry

In this paper we propose that spacetime is an emergent fractal geometry generated by the entanglement structure of an underlying quantum information network. Indeed, it is developed a framework in which spacetime, quantum mechanics, and gravity emerge from the entanglement structure of a universal quantum state. Geometry is defined by an information-theoretic distance $d_{ij}=-\ell_0\log(I_{ij}/I_0)$ on an entanglement graph, producing a scale-dependent, fractal spacetime whose effective dimension flows toward $D\to 2$ near the Planck scale. In this fractal geometry, nondifferentiable trajectories lead to stochastic geodesics and a complex covariant derivative, from which the Schr\"odinger equation follows as an emergent dynamical law. Gravity arises from the time dependence of the entanglement-induced metric, yielding Einstein gravity in the macroscopic limit and fractal corrections encoded in a generalized field equation $G_{\mu\nu}=8\pi G(T_{\mu\nu}+\alpha E_{\mu\nu}+\beta F_{\mu\nu})$. The resulting \emph{Fractal Entanglement Quantum Gravity} (FEQG) framework predicts dimensional reduction, modified gravitational potentials, and possible deviations from standard quantum mechanics at ultrashort scales, offering a unified informational origin for quantum theory and gravitation.