On the Correlations of Space and Time at the Planck Scale

This paper investigates a theoretical framework in which continuous spacetimeemerges from a discrete quantum information network at the fundamental scale.By modeling the foundational layer as a random geometric graph governedby a graph Laplacian and applying a continuum limit via Taylor expansion,the analysis demonstrates how an isotropic spatial Laplacian and emergentLorentz symmetry can arise at low energy regimes. The system dynamics areevaluated through a stochastic Langevin equation rooted in the fluctuation-dissipation theorem, yielding a sinusoidal dispersion relation for the discretenetwork. While low-energy excitations approximate conventional wave opera-tors with microscopic suppression factors, energies approaching the Planck limitlead to vanishing group velocities and Anderson localization effects. This mecha-nism suggests that spatial translation may break down at extreme scales, leavingabsolute time as a fundamental dimension while emergent spatial relativitydissolves.