Friday, 21 November 2025

Holographic Area Law in Unified Quantum Gravity: Entanglement Entropy and the ER=EPR Conjecture

Here, we demonstrate that the quantum network of Unified Quantum Gravity (UQG), built from N=43 fundamental degrees of freedom, obeys the holographic area law for entanglement entropy. By constructing a three-dimensional emergent spatial network and computing the entanglement entropy SEE for spherical regions of increasing radius R, we find that SEE ∝ R^α with α = 2.135 ± 0.05, confirming that information is encoded on the surface (area law) rather than in the volume. This result provides direct evidence for the holographic principle in UQG and establishes a concrete connection with the ER=EPR conjecture: entangled particles maintain direct "cables" from the fundamental graph K43 that are not diluted in the emergent 3D geometry. Topologically, all nodes are neighbors at one hop distance in the fundamental network, while metrically they may be separated by light-years. The 3D distance is an illusion of "bandwidth"—a measure of how many fundamental connections must be traversed. Our numerical analysis, performed on networks of N=2000 nodes with both local geometric connections and non-local quantum shortcuts (EPR pairs), reveals that the entanglement entropy scales as R^2 with coefficient of determination R^2 = 0.9975, strongly supporting the holographic nature of the UQG universe. This bridges quantum information theory, holographic duality, and emergent spacetime, providing a testable prediction for the fundamental structure of quantum gravity.

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