The Emergence of Three-Dimensional Space from Quantum Connectivity in Unified Quantum Gravity

Here, we demonstrate that three-dimensional space emerges from quantum connectivity in Unified Quantum Gravity (UQG), where the fundamental degrees of freedom N = 43 form a quantum network. Through spectral dimension analysis, we show that the effective dimensionality grows from ds ≈ 0.03 at N = 43 to ds ≈ 0.78 at N = 4000, representing a 28-fold increase. Asymptotic extrapolation reveals a logarithmic scaling law with d∞ ≈ 1.34 for N → ∞. More strikingly, spectral embedding (Laplacian eigenmaps) reveals that the three-dimensional structure is inherently encoded in the connectivity matrix from the beginning: the seed network (N = 43) already exhibits 96% sphericity, which refines to 99.9% at N = 1500. This resolves an apparent contradiction: dynamically, space starts as "quantum foam" with low spectral dimension, but topologically, the 3D structure is present from the outset. We interpret this as evidence that N = 43 contains the architectural blueprint of a 3D universe, and cosmic expansion (increasing N) does not create space but rather "fills in" the pre-existing structure, making it a navigable continuum. This explains why the universe has three spatial dimensions: it is the maximum entropy configuration for a random unitary interaction network (GUE ensemble). Our results bridge random matrix theory, spectral graph theory, and quantum gravity, providing a concrete mechanism for dimensional emergence.