A natural explanation for galactic angular momentum bias and void topology, as discussed in later sections.
The fractal dimension DHD_H becomes a dynamical parameter, potentially evolving over cosmic time, reflecting the degree of matter clustering and gravitational complexity. This connects the emergent geometry of the Universe with the informational entropy flow, consistent with the holographic and topological foundations developed in previous sections.
III. Mathematical Formulation
A. Modified Friedmann Equation with Layer Coupling and Topological Phase Interference
To account for the multi-layered structure of the Universe and topological phase interactions proposed in this framework, we introduce a generalized Friedmann equation for each spacetime "layer" or domain ii. This formulation retains the standard terms of general relativity but incorporates interference contributions from adjacent or causally linked layers.
We define the modified Friedmann-like evolution for the Hubble parameter Hi=aiaiH_i = \frac{\dot{a}_i}{a_i} in layer ii as:
Hi2=8G3ikiai2+jiijcos(ij)H_i^2 = \frac{8\pi G}{3} \rho_i - \frac{k_i}{a_i^2} + \varepsilon \sum_{j \neq i} \alpha_{ij} \cos(\theta_i - \theta_j)
1. Terms Explanation
i\rho_i: Effective matter-energy density in the ii-th layer, potentially exhibiting fractal scaling i(r)rDHi3\rho_i(r) \sim r^{D_{H_i} - 3}.
kik_i: Spatial curvature parameter of the ii-th layer (normalized to 0, +1, or --1), possibly varying across layers.
aia_i: Scale factor of the ii-th cosmological layer.
