\varepsilon: A small coupling constant quantifying the strength of inter-layer topological interference.
ij\alpha_{ij}: Coupling coefficient between layer ii and jj, reflecting geometric overlap, causal contact, or shared topological boundary.
i\theta_i: Topological phase angle associated with layer ii, emerging from path integral formalism or Berry-like curvature contributions of the quantum cosmological state.
2. Physical Interpretation
This equation extends the traditional Friedmann dynamics by encoding two key effects:
Layer-Dependent Evolution: Each region or layer evolves with its own curvature kik_i, density i\rho_i, and scale factor aia_i, allowing for a heterogeneous and anisotropic large-scale cosmology.
Phase Interference Coupling: The term
jiijcos(ij)\varepsilon \sum_{j \neq i} \alpha_{ij} \cos(\theta_i - \theta_j)
represents constructive or destructive interference between adjacent cosmological layers, analogous to coupled oscillators or Josephson-like interactions in condensed matter systems. This term can modulate the local expansion rate based on global topological alignments.
3. Emergent Effects
The interference term introduces oscillatory behavior or modulations in expansion history, potentially accounting for:
Apparent Hubble constant variation in different directions (explaining the Hubble tension as a layer-local effect),
Low-\ell CMB anomalies through coupling-induced inhomogeneities,
