To ensure physical consistency, we impose generalized Israel junction conditions across layer boundaries:
[KhK]ij=8GS(ij)\left[ K_{\mu\nu} - h_{\mu\nu} K \right]_{\Sigma_{ij}} = -8\pi G S_{\mu\nu}^{(ij)}
Where:
KK_{\mu\nu} is the extrinsic curvature of the boundary surface ij\Sigma_{ij},
hh_{\mu\nu} is the induced metric,
S(ij)S_{\mu\nu}^{(ij)} is the surface stress tensor arising from field discontinuities.
Such matching conditions ensure that while metric derivatives may jump, the underlying energy conservation and field continuity is preserved.
6. Observable Consequences and Testability
The Multilayer model predicts:
Direction-dependent Hubble constants, even after correcting for local structure,
Large-scale parity violation in galaxy clustering,
Super-voids or over-densities at layer junctions,
Low-multipole anomalies in the CMB as phase interference shadows from adjacent layers.
These predictions are testable by:
Cross-correlating cosmic shear with CMB lensing dipoles,
Mapping void distributions via deep field surveys (e.g., Euclid),
Reconstructing Hubble flow anisotropies with Type Ia supernovae and BAO measurements.
In summary, the Multilayered Topological Architecture generalizes standard cosmology by introducing a stacked spacetime structure, in which local FRW geometries are connected via topological interfaces and governed by layer-specific field dynamics. This layered cosmology provides a coherent framework for addressing Hubble tension, cosmic anisotropies, and void-related anomalies, while offering testable predictions for current and upcoming surveys.
II.3. Fractal Geometry of Cosmic Matter
While the CDM model assumes homogeneity and isotropy on large scales (beyond ~100 Mpc), growing evidence suggests that the distribution of matter---both luminous and dark---exhibits persistent self-similar structures at larger scales than previously anticipated. To reconcile this with observed cosmic anisotropies and early galaxy formation, we propose that the Universe exhibits a fractal-like internal geometry characterized by non-integer effective dimensions, void hierarchies, and clustering patterns obeying scale-invariant laws.
1. Fractal Dimensional Analysis
