Fractality governs intra-layer matter distribution,
Layer transitions encode inter-layer anisotropy and non-uniformity,
The effective dimensionality DH(i)D_H^{(i)} varies with cosmic layer index ii, yielding direction-dependent observational effects.
This naturally explains:
The early appearance of massive galaxies, as denser fractal zones condense earlier,
The asymmetry in void distributions, as layer curvature and expansion history vary,
The fractal nature of entropy growth, as self-similar structure implies hierarchical energy dispersion across scales.
The Fractal Geometry of Cosmic Matter introduces a nonlinear, scale-invariant component to our cosmological model, challenging the assumption of large-scale homogeneity. This fractal structure is encoded both in the spatial clustering of galaxies and the distribution of dark matter halos, and is amplified by our multilayered topological universe. Together, they yield testable predictions---such as persistent anisotropies, non-Gaussian CMB features, and directional scaling laws---that can be verified using high-resolution surveys and lensing data.
III. Mathematical Formulation
III.1. Multilayer Spacetime Metric Tensor
To describe a universe composed of multiple nested cosmological layers---each with potentially distinct expansion dynamics, curvature properties, and topological signatures---we introduce a generalization of the standard Friedmann--Lematre--Robertson--Walker (FLRW) metric. Our Multilayer Spacetime Framework is constructed on the premise that the observable universe is a quasi-3+1-dimensional projection of an internally layered structure, each layer corresponding to a distinct hypersurface of evolution within a higher-dimensional, information-theoretic manifold.
1. Generalized Multilayer Metric
We define the metric for the i-th cosmological layer as:
dsi2=c2dt2+ai2(t)[dr21kir2+r2d2]ds_i^2 = -c^2 dt^2 + a_i^2(t) \left[ \frac{dr^2}{1 - k_i r^2} + r^2 d\Omega^2 \right]
Where:
ai(t)a_i(t) is the scale factor for the i-th layer,
ki{1,0,+1}k_i \in \{-1, 0, +1\} encodes the spatial curvature of that layer,
d2=d2+sin2d2d\Omega^2 = d\theta^2 + \sin^2\theta \, d\phi^2 is the standard spherical element.
This formulation permits layer-dependent curvature and expansion, allowing each shell to evolve with its own Hubble parameter Hi(t)=ai/aiH_i(t) = \dot{a}_i / a_i.
2. Nested Manifold Structure and Layer Embedding
