To represent the ensemble of all such layers, we consider a 5-dimensional embedding manifold M5\mathcal{M}^5, within which the 4D layers i\Sigma_i are embedded as hypersurfaces:
M5=i=1Niwithi:fi(x)=0\mathcal{M}^5 = \bigcup_{i=1}^{N} \Sigma_i \quad \text{with} \quad \Sigma_i : f_i(x^\mu) = 0
Each layer i\Sigma_i is defined by a level set fi(x)f_i(x^\mu), with the induced metric on each layer given by the pullback of the 5D bulk metric gAB(5)g^{(5)}_{AB} via:
g(i)(4)=XAxXBxgAB(5)g^{(4)}_{\mu\nu (i)} = \frac{\partial X^A}{\partial x^\mu} \frac{\partial X^B}{\partial x^\nu} g^{(5)}_{AB}
This structure allows inter-layer causal and informational connectivity governed by bulk geometrical and topological constraints.
3. Inter-Layer Dynamics and Hubble Variation
We postulate a differential Hubble flow across layers:
Hi(t)=Href(t)e(ii0)H_i(t) = H_{\text{ref}}(t) \, e^{-\alpha (i - i_0)}
where:
HrefH_{\text{ref}} is a reference expansion rate (e.g., for the central layer),
\alpha is a Hubble gradient parameter controlling how expansion decays (or grows) across layers,
i0i_0 denotes the central or observer layer.
This ansatz leads to radial Hubble stratification, naturally accounting for variable Hubble measurements across cosmic voids, over-dense regions, or redshift ranges.
4. Curvature Coupling Across Layers
