We extend the Einstein field equations to incorporate cross-layer coupling by promoting the stress-energy tensor T(i)T^{(i)}_{\mu\nu} of each layer to depend weakly on adjacent layer curvatures R(i1)R^{(i \pm 1)}:
G(i)+ig(i)=[T(i)+(F(R(i1))+F(R(i+1)))g(i)]G^{(i)}_{\mu\nu} + \Lambda_i g^{(i)}_{\mu\nu} = \kappa \left[ T^{(i)}_{\mu\nu} + \epsilon \left( \mathcal{F}(R^{(i-1)}) + \mathcal{F}(R^{(i+1)}) \right) g^{(i)}_{\mu\nu} \right]
Here:
1\epsilon \ll 1 is the inter-layer coupling coefficient,
F(R)\mathcal{F}(R) is a functional of Ricci scalar from adjacent layers, introducing non-local corrections to local dynamics.
This yields a multi-shell Friedmann equation:
(aiai)2=8G3i+i3kiai2+Hi2\left( \frac{\dot{a}_i}{a_i} \right)^2 = \frac{8\pi G}{3} \rho_i + \frac{\Lambda_i}{3} - \frac{k_i}{a_i^2} + \epsilon \, \delta H^2_i
where Hi2F(R(i1))+F(R(i+1))\delta H^2_i \equiv \mathcal{F}(R^{(i-1)}) + \mathcal{F}(R^{(i+1)}).
5. Layer Index as a Geometric Qubit
In alignment with the information-theoretic foundation of this theory, we consider each layer index ii to correspond to a quantized topological unit, analogized as a geometric qubit state:
Li=Hi,ki,i|L_i\rangle = |H_i, k_i, \rho_i\rangle
Transitions between layers are interpreted as quantum transitions between macroscopic geometric qubit states, with tunneling amplitudes mediated by non-perturbative topological instantons (see Section II.1).
6. Observational Implications of the Metric
