Derivation of Modified Einstein Equations from the Variation of the Hyperfield Lagrangian L_hyper and the THZ Action in the Framework of FQHP 4.0 (Preliminary Version)

This work presents a rigorous derivation of the gravitational field equations from the variational principle for the hyperfield Lagrangian L_hyper defined on the 5D shell ∂M₅. It is shown that upon compactification over the fifth coordinate and taking into account the fractal measure dμ_D (dimension D = 2.72), variation of the action with respect to the metric gμν yields the Einstein tensor Gμν with an effective cosmological constant Λ_eff. A direct relation is established between the hyperfield coupling constant ξ and Newton’s constant G. A physical interpretation of the hyperfield vacuum expectation value v² as a measure of information manifestness, related to the deployment parameter η = 0.72, is given. It is shown how the Einstein term emerges from compactification, how the cosmological constant arises from the potential V(Φ), and how modifications of GR manifest in gravitational waves. A numerical estimate of G through the hyperfield parameters is performed, confirming the self‑consistency of the theory. The work proves that gravity in the Fractal Quantum Holographic Paradigm is a projection of the tensor stress of the information hyperfield.