Derivation of Modified Einstein Equations from the Variation of the Hyperfield Lagrangian Lₕyper and the THZ Action in the Framework of FQHP 4. 0 (Preliminary Version)

Abstract. This work presents a rigorous derivation of the gravitational field equations from the variational principle for the hyperfield Lagrangian Lₕyper defined on the 5D shell ∂M₅. It is shown that upon compactification over the fifth coordinate and taking into account the fractal measure dμD (dimension D = 2. 72), variation of the action with respect to the metric g⏛⏜ yields the Einstein tensor G⏛⏜ with an effective cosmological constant Λₑff. 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.