Lagrangian of the Hyperfield on the 5D Shell A Unified Action for the Fractal Quantum Holographic Paradigm (FQHP 4.0, Preliminary Version)

Abstract.We propose a unified hyperfield Lagrangian defined on the 5D shell – the boundary hypersurface between our 3+1‑dimensional spacetime and the deep fractal hierarchy in the Fractal Quantum Holographic Paradigm (FQHP). The hyperfield unifies all known fields: the rendering field, the information field (informaton), and the gauge fields. It is shown that after compactification of the fifth dimension and averaging over the fractal measure, the Lagrangian yields the modified Trojan–Hopfield–Zi (THZ) action and the Einstein equations with an effective cosmological constant. The deployment parameter η = 0.72 and the fractal dimension D = 2.72 emerge naturally from the geometry of the 5D shell. This work lays the theoretical foundation for FQHP version 4.0 and opens a path toward unifying quantum gravity, information, and fractal geometry.