Lagrangian of the Hyperfield in 3+1 Dimensional Spacetime A Unified Action for the Fractal Quantum Holographic Paradigm (FQHP 4.0, Preliminary Version)

Abstract.We construct a hyperfield Lagrangian directly in 3+1‑dimensional spacetime, without introducing extra dimensions. The hyperfield unifies the rendering field, the information field (informaton), and the gauge fields. The fractal structure of reality is taken into account through a fractal measure and Caputo fractional derivatives of order α = D – 2, where D = 2.72 is the fractal dimension of deep packing. Variation of the Lagrangian yields the modified Trojan–Hopfield–Zi (THZ) action and the Einstein equations. The deployment parameter η = 0.72 emerges as the ratio of the norms of the real and imaginary parts of the hyperfield. This work provides an economical and physically transparent alternative to models with extra dimensions, laying the foundation for FQHP version 4.0.