Fractal Emergence of Spacetime and Gravity from a Unified Scalar Field: A Quantum and Cosmological Analysis of the Derivative Vacuum

We present a theoretical framework in which spacetime geometry, gravity, and gauge fields emerge as successive derivatives of a single real scalar field Φ defined over a five-dimensional manifold, including a fractal-like internal dimension χ. This model, termed the Derivative Vacuum Framework (DIM), proposes that temporal, gravitational, and gauge phenomena correspond to the first, second, and higher-order derivatives of Φ along χ, respectively. We construct a fundamental action with scale-dependent couplings and demonstrate, via Functional Renormalization Group (FRG) techniques, the existence of a non-Gaussian ultraviolet fixed point. In the infrared, the field dynamically relaxes to a maximally symmetric “derivative vacuum” state. Complementary simulations reveal emergent gauge-like tensor fields, persistent fractal scaling in power spectra, and long-term self-organization under dissipation. These findings are supported by analytical derivations and further correlated with astrophysical data from the Sloan Digital Sky Survey (SDSS), where a nontrivial fractal dimension D2 ≈ 2.5 emerges. The DIM model thus provides a geometrically grounded, renormalizable, and empirically testable pathway toward unification, supported by both numerical and observational evidence. A detailed companion document includes full simulations, derivations, and data analyses.