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

We propose a unified theoretical framework in which spacetime, gravity, and gauge interactions emerge from successive derivatives of a single real scalar field defined over a discrete, fractal-like extra dimension. This approach—called the Derivative-Infinitesimal Matrix (DIM) theory—describes time as the first derivative, gravity as the second, and gauge fields as modulated higher-order derivatives of the field with respect to the fractal coordinate. Using a five-dimensional action with scale-dependent geometry, we analyze the renormalization flow via functional renormalization group methods and identify a non-Gaussian UV fixed point. In the infrared, the system dynamically relaxes toward a symmetric, structureless "derivative vacuum." Numerical simulations support this attractor behavior and reveal log-periodic signatures in the power spectrum. The model leads to testable predictions, including submillimeter deviations from Newtonian gravity and late-time cosmological acceleration without fine-tuning.