Fractal Substrate Theory: Emergence of Spacetime, Holography, Black Hole Entropy, and Complexity

We present a discrete, fractal-driven informational substrate that unifies the emergence of spacetime geometry, gauge symmetries, black hole entropy, holographic dualities, and integrated complexity. Our approach embeds self-similar local update rules within a four-dimensional lattice, enabling Lorentz invariance and holographic entanglement to arise without imposing geometry or gauge structures a priori. By analyzing this lattice through a renormalization group (RG) flow that respects fractal scaling, we demonstrate that black hole thermodynamics and integrated complexity measures, including Integrated Information Theory’s Phi, emerge naturally from the substrate’s causal structure. The theory’s limits reproduce Loop Quantum Gravity (LQG) and causal set frameworks, illustrating how fractal principles may underpin diverse quantum-gravitational models. Additionally, Godelian undecidability arises from the universal computation capacity of our self-similar substrate, highlighting fundamental epistemic constraints at cosmic and cognitive scales. Our results suggest fractal emergence as a guiding principle for unifying quantum field theory, holography, black hole entropy, and complexity science.