Fractals and Finitism: A Computational Foundation for Emergent Spacetime

We present a rigorous computational framework demonstrating that the universe's discrete Planck-scale substrate necessarily generates macroscopic 3D space and 4D spacetime through φ-fractal information dynamics. Through four computational milestones, we prove: (1) the universe achieves a Kolmogorov compression ratio of ~10⁹²:1 by encoding macroscopic geometry via fractal generators rather than explicit voxel storage; (2) topological redundancy (the Topological Redundancy Buffer (TRB)) acts as an error-correcting mechanism, maintaining antifragile entropy Sφ ≈ 0.988 under quantum noise; (3) the 1+2+3 gauge branching rules (U(1), SU(2), SU(3)) physically construct 3D space (V ∝ t³) and 4D spacetime (V ∝ t⁴); and (4) the Fano-φ-1+2+3 topology is uniquely selected as the only configuration satisfying the Triad of Computational Constraints (compression, antifragility, and dimensionality). This work recontextualizes the holographic principle as a local, dynamical process and establishes Computational Finitism as a mathematically rigorous foundation for emergent spacetime, bridging discrete information theory with continuous physical geometry.