This paper establishes the computational and number-theoretic foundations for fractal expansion within the framework of Computational Finitism. Building upon the discrete topology of the Planck voxel (Fano plane geometry, 144 total channels, 137 electromagnetic channels), we define the four rigorous properties of the "Finitism Fractal." To model the expansion of the discrete alphabet into macroscopic continuous geometry, we construct the Golden-Fano Cellular Automaton (Rule GF). Through exhaustive simulation across multiple numeral bases (2, 8, 10, 12, 16) and growth dynamics (Golden, Silver, Bronze, Copper ratios, , and ), we demonstrate that maximal information density is governed by number-theoretic coprimality. We prove that the Golden Ratio () is not an arbitrary aesthetic choice, but the unique computational optimum for a discrete, Fano-constrained universe. Because is the "most irrational" number; it avoids the destructive arithmetic resonance traps that plague more rational growth ratios. These findings provide a rigorous computational complement to Pellis(2025)-Fractal Information Theory (PFIT), grounding its -scaled physical constants in the fundamental mechanics of discrete information packing.
Nestor Ramos (Fri,) studied this question.