The Fractal Horizon: Entropy, Dimensional Collapse, and the Finite Alphabet of Reality

Modern theoretical physics assumes that fractals require infinite recursion and that higher-dimensional algebras (32D, 64D, 256D) offer new physics. Computational Finitism proves both assumptions are false. By applying the finite alphabet {0,1, … ,9} to fractal generation, we demonstrate that complexity emerges from bounded recursion, not infinite depth. We then show that the macroscopic phenomenon of entropy is the signature of octonionic non-associativity at the Planck scale. Finally, we map the Cayley-Dickson tower beyond 16D, revealing that dimensions 32-256 do not contain "new physics"; they represent a pathological collapse of algebraic coherence into structural noise. Reality is finite, self-similar within bounds, and capped at the octonion horizon.