The Finite Horizon: How Computational Finitism Replaces Infinity with Structural Completion

Modern theoretical physics is plagued by singularities, points where the mathematics of the continuum breaks down, predicting infinite curvature, infinite density, or infinite time dilation. Roger Penrose's Conformal Cyclic Cosmology (CCC) relies on one such singularity: the assertion that massless particles traveling at experience infinite time dilation, effectively stopping time and erasing scale, thereby allowing the "end" of the universe to geometrically identify with the "beginning." We refute this by applying the principles of Computational Finitism. We demonstrate that in a discrete, pixelated universe governed by a finite alphabet (= 9), or any other base 2,16, 60, etc., all divergent series saturate at a finite limit. We unify four critical domains of physics and mathematics under this framework: Black Holes, Time Dilation, Chaos, and Number Theory. By regularizing the Lorentz factor and the Schwarzschild metric, we show that the universe does not loop. It is a single, finite computational process that reaches a stable state, not a singularity.