Chaos Without Infinity: A Falsifiable Bridge from Wolfram to Computational Finitism

We acknowledge that in our paper (Ramos 2026b), the resolution of the chaos paradox draws on Wolfram's computational paradigm, but we extend it into a rigorously falsifiable physical framework. Where S. Wolfram(2002, 2020) demonstrated that discrete rules can generate complex behavior, we prove that finite alphabets enforce non-positive Lyapunov spectra, making true mathematical chaos topologically impossible. Using Kaprekar dynamics as a testbed, we derive a discrete sensitivity exponent for all digit lengths , showing that apparent chaos emerges from timescale separation in astronomically large but bounded state spaces. We present three falsifiable predictions that distinguish computational finitism from both continuum chaos theory and Wolfram's unfalsifiable paradigm: (1) finite predictability horizons scaling as , (2) asymptotic decay of effective Lyapunov exponents toward zero, and (3) recurrence signatures in quantized long-term simulations. Chaos is not a violation of finitism; it is finitism operating at the edge of computational irreducibility.