Emergent Quantum Mechanics from a Finite Alphabet: Computational Verification of Mass, Forces, and Gauge Symmetries in a Finite Substrate

Standard Quantum Field Theory and General Relativity rely on continuous mathematics and infinite limits, leading to pathologies such as singularities, renormalization divergences, and the cosmological constant problem. We present a complete computational verification of Computational Finitism, a discrete physics framework where the universe operates on a finite alphabet (Base-arithmetic with states 0 to − 1), and "infinity" is replaced by structural completion at the maximum digit. Through six rigorous numerical simulations, we demonstrate that: (1) massless wave propagation emerges at ≈ 0.9540 from Modulo-9 discrete wave equations; (2) rest mass arises as computational inertia from topological defects with internal state cycling; (3) the particle mass hierarchy (photon > neutrino > electron > muon) emerges from fractional bandwidth allocation of the fundamental 1/137 channel; (4) fundamental forces (Compton scattering, Coulomb repulsion, matter-antimatter annihilation) arise from nonlinear arithmetic interference without continuous fields; (5) gauge symmetries manifest as volumetric information inflation of the active vacuum; and (6) the framework is base-independent, functioning identically across Base-10, Base-16, and other finite alphabets, while Base-2 fails to support fermionic matter, revealing a minimum alphabet constraint. We derive the fine-structure constant ≈ 1/137 endogenously from the geometric constraints of the Fano plane (7 vacuum maintenance channels) and the algebraic 1+2+3 register partition, proving that gauge locality requires exactly 137 active channels. All simulations are reproducible, falsifiable, and require no continuous mathematics, infinite limits, or ad-hoc renormalization. Computational Finitism resolves the singularities of black holes and the Big Bang, eliminates the need for the Higgs field and virtual particles, and provides a complete, computably finite foundation for 21st-century physics.