The Algebraic Origin of Gauge Symmetries and the Bare Fine-Structure Constant in Computational Finitism

In the companion paper (Ramos 2026 b), we established that the universe is a discrete causal graph where singularities are replaced by topological saturation, governed by a maximum spatial connectivity of = 24 (the 4D kissing number) and an internal computational capacity of = 6. However, the physical origin of the 1 + 2 + 3 = 6 register partition, and the derivation of the fine-structure constant () remained numerically suggestive but algebraically incomplete. In this paper, we provide the rigorous algebraic and computational derivation. We introduce the Reserved State Hypothesis, identifying the maximum digit of our Base-10 alphabet ('9') as the discrete analog of infinity (the saturation state), leaving exactly = 9 usable states (0-8) per register. We demonstrate computationally that this specific alphabet size is the minimum required for stable quantum interference, and that the 1+2+3 partition is the only configuration that maintains strict gauge locality. Finally, we derive the bare fine-structure constant as = 1/(× 6) = 1/144, identifying the 5% discrepancy with the experimental 1/137as the exact contribution of QED vacuum polarization