Derivation of the Electron from the Fano-TRB Substrate: Five Rigorous Results

Abstract

We derive five properties of the electron from the specific discrete substrate of Computational Finitism: the fine-structure constant α = 1/137, the 12-vertex icosahedral shell, the 13-fold twist of the Möbius topology, spin-1/2, and the Zitterbewegung frequency. The substrate is a finite causal graph of Planck voxels governed by a Modulo-9 alphabet, a Fano plane of 7 points and 7 lines, and a 9-dimensional Fano-TRB algebra over ℤ/9ℤ whose primitive operation is non-commutative and non-associative. We first document a negative result: twenty scalar Modulo-9 cellular automata across 1D, 2D, and 4D lattices cannot produce chirality or localized defects, and conclude that the substrate is algebraic, not scalar. We then construct the correct algebra explicitly and derive the five results. We distinguish two types of derivation: counting arguments (using native Finitism numbers) and algebraic theorems (using the structure of the Fano-TRB algebra). Both are rigorous; they differ in source, not in strength. We document four remaining candidates: the sub-channel 2/10, the anomalous g-factor, Guynn's angle, and the spinor clock rate, and argue that the specific constants of the substrate are the content of the theory, not its abstract algebraic shadow. Finitism is not a special case of abstract algebra; abstract algebra is a tool used within Finitism.