The Weinberg Angle, Fine-Structure Constant, and Gauge Couplings from the Polynomial

A gauge coupling is a number that tells you how strongly particles interact via one of the three non-gravitational forces. The Standard Model of particle physics has three such numbers — one for each force — and cannot explain their values. The GTE framework derives all three gauge couplings from the arithmetic structure of the GTE polynomial. This tutorial explains how a 19-bit cellular automaton rule encodes the strengths of electromagnetism (the fine-structure constant alpha approximately 1/137) and the weak nuclear force (the Weinberg angle theta_W approximately 28.7 degrees). The derivation proceeds from the number-theoretic properties of the GTE polynomial modulo 7, the F_21 symmetry group structure, and the SRRG fixed point — producing specific numerical predictions that match experiment without any free parameters.