A Deterministic Derivation of the Fine-Structure Constant - Pentagon Path, Renormalization Cascade, and Falsifiable Quasar Prediction

This paper presents a deterministic derivation of the fine-structure constant within the framework of the Arithmetic Theory of Everything (AToE). While the Standard Model of physics treats dimensionless coupling constants as empirical input parameters, the AToE suggests that these values emerge as necessary consequences of discrete geometric symmetries and prime-number-based resonances. The derivation is based on the "Pentagon Path," a structural hierarchy that links the five-fold symmetry of the pentagon and icosahedron to the root system of the E8 Lie algebra via the McKay correspondence. By analyzing the trace of a discrete operator acting on a 12-base arithmetic manifold, the study identifies a unique combination of three geometric levels that yields a value for the inverse fine-structure constant deviating by only 0.0006 ppm from the CODATA 2022 recommendation. Beyond numerical proximity, the paper provides a variational proof demonstrating the global stability and uniqueness of the selected level configuration. Furthermore, it introduces a dynamical renormalization cascade that explains the precision of the constant as the result of a cosmological scaling process from the Planck scale to the current Hubble length. The theory is explicitly falsifiable: it predicts a specific, monotonically increasing variation of the fine-structure constant at high redshifts, establishing a hard asymptotic bound of 2.23 ppm that can be tested by next-generation quasar spectroscopy. This work marks a shift from probabilistic physics to a purely deterministic, arithmetic understanding of natural constants, suggesting that the physical parameters of our universe are not accidental but represent the only stable "accounting" solution for a discretely structured spacetime.