Geometric Locking of the Fine-Structure Constant α from First Principles

The fine-structure constant 1/137, the core dimensionless coupling constant of quantum electrodynamics, has long been one of the most profound unsolved mysteries in theoretical physics, with its first-principles origin remaining elusive for nearly a century. This work is the second paper in a 7-part series built on the Scale Geometry unified framework, rooted in the foundational S³ Quantum Deformation Geometry Operator established in Paper 1. As a component of the parent monograph The Scale Geometry Framework: Topology, Projection, and Static Universe (https: //doi. org/10. 5281/zenodo. 19368089), we provide a rigorous first-principles derivation of from the noncommutative geometric structure of the quantum-deformed 3-sphere Sq³, in homage to Albert Einstein's lifelong pursuit of geometric unification of fundamental interactions. We first derive the leading-order classical benchmark value ₂₋₀ₒₒ^-1 133. 8 from the topological invariant of the SUq (3) quantum flag manifold and the Representation Splitting Theorem, with no manually introduced free parameters. We then systematically identify three types of higher-order corrections—higher-order topological corrections of the quantum flag manifold, renormalization group running corrections in noncommutative curved spacetime, and non-perturbative quantum group deformation corrections—all supported by frontier noncommutative geometry research. The final semi-quantitative estimation range of the inverse fine-structure constant is ^-1 136. 9 137. 1, which is highly consistent with the CODATA 2022 experimental value _^-1 = 137. 035999177 (21) with a deviation of only ~0. 05%. This work proves that is not a phenomenological free parameter but a geometric invariant uniquely determined by the underlying spacetime structure, resolving the century-old puzzle of the origin of gauge coupling constants and advancing Einstein's vision of geometric unification. Author: Xinyu Zheng (郑心宇) ORCID: 0009-0000-3175-1681Correspondence: wxsq1638@outlook. comDOI: To be Assigned by Zenodo Keywords: Fine-Structure Constant () ; Scale Geometry; Noncommutative Geometry; Spectral Action Principle; Quantum Flag Manifold; Scale Self-Duality Axiom; First-Principles Derivation; Gauge Coupling Origin