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

The fine-structure constant α ≈ 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 S^3 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 α from the noncommutative geometric structure of the quantum-deformed 3-sphere S_q^3, in homage to Albert Einstein's lifelong pursuit of geometric unification of fundamental interactions. We first derive the leading-order classical benchmark value αclass⁻¹ ≈ 133.8 from the topological invariant of the SU_q(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 α⁻¹ ≈ 136.9 ~ 137.1, which is highly consistent with the CODATA 2022 experimental value αₑₓₚ⁻¹ = 137.035999177(21) with a deviation of only ~0.05%. This work proves that α 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]