Ultraviolet (UV) divergences in quantum field theory (QFT) have plagued theoretical physics for nearly a century, with conventional renormalization relying on ad-hoc cancellation of infinite bare parameters without a first-principles explanation. This work is the third paper in a 7-part series built on the Scale Geometry unified framework, rooted in the foundational S³ Quantum Deformation Geometry Operator (Paper 1) and the geometric locking of the fine-structure constant (Paper 2). 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 solution to the renormalization problem, in homage to Albert Einstein's vision of a self-consistent geometric unification of quantum and gravitational physics. We rigorously derive scaling relations of cutoff function moments from the core Scale Self-Duality Axiom, proving that all power-law divergences in the spectral action’s heat kernel expansion are automatically canceled by geometric scaling, yielding a fully finite action at all orders. We demonstrate full-order finiteness of loop corrections in quantum electrodynamics (QED), reproducing low-energy results of conventional renormalization without infinite bare parameters or counterterms. We further resolve the U (1) gauge coupling’s Landau pole via UV-IR duality, with the coupling saturating at the Planck scale instead of diverging. The framework yields a UV-complete, falsifiable geometric finite QFT, with testable predictions including high-energy coupling saturation and tiny low-energy atomic corrections. This work solves the century-old renormalization problem, laying a self-consistent finite foundation for the Standard Model and quantum gravity within Scale Geometry. Author: Xinyu Zheng (郑心宇) ORCID: 0009-0000-3175-1681Correspondence: wxsq1638@outlook. comDOI: To be Assigned by Zenodo
Xinyu Zheng (Thu,) studied this question.