Spectral-Geometric Origin of the Muon g−2 Anomaly

This research establishes a spectral-geometric framework to address the persistent tension in the muon anomalous magnetic moment (Δ aμ). Departing from traditional perturbative QED approaches, we derive the correction to aμ from the operator-theoretic structure of the vacuum, specifically utilizing the heat kernel expansion of a Dirac-type operator defined on a gauge-invariant geometric structure. The model provides a rigorous mathematical origin for both ultraviolet (UV)-suppressed contributions and infrared (IR)-enhanced mechanisms mediated by topological zero-modes. Crucially, the framework generates a hierarchy of falsifiable signatures—including multipole field correlations, beam-profile dependencies, and azimuthal modulations—that are experimentally testable in current and future storage ring facilities. This work offers an alternative theoretical perspective on the $g-2$ anomaly, emphasizing the fundamental spectral geometry of the vacuum over standard model phenomenological fitting.