Spectral Geometry and the One-Loop QED β-Function on S 3 × S 1

We compute the one-loop QED Formula: see text-function coefficient directly from heat kernel data of the twisted Spin c Dirac operator on Formula: see text. Using Formula: see text-function regularization, the logarithmic scale dependence is encoded in the Formula: see text coefficient of the spectral expansion. The Formula: see text term in Formula: see text yields exactly Formula: see text, independent of Formula: see text, Formula: see text, or background, verifying spectral RG flow without flat-space propagators. The result is independent of the radii of Formula: see text and Formula: see text and of the choice of gauge background, providing a parameterfree consistency check that spectral data on compact manifolds encode renormalization group information. Beyond a mere verification of the coupling flow, this result serves as a non-trivial consistency check of the Spectral Action Principle in a curved background. It demonstrates that universal quantum corrections can be extracted purely from geometric spectral invariants, distinguishing this geometric spectral derivation from momentumspace propagator methods.