Spectral Geometry and the One-Loop QED $β$-Function on $S^3 \times S^1$

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.