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

We compute the one-loop QED $\beta$-function coefficient directly from heat kernel data of the twisted Spin$^c$ Dirac operator on $S^3 \times S^1$. Using $\zeta$-function regularization, the logarithmic scale dependence is encoded in the $a_4$ coefficient of the spectral expansion. The $F_{\mu\nu} F^{\mu\nu}$ term in $a_4$ yields exactly $\beta(e) = e^3/(12\pi^2)$, independent of $r$, $L$, or background, verifying spectral RG flow without flat-space propagators. The result is independent of the radii of $S^3$ and $S^1$ and of the choice of gauge background, providing a parameter-free 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 momentum-space propagator methods.