Helical Spacetime Topology and the Spectral Origin of the Fine-Structure Constant A Topological Mode Expansion Approach to Fundamental Constants

We establish a parameter-free geometric-topological framework in which dimensionless fundamental constants emerge as degeneracy-weighted spectral sums over discrete eigenmodes of helical spacetime. Modeling spacetime trajectories as light-speed helical curves yields a universal $\pi$-series expansion for the inverse fine-structure constant:
\[
\alpha^{-1} = \sum_{n=1}^{\infty} d_n \pi^n
\]
where the integer topological degeneracies $d_n$ are determined solely by the geometric degrees of freedom of 3D helical structures. Rigorous counting gives $d_1 = 1$, $d_2 = 1$, $d_3 = 4$, producing the leading-order approximation
\[
\alpha^{-1} \approx 4\pi^3+\pi^2+\pi \approx 137.0363038.
\]
This agrees with the CODATA 2022 recommended value $137.035999177(21)$ to within 2.22 ppm. The residual $\Delta \approx 3.048 \times 10^{-4}$ is attributed to truncated higher-order modes ($\mathcal{O}(\pi^4)$), with the series exhibiting rapid convergence due to hierarchical geometric suppression. No variation of $\alpha$ is required, ensuring full consistency with precision measurements. The framework predicts a universal $\pi$-spectrum for all dimensionless constants, offering a first-principles geometric pathway toward unification.