Cosmological Constant from Compact Three-Sphere Topology: A Zero-Parameter Framework with Falsifiable Predictions

This work initiates the Spectral Bifurcation Cosmology (SBC) series. We argue that the well-known 10¹²² vacuum energy fine-tuning puzzle originates from the conventional assumption of non-compact flat spacetime. Instead, we adopt a compact, simply connected 3-sphere $S^3$ as the spatial vacuum manifold. The discrete spectrum of the Laplace operator on $S^3$ naturally eliminates ultraviolet divergences without artificial cutoff. The cosmological constant $\Lambda$ emerges purely as a topological quantity $\Lambda=1/R^2$, where $R$ denotes the global radius of the $S^3$ geometry. All free parameters are removed from the static vacuum framework. We derive observable predictions, including the suppression of CMB power at low multipoles $\ell=2$ and the dark energy equation of state $w=-1$. This paper focuses on the static vacuum foundation; cosmic dynamical evolution, CMB perturbation calculations and particle physics implications are developed in companion papers Series II and Series III.