Spectral Duality and the Cosmological Scale: A Rigorous Derivation of the Vacuum Energy Unit from \(S^{3}\)

his paper is the third installment of the Spectral Bifurcation Cosmology (SBC) research series. Based on the compact $S^3$ closed manifold geometry established in the preceding works, we introduce spectral duality and provide a rigorous mathematical derivation linking topological geometry to vacuum energy scale. The calculation yields a natural vacuum energy unit without artificial fine-tuning. This result bridges static vacuum solutions and cosmic dynamical evolution in the SBC framework, and establishes a foundation for subsequent investigations into fundamental physical constants and quantum geometric effects.