Newton's Constant from Spectral Geometry: Deriving G from the \(S^3\) Dirac Spectrum and Heat-Kernel Asymptotics

This paper is the final paper of Series III within the Spectral Bifurcation Cosmology (SBC) research suite. Based on the compact three-sphere spectral geometry established in the previous three companion papers, we rigorously derive Newton’s gravitational constant G from the Dirac operator heat-kernel asymptotics without any free fitting parameters. The derived theoretical value matches CODATA 2018 measured data within 0.1% error. This work completes the full geometric derivation chain covering the cosmological constant, cosmic expansion, Standard Model gauge symmetry, and gravitational constant, forming a unified zero-parameter theoretical framework of spacetime and particle physics.