Neutrino Mass Splittings from E8 Boundary Geometry

The atmospheric and solar neutrino mass splittings are derived from the E8 root lattice, completing the Standard Model prediction programme. The seesaw right-handed neutrino mass at the E8 boundary is M_{R}=m_{Pl}\Lambda_{Pl}^{\epsilon_{E8}}=5.40\times10^{14} GeV. A two-loop E8 WZW correction by the factor e_{2}/rank(E_{8})=7/8 gives the atmospheric mass: m_{\nu_{3}}=(e_{2}/rank)\times v^2/(2M_{R})=49.16 meV, with \Delta m_{31}^2=2.417\times10^{-3} eV^2 (1.09 \sigma; PDG: 2.453\pm0.033\times10^{-3} eV^2). The ratio m_{\nu_{2}}/m_{\nu_{3}}=e_{2}/(rank\times5)=7/40 gives \Delta m_{21}^2=7.40\times10^{-5} eV^2 (0.71 \sigma) and the ratio \Delta m_{21}^2/\Delta m_{31}^2=(7/40)^2=49/1600=0.030625 (0.086 \sigma; PDG: 0.030697). The prediction for the lightest neutrino is m_{\nu_{1}}=m_{\nu_{2}}\cdot7/152=0.396 meV, testable by future neutrinoless double-\beta decay experiments. All inputs from prior papers in this series; zero free parameters.