Neutrino Mass Hierarchy and PMNS Angles from E₈ Spinor Dimension: A Geometric See-Saw Mechanism

We derive the neutrino mass hierarchy and PMNS mixing angles from the exceptional Lie algebra E₈ using a novel Spinor Dimension Hypothesis: neutrinos acquire mass through coupling to the 128-dimensional spinor representation of SO(16), the maximal subalgebra of E₈. Unlike charged fermions which occupy vector channels (112_v), neutrinos couple to pure spinor channels, leading to exponential mass suppression. The key innovation is identifying k=64 — the spinor dimension reduced by factor 2 — as the geometric channel controlling neutrino mass generation. Combined with channel k=13 (the sterile neutrino locus with pure spinor multiplicity N_s = 4, N_v = 0), this produces: Total neutrino mass Σm_ν = 0.0634 eV (geometric see-saw with GUT-scale right-handed masses) Mass hierarchy (Normal Ordering): m₁ = 0.0041 eV, m₂ = 0.0090 eV, m₃ = 0.0503 eV PMNS mixing angles from unused E₈ channels: θ₂₃ = 49.22° (k=6), θ₁₂ = 34.56° (k=0), θ₁₃ = 8.16° (90° - k=24) All predictions achieved with zero free parameters using only the established E₈ framework constants: Ω = 51.9 GeV, ε = 0.6366, W_E8 = 696,729,600. The framework explains why neutrino mixing angles are O(30°-50°) while quark mixing angles are O(1°-13°): quarks utilize occupied resonant channels, while neutrinos utilize the complementary "negative space" — the unused channels that define the PMNS topology. Keywords: neutrino mass, PMNS matrix, E₈ geometry, spinor representation, SO(16) decomposition, see-saw mechanism, sterile neutrino