Geometric Origin of Neutrino Mixing Angles from E₈ Root Lattice Projection

We present a complete geometric derivation of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix from the projection structure of the E₈ root lattice onto its Weyl vector. Unlike the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix, which arises from Dynkin diagram distances yielding small mixing angles, the large neutrino mixing angles correspond to specific unused geometric channels in the 29-channel E₈ mass spectrum. We demonstrate that: The atmospheric mixing angle θ₂₃ = 49.22° corresponds to channel k=6 (the first "ghost" channel between top and bottom quarks), matching experimental values to 0.24% precision The solar mixing angle θ₁₂ = 34.56° corresponds to channel k=0 (the "top axis" where maximum mass and maximum mixing unify), with 3.44% precision The reactor mixing angle θ₁₃ = 8.16° corresponds to the complement of channel k=24, with 4.4% precision Channel k=13 is identified as the geometric locus of the sterile (right-handed) neutrino, being composed entirely of spinor roots (128ₛ of SO(16)) with zero vector component overlap The framework explains why neutrino mixing angles are large (O(30°-50°)) while quark mixing angles are small (O(1°-13°)): quarks utilize occupied resonant channels for mass generation, while neutrinos utilize the complementary "negative space" — the unused channels — for mixing topology. This duality provides a unified geometric origin for both CKM and PMNS matrices with zero free parameters. Keywords: neutrino mixing, PMNS matrix, E₈ geometry, exceptional Lie algebras, sterile neutrino, root lattice projection, SO(16) decomposition