Research Note: Geometric Origin of the PMNS Matrix Deriving Large Neutrino Mixing Angles from the FCC Bulk Lattice

The Standard Model of particle physics offers no explanation for the striking disparity be- tween the Quark Mixing Matrix (CKM), which is characterized by small mixing angles (θC ≈13◦), and the Neutrino Mixing Matrix (PMNS), which exhibits large, near-maximal angles. We present a rigorous derivation of these values within the Unified Geometric Lattice Theory (UGLT) [1]. By identifying neutrinos as unanchored bulk defects in a Face-Centered Cubic (FCC) vacuum lattice, we demonstrate that their mixing angles are determined by the fundamental symmetry vectors of the Cuboctahedron unit cell. We ana- lytically derive the Atmospheric angle θ23 = 45◦ from the diagonal symmetry of the lattice faces, and the Solar angle sin2 θ12 = 1/3 (θ12 ≈35.3◦) from the projection of the bulk Body Diagonal onto the interaction Face Diagonal. We resolve the tension between this geomet- ric prediction (0.333) and the global fit (0.307) by applying a unitarity constraint with the measured reactor angle θ13 as an input, which corrects the solar angle to sin2 θ12 ≈0.318. This framework resolves the ”Flavor Puzzle” as a consequence of topological dimensionality: quarks are confined to 2D surface geometries, while neutrinos traverse the 3D bulk.