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.