Predictions — CKM, PMNS BCT Letter 8: The Complete CKM and PMNS Mixing Matrices from Tetrahedral Void Symmetry

The quark and lepton mixing matrices—the CKM and PMNS matrices respectively—encode the flavour structure of the Standard Model. Their elements are among the most precisely measured quantities in particle physics, yet no principle explains why they take the values they do. We derive both matrices from the tetrahedral void symmetry group Td (order 24) of the Body-Centred Tetragonal (BCT) vacuum lattice. The Cabibbo angle is θC = arctan(rtet/roct) = 12.87- (observed: 13.04-, error −1.3%). The CKM CP-violating phase is δCKM = arccos(1/3) = 70.53-(the tetrahedral vertex angle). The PMNS CP-violating phase is δCP = −π/2 exactly, from Td quaternion structure. The reactor mixing angle satisfies the identity sin2θ13 = 3α (observed: sin2θ13 = 0.0218 ± 0.0007; BCT: 3α0 = 0.0222, error +1.0%), linking the neutrino sector directly to the electromagnetic fine structure constant through the factor of 3 arising from D4 triality. All CKM and PMNS elements are derived with zero free parameters.