CKM and PMNS mixing matrices from discrete subgroups of SU(2)

Remaining within the realm of the Standard Model(SM) local gauge group, this first principles derivation of both the PMNS and CKM matrices utilizes quaternion generators of the three discrete (i.e., finite) binary rotational subgroups of SU(2) called [3,3,2], [4,3,2], and [5,3,2] for three lepton families in R³ and four related discrete binary rotational subgroups [3,3,3], [4,3,3], [3,4,3], and [5,3,3] represented by four quark families in R⁴. The traditional 3x3 CKM matrix is extracted as a submatrix of the 4x4 CKM4 matrix. If these two additional quarks b' and t' of a 4th quark family exist, there is the possibility that the SM lagrangian may apply all the way down to the Planck scale. There are then numerous other important consequences. The Weinberg angle is derived using these same quaternion generators, and the triangle anomaly cancellation is satisfied even though there is an obvious mismatch of three lepton families to four quark families. In a discrete space, one can also use these generators to derive a unique connection from the electroweak local gauge group SU(2)_L x U(1)_Y acting in R⁴ to the discrete group Weyl E₈ in R⁸. By considering Lorentz transformations in discrete (3,1)-D spacetime, one obtains another Weyl E₈ discrete symmetry group in R⁸, so that the combined symmetry is Weyl E₈ x Weyl E₈ = "discrete" SO(9,1) in 10-D spacetime. This unique connection is in direct contrast to the 10⁵⁰⁰ possible connections for superstring theory!