M Theory Lesson 264

Recall that circulants are always magic as well as square magic in the sense that the sum of squares along a row or column is a fixed constant. In particular, any Koide mass matrix $M$ has this property.

But MUB operators such as $F_3$ are not magic. For example, the action of $F_3{F}_2$ (the neutrino mixing matrix) on $M$ results in a $1\times 2$ block matrix in terms of the square roots of the masses, because $F_3$ diagonalises, and $F_2$ then acts on a pair of mass eigenvalues. The fact that this matrix is not magic is the same as the statement that $m_1\ne m_2$. The fact that it is not square magic follows from the statement that in

$\text{cos}\delta =\frac{s-6v}{s}$,

where $\delta$ is the angle shared by all three masses. This property is shared by the hadron fits.