### M Theory Lesson 258

A general unitary magic 1-circulant may be written as the sum of two magic 1-circulants, as in $(a,b,b) + (0,c,0)$.

The $n$-th power of this sum has a binomial expansion for which at least one matrix factor in each product has a power greater than or equal to $n/2$. Since $DM = D$ (where $D$ is the unitary democratic matrix), for any such 1-circulant $M$ it follows that the limit of the power as $n \rightarrow \infty$ must also be $D$. These arguments apply to matrices over restricted domains for the rationals or reals. Similar arguments apply to 2-circulants.

Now general magic unitary matrices that are written as sums of two circulants, such as the approximate norm square of the CKM matrix, may also be expanded binomially to a sum of products that converges to $D$.

The $n$-th power of this sum has a binomial expansion for which at least one matrix factor in each product has a power greater than or equal to $n/2$. Since $DM = D$ (where $D$ is the unitary democratic matrix), for any such 1-circulant $M$ it follows that the limit of the power as $n \rightarrow \infty$ must also be $D$. These arguments apply to matrices over restricted domains for the rationals or reals. Similar arguments apply to 2-circulants.

Now general magic unitary matrices that are written as sums of two circulants, such as the approximate norm square of the CKM matrix, may also be expanded binomially to a sum of products that converges to $D$.

## 2 Comments:

This also has the feeling of being a clue.

I'm to be back in Moses Lake tonight. Historically, this means I'm more likely to be inclined to program java and could make progress on what you're looking for in terms of CKM modeling.

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