M Theory Lesson 258
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 $D$.