### CKM Recipe II

In the approximation of a cubed root cosine by a square root, if one starts with the exact number $26/27$ (for the CKM row sum), then equality implies a numerator not of 723, but of 722.9792412. On the other hand, assuming an exact numerator of 723 results in the number $26.003436/27$. Anyway, $723 = 697 + 26$ and these integers appear in the decomposition of the symmetric magic matrix $U$ into 1-circulant and 2-circulant integer matrices: Observe that the small value of 3 (off the $U(2) \times U(1)$ block) limits the number of positive integer decompositions to four. The 2-circulant piece always takes the form $(26,0,0) + kD$, where $D$ is the democratic matrix and $k \in 0,1,2,3$.

## 4 Comments:

Lots of number crunching ideas here ... putting the near-26 value back into the 723 numerator gives another value (for the cubed row sum) of 26.004005. Putting this in gives the next value 26.004099, which in turn gives 26.004114, which in turn gives 26.004117. The exact relation seems to converge to roughly this value. That is, 26.004117 can be used in both the near-723 and also in the CKM row sum.

The question is, does this value give us better errors for the small entries?

Remote KeaThis number x = 26.004117 satisfies another, simpler relation:

27 - x = sqrt(697+x/729)

which is to say that

530744 - 39367 x + 729 x^2 = 0

and this puts x into the exact form

1458 x = 39367 - sqrt(2111185)

The positive sign solution to this quadratic is 27.9973.

Remote KeaAh, OK. The (27 - x) quadratic relation follows directly from the condition that

[V] + (1/27)[V^(1/3)] = 1

where square brackets denote real row sum and V is the CKM matrix. That is, there is some combination of quarks and preons that results in a unitary magic matrix. This might actually fully determine V.

Remote KeaFor a while, my intuition was saying to me that a big clue is that the MNS matrix can be written uniquely as the sum of two "real unitary" matrices, a 1-circulant and a 2-circulant, if you know what I mean. This is due to its having an entry that is exactly zero.

The same thing applies to the CKM matrix if you let those two pesky corner entries be not just equal, but equal to zero.

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