### M Theory Lesson 260

This CKM business is getting a bit messy, so let us recall that the Fourier transform of a complex circulant sum takes the form Carl Brannen's values for the CKM matrix (perhaps slightly inaccurate) are

$I = 0.973313$

$J = -0.008577$

$K = 0.000466$

$R = 0.040013$

$G = 0.225762$

$B = -0.004273$.

In particular, $(I + J + K) + i(R + G + B) = 0.965202 + 0.261502 i$, which has norm 1. The point is that, focusing on the real parts, it is better to think of these numbers in the form $a/27$. No doubt Carl will fix the numerical fit and blog about it shortly. The number 27 (or rather its square, $729 = 3^6$) is a natural normalisation factor for products of MUB type matrices.

$I = 0.973313$

$J = -0.008577$

$K = 0.000466$

$R = 0.040013$

$G = 0.225762$

$B = -0.004273$.

In particular, $(I + J + K) + i(R + G + B) = 0.965202 + 0.261502 i$, which has norm 1. The point is that, focusing on the real parts, it is better to think of these numbers in the form $a/27$. No doubt Carl will fix the numerical fit and blog about it shortly. The number 27 (or rather its square, $729 = 3^6$) is a natural normalisation factor for products of MUB type matrices.

## 2 Comments:

One of the things I was exploring the other day was what the Koide translation of (I,J,K) and (R,G,B) looked like. It seemed a little interesting but nothing outstanding.

This morning, I realized that when I did the calculations, I used the squares of these values. In doing this, I miss the minus signs in J and B. To get those minus signs (which reminds me of how the neutrino mass formula needs a minus sign for the square root of the lowest mass neutrino), I need to modify my program. Will talk more about this later. Right now things are kind of busy.

The IJK values are:

valence = 0.3217343717

sea = 0.6515997325

delta = 0.459091 degrees

RGB are:

valence = 0.087167109

sea = 0.140933472

delta = -10.45260 degrees

Only interesting thing is that the two angles seem close to 0 and 10 degrees, and differ from these by the same amounts.

In terms of topological phases, of course the angles need to be tripled so they are close to 0 and -30 degrees. On doing this, one finds that the average of the errors from 0 and 30 degrees is about 1.3675 degrees. A numerologically interesting thing I noticed when computing this is that twice this value, i.e. 2.735, is just 10 degrees less than 12.73 degrees, the familiar value for 2/9 radians.

I think that this is a dead end, and will now go look at the stuff you've most recently posted.

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