### M Theory Lesson 300

Last time we considered the relation between $\Omega_{B}$, the baryonic mass fraction, and two phases appearing in Carl Brannen's mass matrices. A more natural choice involves the angles

$\phi = \frac{1}{9 \pi}$

$\alpha = \frac{1}{24}$,

where $\alpha$ comes from the phase $\pi / 12$. The difference $( \alpha - \phi )$ appears in mass relations for neutrinos, charged leptons and hadrons. We now have

$24 \alpha - 27 \phi = \Omega_{B}$

where $24 \alpha = 1$, the total of baryonic and dark matter components. Both the numbers $24$ and $27$ have a multitude of interesting number theoretic properties which M theorists enjoy. For example, $27$ is the dimension of the octonionic exceptional Jordan algebra, which in terms of $3 \times 3$ Hermitian matrices has a $24$ dimensional component coming from three octonion elements. The number $24$ decomposes into $2^3 \cdot 3$, and in Brannen's path integrals the $3$ indexes particle generation number, and $2$ stands for the fermionic spin quantum number.

$\phi = \frac{1}{9 \pi}$

$\alpha = \frac{1}{24}$,

where $\alpha$ comes from the phase $\pi / 12$. The difference $( \alpha - \phi )$ appears in mass relations for neutrinos, charged leptons and hadrons. We now have

$24 \alpha - 27 \phi = \Omega_{B}$

where $24 \alpha = 1$, the total of baryonic and dark matter components. Both the numbers $24$ and $27$ have a multitude of interesting number theoretic properties which M theorists enjoy. For example, $27$ is the dimension of the octonionic exceptional Jordan algebra, which in terms of $3 \times 3$ Hermitian matrices has a $24$ dimensional component coming from three octonion elements. The number $24$ decomposes into $2^3 \cdot 3$, and in Brannen's path integrals the $3$ indexes particle generation number, and $2$ stands for the fermionic spin quantum number.

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