#### arivero

Gold Member

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As for the relationship between the above folding and the S4 generalisation of Koide, I find that they are two solutions of the eight S4 simultaneus equations that seem relevant:

[tex]

\begin{array}{|ll|}

\hline

3.64098 & 0 \\

1.69854 & 1.69854 \\

0.12195 & 0.12195 \\

\hline

\end{array}

\dots

\begin{array}{|ll|}

\hline

b & u \\

d & c \\

s & t \\

\hline

\end{array}

\dots

\begin{array}{|ll|}

\hline

3.640 & 3.640 \\

1.698 & 1.698 \\

0.1219 & 174.1 \\

\hline

\end{array}

[/tex]

The one on the left appears when looking for zero'ed solutions; the one on the right appears in the resolvent of the system when looking for zero-less solutions; so both of them are singled-out very specifically even if, being doubly degenerated, they are hidden under the carpet of a continuous spectrum of solutions.

To be more specific: a S4-Koide system on the above "folded" quark pairings should be a set of eight simultaneous Koide equations, for all the possible combinations: bds, bdt, bcs, bct, uds, udt, ucs, uct. A double degenerated solution of such S4-Koide system lives naturally inside a continuum: the equation K(M1,M2,x)=0, with M1 and M2 being the degenerated masses, has multiple solutions for x, and any two of them can be used to build the non-degenerated pair of the folding.

The solution in the left is one of the possible solutions having at least a zero; up to an scale factor, there are only four of them. I have scaled it to match with the solution in the right.

The solution in the right is one of the solutions obtained by using the method of polynomial resolvents to solve the system of eight equations (actually, we fix a mass and then solve the four equations containing such fixed mass). It is scaled so that its higher mass coincides with the top mass.

For details on the calculation of the solutions, please refer to the thread on Koide.

[tex]

\begin{array}{|ll|}

\hline

3.64098 & 0 \\

1.69854 & 1.69854 \\

0.12195 & 0.12195 \\

\hline

\end{array}

\dots

\begin{array}{|ll|}

\hline

b & u \\

d & c \\

s & t \\

\hline

\end{array}

\dots

\begin{array}{|ll|}

\hline

3.640 & 3.640 \\

1.698 & 1.698 \\

0.1219 & 174.1 \\

\hline

\end{array}

[/tex]

The one on the left appears when looking for zero'ed solutions; the one on the right appears in the resolvent of the system when looking for zero-less solutions; so both of them are singled-out very specifically even if, being doubly degenerated, they are hidden under the carpet of a continuous spectrum of solutions.

To be more specific: a S4-Koide system on the above "folded" quark pairings should be a set of eight simultaneous Koide equations, for all the possible combinations: bds, bdt, bcs, bct, uds, udt, ucs, uct. A double degenerated solution of such S4-Koide system lives naturally inside a continuum: the equation K(M1,M2,x)=0, with M1 and M2 being the degenerated masses, has multiple solutions for x, and any two of them can be used to build the non-degenerated pair of the folding.

The solution in the left is one of the possible solutions having at least a zero; up to an scale factor, there are only four of them. I have scaled it to match with the solution in the right.

The solution in the right is one of the solutions obtained by using the method of polynomial resolvents to solve the system of eight equations (actually, we fix a mass and then solve the four equations containing such fixed mass). It is scaled so that its higher mass coincides with the top mass.

For details on the calculation of the solutions, please refer to the thread on Koide.

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