The three lepton generations and Koide's relation

The electron has two heavier cousins: the muon, about 207 times heavier, and the tau, about 3,477 times heavier. RSI proposes that all three are the same twice-winding loop in three distinct states, one for each independent axis of space. Seen along the diagonal that joins the three axes, the states sit 120 degrees apart, the same three-fold geometry that gives the proton its three loops and charge its thirds. The electron is the ground state. The muon and tau are excited states, which is why they decay.

Koide's relation. In 1982 Yoshio Koide found that the three masses, added together and divided by the square of the sum of their square roots, give 2/3, to within about 3 parts per million. Any three masses can be written as three points 120 degrees apart on a circle, so that form by itself fixes nothing. The value 2/3 needs one more fact: the swing of the three square roots around their common value must be exactly √2 times that value, which places the square roots of the masses 45 degrees from the diagonal of the three axes. Three-fold symmetry alone does not supply that. RSI proposes that the √2 is a balance between the part of the winding shared by all three axes and the part particular to each, and deriving it from the medium is open.

What it predicts. From the electron and muon masses, Koide's relation gives a tau mass of 1776.97 MeV, against the measured 1776.93 ± 0.09 MeV. In 2006 Carl Brannen noticed that the phase of the three points on that circle is very close to 2/9. Taking √2 and 2/9 as given, the electron's mass alone yields the muon to within 10 parts per million and the tau to within 0.007 percent. For now both numbers are taken from the data, so these are not yet genuine predictions. Deriving √2 and 2/9 from the geometry of the 720-degree loop would make them so, and a derived tau mass that disagreed with the measured one would falsify this part of the framework.

Decay. The muon decays almost always into an electron and two neutrinos. The tau decays into an electron or a muon with two neutrinos, or into mesons with one neutrino. In RSI's reading, an excited state relaxes into a lower one. The tau lives about 290 femtoseconds, the muon 2.2 microseconds, and the difference comes mainly from the tau's greater mass, since these decay rates grow roughly as the fifth power of the mass.