Now we can put the $9$ edge hexagon into the tetractys of commuting three qutrit paths, by placing the three chords on an outside edge.
This leaves room for the neutrinos and the charged leptons. That is, the $10$ vertices of the tetractys expand the $8$ vertices of an ordinary parity cube.
The charged lepton words ($XXX$, $YYY$ and $ZZZ$) should all have three charged strands for a total charge of $3$. Recall the embedding of the two qutrit paths into the tetractys. Usually we place the two qutrit hexagon in a corner of the tetractys, but we can choose instead to map it to the quark hexagon as shown. Then the path counts on the original hexagon, namely $1$ (for $XX$) or $2$ (for $XY$ and $YX$), are attached to the quarks as charges.
6 years ago