We now have two routes to the theory. In Section 3.2 we saw how to reach it via the theory:
Our work in that section and in Section 3.1 showed that this diagram commutes, which is a way of saying that the theory extends the Standard Model.
In Section 3.4 we saw another route to the
theory, which goes through
Drawing these two routes to
together gives us a cube:
We have already seen in Sections 3.1-3.4 that the
vertical faces commute. So, we are left with two questions involving the
horizontal faces. First: does the top face of the cube
In fact they both do, and we can use our affirmative answer to the
second question to settle the first.
As we remarked in Section 3.4, applying the map
to the Pati-Salam binary code given in Table 6, we get the
binary code given in Table 4. Thus, the linear maps
and agree on a basis, so this triangle commutes:
The work of Section 3.1 through Section 3.4
showed that the vertical faces of the cube commute.
We can thus conclude from diagrammatic reasoning that the two maps from
going around the top face are equal: