After the night I found another way to look at the dual of the extended beta move. Recall that in the previous post was proved that the dual of the extended beta move is equivalent with this mystery move.
The graph from the left hand side can be expressed by using the (emergent algebra) crossing macros:
This form of the graph makes obvious where to apply the extended beta move. Let’s see what we get.
Therefore it looks like the mystery move is a combination of the extended beta move and elimination of loops. There is no need to replace the termination gate by a trivalent graph, as suggested in the previous post, although that is an idea worthy of further exploration.