Mapping Markov points on the Lie graph

  This equation tells us that we have three ways to count items using x,y,z as 'digits' .

For example, an item a,b,c can be counted c,a,b or b,c,a depending on the starting point.  There is a symmetry. Counting on item automatically defines the two other ways it can be counted.

So we take our hyperbolic geodesic and squash it down to the Lie graph. The angles taken around the Lie graph are three times what is needed to count all independent events.  But we have a commuter due to symmetry. That commuter is essentially a pit boss, it can exchange the order of X,Y,Z (now angles) and  count one item with one angle having bounded error.

This. Z on the outer edge, X on the inner. As we count angles about the unit circle, using the commuter, we can skip two intermediate angles and deliver the smallest error, which is bound.

We are throwing away two of three sample. If we take this and expand it back to the geodesic, we are adding two of three samples.