Think Lie group

 

Think of this a multiple circles, he resolution less as the radius decreases.  They are independent, or semi independent circular counters on the hyperbolic angle, each angle rotation of the same resolution but different points.

This machine will implement a Marfov deviation counter, count around the ring of hyperbolic angles, by geodesics.  There will be an ordered count of some finite deviation counter from 111 to xyz, with resolution maximum and rotation jitter minimum, in three dimensions.

We are concentric because relative primeness makes the long counter more than sufficiently accurate.

Then this has focus, it is curved and aims at some start of a knot in the center and counts along the knot axis. Look closely. This is a thinned factorial tree, it is combining the symmetric moves the knot can do.

I make a prediction. To keep the Lie groups in stability we need to run the loop one and a half times per group. That is what is needed to just keep deviations bound. This is efficient, it utilizes the higher precision of the outside rings. We are counting sequential integers around the ring, aggregate  The disequilibrium in N spirals out. There is a theorem on this, Markov counting hyperbolic angles. But there will be a maximum N. some N which cannot equally dissipate the spirals at the edge.