Martingale paths

 It occurs to me.   The Markov condition says that one of the three axis will be sampled twice at the start of a path.  Sampled twice means some color placed two geodesic points on the salad bowl. This happens for each path.  The color operator also cannot have common paths, meaning the factorial would equivalent paths. 

So, sampling twice is like an extra coin toss.  It makes the current binomial closer to a fair coin.  The paths sometimes end and then start again in the same color.   The goal is to keep the layers all constantly separated, keeping their axis are relative prime. 

The condition is caused by a bunch of the same thing under some density.  If there are too many collisions, and the things are a bit elastic, then they adopt a new dimension for kinetic flow, aliasing solution when under sampled.   The slight  inelasticity causes momentum after collisions. If congested places the momentum vector should be more perpendicular to the collision.  

What is the accuracy of the new dimension?  M^M times the accumulated elasticities of the prior dimensions.   That product is the sum total ways one can mix paths and have the total length ordered.  Dimensionality allows the system to partition kinetic energy.  Since it is never the case, in balance, that any thing has to process more than one partition at a time, then the whole system is limited only by the elasticity of the small thing.

Note, in 3D the number sampled needed is six, at two per, the number taken is four.