Our three color pit

 ${\displaystyle f(x)+f(y)=f(z)}$ with f(t) = arcosh(3t/2)

What does the arc function give us? The rank of our generators.  So we have a simple condition when do the three color, the ranks add going up the tree.  The three sequences windows,x,y,z; are compact generators, they have the arc function, to an approximation.  t tells us where abouts on the tree we would be.

This algebra makes sense,  If we didn't get an algebra then the system would do a two color decomposition.

For central banks, this means keeping government  interest charges within bounds means the pit boss will be moving up and down the Markov tree, and we need thousands of accounts in the window, not a mere 30.  All accounts have to be auto-traded.  

When government is out of balance, the Fed bot will inentivize a million depositors to increase deposits and decrease loans as it attempts to lengthen the window on deposits and cover government over the longer haul. The pit boss does extend, but it is not pretending.