Restrictions on the Markov triples

(x^2 +y^2 +z^2) = 3*(xyz)

This is a measure space composed of three relative primes. Let us lose some generality by selecting 1,2,5; three relative prime number.

Divide by N, and we see that these are the integer fraction counts with which these counters have this state over the whole colored beach ball.

Ignore the rules a moment and we can see that the counters at 1,1,1 have three possible next states.  Each of these states requires at least  two pending in one of the counters.

Then let us eliminate some possible.  The ones  counter never has two pending before it emits a color. The to counter has one possible moment for two pending, but the fives count has two. That counter has the possible states: 0,1,2,3 but then it must roll over on the next count.  the states with 2 or 3 pending updates merge since the 3 count includes the 2 count. So we have two possible states of the five counter which meets the conditions.

That conic, with x,y,z fixed, describes all, finite  states of the round off errors each axis. The actual space is allocated, but never actually met. 1,1,1 cannot occur look at the previous states, to be the case, the previous counter must have colored the previous button three times. No two counter can roll over at once. Every counter get a new update request per step.

Because of path merging, there are multiple paths that can count an error of 1,1,1 in the three counters.  Half of the two counter paths generate fifteen out of the thirty combinations, but they merge with the 1/3 counter, both are multiple of the factorial count. Commutative property holds since we are counting almost independent arrivals. Hence the error value is not unique and arrivals are not independent. By contradiction, there must be coloring prior in the sequence which cancel those paths.

The general proof on these things should be to treat those counters as Poisson queues and treat the Markov equation as a probability statement.  Then let N get smaller (from infinity) and some constant N keeps that probability equation within an integer. This is the condition that -iLog(i) for all messages in a Shannon channel be within one.