Room to unravel

So we are counting deviations from the minimum.  The minimum is a knot and needs that amount of round off error to resolve. When x,y,z get large, the condition still holds inside of the larger color operator..  The error counter is a fixed, reduced factorial tree, and deviation count never happen more than twice for any of the relatively prime deviation counters. The solutions force curvature in the integers because the dimension is three, it won't fit on a plane. His operator is not always at minimum at the boundary. Each of those counts have an independent symmetry and cannot be minimal at the boundary.

The integers x,y,z are really rates. The number of ways we can make a pair from x, is ultimately a relative occurrence from y or z.  The xyz are the orders system of counts along the knot. Your color operator is like one complete move with this device:

Another version of Chinese remainder.  The color operator ios a curve imposed on a geo disc on the sphere surface. It can undergo a series of complete shifts along two axis and a shift. In other words, it makes three orthogonal moves, rotations inside this geodesic.

That move sequence can be modeled as a rope, kinking around itself, inside the sphere.  We are coloring the surface to make make room for each of the knot crossings.  Each time the operator moves, it counts one more notch up the knot central axis of spiral.

The deviation count must be at the resolution of the knot crossing rank, coherent to he various rotations of the knot about is spiral axis. Both tri colored.

One can play a game of whack a mole if we converted a colored beach ball into a geodesic hand toy. The idea would be to find two legal rotations that makes the beach ball separated by color. There would be some finite number of geodesics, and each geodesic has independent rotation of its like color operators.

In the sandbox.

The pit boss is managing the deviation counts, the independent agents generate their own savings and loans; except the pit boss, separately, keeps y,z Huffman encoded to represent a valid integer sets x. and y. The pit boss operate on the 1,x,y nodes. Then he sets interest charge to make himself within 1. The result is a liquidity swap, and liquidity is counted in integer deviations from resolution adjusted samples. Liquidity counts as if it were uni of denominations, like money, along the 1,y,z Markov axis.

If the pit boss takes a profit it has wandered to one of the other Markov axis.