Representing cube, and square, root solutions of polynomials require complex values of a Z plane, in general. However, in a system where the set of solutions is limited and quantized, a decoder works much better. Consider:
ax^3 + bx^2 + cx + d = 0. Being cube roots, they count out solutions by three. If I know my solution set is limited to, say 27 solutions, I can make a decoder tree and step through my roots in sequence. Is this fair? Sure, when we have hyperbolic differentials with local solutions, they have phase shift and the solutions will be in sequence.
I also know that I can orthogonalize my solution set so each of: log3(b)+log3(x), where the b are coefficients, can be accurate to my required precision. And, for 27 solutions, I get a three level tree with three branches. I also claim that if we are limited to 27 solutions. The equation above is over specified, and will be reduced at minimum redundancy, they are not wave equations.
The bubbles are simply stepping through a spiral stair case, each stopping point separated by a noise band. Cube roots and square roots and linears are all separable. I further claim that there will always be a 'baud' path that identifies the next solution in sequence. And there should exist two dual solutions, marching in tandem, one inside the unit spheres and one outside.
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