A solution method?

The equation above says that bandwidth, f^3 - f, is the second derivative of f. That is, the gluons are band limited.

The solution is tanh.  We treat tanh(r) as a finite power series in r, and put that in the equation. We get, after much polynomial division, a cubic in r.

The various qualities of quarks, spin, color, charge, etc, are recursions among the quarks roots together, they rotate through color according to the rules. recursion.  Then the recursion for each quark alone is a combination if its generated quantum state. So there is a multiply of sequences, , matching of coefficients, and the initial values set according to the quark rules.

And every solution S(x) = r^3 + r^2 + r^1 +1, then r*S(x) is a solution. We can make are initial power series in r up to some appropriate n, the total number of things counted. But, ultimately is should resolve to a cubic polynomial. This seems like a general method for finding a finite base.

N body problem using this general approach.

If we wrote the finite element solution to the n body orbital problem, for example, as a solution set in some base r. Then  we have 2*n values, an x and a y for each body. We assume quants count N=1,2,3,4. And we write each x and y as the r^(N+1), relative to the other bodies. We get a matrix of 2*N coefficients and diagonalize. Then we cover those to coefficients in our tanh(x) power series, keep the series 2*N long. Then we divide thru using the original differential equation above.  We solve for the base, r, it will be a root of the same order as the equation of motions, likely an r^2. We solve for the root given the differential equation.  We asign initial x and y positions for each body, then just crank N from one to infinite and byond! Does thsi work? Avoid transcendental, power and ratios, add and subtract are OK. It should work, the bodies should cycle through their x and y at the delta rate used in the equations of motion.