Bounded, finite order polynomial wave equations

This chart has tanh(n)^k, for several k with n on the x axis.  I use this chart because I only consider polynomial wave equations with Tanh(n) solutions. Polynomials meet the condition when their differential goes to zero within some bounded range. The range is specified by a Tanh(x)^k, where k is one more than the order of the polynomial. Stability requires that the nth derivative go to zero faster then the nth +1 derivative. Thus, the range of n, the argument in tanh(n) will look like:
n1 < n2 < n3 <... where each ni is the bandstop for the ith derivative.  It is the coefficients of the polynomial terms which limit the bandwidth, but the ranges are easily determined.

So, there is some range of quant,n, which satisfies the differential, But the solutions are linear, so if S(Nmax) solves the equation for some n < Nmax, then we know that the argument is periodic. Thus  S(j*Nmax)  will solve it where j >= Nmax.  (I leave out the proof) So, we get a hierarchical system of digits. Further, we know there is a range of rational bases such that abs(e - (p/q)) is bounded, so the solution need not use transcendental symbols.

In other words, this is the general solution method for systems like the proton. Once we know the arrangement of allowable quants, then we can derive the equations of motion in quant variables, without reference to time or space.

Differentials of x = Tanh(n)^i, for example when i=4 we get:  5*x^4*(1−x^2)

So we have a relationship between the coefficients of the polynomial in powers of Tanh(n), and 1-tanh(x)^2. We can use the relationship to directly derive the bounds on n in tanh(n), for each term of the polynomial.

Games to play

 If two ranges of quants overlap, then we know we have an unfactorable polynomial in the system, and the quants have to have a relationship over some degree of freedom. We can estimate the roots because we know the quant n when the differential goes to zero.

We can generate local time. If polynomial has few factors, then its operation count is smaller and so we can compute time. If we know the density of the vacuum, then we can derive time/space equation by using time and space as output.  If we know, for example, the wave equations for light in one of the atomic orbitals, then we can derive a momentum operator that removes one degree of freedom and generates the free light wave equation.

This method is the standard tool for quantum physics. It works because quants tell us the number of actions possible from N total, taken k at a time over m recursions.  That is how the Tanh(n) is constructed. So we have an algebra to do combinatorics and differentials.

 Example. This is an F(n)*Tanh(n)^3, ot something like that from my spreadsheet.  It is the quantum solution to something, I can tell because the quants are sparse and the lines not smooth.  Could this be a particle in a box? Sure could, or a dog in a doghouse, or deliveries of eggs. Who knows since it was a constructed example.  But it is all going to work this way, start with an estimation of the quants and their relationships, because, I think, that is how nature works the problem.  Time and space are not really part of the deal.

Sphere packing:

I think sphere packers are bound to a fourth order spectrum, so the solutions are all third order polynomials in the variable quant.  But, at some point, subdividing the quant resolution will not exceed the bounds on p/k, the rational approximation to the natural base.  So, I have sort of hand waved the proof of existance for a maximum Avogadro's number.  There is a maximum finite integer in the sphere packing world.

Anyway, I call this part a wrap, mathematicians all know what to do from here, and most of them smarter than I.