Quantizing tanh and measuring stuff

Using any recursive series which has an expression for powers of a base.  So tanh become a ratio of two irrationals. Many differential equations can be recast as power series of Tanh because any differential of tanh is a power series in tanh.  But with a recursive quantized solutions for the powers of an irrational root, any power of tanh resolves in a ratio of two first order combinations in the base.

 Simplifying tanh requires some base such that:
b^n = kn*b + kn-1 where the kn are a sequence that is recursively known. Then the variable x becomes ln(b)*n, where n is the quantizer.

Its pretty sticky, dealing with both b and its log.

What's the point?
The idea is to make the optimum analog to digital converter, or a digit system, that is tuned to measure the third derivative of some signal that is partitioned.
The equation is: F''' = a + b*F + c*F^2 + d*F^3

So the power series in F measures the third derivative of F.  We do not want to know F, we want to know a good digital measurement of F which uses a base that optimally partitions F into its short to long wave components.

The hyperbolics have this characteristic: deviation^2 = mean^2 -1, which says that any hyperbolic line guarantees that there is enough space to measure the mean value plus a gap and not exceed the deviation. So the idea is to pick off a subselected set of the tanh values that holds an optimum deviation/mean value matching the partition of the triple derivative.

Instead of the original equation consider:

a+b*base+c*base^2+d*base^3 = 1

This measures one unit when for a point solution at base. But it spreads the signal optimally over all the digits in base, measuring the smallest unit of F encountered. But we can scale this equation, multiplying by b^[1], n from 1 to N. Each time we scale, we move the optimal four digit system to the level of the signal, each time scaling the digit system to match the signal strength. Base, then is the partition function for the signal. The b^n will not uniformly increase but form an expandinging wave.

So its all about designing the optimum analog to digital conversion system that matches the signal.