Using tanh when F is F(x,y)

Last post I outlined a possible method to using tanh to define a set of quantixed grid sizes such that a digit system iw optimum in measuring:
 F'''= a +b*F + c*F^2 + d*F^3
The was not to find F but to design a decoder that decomposes signals from F into a small set of digits optimally weighted to measure the structure of F, the optimum analog to digital converter for an equation of the form above.

I used tanh(n) with N integer, because tanh is the deviation over the mean, it gives me an individual grids in space, each grid size matching some n,
n= 1..Nmax.
What is the proper grid size when F = F(x,y)?

The trick is to only work along lines of symmetry where the result F''' can be separated. Thus, the grids, tanh(n), can all be assigned in the m dimensional space, along lines of symmetry.  In two dimensional space, the set of exponents, n, will be of the form n[i,j].  So we quantize x to i, and y to j.  y and x must have a relationship along the line of symmetry, x = g(y).

If we do this right, normalize the total of grids to the unit sphere,  assuming spheric symmetry, then the power series b^[-k*a], where a is the computed efficiency, and dependent on n, the digit number. We get this result when the base is not optimum.  F'' is, in general, a third degree and not be square symmetrical.

And k is a possibly repeated series of n*[1..3], then we should be maximum entropy and meet the Shannon condition.  So we can draw the probability distribution of the grid sizes. Without even knowing F or its symmetry, we can try matching function to distribute the grids over the sphere volume and meet the distribution shape.

But isn't deviation/mean a two dimensional? Yes, and I have not thought that through, but it would introduce some slight error.  What are the limts on F'''? Well, tanh^n, n=1,..4; say, they are even odd functions with varying slopes; and not positive definit. I am working the rest of the requirements as we speak.