Some corrections on Log(x)

That is defined as the sum of path (length * length from 0 to x), for each baud on the number line. I mis-spoke a few times on the issue. This is still not completely clear, but I continue.

Log(1) in the discrete system is the sum of path lengths from zero to one, should be less than  e  the multiply precision. The zero function clears error, mainly by just bumping the fractional baud counter. So, within one complete sequence, fractional error need to stay below the system allowance. Each quant goes to complex angle zero once in a cycle and will clear errors at that quant level. At maximum entropy, there is a one to one between the ordered complex numbers and the baud index in a complete sequence.
Lets try something, again:
Finite log(1+y^X/(1+e)^2) =d, where e is the multiply error. This tells you how much error can be supported by dimension X, at maximum entropy over one complete sequence, before it has to go to zero. d is the supported error at X, and :
X^d - 1 is the total entropy supported by X, and includes the zero function. X is the dimension of the Reimann discrete complex plane at  X.

I am still working on all this, but the problem being solved is this. When complex number goes to 3+2k, and carries, that is the multiply. Then it goes to 0+0k, and stays there until its accumulated error is less than 1/3, and then goes to 3+1k. The dimension of X is X-1. The rule of conservation ensures we are finite, cyclic, and do not slip counts. All of the irrational roots become rational ratios in the finite system, and need error bounds. And we need to determine stability by band width functions. This  will all work out, and likely has already been done.

Consider 2^(1/2) and 3^(1/3).  2, a complex number of dimension 2 has to carry to 3, of dimension 3.  It cannot overflow.  This is the stuff we need.

Entropy is a big concept because if the system is not at maximum entropy, then we have to worry incompleteness, some quants may not connect up. And, at maximum entropy, the spectra is divided between less than zero and greater than zero. Ergodic, I think means minimum phase, the power spectra continually rises up toward the center of the band.