Doing Shannon as an adapted quantized system

This is unfinshed work, I ain't figured it all out.  There will be another post on fermions and bosons, likely unfinished. In that work I construct the fermion problem as a queue with only 0 or 1 item; and the boson as a queue with 1 or 2 for the bosons, then construct the queues as a series sum in (1/3^n) and (2/3^n), getting an aggregate sample rate (Temp, in the physics) or 2 and 1/2.  Stuff like that. I am fishing for those friggen mathematicians, as usual. But I post unfinished work, that is what I do, sorry.


Making Shannon a simple spectral allocation

Let's consider all following as variance or power spectra:

C - Capacity
B - Total power availailable, energy
S -  Available spectra, signal spectra as an output
N - Allowable quantization error

Now I have:

C/B = 1+ S/N or the proportion of capacity variance, as a proportion of the available variance must be the variance of the sepataror (Ito's dx unitized) plus the proportion of the signal space available relative to error, this ratio is precision.

In other words, given the allowable error I the system needs, and the precision available we have the capacity needed to stay within the total energy.  Its a kind of quantize Hamiltonian.

Not I want to adapt the system do get the dynamics.  The system always makes exchanges to maintain the balance above, especially precision, S/N, whihc is fixed.  To make it adapted I need to ensure the system maintain exchanges within the bandwidth allows for symmetric precision.  So I convert all my power spectra into exponetial growth projected two period ahead so the exchanges sample at twice the rate of the projection. For simplicity let the log of the sqrt variable above be its lower case. We get, after moving thins around:

1/e * [e^[2*(c-b] - e^[2*(s-n)] = 1
or
   {e^[c-b+1/2]}^2 -{e^[s-n+1/2]}^2 = 1

Is this legal? It is if if I have two independent exchange systems at the edapt equilibrium, but it says nothing about dis-equilibrium. Ons solutiion set are the hyperbolics if we have a functions that maps c-b+12/ into s-b+1/2, which are the logs of cosh and sinh respective.

Solution:

Anyway, make a short story here, we get two roots for Y in the Y + 1/Y  and Y-1/Y, one from B/S and one from S/N.