Some mathematical physics (and updates)

I re-enter the zone of minimal redundancy, integrating a function such that precision is maintained and operations are minimized.

Consider f'/f, the rate of change relative to the size. Below I change that to Tanh, but for now I handwave.  I wish to do an integrate of f with the least number of operations, but I am limited to a fixed number of bits, the number of bits is my dimensionality. My function,f is broken into discrete chunks, and the total chunks is some finite number T.  So the portion with a large f will add to my integral in the weight f/T. But, if I am hyperbolic, f'/f approaches 1.0, it adds more information.  When f'/f is low, it adds less information.  These results are because I am using hyperbolics, I force the function to be modelled as a series of cosh(x), for some set of x, angles which will be spaced. So cosh(x) is the share of the integral allocated for some x, and -log(cosh(x)) are the number of bits allocated.  Hence I want cosh(x)*ln(cosh(x)), the entropy, to be almost equal for each sum in my integral.  Partial sums that do not add much information will have more bits, but will be included less often. Hence, for some total precision, the total integral will be encoded to have the smallest number of operations, its encoding graph will be minimal. So a Lebesque integral is about minimizing the number of operations. The encoding process determines the sets the error function and the quantization sizes. In other words, this is not encoding to a given error function, creating both the error function and quant size to match a particular total precision.

Let me attempt a handwave (with errors)  using hyperbolics:


Consider tanh(n*a), a discrete set of values for angles a times the integer set n.  Tanh is cosh'/cosh or the rate of change of some measure divided by  Cosh. So we have a ntural process and we model it as a discrete set of cosh function.

Ln(cosh(n*a)'/cosh(n*s) is the integral of n*a * tanh(n*a).

Now consider a probabilistic integral in which the summation is composed of integrands summed such that the error introduced by each integral is approximately equal, bound by an error term.  This is an integral of a natural stationary process. so we apply the Shannon condition:

-Cosh(n*a) * ln(Cosh(a*s) which becomes -cosh(n*s) *n*a * int(tanh(n*a)).

But we are discrete and we hwant the integral into sums of:
Tanh(n*s) * a.  If the n and a are spaced properly, then the integral is accurate in the stationary sense.

From there I think we can get to the Schremm-Loewner condition.  Hence, I do believe that the hyperbolics are the key functions in measure theory. The N is the dimensionality of the system, and the a must match such that -cosh*log(cosh) are maximally equal for any given n.

So we model our natural function in the form P(n,a)*Tanh(n*a), P is a polynomial meeting the Schramm-Loewrner conditions and modelling the natural process.