Taylor series, proton stability, and group theory, yet again

When I set up the quant table, all 108 *2 log values, in twops bit, each quant looks like:

val * 2**[index].  The indices go from 0 to 4095, typically, and the reason is that I want the quants separated so I can compute what happens in the gaps between them.  They look like a 4096 bit positive number of the form:
1000010001001000100010100001... and each set of zeros is the fractional gap between quants.  That is what I am after, that is where the groups are separated.

But I know the gaps before hand, so each quant velue is:

val * 2**[index-gap] relative to the previous quant value, because I know the gap distance. That lets me work with small fractions as the quantization macro climbs the quant chain.

How did I scale the compression force? Well, it starts as a standard double in basic, but each time I move upo the chain it is re-normalized as follows:

F* 2**[index] becomes [(1/2**gap) * F] * [2**(index + gap)]. That allows simple arithmetic at each level. So I do a relative shift of bits, essentially.  But, if this works, and I think it will, then I am essentially doing a Taylor series expansion of the log2 value of force as the macro moves force up the chain. 

Why do I care?  Because at some point, this change of scale in force starts a 'carry the one' all the way up to the proton level, and converts a whole series of ones into zeros, making the proton stable (having large Shannon separation from its neighbor).  In other words, the proton is stable because their is a certain force which, at some point, flips the bits.

What does this have to do with group theory?  Well, groups are separated in the relative log base one is working with, so there must be stable points about which the Taylor series hits nulls points.

Where is this for the proton?
Actually, it should be easy to compute if we knew for certain its stability time, because we can convert that stability time into a logarithm, then count down the orders from the proton and find out where the bit got flipped. We will find it near the electron order, I am sure.

But, the whole point is that, 1) Computing in logarithms makes one insane, and 2) The Taylor series expansion must somehow be connected intimately with group theory. We need an expert here, someone has to step up to the plate who understands all this.
And, a slightly related point. If groups are separated by a phase alignment, then is that alignment a separation of positive and negative phases? The answer would be yes, and it is generally negative, with the positive half on the outside. And for me to understand why I would need to understand the possible topology solutions for how a vacuum might work. So we need a topology expert here also.

But, more to the point.  The vacuum is likely doing a Taylor series expansion.But the two phase volumes and the null are acting like a Fibonacci sum, with the positive negative being the smaller of the three volumes.