Look at log(1+entropy of Pi), a version of Shannon. A slice of the entropy is 1-tanh(n*a)^2 where a is log(phi). The idea is that the electron moves such that is separates out the integer values of Lucas at the surface (or center?) of the orbital shell, where the Lucas polynomial is at x=1. We want to know how the Lucas numbers are paired to make orbitals. The electron angular motion would effect a transfer to balance out the phase and match motion to phase imbalance.
So we get:
log(2-[sinh^2/cosh^2]) or log(2*cosh^2-sinh^2) - log(cosh^2).
The first term becomes: log(sinh^2+2)
Then converting to the Lucas system which scales the unit circle we find that sinh^2 = L(2n) because the integer 2 goes away with the square of sinh. But L(2n) is cosh(2n)* sqrt(2) so we end up with:
log(cosh(2n*a) - 2*log(cosh(n*a)) + log(sqrt(2), and these come out to integrals of tanh, which is a sum in out discrete system. So it may be that the system is working with copies of Lucas in n and 2n to balance Pi.
I think I have most of this, but errors abound, so be careful.
Another way to check this is go to the spread sheet and use irrational Phi, and take the logs of 1-tanh(n)^2 for the power series and directly construct a two binary digit system.
This is not something I should be working on, it is a critical piece of the puzzle and the pros need to be on this.
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