How close can Fibonacci get to the fine structure constant?

Well the maximum deviation Fibonacci has from Pi/2 is -0.0072908055, and its inverse is -137.1590565066, the physicists have:

I call that close. We need to dig up Prof Lagrange and Lucas for the Swedish Banana. This had to be true, the degrees of freedom are equally utilized, so Lagrange estimation will be spherical and that makes Phi a winner. The Fine Structure is simply t the variance of light exchange rate and given the size of the universe, the quasars simply got the vacuum optimum to the degrees of freedom that match.

It came out to a power series in tanh', the derivative, because tanh is the allowable motion, and the variance in that has to be split up to minimize redundancy.  The variance in that value is of course the spectrum of fundamental light, its sample rate noise.

This number is theoretical based on using Phi to the limit of my spread sheet.  When a finite rate is substituted the number is between 0.0002269164 and -0.0045091191. My finite ratio for Phi is 89/55. So the universe can compute Pi and implicitly Ln(Phi), so it has enough know how to do Isaac's grammar to a finite limit of .00729 = dx. Is this the limit? I dunno if the quark system is more accurate, this only applies to outside the proton shell as near as I can tell, but do not quote me, check with Weinberg and Higgs.

How does this fit with my spectral chart? The chart was just a computation about how much would fit if there was no structure, no electron, no quarks.  The came me trying to find applicable recursive power series which the vacuum could do with only local knowledge and maximum entropy. Most of my blundering about was ignorance and application of things like primes and power series, most of which I barely understood.  Lucas did the real work here when I realized his integer series was cyclotomic.

There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!
—Richard Feynman, Richard P. Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129. ISBN 0-691-08388-6.

Packing spheres, Richard, packing spheres. 

Charge is offset, is the electro dynamic entropy in a sphere, composed of what seems to be a fifth to sixth order power series. By offset I mean the exponent in the power series when phi is the base. It is the contribution to the entropy of pi from each of the 'digits'. Free light in 'space' is the spectrum of light without enough Higgs density to be meet the Compton limit, and that is conductance. {ut in other words, Conductance is the ability to move charge without moving Higgs elements.

There should be up to six configurations in which the elements of light can be packed against Higgs elements. But, like I say, I am not sure how the fine structure is split between the proton and the electron. But all the measurements of the constant seem to be along a single axis, but do not quote me, I am still reading up.

What about magnetism? Not sure, but I am looking at the cosh(n*w) plus i*Sinh(nw-n), the two being one delta angle separated.The motion of the unit sphere needing to avoid zeros when an electron is involved. Look at the Lucas sequence and note the alternation between cosh and sinh.  The quant, the ln(Phi),  is built into the cross product I think.

What about size difference?
One of my incomplete ideas.  Light (or the Higgs) can change shape and that gives us the seven degrees of freedom.  That seven determines the degree of the power series.  So that problem melted away.

Who is doing numerology here?
The physicists who assume the number line goes to infinity. Not the finite element, finite order wave folks like me.