My numbers

These numbers are me using the rational ratio for Phi. If I use the limit of Lucas  and actual Phi, I get -137.1590861574, which about right. Using rational ratio, my entropy in pi is a lower than the actual fine structure, .006 instead of .007.  However, that is likely a selection error on my part. I have the titles and Fibonacci  numbers both in the chart. I think there is a trade where adding more degrees of freedom and suffer a bit of error in Pi yield a better stability for the atom. The odd orbital shapes happen because the entire power series is not used everywhere. The Lucas number do not actually divide out a rational ratio, they count up to the maximum degree, (L number)  then count down, in a quantum oscillation. Everything is done by n, the Lucas quant, no time or space. It looks like everything in the nucleus and orbitals is composed of just 16-20 integer counts.  Separability by primes insure that they combine in the proper two or three integers near the surface of a sphere.

	1.619047619	rational ratio.
n	Ph()^n	p^n-p^-n/p^n+p^-n	1- ratio^2
1	1.619047619	0.4477144646	0.7995517582
2	2.6213151927	0.7459121502	0.4436150642
3	4.2440341216	0.8948023173	0.199328813
4	6.8712933397	0.958518851	0.0812416123
5	11.1249511214	0.9839698039	0.031803425
6	18.0118256252	0.993854207	0.0122538153
7	29.1620033931	0.9976509898	0.0046925025
		sum of	1.5677944879
	Rational phi	sum of	1.5724869905
1	2
2	1.5	error from pi/2	0.0030018388
3	1.6666666667	error from pi/2	-0.0016906637
5	1.6
8	1.625		0.0060036777
13	1.6153846154
21	1.619047619
34	1.6176470588
55	1.6181818182
89	1.6179775281
144	1.6180555556
233	1.6180257511
377	0