Force coupling constant and sphere packing

Alpha here is the called the fine structure constant in physics, e is electric charge. As you can see, its purpose is to estimate the perfect sphere by estimating pi.
This is easy to understand from maximum entropy considerations. The proton needs to match the band limit of light to the quant ratio so that pi is maximally accurate.  It only knows pi as a rational fraction.

The way I work the problem is that there exists an angle x such that tanh(n*x),  are linear for n=1,2,3.

Anyway, these angles are multiples of the log of the exponent in the quant ratio.  The maximum Compton matchi is at wave 91, and 3/2 * 91 = 137.5.   And
4/3 * 137 gets you to 182 where I find a near perfect Compton match with ratio 3/2 at 216.  Coincidence?  Well, I am using the first Lagrange,Phi, and 3/2 is a known ratio for sphere packing. So I am going to find these matches.  The question is, does the vacuum count the same way my spread sheet does? This makes 182 the band limit of light, as an exponent of phi, much higher than I would thing.

As an angle on the hyperbolic then the universe counts from pi/2 (not including pi/2)  by sixths, a quadratic root for each of three cubic roots.

Why I use exponents:

 I cheat, I approximate values using Phi, which nature does not know.  But I cheat and get the property to the left. So when the physicists report a prime number, then they have cancelled common terms, and I know the number they report is a common multi0ple, so I know how far up the Fibonacci chain the universe has gotten.  I do not find Fibonacci numbers, but I look for Fibonacci exponents.  This is primitive, yes, but good enough for me.  The real step is finding the ultimate rational ratio that the unioverse has settled on to approximate pi and e. That is not a Phi thing, but a sphere thing, and the numbers keep coming back to about 2/3 or 3/4, I dunno. The length of the power ies the universe uses seem to be about 6, but as combi8nations of two and three, that comes to a 12-16 bit computer.

How is the universe so stable then? It has matches the relative bubble sizes in the salad bowl to a 12 bit computer. The real stability comes from having the bandwidth of light match the sphere, allowing motion between unit spheres. The real trick of the universe is to recycle the vacuum bubble, polish them up, resize them, and recycle them.