The magic seems to be 17

Phi, like Pi, is something that allows mathematicians to use some grammar, in the limit.  But at the end of the day, there is no magic in the vacuum, except, I think 17 and spherical symmetry.  17 seems to be the number of different things that can be packed into a sphere and still make Stokes work within the error of the Fine Structure. The Fine Structure seems to be an outcome of the finite sphere problem.  Each time I look at any power series that counts up to the Fine Structure, I get a limit in its order, and that seems to be 17.  The limit can be measured using a recursive power series and decoded into a finite rational ratio. But even ratio is a bad term, there seems to be an actual set of integers that maximally pack a sphere. And even 'integer' is an improper term since that smallest elements that are counted up can vary in their counts up to the finite structure. 17 is the open limit, the actual degrees of freedom end up being 16, or 8 if we count fractions and wholes.

 This seems to be the ruling equation, where f is radius(n), n being the count. At the limit, r(n) becomes r(16) and has a value that cancels Pi from the volume. Two radii rule, sinh and cosh, and when they maximally identical we get 1/sqrt(pi) and the warping of the grid just cancels out Pi from the equation. When their angular separation is moderate, we get an even mix  angular motion and radial  motion, the peak of the Planck's curve for the sphere, otherwise known as optimal queueing. It is mainly combinatorics, the binomial coefficients have an optimum integer order for the sphere. The limit is based on allowable error in identifying the sphere center. Too many combinations exceed that limit, the sphere degenerates.  Too few combinations and the local grid will renomralize to restore the correct degrees of freedom.

Here is a measure model:

Any unit sphere will follow a density gradient when the change in density is larger than  its own curvature.  When the condition is not met, the unit sphere will engage in angular motion because its own density dominates the gradient. So the electron at the edge of its shell sees a dominate gradient toward the proton, and the electron will move imbalance through itsels causing motion toward the center.  At some the the gradient becomes to steep that is cannot carry charge and it begins motions along an angle to the proton radius. The net is a fixed uncertainty about the actual proton center. When the gradient toward the center goes as 3/r and the gradient orthogonal to the radius goes as Pi * r then they match at r^2 = 3/Pi. At 1/cosh(a)^2 = FS (the fine structure then the hyperbolic angle  a =  Pi - FS/2. So it all boils down to:

• 1) We are limited to the spherical model mainly because of combinatorics packing the 1,2,3 together at the low end. 
• 2) Combinatorics are finite so our degrees of freedom is limited and maximum entropy applies. 
• 3) From these two, hyperbolics follows, as does 16/2  and Pi/2 falls out automatically.

Note, that Phi makes the shallowest angle across the binomial pyramid, up to 8 across, I think.  It is the maximum entropy solution for sphere packing in Log2, mainly because the radius, area and volume have to be balanced and there is a matched set of possible combination for that.

And, minimum redundancy theory is mostly setting the boundary conditions, not about mechanism.
And, this is not a gimmick with the hyperbolic function, rather the reverse.  The hyperbolic system is a gimmick that bridges the gap between finite and infinite systems.  Past the hyperbolic angle Pi/2, calculus becomes  accurate faster than finite systems can grow.  When the angle is between zero and Pi/2, the differential equation above is more accurate with certain finite number sets than with calculus approximations.

At some point the space for a next generation of spheres is not available, and crowding makes counting by five possible and we get organic and bio chemistry.  Up to that point, its a spherical world. I think the quarks have managed a log base 3 system, a 3-ary network. It is when that is squeezed against the orbitals that chemistry evolves with log base 5. The total theory seems to be minimum redundancy combinatorics.