Sphere packing, Ito's Calculus, Weiner processes and Lucas polynomials

I am going to take a shot at the general solution. I start with what I think is he sphere packing equation:

The solution is tanh, but a change of boundary conditions makes coth another solution. Rather imporant since the Lucas numbers give cosh and sinh alternatively. I add the class of martingales:

P(x,t) will be the Lucas polynomial or order n, Ln and tanh(n*a) as Ln*tanh(n*a), a derived from the Lucas number.  t is a curcular angle about a local radial from the unit sphere to the center of the current shell, having derivative being the variance in density curvature.

I have the density gradient of the containing shell counting up from the perimeter toward the center, so the curvature of any concentric shell becomes more accurate toward the center. The Lucas polynomial restricts the range of motion such that the curvature is made accurate by motion. The problem is solve for each of the n separately, n going from 1 to Nmax, Nmax giving the energy level.  As n increases toward the center, the density increases by Lucas number, but the number of zeros in the Lucas polynomial increase at a circular angle relative to the line of site toward the shell center, for the particular n.  I think that under these conditions, the polynomial and tanh are separable.

So, the spiral motion, executed by the unit sphere realizes the variance in curvature. The motion is increasingly restricted as n increases. Tanh and coth are the almost invertible gradients at any phase change from n to n+1. Lucas motion is solved relative to the current centered radial, from the unit sphere, and perpendicular to it, doing the motion. But the motion of the unit sphere is never parallel nor perpendicular to the radial toward the center, the zeros of Lucas force the spiral motion.

So I have moved the uncertainty from the center of the outer shell to at density n, although it may be reflected back toward an uncertain center and solved equivalently.

Anyway, this look like it gets the orbitals. The remaining issue is how does the vacuum do a divide to get tanh and coth from the gradients? One way or the other, I still have a divide function  in here, though Pi is gone. However, I would not be surprised if a= ln(Phi) cancels the divide when second derivative of tanh is taken.