So how am I doing the Lucas atom?

Still a work in progress, but right now I have Lucas order counting up from the edge of the orbital toward the proton.  So a small hyperbolic angle, with respect to the electron reference, which grows large as the electron moves radially toward the proton center.  Moving toward the proton, the radial gradient decreases and the transverse gradient increases. Why increase? Lucas numbers grow! Yes, but derivative Tanh decreases. I have likely inverted entropy in prior posts, always be careful with me and sign dyslexia.  But when the Lucas angle grows, gradients become fuzzy. The electron does not know radius, it only knows its own surface curvature relative to the immediate environment.


So the electron transfers radial motion to transverse motion going down, as kinetic energy increases. The transverse motion become circular as the electron creates its own center of motion. Or, one can say, as the electron moves toward the proton, the proton center becomes ambiguous and its uncertain location is distributed by angular rotation. The recursive nature of uncertainty causes an obliqueness in the angular movement, allowing the electron to move from orbital to orbital.


So, in this model, the orbitals are centered by Lagrange points, centers of motion.  The electron is guided by a hyperbolic compass and will move toward the most certain Lagrange point.  But the motion of the electron, itself, makes the Lagrange points fuzzy, and ambiguous, causing angular motion.  So, in the end, we have the unit spheres moving in a dense shell of multiple Lagrange points.  The unit spheres constantly distribute 'fuzziness' among these points, which is the minimum redundancy condition.  It is a sort of spectral optimization in the sphere.  The number of Lagrange points increase with energy, and seem to be limited to about 16.


Being careful with terms.  Certain is the opposite of fuzzy. When the electron has a small angle, it has high variance, 1-Tanh^2 is large, and that means it has a radial direction.  Fuzzy is when the electron approaches Pi/2 in the angle and 1-tanh^2 is near the fine structure constant.     I am learning about this along with everyone else, so always expect errors, and do not be surprised if professional mathematicians switch directionality.


The math:

 I would give the electron one standard unit of fuzziness which it distributes among the selected number of Lagrange points.  The electron can deal with no more than three L points at a time, and I would give it a separate tanh angle set for each point.  The electron moves in units of hyperbolic angle quants, it take one step toward the least fuzzy L point by increasing the angle, which increases the L point fuzziness. Then it takes the next quant step toward the second least fuzzy L point and performs the same step.   If the electron exceeds its fuzzy budget, it degenerates.  The goal of the electron is to avoid calculus, so it must operate from 0 to pi/2, generally this would be the Pi/2 square angle.  If the electron is doing log 3, then each of the angle changes mean a net change in the electron intertness sum network, it takes in and emits inertness, the opposite of fuzzy. Hence the electron leaves intertness balanced in the shell and L points. I have not done this yet, but the method is close. At equilibrium, the electron is suspended in adiabatic motion, always staying below its fuzzy budget and the atomic shell has balanced inertness. When the electron is near the edge of an orbital shell, it likely dumps some intertness, essentially defining the Lucas zero points and setting the grid. When it gets too much fuzziness it will take in a batch of interness. I think the dumping and gulping of interness causes the tanf to scale a bit, subtracting or adding separation between angles. There is no distance or time in the thing, one has to create an engineering unit  standard and compute these as functions of the quants.

With three hyperbolic navigators, I think the electron will solve cubic roots along the way by noting the sequence of quant jumps it takes. I have not done this, but this just seems like a simple extension of finite difference computation incorporating entropy. For example, when the computational system arranges the grid for maximum entropy, then they can run the Lucas equations from zero to one and draw or derive their motion.


Finite Log:

This should work. Put N bodies in the system with enough fuzziness to prevent degeneration. Then step them through, removing excess inertness until they stabilize.  The stable angles will be the finite log rotation.

Quarks?
Dunno, but they definitely do not have a dominant Lagrange.  All their motion seems to be angular. But I am still not sure.