Lets make a Fibonacci polynomial group

I will hand wave it.
I take the Pell equation and let my short but order polynonial, R and Q. R,Q is a smaller polynomial in Phi irrational number. It is two Finacci sequencse powers  of the form :   mR + n. I have to make a :

u^2 - x^2 (u)^2 = 4

Which looks like this from the Wiki  hyperbolic functions. Except that Cosh(u) is compressed by the factor ln(r). And the unit circle has become the unit oval.

I like unit circles. I resolve the polynomials into a:  R^k =  mR + n, a resolved finbonacci polynomial, of a number of digits  acually, Some shift to get mR + nR  which makes a circle.  I get a Large Lagrange, in exchange for a large shift in the two polynomials.  So, we make a set of these long shifts. But we get a difit for each shift. Lagrangerions have confirmed that the large Lagrange will be less than some O(n+1) error., so I know my digits length. the error is the smallest bag of nulls that stabilize, having the least kinetic energy. Then we thro in the null bubble, the polynomial is composed of bubble 1, the bubble 2 and bubble zero, our favorite. b1 and b2 exchange with b0. In the process they leave little chunks of null bubbles b0.  The b0 out number b1 and b2 inside the unit circle. Outside, from the surface of the unit sphere, the noise is some radially invariant set of conics, rotationally invariant, perpendicular to the sphere surface, and from the outside surface, they count null bubbles by exchange. The mass of nulls is the constant in our Rk, n.  R is exchanged alternating between b1 and b2. The small m, is the shift or charge, spin thing. It tells us when the b1 out number the b2.  But big n makes longer digits systems.

 Bags of packed nulls have surface deformations, noise plus offset. Phase offsets make the b0  packed slightly oval perpendicular, flat  side to the line of motion the sphere. Packed nulls scoot around.

Two bubbles will attempt to cancel, wave moves toward a line of symmetry where Nulls increase in number, counting in our digit system.. Three quarks, havine large nulls and long digit systems will have large offsets cancelling in the center and a tubs of b0 sloshing around  The three unit spheres, do what units spheres do, but are anchored to the tub of b0. Out side the three, the quarks have much more b1 and b2, and much fewer b0.

Rotations

Rotations of the center 'angle' the point Phi^N, about which the fractions 1/Phi^N and set, that point has to rotate in space to preserve volume and orientation and maintain the special linear group.  When N goes as 1,2,3; then U changes by quants, and each kU represent a digit in the system of the form: Phi^kU + 1/Phi^kU.  Changing the Lagrange, the parameters r, takes us from one, two and three degrees of freedom, each degree representing a rotation about the center line of U. The third Lagrange gives us three roatations, hence the quarks. Each roatation should cause a spiral up the center line, I think.

We end up with this compound, spherical worm gear system which maintains gear teeth using batches of Nulls, some packed into matter.  The system goes out to about 180 meters, and the net effect is to keep the proton centers inside a slightly curved salad bowl. When protons coordinate, they tend to curve space toward the quasars a bit, communication space curvature thru a broadly distributed network of protons. Eventually, the protons in direct management of the quasar center, tilt the local gradient to adjust the spew of new baryons for local galactic debris. The universe becomes a huge salad bowl.