Games I am playing with recursive sequences and hyperbolics

Consider r1=3/2+sqrt(5)/2 and r2=3/2-sqrt(5)/2.   The sum is 3 and their product is one.  I get these values from Phi^2 + Phi^-2.

Take the log(r1) = a and then sinh(a) = sqrt(5)/2. Cosh(a) = 3/2. Tanh(a) = sqrt(5)/3. Hence Phi = 1/2+3*tanh(a)/2 =  1/2 + sinh(a).
Then (Phi)^n = Fn*(1/2+3*tanh(a)/2) + Fn-1 for example. and any (3/2)^n = Cosh(a)^n.

Also the derivative of the tanh will  be 1/cosh(a)^2. Neat stuff.  Is this ratio the reason I got the match at Phi^91 and (3/2)^108? Yes, but I caused the ration because I sorted the powers of Phi and 3/2. Hence I forced this angle, I did a Higgs mechanism on my spread sheet. Now I can work all the orbitals using shifted powers,  just like Higgs, and I can change the ratios r1 and r2.

Thuis idea that light is a bubble exchange with two bubble type swapping places with Higgs nulls is correct.  The shift  I see to get the ratio is charge and spin, and all the properties the Standard model has.

So think of it this way.  Light knows a match radius must be three, it is simply working with Higgs to find a grid size agreement that make the radius three, which it must be at the match point. Higgs and light doing the Lattice QCD for real. These shift in the phase difference between the light bubbles will stabilize then their rate of change can be contained by Higgs bubbles. That determine a unit circle. They can all be calculated using hyperbolic differential wave equations, I think. The phase shifts increase toward the center of  the packed sphere and that add angular motion to the degrees of freedom.

This angle is log(sqrt(5)/2 + 3/2.  That is about ,96 radians. Cannot find it on any map, I am still a little befuddled about where these angles are measured from. And I still think my choice for the Higgs bubble size was mostly luck and guessing. I never actually measured one of them.

There is one better recursive relationship

Phi makes the small and large light bubbles to distant. So I tried this:

r -1/r = 1/2  or r = 1/4+SQRT(17)/4 . This actually worked and makes the proton.electron mass much more accurate, 17*108 = 1836.  It also makes sense since a half shift is r^2 + r.  Log(1836,(r^2+1/r)) = 17/2. Sort of a bingo and a half shift is spin. And the Higgs bubbles do not come out so large, and the light bubble are a reasonable  So, I am looking at these bubble sizes for sphere packing. The hyperbolic angle is .2475, and sinh(a) = 1/4 and cosh(a) = sqrt(17)/4.

So the current plan. We have a measuring stick on the left, measuring the two active bubbles. We design it to count in a way that matches the inactive bubble on the right.  I can make any form of 2/3, 3/2, 3/4; to match some criteria represented by a ratio of grid sizes, using sinh(a). Then have fun with tanh, and read lots of papers by mathematicians doing tanh solutions.

Constraints are boundaries on the null grid, written in cosh as fraction and define null quants at the hyperbolic angle specified by the measuring stick. So my constraint is that null bubble be 3/2 times the measuring stick. We can still formulate (3/2)^n, easy enough, so we still get the dandy match at 108, and now I got a 17, whoopie doo.  The measuring stick is not more accurate at angle .24 then Phi at .96.  But we know Phi needed a shift four to match the density.  Its dense, volume to area favors highly curled action by the measuring stick.  The new ratio will need a 16 shift, is my lucky guess. Phi had the 17s if I remember.