suma de cosh en ángulos Lucas

I am looking at this: 4^Phi = 3Pi, increasing accuracy along some integer set.

So I know that Phi can be a sum of Phi(-n).  And sum of Phi^-n becomes sum cosh(an), a = ln(Phi), at the Lucas angles, with the usual common factors lost by me.

I take the Taylor series of cosh, for each of the angles, and align powers vertically, re-sum and I get sums of integer powers, and posers of a,, along an integer set.  That is a finite set of polynomials with order equal to the finite dimension, Nmax.. The sum of powers, k; 4^k,4^(k-1),4^(k-2), k < Nmax = 12.

The powers,k, tell use the expansion of the complete set by taken k at a time, k increasing.  These come as powers of the integer set, but that is just a mapping back to the standard integer line, under the condition that each additional prime has size equal to its ordering in the finite set of primes. It is the optimal counting of the number of ways to combine things when you add one more prime symbol to each naming.

So we consider any natural system with local addition only, and each local combination attempts to minimize dimension. Then (string, bubble, beans, lottery tickets), can be viewed as a sphere which rotates at quantum steps depending its position in the naming. That is, by serendipitous stumble along the equation above indicates Phi does sphere packing. It is a maximum packing along the x integer line, a packing modelled as an exchange between like items, but each having to maintain its own rotation cycle. These things are the stuff, and have a limited set of add modes they can perform and maintain the integer set. These like items can warp their sphere only so much to accommodate fractions.

So these elements organize such that the integers are maximally used by enclosing integer space with fractional counts.  The surface being the composition of the Lucas prime, the point where some local addition produces that prime. Between two surfaces the divergence rule requires Cosh^2 - Sinh^2 = 1, this conditions maps to the integer x number line.

Simplification of Lucas angles. They come in Cosh(a*n),Sinsh(a*(n+1),...  But you can see that the one is composed of its neighbors, So we can use some sum of cosh, with additional factors. a comes in powers when computing the number of combinations.  So one unit of the thing is held between two surfaces. That must be the x number line. That must be the condition of maximum divergence.

The unit sphere is a collection of thing where the things that do not change shape, prime = 1,  is at vacuum, ground state zero. These regions where, null rotations take place, exist between surfaces.

Anyway, the elements have a limited set of shapes (vibrations, rotating facets,...). That set or angles is the maximum congestion of free space. So a compressed vacuum is limited in the maximum prime size. The next most rational approximation to Phi requires greater spectra in the set of shapes.  But a lest rational, but more accurate approximation can be made from a cube root. The polynomial giving the number of new sets for the next prime can be factored; and estimate something not Pi but a close approximation. Likely 2,1,3,4,7,11,18; the 18.  7+6 = 13 + 6 = 18.  Angle 6 is used, 2+2+2. That is when the quarks get three way exchange, I think.  All the angles have to shift. Angle shift is relative to vacuum, and I think vacuum gets 2,1,3.

Let me conjecture.  The math simply shows a march across the Wythoff array.  Starting in the upper left, top row, we march right until we can do a sqrt root.  We then drop a row and get the Lucas.  From there we march right until we get a cube root.  We make quarks, and drop a ros.  Next comes the eight root and we get Oxygen,Carbon, and galaxies. Every row drop is a scale up, and it means greater densities, thus a more precise quasar recycler. The dimensionality of light increases.  I think the univere has been working this problem for a very long time.