Then why this relationship: 2^(2*Phi) = 3*Pi?

There would be no reason for this to be true unless it was a result of Newton's grammar. That result tells me that number line work because notches are spaced apart to support local sums of combinatorics. And Newton just carried over that relationship with symbolic transcendentals.

Pi*r^2/r gives us circumference to radius, that is the maximum divergence about an error for some uncertain radius.  So apply the Shannon information theory :

B/C = log2(1+SNR)

and that gives you the channel to bandwidth.  Work it two or three ways, add the fine structure as the noise, let the channel be the speed of light, try it with volume to area, invert things. But the result is always the same, numbers need to find the maximum divergence, multiply does not really exist in nature, and it is only approximate in the grammar as the number line heads to infinity.  In any finite system, multiply is always log additive, and combinatorics locally add. So you get the same result, 2^(2*Phi), or versions of it.

Newton;s calculus, in its pure form, is a story about very tiny things estimating the three transcendentals.  That is, stop the process as dx goes to zero and look at what the tiny components of dx are doing. What are they doing? Walking a graph to keep the trenscendentals optimally estimated to the nearest whole number. That is the story of Schramm-Loewner, they are stopping dx on the way to zero and they see what the tinies are up to.

Let your channel bandwidth be the sample rate of light and you get the number of tinies on the surface of the atomic sphere. Higgs got it right, he just worked it from infinity down to the tiniest.  I worked it from the tiniest up to the transcendental. Oded Schramm worked the middle. Einstein just wrote the Poincare group for a sphere.

And implicitly, or explicitly, Newton's  grammar was nothing more than a symbolic multiply using the three transcendentals.  We have to live with the shock, of it all. The world is made of the tiniest thing and there is nothing more we can see.  We are stuck, the theory of everything ultimately is quite simple.