How accurate is the finite approximation to log(Phi)?

The Taylor series expansion of ln(Phi) is an expansion in 1/Phi^2, from the fact that Phi and Phi-1 are inverses. So I am interested on the order of the expansion when ln(phi) is within the fine structure.

Here is the graph:

And we see log(Phi) is within the Fine structure when the order is 16. I take the expansion up to some order, and correct by the Fine Structure. Then I see how close we are.  The numbers are here:

Order  Final error
  15    0.0005046831
  16   1.214354E-005
  17  -0.0004219024

So, nothing special here except the optimum choice is order 16.  For any other value of Fine Structure, there will some optimum expansion order, but what seems interesting is that the expansion order come to 16.  16 is a canonical set of binary values, or a set of values in which the Shannon encoding is fully non redundant.  This is to be expected in adapted systems, and one might expect this number to be 8 4,16,32 ,64 (I think), depending upon how accurate the quasars are in pressing the vacuum into electrons.  So, the fine structure is really an output from the Theory of Everything. 

The principles:

This is all about aggregates confined in a compression. The solution is always some elasticity in collisions such that collisions are minimally redundant, connected, and equally precise across the  dimensions of actions.  The three principles boil down to the aggregate finding rational approximations to phi,e, and pi. It is Newton done backwards, the chaos needs to find the best approximation to Newton's rules of grammar. We end up with one or more independent additive sequences in Phi that superimpose.  near tanh(0) there is a linear region where the center is adjusting energy so the aggregate is adiabatic to the compression force exterior. So we get an uncertainty, the fine structure, that separates the spectral modes and matches the uncertainty of the exterior environment.