I think one of the advantages of writing blog posts rather than academic articles is that they are often more readable to a wider audience as the authors can be a little less formal in tone, expand on things (including copious illustrations), without worrying about space constraints.
Quite often I will plough through papers and some of the more challenging blog posts that are linked here. A post like yours is challenging but only for the right reasons. You avoid a too "chatty" and "pally" style and present facts concisely but with a bit of context - enough to point amateurs in the right direction.
These produce the largest d for N=4,8,6,12 and 20 resp.
Thus, it is presumed by almost everyone that d=3.64 is the global upper bound.
Unfortunately, I believe it is still an open problem to to prove this or to describe a general upper bound for specific N not equal to any of these five values.
One of my other references is "Distributing many points on a sphere" as it is written by E.B. Saff who is basically a legend in this field.
Hope that helps!
Technically d* does not exist, because as N-> infinity, d* alternates between 3.03 and 3.07 (depending on if k is odd or even).
Compare this to the canonical fibonacci sequence gives a value of d_N = 3.07 for all values of N, and so d* = 3.07
For an excellent commentary on the latest for optimal Riesz energy (which includes Coulomb potentials) configurations can be found in the first paper that I reference: "A comparison of popular point configurations on S2", by Hardin, Michaels and Saff.
You're going to get a kick out of: https://news.ycombinator.com/item?id=17765388