Earlier quoted context omitted.
Not sure. For n points on a sphere, you're trying to prove a global minimum a very bumpy function over 2n-dimensional space. It may become computationally intractable.
And even if you can find a numerical solution to arbitrary precision, at best you only get arbitrarily close to the mathematically optimal solution ( at best because a purely numerical approach cannot rule out that it misses a very small, but high peak in the function to be optimized) As an extreme example, for n=2 and using binary floating point, you probably will find the mathematically correct solution, but there'…
For N=8 the solution listed on the Wikipedia page is a square antiprism (https://en.m.wikipedia.org/wiki/Square_antiprism) which seems intuitively reasonable. But, I don't see anything listed for N=20. Look at the table in S.5 of the article (https://en.m.wikipedia.org/wiki/Thomson_problem#Configuratio...) for more links to visualisations in the rightmost column, if one exists.