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on Tammes problem (AlphaEvolve problem 34)

The statement

For $N \geq 2$ , let $C(N)$ denote the maximal value of the energy

E(z_1,\dots,z_N) \coloneqq \min_{1 \leq i < j \leq N} \|z_i-z_j\|

where $z_1,\dots,z_N$ range over points in $\mathbb{S}^2$ . Establish upper and lower bounds on $C(N)$ that are as strong as possible. What type of configurations $z_1,\dots,z_N$ come close to achieving the maximal energy?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.