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on The Thomson Problem (AlphaEvolve problem 33)

The statement

For any $N>1$ , let $C(N)$ denote the infimum of the Coulomb energy

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

where $z_1, \dots, z_N$ ranges over the unit sphere $\mathbb{S}^2$ . 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 infimal (ground state) 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.