Elliptic Fekete / log-energy points on $S^2$

• Slug: fekete-s2
• List: Smale 7 (1998)
• Solver: Cursor Grok 4.6 cloud agent
• Status: open. Finite-$N$ records are computational. The conjecture is not claimed.
• Area: Discrete energy / spherical points
• Sources: Smale, Math. Intelligencer 20 (1998); Beltrán; Brauchart–Hardin–Saff; Cohn–Kumar; Womersley; Ridgway–Cheviakov; Rathbun–Ridgway
• Started: 2026-08-19

Statement

Place $N$ distinct points $X=\{x_1,\dots,x_N\}$ on the unit sphere $S^2\subset\mathbb R^3$ so as to minimize the logarithmic energy

Equivalently, maximize the product of pairwise Euclidean distances. Minimizers are elliptic Fekete points. Smale's 7th problem asks for a polynomial-time algorithm that returns an $N$-tuple whose energy is within $c\log N$ of the minimum, for a universal $c$. That algorithmic question is not the target of this folder.

Exact global minimizers are known for $N=1$–$6$ and $N=12$ (regular icosahedron). For every other $N$ the record is computational.

Two conventions appear in the literature and must not be mixed:

• this folder, Ridgway–Cheviakov (CPC 2018), Rathbun–Ridgway (arXiv:2008.04880): $E=\sum_{i<j}\log(1/|x_i-x_j|)$;
• Beltrán–Lizarte (arXiv:2502.10152) and some potential-theory papers: $\sum_{i\neq j}=2E$.

Tonight

A finite new bound, not the conjecture. Success is one of:

(A) An explicit $N$-point configuration on $S^2$ whose independently recomputed log-energy strictly beats a documented published record for that $N$, with points + energy + a from-scratch verifier, and a citation of the beaten number.

(B) If no beat: independently replay a published table for several $N$ (recompute energies from published coordinates, or regenerate known optima for $N=2$–$6,12$), write the replay, and the best leftover. That is not a new bound.

A candidate is nothing until the independent verifier agrees and the published number being compared is from a paper. Forum numbers are leads, not citations.

Outcome (2026-08-19)

No improvement. Independently replayed Ridgway–Cheviakov 2018 Table 3 for $N=2$–$65$ from Rathbun–Ridgway coordinates (arXiv:2008.04880, Zenodo 5595366): every global matches the printed 8 decimals, and matches the file ener= to $\sim 10^{-13}$. Regenerated $N=2$–$7,12$ from closed forms / platonic coordinates. Multi-start descent recovered the same energies at $N=7,8,9,10,14,19,24,32,33,46,48$. Last-digit undercuts of the 8-decimal print are rounding, not a new configuration.

Best leftover: $N=33$ (random starts stall $\sim 10^{-6}$ above the global; the paper's own hard case) and $N=46$ (spiral/rng recover the 2018 new global; many rng land on the published local $-249.454650$). Did not beat any published record. Smale 7 remains open.

Replay: cd problems/fekete-s2/compute && sh run_all.sh