Attack log — elliptic Fekete points on $S^2$

2026-08-19 — start

• New folder problems/fekete-s2. House rules: write only here; do not touch problems/covering/ or problems/landau-n2-plus-1; do not invent a dent; do not claim Smale 7. A configuration is nothing until the independent verifier agrees and the beaten number is from a paper.
• Energy used throughout: $E=\sum_{i<j}\log(1/|x_i-x_j|)$ after projecting to $S^2$.
• Tonight: either (A) a certified energy beat of a published record, or (B) an honest replay plus residue.

Published records fetched tonight

Sources actually opened (arXiv HTML/PDF or author PDF), not forum posts:

• Smale, Math. Intelligencer 20 (1998): problem 7 is the $c\log N$ algorithm, not a table.
• Cohn–Kumar, arXiv:math/0508114: $N=2,3,4,6,12$ on $S^2$ are universally optimal, hence log-optimal. Icosahedron is the $N=12$ case.
• Dragnev–Legg–Townsend, Pacific J. Math. 207 (2002): $N=5$ is the triangular bipyramid $\{1,3,1\}$.
• Beltrán–Lizarte, arXiv:2502.10152v3 / J. Symbolic Comput. (2026): complete critical-point classification for $n\le 6$ in every dimension; $N=7$ on $S^2$ is 1:5:1 among dipole configurations, printed $E_{\log}=\sum_{i\neq j}=-16.3649557$ (twice this folder's $E$).
• Ridgway–Cheviakov, Comput. Phys. Commun. 233 (2018), Table 3: putative global log-energies for $N=4$–$65$ to about 8 decimals. New globals claimed at $N=19$ and $N=46$ versus Bergersen–Boal– Palffy-Muhoray 1994. This is the printed table we replay and try to beat. Author PDF: https://researchers.usask.ca/alexey-shevyakov/papersetc/2018_wr_ac_cpc_iterative_optim.pdf
• Rathbun–Ridgway, arXiv:2008.04880: 77-digit coordinates for the same $N\le 65$ log-minima (Zenodo 5595366). They match Ridgway–Cheviakov to $4.95\times 10^{-8}$. Algebraic energies for several small $N$.
• Amore–Figueroa–Ramos, arXiv:2512.12416 (Dec 2025) / J. Stat. Phys. (2026): landscape to $N=160$, 14142 local minima at $N=160$. No printed $E_{\min}(N)$ table. Zenodo 17923310 is a 591 MB dump of all local minima, not a record table.
• Brauchart–Hardin–Saff, arXiv:1202.4037: next-order log-energy asymptotics. Notes that Rakhmanov–Saff–Zhou 1994 used half this folder's $E$ (i.e. $\sum_{i<j}$).
• Hardin–Michaels–Saff, arXiv:1607.04590: comparison of constructions; Womersley supplied near-min log points for $N<500$. Mesh-ratio table, not an $E_{\min}$ table.
• Womersley energy page https://web.maths.unsw.edu.au/~rsw/Sphere/Energy/: Coulomb / $s=2$ ME points (2003), not a log-energy record table.

No later accepted table with a lower $E_{\min}$ for any modest $N$ was found tonight. The number to beat for $4\le N\le 65$ is the first (global) entry of Ridgway–Cheviakov Table 3, refined by Rathbun–Ridgway to many more digits.

Plan

1. Independent verifier: project to $S^2$, print $E$. Sanity: tetrahedron, octahedron, icosahedron.
2. Replay Table 3 for several $N$ by regenerating known optima and by multi-start Riemannian descent with deterministic seeds.
3. Search at $N$ with slack: 7, 8, 9, 10, 14, 19, 24, 32, 33, 46, 48. $N=33$ is the paper's own hard case (saddle first, then $10^6$ iterations).
4. A printed 8-decimal that we undercut only in the last digit is rounding, not a beat. A new configuration whose $E$ is below the printed global by more than $10^{-7}$ (Rathbun's comparison tolerance) would be a candidate.

Files

• compute/energy.py — from-scratch verifier.
• compute/known.py — exact $N=2$–$6,12$ and 1:5:1 for $N=7$.
• compute/optimize.py — deterministic multi-start descent.
• compute/ridgway2018.json — Table 3 globals extracted from the author PDF (checked against pdftotext of that PDF).
• compute/replay_published.py — parse Rathbun GP-Pari files.
• compute/replay_rathbun.json — $N=2$–$65$ recomputed energies.
• compute/checkpoints/N*.json — search points + start log.
• compute/run_all.sh — replay.

2026-08-19 — verifier sanity

compute/known.py + energy.py:

Beltrán–Lizarte's printed $E_{\log}=-16.3649557$ for $N=7$ is exactly $2E$ (their $\sum_{i\neq j}$). Convention check, not a beat.

2026-08-19 — full Table 3 replay from published coordinates

Zenodo 5595366 log.3.N.80 parsed and run through energy.py. Every $N=2$–$65$ matches the file ener= to $\sim 10^{-13}$ (N=34 has no ener= line; we still recompute $E=-142.3758522709$ vs Table 3 $-142.3758523$). Every Table 3 global matches to the printed 8 decimals ($|\Delta|<5\times 10^{-8}$, the rounding of the last digit). Table: compute/replay_rathbun.json.

This is a replay of the published record, not a dent.

2026-08-19 — search at slack $N$

Deterministic seeds (Fibonacci, RSZ spiral, square-antiprism heights for $N=8$, 32 RNG seeds with base 4102, plus Rathbun + four kicks). Riemannian Armijo descent. Independent verifier on stored points.

No $\Delta < -10^{-7}$. The tiny negatives are extra digits of the same 8-decimal print. Not a beat.

$N=46$ independently recovers the 2018 new global from spiral and several rng starts. Many rng land on the published local $-249.454650$, which is Table 3's second line. Fibonacci stalls at $-249.44978$, a still-worse basin. That is the residue: the landscape has the old and new globals close together, and we found both, not a third.

$N=33$ is the paper's saddle/global struggle. Kicks from the published point stay at the global; 32 random starts do not reach it within $10^{-6}$. We did not improve it.

Stop

No certified beat. Outcome (B): replay + residue. Smale 7 not claimed.