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

Citations only. No paper PDFs in the repo. Forum numbers are leads.

Problem

• S. Smale, Mathematical problems for the next century, Mathematical Intelligencer 20 (1998), 7–15. Problem 7: find $x_1,\dots,x_N\in S^2$ with $E(X)-E_{\min}\le c\log N$ in time polynomial in $N$.

Exact optima

• H. Cohn, A. Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007), 99–148, arXiv:math/0508114. $N=2,3,4,6,12$ on $S^2$ are universally optimal.
• P. D. Dragnev, D. A. Legg, D. W. Townsend, Discrete logarithmic energy on the sphere, Pacific J. Math. 207 (2002), 345–358. $N=5$ is the triangular bipyramid.
• C. Beltrán, F. Lizarte, Characterization of logarithmic Fekete critical configurations of at most six points in all dimensions, arXiv:2502.10152v3, J. Symbolic Comput. (2026), doi:10.1016/j.jsc.2026.102570. Critical-point classification for $n\le 6$; $N=7$ dipole case is 1:5:1.

Computational tables (the numbers we replay)

• W. J. M. Ridgway, A. F. Cheviakov, An iterative procedure for finding locally and globally optimal arrangements of particles on the unit sphere, Comput. Phys. Commun. 233 (2018), 84–109, doi:10.1016/j.cpc.2018.03.029. Author PDF fetched 2026-08-19: https://researchers.usask.ca/alexey-shevyakov/papersetc/2018_wr_ac_cpc_iterative_optim.pdf Table 3: log-energy locals and globals for $N=4$–$65$. New globals at $N=19,46$ versus Bergersen–Boal–Palffy-Muhoray 1994.
• R. L. Rathbun, W. J. M. Ridgway, Some spherical codes in $S^2$ and their algebraic numbers, arXiv:2008.04880. 77-digit log-minima for $N\le 65$. Coordinates: Zenodo 10.5281/zenodo.5595366. Match Ridgway–Cheviakov to $4.95\times 10^{-8}$.
• B. Bergersen, D. Boal, P. Palffy-Muhoray, Equilibrium configurations of particles on a sphere: the case of logarithmic interactions, J. Phys. A 27 (1994), 2579. Earlier $N\le 65$ table that Ridgway beat at 19 and 46.

2025–2026 landscape (fetched; not a record table)

• P. Amore, V. Figueroa, R. Ramos, Exploring the energy landscape of the logarithmic potential: local minima and stationary states, arXiv:2512.12416v1, J. Stat. Phys. (2026), doi:10.1007/s10955-026-03629-8. $N\le 160$ local minima; $N\le 24$ stationary states. Zenodo 10.5281/zenodo.17923310 (591 MB of spherical-coordinate dumps). No printed $E_{\min}(N)$.

Asymptotics and constructions (not records)

• E. A. Rakhmanov, E. B. Saff, Y. M. Zhou, Minimal discrete energy on the sphere, Math. Res. Lett. 1 (1994), 647–662. $N\le 200$ experiments; $E=\sum_{i<j}\log(1/r)$. Fitted formula, not a per-$N$ table we could beat line-by-line.
• J. S. Brauchart, D. P. Hardin, E. B. Saff, The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere, arXiv:1202.4037.
• D. P. Hardin, T. Michaels, E. B. Saff, A comparison of popular point configurations on $S^2$, arXiv:1607.04590, Dolomites Res. Notes Approx. 9 (2016). Womersley near-min log points for $N<500$; mesh ratios, not $E_{\min}$.
• D. Armentano, C. Beltrán, M. Shub, Minimizing the discrete logarithmic energy on the sphere: the role of random polynomials, Trans. Amer. Math. Soc. 363 (2011), 2955–2965.
• C. Beltrán, U. Etayo, The Diamond ensemble: a constructive set of spherical points with small logarithmic energy, J. Complexity 59 (2020).
• C. Beltrán, The state of the art in Smale's 7th problem, in Foundations of Computational Mathematics, Budapest 2011, Cambridge Univ. Press, 2013. Survey; exponential-time algorithm, no table.
• R. S. Womersley, Minimum energy points on the sphere, https://web.maths.unsw.edu.au/~rsw/Sphere/Energy/ (last updated 16-Jan-2003). Coulomb / Riesz $s=2$, not log.

Energy convention

This folder and Ridgway–Cheviakov Table 3:

Beltrán–Lizarte print $\sum_{i\neq j}=2E$. For the $N=7$ pentagonal bipyramid they give $-16.3649557$, which is $2\times(-8.18247785)$, matching Table 3's $-8.1824779$.

What would count as a beat

A configuration whose independently recomputed $E$ is strictly below the first (global) Table 3 entry for that $N$, by more than rounding ($\gtrsim 10^{-7}$), with the points stored and the verifier agreeing. Rathbun–Ridgway's 77-digit refinement of the same table is the sharper form of the same record for $N\le 65$.

Independent checks (this folder)

• compute/known.py — PASS. Closed forms for $N=2,3,4,5,6$ match energy.py to $10^{-15}$. $N=7$ 1:5:1 and $N=12$ icosahedron match Table 3 to the printed 8 decimals.
• compute/replay_published.py — PASS on Zenodo 5595366 log.3.N.80 for $N=2$–$65$. Recomputed $E$ equals the file ener= to $\sim 10^{-13}$ (N=34 has no ener= line). Table 3 globals match to $|\Delta|<5\times 10^{-8}$.
• compute/optimize.py — recovered the same globals at $N=7,8,9,10,14,19,24,32,33,46,48$. Fibonacci / spiral / rng recover several of them without published seeds. No energy below the record by more than rounding.
• Beltrán–Lizarte $N=7$ printed $-16.3649557$ is $2E$. Convention, not a second table.

What we did not beat

• Any Table 3 global, or the Rathbun–Ridgway 77-digit refinement.
• Smale 7 (polynomial-time $c\log N$ approximation).
• A published $E_{\min}$ for $N>65$: Amore–Figueroa–Ramos 2025 explores the landscape to $N=160$ but does not print a record table, and their 591 MB Zenodo dump was not treated as a citation number.

How to replay

cd problems/fekete-s2/compute
sh run_all.sh

Expected: PASS from verify_replay.py, closed-form matches for $N=2$–$6$, and checkpoint energies agreeing with energy.py. If /tmp/fekete-data/rathbun is present, the full $N=2$–$65$ coordinate replay is re-run; otherwise use the stored replay_rathbun.json.