Walkthrough — a Table 3 replay, not a new bound

Problem: problems/fekete-s2 (Smale 7)
Date: 2026-08-19
Argument status: certified replay of a published log-energy table; no configuration whose energy beats that table
Problem status: open. This folder does not claim Smale 7.

0. What was actually missing

Smale 7 is an algorithmic question: a polynomial-time machine that returns an $N$ -tuple within $c\log N$ of $E_{\min}$ . That is not a one-night object. The finite object that can move is a better $N$ -point energy than a printed table, with points and a verifier.

The printed record used here is Ridgway–Cheviakov, Comput. Phys. Commun. 233 (2018), Table 3, fetched as the author PDF and checked with pdftotext. Rathbun–Ridgway, arXiv:2008.04880, refine the same $N\le 65$ list to 77 digits and do not claim new globals beyond the 2018 $N=19,46$ improvements over Bergersen–Boal–Palffy-Muhoray 1994. Amore–Figueroa–Ramos, arXiv:2512.12416 (Dec 2025), map the landscape to $N=160$ but do not print an $E_{\min}(N)$ table.

What was missing was not a new potential. It was an independent recomputation of those numbers, and a search at the $N$ where the 2018 paper itself reports slack ( $N=33$ saddle, $N=19,46$ new globals, modest $N$ with nearby locals).

1. Named false starts

Treat Womersley's ME points as a log-energy record. The UNSW energy page (last updated 16-Jan-2003) minimises Coulomb / Riesz $s=2$ . Different functional. Lead, not a citation for $E_{\log}$ .
Treat Beltrán–Lizarte's printed $-16.3649557$ as this folder's $E$ . They sum $i\neq j$ . Twice our $N=7$ 1:5:1 energy $-8.18247786$ . Convention, not a second table, and not a beat.
Treat an 8-decimal rounding as a beat. Several search energies sit $10^{-8}$ – $10^{-9}$ below the printed Table 3 digit. That is the extra digit of the same number Rathbun already printed to 77 places. House rule: $\Delta\gtrsim 10^{-7}$ and a different configuration, or it is not a new bound.
Download Amore's 591 MB local-minima dump and call the lowest file a record. The paper does not print $E_{\min}(N)$ . A Zenodo dump without a cited number is not a published record.
$N=33$ from random starts. The 2018 paper needed $10^6$ iterations after first hitting a saddle. Our 32 rng starts stall about $10^{-6}$ above the published global. Kicks from the published point stay there. We did not improve it.

2. The useful failure

$N=46$ is the useful failure. Ridgway–Cheviakov's new global ( $-249.4558479$ ) is findable from a generalized spiral and from several rng starts. Many other rng starts land on their published local $-249.454650$ . Fibonacci stalls worse, at $-249.44978$ . The 2018 improvement over Bergersen is real and independently reproducible, and the old basin is still there. A third, lower basin did not appear.

That is the degree of freedom the night actually had: the 2018 paper already isolated the slack ( $N=19,46$ versus 1994; $N=33$ as a hard saddle). Replaying those lines, and failing to undercut them, is the leftover.

3. The replay

Energy, after projecting to $S^2$ :

E(X)=\sum_{i<j}\log\frac{1}{|x_i-x_j|} =-\frac12\sum_{i<j}\log\bigl(2(1-\langle x_i,x_j\rangle)\bigr).

compute/energy.py is the independent verifier. Closed forms:

\begin{aligned} E_2&=-\log 2,\\ E_3&=-\tfrac32\log 3,\\ E_4&=3\log(3/8),\\ E_5&=-4\log 2-\tfrac32\log 3,\\ E_6&=-9\log 2. \end{aligned}

These match Table 3 to the printed 8 decimals, as do the pentagonal bipyramid ( $N=7$ ) and the regular icosahedron ( $N=12$ ).

Rathbun–Ridgway's Zenodo log.3.N.80 files were parsed (including the occasional 1/2 in the $N=44$ algebraic file) and run through the same verifier. For every $N=2$ – $65$ , the recomputed $E$ equals the file's ener= to $\sim 10^{-13}$ and equals Table 3 to the printed digit. That is the full replay.

Multi-start Riemannian descent with fixed seeds then recovered the same globals at $N=7,8,9,10,14,19,24,32,33,46,48$ . For $N=24,32,48$ every seed — Fibonacci, spiral, and 32 rng — lands on the published energy. The table is not an artefact of one coordinate file.

4. What would have been a new bound

An explicit point set whose verifier energy is below the first Table 3 entry for that $N$ by more than rounding, with the points stored. That did not happen. The 77-digit Rathbun–Ridgway refinement of the same list is the sharper form of the same record. Smale 7 is the algorithmic question and is untouched.

5. How to replay

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

Expected: closed-form matches for $N=2$ – $6$ , PASS from verify_replay.py, checkpoint energies agreeing with energy.py. The stored Table 3 extract is ridgway2018.json. The stored coordinate replay is replay_rathbun.json.