A machine record around Smale's 6th problem (finiteness of planar central configurations of the Newtonian n-body problem). Four contributions, all exact or certified: (i) an exact calibration of the Albouy-Chenciner mutual-distance equations in open tooling, including a machine rediscovery of their dimension-blindness (the regular tetrahedron coexists with the square in the equal-mass rhombus stratum of the 4-body distance system until the Cayley-Menger equation is adjoined; the square's side satisfies 32a⁶ - 32a³ + 7 = 0) ; (ii) the enrichment law, measured: adjoining the asymmetric (Roberts) equations and the energy-inertia relation makes the n = 3 distance ideal zero-dimensional directly, removing the coordinate-line components of the bare symmetric system; (iii) an independent exact reproduction of the Jensen-Leykin tropical prevariety certificate of generic-mass finiteness for n = 5 (arXiv: 2301. 02305v2): both published f-vectors match digit for digit, the pointedness of the recession cone is verified by an independent exact parser, and the published 257-component count is reproduced; and (iv) a screening study over mass-valuation families and equation-system variants at n = 4 and n = 5, which finds two new working valuation families beyond the two published, replicates generic-mass finiteness for n = 4 purely polyhedrally in seconds, and shows that removing the ten dependent symmetric Albouy-Chenciner equations destroys the certificate at every tested valuation, so the algebraically redundant equations are tropically load-bearing. This is a replication-and-instruments record: the finiteness theorems replicated are due to Hampton-Moeckel, Albouy-Kaloshin and Jensen-Leykin, and the equal-mass censuses to Moczurad-Zgliczynski; the measurements are machine facts about known objects, stated with their provenance. Negative results are reported as absence of a certificate, never as a proof of the negation. All scripts, artifacts, hypotheses declared before each run, and verdicts including refutations: https: //github. com/fsantibanezleal/CAOSRESEARCH (problems/dynamical-systems/central-configurations).
Felipe Santibañez-Leal (Sat,) studied this question.