Exact replication and screening of tropical finiteness certificates for central configurations

Abstract

We report an exact, fully reproducible replication and screening study around Smale's 6th problem (finiteness of planar central configurations), with the following contributions: (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, after which the square, with side a satisfying a^3 = (4 + √2)/8, the larger of the two positive roots of 32a^6 - 32a^3 + 7 = 0, is the unique positive point of the stratum); (ii) a machine measurement of the enrichment law: adjoining the asymmetric (Roberts) equations and the energy–inertia relation renders the n = 3 distance ideal zero-dimensional directly, eliminating 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 and the pointedness of the recession cone is verified by an independent exact parser; and (iv) a screening study over mass-valuation families and equation-system variants at n = 4, 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: the "redundant" equations carry essential tropical information. Every comet of every screened cell is then decided exactly by a rational-arithmetic feasibility certificate, which upgrades the negative half of the screening from absence of a certificate to proof; adding further valid equations is measured separately, showing Dziobek's equations to be tropically active but not decisive at the hard case and the energy–inertia relation to be tropically inert; and two censuses that defeated our own engine are closed by an independent solver with exact cross-verification. All computations are exact (rational or integer arithmetic, or certified polyhedral computation); every stated number is reproducible from the released code and recorded hashes. The finiteness theorems replicated are due to Hampton–Moeckel, Albouy–Kaloshin, and Jensen–Leykin; the measurements in (i), (ii) and (iv) are, to our knowledge, new at this level of explicitness. Source code, data and computational records: https://github.com/fsantibanezleal/CAOS_RESEARCH (problems/dynamical-systems/central-configurations).