The number of planar central configurations of four bodies

Abstract

For four positive masses in the plane we determine the number N of central configurations, counted up to rotation, translation and scaling, in terms of one fold surface, whose certified enclosures decide N for all masses outside a closed null set. Of these configurations, 12 are collinear, by Moulton's theorem, and 6 are convex, by the existence theorem of MacMillan and Bartky and the uniqueness proved in the author's preprint arXiv:2609.35632. Through Dziobek's relations, the concave configurations with inner body i form the fibre of a proper mass map of degree 2. Their number c_i, counted up to reflection, is 1, 2, 3 or 4, and we read it off from the position of the mass ratios relative to this surface. Altogether N = 18 + 2(c₁ + c₂ + c₃ + c₄), an even number with 32 ≤ N ≤ 50. Moreover N = 32 exactly at Palmore's masses, and every even value in this range occurs. For four equal masses we recover Albouy's count N = 50. The proof shows with interval certificates that the mass map is injective where its Jacobian is not positive, obtains its degree from the restricted problem, and analyses its folds, cusps and Palmore's umbilic. A formalization in Lean also proves these results.