Finiteness of central configurations of the five-body problem at exceptional mass values (the remaining core of Smale's 6th problem for N=5)

For the Newtonian five-body problem, finiteness of the number of equivalence classes of central configurations is known for all positive masses outside explicitly described exceptional algebraic sets in mass space: the planar exceptional set of Albouy and Kaloshin (2012) and the spatial exceptional cases of Hampton and Jensen (2011). A 2026 computer-assisted study by Moczurad and Zgliczyński verified finiteness at discrete sample mass points on these exceptional families — with exact counts in the planar samples and only symmetry-based upper bounds in the spatial samples — and explicitly left the general question open, together with a supporting conjecture (their Conjecture 21) on permuting bodies to make collinear spatial central configurations non-degenerate in a reduced system, and the certification of bifurcation loci. The problem is to decide, and where true prove, finiteness for every positive mass vector lying on these exceptional sets, in both the planar and the spatial problem, thereby completing the five-body case of Smale's 6th problem for all positive masses.

Background

In the Newtonian $n$-body problem, a central configuration (CC) is a configuration of the $n$ bodies, with positions $q_i$ measured from the center of mass, such that each body's gravitational acceleration satisfies $\ddot q_i = \lambda q_i$ with a common constant $\lambda$. Central configurations generate the relative equilibria — solutions rotating rigidly without deformation — and the homographic solutions, so their count controls the bifurcation structure of the reduced dynamics. Two CCs are considered equivalent if one is obtained from the other by a rotation about the center of mass, a rescaling, and a relabeling of the bodies that permutes bodies of equal mass; the object of study is the number of equivalence classes for a fixed vector of positive masses. Since Wintner's classical treatise, the finiteness of this number has been a standing question, and Smale's 6th problem for the 21st century (1998) asks whether it is finite for every $n$ and every choice of positive masses. Collinear CCs are always finite in number — a classical theorem of Moulton (1910) gives exactly $n!/2$ classes — so the difficulty lies in the genuinely planar and spatial configurations, where a continuum of classes is not a priori excluded. The problem is settled for $n \le 4$ in the plane (Hampton and Moeckel, 2006) and for $n=5$ outside exceptional sets: Albouy and Kaloshin (2012) proved finiteness of planar five-body CCs for all positive masses except a codimension-2 algebraic exceptional set in mass space (with cases such as $m_1=m_2$ and $m_3=m_4$), and, building on Moeckel's 2001 generic finiteness result for Dziobek configurations (spatial central configurations in which the five bodies affinely span three dimensions, i.e., the genuinely non-planar ones), Hampton and Jensen (2011) proved finiteness of non-planar spatial five-body CCs apart from explicitly listed exceptional mass cases, each described by concrete polynomial relations such as $m_1m_2 - m_3m_4 - m_3m_5 = 0$. These exceptional sets are exactly where the transversality and rank arguments of the generic theory degenerate, and they are the surviving open core of the $n=5$ finiteness question. Computer-assisted proofs (CAP) attack the problem by covering the (compactified) configuration space with interval boxes and certifying, via the Newton–Krawczyk interval-Newton operator, that each box contains either no CC or a unique one; one works with normalized central configurations (rescaled so that $\lambda=1$ with the center of mass at the origin) and quotients the rotational symmetry by passing to a reduced system (RS) obtained by fixing one body on a coordinate axis (in space, additionally fixing a second body in a coordinate plane) and eliminating the center-of-mass and rotation variables, switching between body permutations and axis choices (RS-switching) to avoid degenerate Jacobians. Moczurad and Zgliczyński used this method for the equal-mass five-body problem (354 planar and 307 spatial classes — note that equal masses already lie on the exceptional varieties; the spatial count 307 is the value corrected in the 2026 source paper), and in the 2026 source paper extended it to several unequal-mass sample points on the exceptional families: planar counts between 294 and 450 on the $m_1=m_2$, $m_3=m_4$ family and two related 'difficult' one-relation families (the product relation $m_1m_3=m_2m_4$ and the square-root relation $1/\sqrt{m_3}=1/\sqrt{m_1}+1/\sqrt{m_2}$), and spatial upper bounds of 376 and 785 for the mass vectors $(2,1,1,1,1)$ and $(2,2,2,1,1)$ satisfying the Hampton–Jensen exceptional polynomials. The authors stress that they treat only sample points, that their certification necessarily fails at bifurcation loci where the Jacobian is singular (the varying counts between samples suggest such bifurcations exist on the exceptional manifolds), and that the spatial enumeration is computationally delicate; they also state, without proof, Conjecture 21: that for $d=3$ every collinear normalized central configuration satisfying the reduced-system normalization condition $y_{n-1}=0$ becomes a non-degenerate solution of an induced reduced system after a suitable permutation of the bodies — the property their RS-switching technique needs in order to handle collinear spatial configurations in general.

Problem Statement

Decide, and where true prove, the following: for every vector of five positive masses lying on the exceptional sets left open by the generic finiteness theorems — namely (a) the planar exceptional set of Albouy and Kaloshin: the codimension-2 algebraic loci of their Figure 11, notably the family defined by $m_1=m_2$ and $m_3=m_4$ (equation (32), case 8.6 of that paper), together with the codimension-2 loci obtained, per their Theorem 6, by adding a further polynomial relation to the 'difficult' one-relation families $m_1m_3=m_2m_4$ (case 8.3) and $1/\sqrt{m_3}=1/\sqrt{m_1}+1/\sqrt{m_2}$ (case 8.1); and (b) the spatial exceptional cases of Hampton and Jensen (Table 1 of that paper, including ray [59] defined by $m_1m_2-m_3m_4-m_3m_5=0$ and ray [59,72] defined additionally by $m_3-m_4-m_5=0$ and $m_4^2+2m_4m_5+m_5^2-m_1m_2=0$) — the number of equivalence classes of central configurations of the Newtonian five-body problem, planar and spatial respectively, is finite. Finiteness fails exactly when some mass vector in these exceptional sets admits infinitely many pairwise inequivalent central configurations. This completes the five-body case of Smale's 6th problem for all positive masses. The source paper also records two further open technical questions of its own computer-assisted framework — Conjecture 21 of Moczurad–Zgliczyński (for $d=3$, every collinear normalized central configuration $Q_n^m$ satisfying the reduced-system normalization condition $y_{n-1}=0$ can be made a non-degenerate solution of the induced reduced system after a suitable permutation of the bodies) and the rigorous localization and classification of the bifurcation loci on the exceptional mass sets, at which the Newton–Krawczyk certification used by that approach currently fails. These are intermediate needs of the reduced-system interval-arithmetic program, not components of the target stated above, which may be settled by any valid means.

References

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