Finiteness of spatial central configurations of the Newtonian five-body problem for all positive masses: the exceptional mass loci of Hampton and Jensen

For the Newtonian five-body problem in three-dimensional space, it is known (Hampton and Jensen, 2011, computer-assisted) that the number of central configurations is finite for every positive mass vector except possibly those lying on an explicitly listed finite union of algebraic loci in mass space (their Table 1, including the 'troublesome' rays 270 and 275). Whether finiteness also holds on these exceptional loci — equivalently, whether the spatial five-body problem has finitely many central configurations for all positive masses, the spatial N=5 instance of Wintner's finiteness conjecture — remains open. Only isolated mass points on the loci (equal masses; (2,1,1,1,1); (2,2,2,1,1)) have been settled since, by interval-arithmetic computer-assisted proofs.

Background

A central configuration of the Newtonian n-body problem is a spatial arrangement of n point masses $m_1,\dots,m_n$ at positions $q_1,\dots,q_n \in \mathbb{R}^3$ such that the gravitational force on each body is proportional to its displacement from the center of mass: $\sum_{j\neq i} m_j (q_i-q_j)/r_{ij}^3 = \lambda (q_i - c)$ with a common constant $\lambda$, where $r_{ij}=|q_i-q_j|$ and $c$ is the center of mass. Configurations are counted up to the natural equivalences: translation, rotation, scaling, and relabeling of bodies. Central configurations started from rest collapse homothetically to the center of mass, and in the planar case they generate relative equilibria — rigidly rotating solutions in which every body moves on a circle at a common angular speed. A conjecture of Wintner (1941), appearing as Problem 6 on Smale's list of problems for the twenty-first century in its planar form, asserts that for any n and any positive masses there are only finitely many central configurations. It is known for n=3 (Euler's collinear and Lagrange's equilateral configurations, with Moulton's n!/2 count for collinear configurations generally) and for n=4 in all dimensions (Hampton and Moeckel 2006, computer-assisted); Albouy and Kaloshin (2012) proved it for the planar five-body problem except for mass vectors in an explicit codimension-2 subvariety. Roberts (1999) showed that a continuum of planar central configurations does exist for five bodies if one mass is negative, so positivity of the masses is essential.

Spatial central configurations of five bodies are Dziobek configurations: configurations of n bodies whose affine span has the minimal dimension n-2 (=3 here). Moeckel (2001) proved generic finiteness for such configurations: finiteness holds for all mass vectors outside a proper algebraic subset of mass space. Hampton and Jensen (2011) strengthened this to an effectively checkable exception set. Writing the central-configuration equations in mutual distances $r_{ij}$ (the Albouy–Chenciner equations, supplemented by Dziobek equations and a Cayley–Menger determinant enforcing embeddability of the distances in $\mathbb{R}^3$), they studied the system with tools from tropical geometry: each polynomial is replaced by its initial form (its lowest-weight part with respect to a weighting $\omega$ of the variables), and the common initial forms define a polyhedral fan of candidates — the tropical prevariety — computed with the software Gfan. If for a given cone of the fan the saturated initial ideal (the initial ideal divided by all products of the variables) contains a monomial, the corresponding piece of the variety cannot carry positive-dimensional families of positive solutions, and finiteness follows for the masses not excluded along the way; saturations and eliminations were carried out with Singular and Sage in exact integer arithmetic.

All but finitely many cone types of the prevariety (576 rays up to symmetry, plus higher-dimensional cones) were eliminated this way. The surviving 'exceptional cases' are mass vectors on which explicit polynomials vanish (Table 1 of the paper): the hypersurface $m_1m_2 - m_3m_4 - m_3m_5 = 0$ (row [59]); a further hypersurface (row [72]); the locus $m_3 = m_4 + m_5$, $(m_4+m_5)^2 = m_1m_2$ (row [59,72]); the two-parameter family $m_1 = m_4$, $m_2 = m_3$ (rows [210], [210,275], [193,210], [210,453], [268,453], where a sum-of-squares decomposition of an eliminated polynomial shows the condition is exactly these equalities); and two 'troublesome' rays. Ray 270 leads to an explicit finite system (equations (8)–(11) of the paper) whose positive-mass solution set is a one-parameter family of exceptional mass vectors (parametrized by $m_1/m_5$, with $m_2 = m_3 = m_4$ always a solution), whose eliminated defining polynomial is described as 'rather large and unwieldy'. Ray 275 splits by a factorization of the initial Cayley–Menger form into two elimination ideals, each yielding a single homogeneous polynomial of degree 12 in the cubed masses $M_i = m_i^3$: P1 and P2. Lemma 1 of the paper shows P1 vanishes on positive masses only when a stated relation holds together with $m_1 = m_2$ or $m_3 = m_4$; Lemma 2 shows P2 vanishes only when three masses are equal. For P2 the authors gave a constructive integer sum-of-squares certificate (an explicit matrix identity inspired by the Real Nullstellensatz and semidefinite-programming experiments with SOSTOOLS); for P1, whose symmetry group is smaller, they could not find an analogous integer decomposition and posed it as 'an exciting challenge'. Their conclusion states: 'It seems likely that the spatial five-body central configurations are always finite for positive masses' — presumably the unresolved cases reflect additional syzygies (algebraic dependencies among the leading forms) not captured by their method.

Subsequent progress has settled only specific mass points on the exceptional loci. Moczurad and Zgliczyński (2020) gave an interval-arithmetic computer-assisted proof (a rigorous branch-and-bound enclosure of all solutions using machine-verified interval enclosures) that for five equal masses — which satisfy the $m_1=m_4$, $m_2=m_3$ exceptional relations — all spatial central configurations are nondegenerate and completely listed; the equivalence-class count under the full rotation group, corrected in their 2026 sequel (the class count printed in the 2020 paper was wrong), is exactly 307. Their 2026 sequel also proved nondegeneracy and finiteness, with exact counts and upper bounds, for the exceptional mass vectors $(2,1,1,1,1)$ (on the row-[59] locus; 94 distinct configurations, at most 376 classes) and $(2,2,2,1,1)$ (satisfying the row-[59,72] relations; 157 distinct configurations, at most 785 classes), and observed that the exceptional character of these cases likely reflects limitations of the algebraic techniques of Albouy–Kaloshin and Hampton–Jensen rather than intrinsic degeneracy. Related but logically distinct results — Deng and Hampton (2023) on finiteness of extensions of fixed subconfigurations, Jensen and Leykin (2025) on generic planar finiteness via tropical computations, and Dias (2026) on generic finiteness of Dziobek configurations for homogeneous potentials with a uniform Bézout-type bound — all leave the exceptional spatial loci untouched.

Problem Statement

Determine whether the number of equivalence classes of spatial central configurations of the Newtonian five-body problem is finite for every vector of five positive masses — the spatial $N=5$ instance of Wintner's finiteness conjecture. Concretely, decide whether finiteness still holds for the mass vectors on the exceptional loci explicitly listed in Table 1 of the cited paper: the hypersurfaces $m_1m_2 - m_3m_4 - m_3m_5 = 0$ and $4m_1m_2 - (m_3+m_4)^2 - (m_3+m_5)^2 + m_3^2 - 2m_4m_5 = 0$; the locus $m_3 = m_4 + m_5$, $(m_4+m_5)^2 = m_1m_2$; the two-parameter family $m_1 = m_4$, $m_2 = m_3$; the one-parameter family of positive solutions of the ray-270 system (equations (8)–(11) of the paper, including $m_2 = m_3 = m_4$); and the ray-275 loci where the eliminated polynomials $P_1$ or $P_2$ (in the cubed masses $M_i = m_i^3$) vanish. An answer must either prove that on each of these loci the set of spatial central configurations is still finite — or exhibit a positive mass vector carrying a positive-dimensional continuum of inequivalent spatial central configurations, which would refute finiteness.

References

1. Marshall Hampton, Anders Jensen, 'Finiteness of spatial central configurations in the five-body problem', Celestial Mechanics and Dynamical Astronomy 109, no. 4 (2011) 321–332, DOI 10.1007/s10569-010-9328-9, https://doi.org/10.1007/s10569-010-9328-9
2. Małgorzata Moczurad, Piotr Zgliczyński, 'Central Configurations with Unequal Masses: Finiteness in Several Exceptional Cases of Five Bodies', arXiv:2601.01165 (2026), https://arxiv.org/abs/2601.01165
3. Małgorzata Moczurad, Piotr Zgliczyński, 'Central configurations in the spatial n-body problem for n = 5, 6 with equal masses', Celestial Mechanics and Dynamical Astronomy 132, article 56 (2020), DOI 10.1007/s10569-020-09993-1, arXiv:2006.06768, https://arxiv.org/abs/2006.06768
4. Alain Albouy, Vadim Kaloshin, 'Finiteness of central configurations of five bodies in the plane', Annals of Mathematics 176, no. 1 (2012) 535–588, DOI 10.4007/annals.2012.176.1.10, https://doi.org/10.4007/annals.2012.176.1.10
5. Yiyang Deng, Marshall Hampton, 'Finiteness of Spatial Central Configurations with Fixed Subconfigurations', SIAM Journal on Applied Algebra and Geometry 7, no. 3 (2023) 610–622, DOI 10.1137/22M1490788, https://doi.org/10.1137/22M1490788
6. Anders N. Jensen, Anton Leykin, 'Smale's 6th problem for generic masses', Journal of Experimental Mathematics 1, no. 2 (2025) 306–314, DOI 10.56994/JXM.001.002.006, arXiv:2301.02305, https://arxiv.org/abs/2301.02305
7. Richard Moeckel, 'Generic Finiteness for Dziobek Configurations', Transactions of the American Mathematical Society 353, no. 11 (2001) 4673–4686, DOI 10.1090/S0002-9947-01-02828-8, https://doi.org/10.1090/S0002-9947-01-02828-8
8. Marshall Hampton, Richard Moeckel, 'Finiteness of relative equilibria of the four-body problem', Inventiones mathematicae 163, no. 2 (2006) 289–312, DOI 10.1007/s00222-005-0461-0, https://doi.org/10.1007/s00222-005-0461-0
9. Thiago Dias, 'The Veronese Geometry of Dziobek Configurations and Generic Finiteness for Homogeneous Potentials', arXiv:2601.07962 (2026), https://arxiv.org/abs/2601.07962
10. Tsung-Lin Lee, Manuele Santoprete, 'Central configurations of the five-body problem with equal masses', Celestial Mechanics and Dynamical Astronomy 104, no. 4 (2009) 369–381, DOI 10.1007/s10569-009-9219-0, https://doi.org/10.1007/s10569-009-9219-0
11. Gareth E. Roberts, 'A continuum of relative equilibria in the five-body problem', Physica D: Nonlinear Phenomena 127, no. 3–4 (1999) 141–145, DOI 10.1016/S0167-2789(98)00315-7, https://doi.org/10.1016/S0167-2789(98)00315-7