Newtonian celestial mechanics and real algebraic geometry

Finiteness of planar central configurations

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For any fixed positive masses, are there only finitely many planar Newtonian central configurations up to similarity? The source constrains possible continuous families and complex boundary clusters, but a universal exclusion of physical families remains open.

\forall n \geq 2 \forall (m_{1}, \dots, m_{n}) \in ℝ_{> 0}^{n} : # C_{n, m}^{phys} < \infty

Smale's sixth problem

Known results and sources

Illustration: An equal-mass equilateral example introduces the question: for fixed positive masses, are there only finitely many central-configuration shapes?

Research problem

Exact mathematical statement

For every integer $n\ge2$ and every fixed labelled mass vector $m=(m_1,…,m_n)\in\mathbb R_{>0}^n$ , prove that the collision-free planar Newtonian central configurations form finitely many similarity classes. After translation to the mass center, these are $q_i\in\mathbb R^2$ , with $q_i\ne q_j$ for $i\ne j$ , satisfying

\sum_i m_iq_i=0,\qquad \sum_{j\ne i}m_j\frac{q_i-q_j}{|q_i-q_j|^3}=\lambda q_i \quad(1\le i\le n),\qquad \lambda>0.

Equivalently, scale to $\lambda=1$ , fix rotation by $y_1=y_2$ and $x_1>x_2$ , and prove that the resulting physical solution set is finite:

\forall n\ge2\ \forall m\in\mathbb R_{>0}^n: \quad \#\mathcal C^{\rm phys}_{n,m}<\infty.

Reflection and any equal-mass permutations are finite ambiguities and do not affect finiteness. The target retains fixed labels, strict positivity, planar positions, and Newtonian potential $r^{-1}$ . Generic mass vectors, signed or zero masses, spatial configurations, and other exponents do not replace it.

Problem infographic

Problem at a glance

Illustration: An equal-mass equilateral example appears beside the positive-mass planar conditions, similarity convention and central-configuration equations. — Illustration: For every fixed labelled positive mass vector in the plane, the open question asks for finitely many collision-free central configurations up to similarity.

Current mathematical picture

Where work on Finiteness of planar central configurations stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureTreating external moments as full force compatibility

Both roots of the displayed parameter quadratic give admissible external order-four scalar data, showing that moments alone cannot finish the residual exclusion. Impose all signed internal force equations and the full normal-Hessian or conormal constraints on the same external candidate.

Route status · Narrowed route

Main reductionInfinitude produces a physical Nash arc

For fixed positive masses and multiplier one, the source reports compactness with positive pair separation and uniqueness in each collinear ordering chamber. The positive reciprocal-distance polynomial model makes infinitely many physical solutions equivalent to a nonconstant regular Nash arc with a noncollinear subarc.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeExclude or fully characterize attachment of the surviving five-body zero-force residual at the prescribed outside mass one.Task status · Ready to work on

Work mapped so far

Finiteness of planar central configurations in numbers

1.1kretained lines of mathematical investigation1,148 in the current working snapshot

Argument development: 825 · 72%
Explored or eliminated routes: 60 · 5%
Computational analysis: 48 · 4%
Open obligations: 72 · 6%
Definitions and setup: 143 · 12%

9selected mapped statements2routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedSep 7, 2026

Source recordedSep 7, 2026

Recommended next task

Exclude or fully characterize attachment of the surviving five-body zero-force residual at the prescribed outside mass one.

Suggested move: Suggested next step: Choose external order two, three, or four, impose all signed force and noncollision equations, compute normal rank and inertia, then apply the complete regular-Hessian or full-residual conormal conditions without discarding resonance alternatives.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

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Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

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Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeTreating external moments as full force compatibility

Route status · Narrowed route

Narrowed routeInferring isolation from normal Hessian corank at most two

The source explicitly distinguishes spatially separated clusters, algebraically isolated leading points, and invertible normal Hessians. Establish isolation where used or analyze the nonisolated locus directly; even a finite exceptional-mass polynomial requires a verified test at mass one.

Route status · Narrowed route

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Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Exclude or fully characterize attachment of the surviving five-body zero-force residual at the prescribed outside mass one.Suggested move: Suggested next step: Choose external order two, three, or four, impose all signed force and noncollision equations, compute normal rank and inertia, then apply the complete regular-Hessian or full-residual conormal conditions without discarding resonance alternatives.

Ready to work on

Find a mandatory boundary or divisor pattern for every physical distance-field arc, including arbitrary genus and mixed nonreal degenerations.Suggested move: Suggested next step: Prove a precisely delimited coverage theorem that treats multiple pole pairs, actual coincidences, isotropic separation, and multiple outside positions before invoking any existing chart exclusion.

Ready to work on

Resolve the surviving degree-six rational denominator models with full planar force and Gram compatibility.Suggested move: Suggested next step: Analyze F²G and FGH while retaining shared square factors, both moment identities, Gram rank, and differentiated moving-stress equations; state the resulting scope without extrapolating to higher degree or genus.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026

Current statusOpen problem

The selected sources leave open finiteness of planar central configurations, modulo the stated geometric equivalences, for every number of bodies and every choice of positive masses. Exceptional-mass qualifications and non-degeneracy hypotheses cannot be discarded.

[1][3][4]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025-02-27
PreprintIn the preprint submitted on this date, Natrup, Wang and Wang report bounds for non-degenerate planar central configurations with a homogeneous potential. The non-degeneracy condition limits the scope.[4]
2012
Peer reviewedAlbouy and Kaloshin proved planar five-body finiteness outside a specified possible exceptional set of positive masses of codimension two.[3]
2006
Peer reviewedHampton and Moeckel proved finiteness of relative equilibria for the four-body problem with positive masses, using symbolic and exact-integer computation.[2]
1998
Historical sourceSmale designated the planar positive-mass finiteness question as Problem 6, explicitly tracing the question to earlier work of Wintner.[1]

Mathematical neighborhood

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction2 of 92
lemma5 of 95
equivalence1 of 91

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included

retained route statementAre there finitely many planar central configurations for every fixed positive mass vector?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementInfinitude produces a physical Nash arcintermediate
retained route statementA theoretical finite-jet criterionintermediate
retained route statementFive-body zero-force Hessian corank boundintermediate
retained route statementExternal moment orders and strict physical domainintermediate
retained route statementQualified finiteness of exceptional outside massesintermediate
retained route statementSingular residuals need the full conormal conditionintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureTreating external moments as full force compatibilityreported failure
Useful failureInferring isolation from normal Hessian corank at most tworeported failure
Research targetExclude or fully characterize attachment of the surviving five-body zero-force residual at the prescribed outside mass one.open
Research targetFind a mandatory boundary or divisor pattern for every physical distance-field arc, including arbitrary genus and mixed nonreal degenerations.open
Research targetResolve the surviving degree-six rational denominator models with full planar force and Gram compatibility.open
Narrowed routeTreating external moments as full force compatibilityBoth roots of the displayed parameter quadratic give admissible external order-four scalar data, showing that moments alone cannot finish the residual exclusion. Impose all signed internal force equations and the full normal-Hessian or conormal constraints on the same external candidate.
Narrowed routeInferring isolation from normal Hessian corank at most twoThe source explicitly distinguishes spatially separated clusters, algebraically isolated leading points, and invertible normal Hessians. Establish isolation where used or analyze the nonisolated locus directly; even a finite exceptional-mass polynomial requires a verified test at mass one.

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A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeExclude or fully characterize attachment of the surviving five-body zero-force residual at the prescribed outside mass one.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

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Prepared starting pointExclude or fully characterize attachment of the surviving five-body zero-force residual at the prescribed outside mass one.

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Sources and references4 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Mathematical Problems for the Next Centuryoriginal source · Steve Smale · The Mathematical Intelligencer 20, 7–15 · 1998 · DOI 10.1007/BF03025291 · accessed Sep 7, 2026
2
Finiteness of relative equilibria of the four-body problempeer reviewed result · Marshall Hampton, Richard Moeckel · Inventiones Mathematicae 163, 289–312 · 2006 · DOI 10.1007/s00222-005-0461-0 · accessed Sep 7, 2026
3
Finiteness of central configurations of five bodies in the planepeer reviewed result · Alain Albouy, Vadim Kaloshin · Annals of Mathematics 176, 535–588 · 2012 · DOI 10.4007/annals.2012.176.1.10 · accessed Sep 7, 2026
4
Finiteness of non-degenerate central configurations of the planar n-body problem with a homogeneous potentialpreprint · Julius Natrup, Qun Wang, Yuchen Wang · arXiv · 2025-02-27 · ARXIV 2502.20228 · accessed Sep 7, 2026

Important qualifications

This is a scoped administrative literature search, not an exhaustive survey or an independent mathematical review.
Submitted packets are not external evidence. Their novel claims, corrections and proof obligations retain their separate source-reported posture.
No external code, proof development, data file or certificate was executed or independently reproduced during collection. Public links confer no acceptance or redistribution authority.
An empty formalization or recognition list means no suitable item was established in this search; it does not establish nonexistence. Unknown origin dates and original attributions are left unresolved.
The 1998 Smale designation is not asserted to be the question's original proposal date. The local title is a descriptive workspace label; the source explicitly supplies the numbered designation.

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Finiteness of planar central configurations

Exact mathematical statement

Problem at a glance

Where work on Finiteness of planar central configurations stands

Finiteness of planar central configurations in numbers

Exclude or fully characterize attachment of the surviving five-body zero-force residual at the prescribed outside mass one.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formal and computational footholds

Claims, milestones, and routes in the current map

Conditions that would advance the current route

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