Smale’s sixth problem claimed for five bodies in the plane
Finitely many central configurations for all positive masses, AI-found and proved in Lean (Tomar)
Awaiting review
Importance: major (4 of 5)The takeaway
An independent researcher claims the planar five-body case of one of Smale’s 18 problems for the 21st century, removing the exceptional masses left by Albouy and Kaloshin in 2012. He states that generative AI found and proved the result and wrote the Lean formalization; the paper does not name the models.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Peer-reviewed
- Importance
- Major (4 of 5)
- Last verified
- 12 October 2026
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None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 101 days before it.
Key facts
- Paper: ‘Finiteness of planar central configurations of five bodies for all positive masses’, arXiv 2610.13100 [math.DS], 9 Oct 2026; Tejasvi Singh Tomar, ‘Independent researcher’; 91 pages
- Theorem 1.1: ‘For every m ∈ (R>0)^5 the planar five-body problem with masses m has finitely many central configurations up to similarity.’
- Smale’s sixth problem (1998) asks whether the number of central configurations is finite for every choice of positive masses. Known before: n = 3 (five configurations, Euler and Lagrange), n = 4 (Hampton–Moeckel 2006), n = 5 for masses outside a closed algebraic subset of codimension two (Albouy–Kaloshin 2012)
- Method: follows Albouy and Kaloshin’s singular-sequence diagrams; determines the possible valuation vectors at all positive masses, excludes diagrams by expanding the potential at a place, then an exact computer-algebra analysis with ‘exact rational certificates’
- Special masses: on the rays through (1,4,4,4,4) and (1,1,4,4,4) the complex equations can have a continuum of solutions, so the real configurations are counted directly: 306 and 294 up to orientation-preserving similarity
- Formal proof: ‘Theorem 1.1 is proved in Lean 4 with Mathlib … It is also accepted by nanoda and lean4lean, two independent implementations of the Lean kernel.’ The paper says re-running the computations from a fresh copy takes about 860 CPU-hours
- AI disclosure (‘Use of AI tools’): ‘Generative AI tools were used to find and prove the results, to write the programs and the Lean formalization, and to draft the text.’ No model is named
- Certificates, checking programs and the Lean project are archived at doi.org/10.5281/zenodo.23202861
Show 2 more
- The same author posted a second AI-generated, Lean-checked result the day before: the rank of 3×3 matrix multiplication over F_2 is 23 (arXiv 2610.11823)
- Scope: the plane and five bodies only; Smale’s problem for n ≥ 6 stays open
What happened
A central configuration is an arrangement of point masses in which the gravitational pull on each body points to the common centre of mass and is proportional to the distance from it. Such arrangements are the shapes that can rotate rigidly or collapse homothetically. Whether there are only finitely many of them for given masses is the sixth problem on Steve Smale’s 1998 list of problems for the 21st century. Four bodies were settled in 2006. For five bodies in the plane, Alain Albouy and Vadim Kaloshin proved finiteness in 2012 except for masses in an explicit exceptional set.
Tomar’s paper claims to remove that exceptional set. It keeps the Albouy–Kaloshin framework, adds expansions of the potential that rule out more limiting diagrams, and settles what remains by exact computer algebra with rational certificates. For two special families of masses, where the complexified equations genuinely have infinitely many solutions, it counts the real configurations directly.
The author states that generative AI tools found and proved the results and wrote the Lean formalization. The Lean proof of the main theorem is reported to pass the standard kernel and two independent re-implementations.
Why it matters
If it holds, it closes a case of a famous problem with a machine-checked proof produced largely by AI, by someone outside a university. A Lean proof shifts the question from “is the argument right” to “does the formal statement say what the theorem says”. The paper points to the file with the formal statement so readers can compare, but nobody else is known to have done that yet.
What is disputed or not yet verified
| Verification | Lean 4 proof reported, accepted by the Lean kernel and by nanoda and lean4lean; unrefereed preprint, the formal statement has not been audited by others |
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The Lean project was not rebuilt for this entry. The paper does not say which models were used. No reaction from specialists in celestial mechanics was found on 12 Oct 2026.
Sources
3 sources from 2 sites. Numbers match the chips in the text.
3 sources: 3 primary
Primary
- arXiv 2610.13100: Finiteness of planar central configurations of five bodies for all positive massesarxiv.org, paper
- Zenodo archive: certificates, checkers and Lean projectdoi.org, code
- Albouy & Kaloshin, Finiteness of central configurations of five bodies in the plane (Annals of Mathematics, 2012)doi.org, paper
Changes
- Filed from the arXiv PDF (found by a manual arXiv scan)