Four-body problem: uniqueness of convex central configurations for all masses claimed (Simó–Yoccoz conjecture), AI-found and formalized in Lean (Tomar)
Awaiting review
The takeaway
On 28 Sept 2026 Tejasvi Singh Tomar (independent researcher) posted arXiv 2609.35632: for every four positive masses and every cyclic ordering there is exactly one strictly convex planar central configuration up to similarity.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 13 October 2026
Your AI and this story
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None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 90 days before it.
Key facts
- Paper: ‘Uniqueness of four-body convex central configurations for all masses’, arXiv 2609.35632 [math.DS], 28 Sept 2026, 34 pages, 1 figure; Tejasvi Singh Tomar
- Claim (abstract): ‘for every choice of four positive masses and every cyclic ordering of the bodies there is exactly one strictly convex planar central configuration with that ordering, up to similarity. This answers Problem 10 in the list of Albouy, Cabral and Santos, which Santoprete calls the Simó-Yoccoz conjecture.’
- History per the paper: MacMillan and Bartky proved existence in 1932; Albouy, Fu and Sun conjectured uniqueness in 2008; before this paper it was known for equal masses (Albouy 1995–96), equal masses at the ends of a diagonal, two pairs of equal adjacent masses (Fernandes, Llibre and Mello 2017), restricted classes (Santoprete), three small masses, and a region of masses by interval arithmetic (Sun, Xie and You 2023)
- Method: a uniform lower bound for the Hessian at convex central configurations (‘at least one quarter of its radial part’); Dziobek’s relations and a Cauchy–Schwarz argument reduce it to showing that the trace of an explicit 2×2 matrix is below a fixed bound, proved by interval arithmetic on the three-dimensional set of normalized convex configurations; uniqueness follows by a covering argument from the equal-mass case
- Checks: ‘The computation has been repeated with a second, independently written program. The Hessian bound and the uniqueness theorem, including the computation, have been formalized in the Lean proof assistant and checked by its kernel, using only the standard axioms.’ The paper says the proof was also checked by nanoda, a second type checker for Lean
- Consequences stated: every degenerate four-body central configuration is concave; the convex configuration depends analytically on the masses; the known symmetry theorems for kites, isosceles trapezoids and rhombi follow in a few lines
- AI disclosure (‘Use of AI tools’): ‘Generative AI tools were used to find and prove the results, to write both programs of Section 5 and the Lean formalization, and to draft the text.’ No model is named
- Code, certificates and Lean project: github.com/tejasvi/lean4body, archived at doi.org/10.5281/zenodo.23004297
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- Status on 13 Oct 2026: unrefereed; no public comment found from specialists in central configurations (web and Hacker News searched; X not searched)
What happened
A central configuration is an arrangement of point masses that can rotate rigidly under Newtonian gravity. For four bodies in the plane, a non-collinear one is either convex (the bodies are the corners of a convex quadrilateral) or concave (one body inside the triangle of the others). It has been known since 1932 that a convex one exists for every choice of masses and every cyclic order of the bodies. Whether it is the only one was an open question that specialists had settled only for special families of masses.
Tomar’s paper claims the general answer: it is unique. The argument bounds the Hessian of the relevant function at every convex central configuration, which makes each one non-degenerate, and then continues from the equal-mass square, where uniqueness is classical. One inequality is checked by interval arithmetic over a three-dimensional region, and the whole chain, including that computation, is reported as formalized in Lean.
The author states that generative AI tools found and proved the results and wrote the programs and the formalization. This was the first of three such papers by the same author in under two weeks.
Why it matters
The problem is less famous than Smale’s sixth, but it sat on the standard list of open problems in the field for 14 years and had attracted steady partial results. If the proof holds, it is another case where an outsider with AI tools and a proof assistant closed a problem that specialists had worked on case by case.
What is disputed or not yet verified
| Verification | Lean formalization reported, checked by the Lean kernel and by nanoda; computation repeated with a second program; unrefereed preprint, formal statement not audited by others |
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Nobody else is known to have re-run the Lean project or compared its formal statement with the theorem. The paper does not name the AI tools. We found no response from the authors of the earlier partial results.
Sources
3 sources from 3 sites. Numbers match the chips in the text.
3 sources: 3 primary
Primary
Changes
- Filed from a lead; abstract, introduction and AI statement read in the arXiv PDF (previously a key fact in the Smale entry)