{"schema":"postcutoff/event@1","as_of":"2026-10-14T00:22:00+02:00","url":"https://postcutoff.com/e/2026-09-28-four-body-convex-central-configurations-unique-tomar/","md":"https://postcutoff.com/e/2026-09-28-four-body-convex-central-configurations-unique-tomar/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-09-28-four-body-convex-central-configurations-unique-tomar","date":"2026-09-28","date_precision":"day","short_title":"Four-body problem: uniqueness of convex central configurations for all masses claimed (Simó–Yoccoz conjecture), AI-found and formalized in Lean (Tomar)","deck":null,"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2609.35632: Uniqueness of four-body convex central configurations for all masses","url":"https://arxiv.org/abs/2609.35632","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GitHub: tejasvi/lean4body (programs, certificates, Lean formalization)","url":"https://github.com/tejasvi/lean4body","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"Zenodo archive of the repository","url":"https://doi.org/10.5281/zenodo.23004297","type":"code","group":"primary","domain":"doi.org"}],"official":3,"filed":"2026-10-13","updated":"2026-10-13","orgs":["independent"],"title":"Four-body problem: uniqueness of convex central configurations for all masses claimed (Simó–Yoccoz conjecture), AI-found and formalized in Lean (Tomar)","summary":"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. This answers Problem 10 of Albouy, Cabral and Santos's 2012 list of open problems on the n-body problem, conjectured by Albouy, Fu and Sun in 2008. The proof uses interval arithmetic and is formalized in Lean; the paper states that \"Generative AI tools were used to find and prove the results\". Unrefereed, no model named, no expert reaction found. It preceded the same author's claims on Smale's sixth problem and on 3×3 matrix multiplication by ten days.","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","Status on 13 Oct 2026: unrefereed; no public comment found from specialists in central configurations (web and Hacker News searched; X not searched)"],"key_numbers":[],"tags":["math","celestial-mechanics","n-body-problem","central-configurations","lean","computer-assisted-proof","interval-arithmetic","ai-generated-proof"],"science":{"field":"mathematics","subfield":"celestial mechanics","problem":"Is the convex central configuration of four bodies with a given cyclic order unique up to similarity for all positive masses? (Albouy–Fu–Sun conjecture 2008; Problem 10 of Albouy–Cabral–Santos 2012; 'Simó–Yoccoz conjecture')","result":"Claimed: exactly one strictly convex planar central configuration for every four positive masses and every cyclic ordering.","open_since":"2008","ai_system":["unnamed generative AI tools"],"human_role":"AI-generated per the author's disclosure (AI found and proved the results, wrote the programs, the Lean formalization and the draft); the human role is not described","verification":{"key":"preprint","label":"Preprint only","text":"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"},"status":"pending","shock":"","short":"Four-body convex central configurations are unique","verified_by":["preprint"]},"body_md":"## What happened\n\nA central configuration is an arrangement of point masses that can rotate rigidly under Newtonian gravity. For four bodies in the\nplane, a non-collinear one is either convex (the bodies are the corners of a convex quadrilateral) or concave (one body inside\nthe triangle of the others). It has been known since 1932 that a convex one exists for every choice of masses and every cyclic\norder of the bodies. Whether it is the only one was an open question that specialists had settled only for special families of\nmasses.\n\nTomar's paper claims the general answer: it is unique. The argument bounds the Hessian of the relevant function at every convex\ncentral configuration, which makes each one non-degenerate, and then continues from the equal-mass square, where uniqueness is\nclassical. One inequality is checked by interval arithmetic over a three-dimensional region, and the whole chain, including that\ncomputation, is reported as formalized in Lean.\n\nThe author states that generative AI tools found and proved the results and wrote the programs and the formalization. This was the\nfirst of three such papers by the same author in under two weeks.\n\n## Why it matters\n\nThe 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\nattracted steady partial results. If the proof holds, it is another case where an outsider with AI tools and a proof assistant\nclosed a problem that specialists had worked on case by case.","disputed":[{"label":"Verification","text":"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","sources":[]}],"related":[{"id":"2026-10-09-smale-sixth-problem-five-bodies-lean-tomar","url":"https://postcutoff.com/e/2026-10-09-smale-sixth-problem-five-bodies-lean-tomar/","date":"2026-10-09","date_precision":"day","short_title":"Smale's sixth problem claimed for five bodies in the plane","deck":"Finitely many central configurations for all positive masses, AI-found and proved in Lean (Tomar)","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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-12","updated":"2026-10-13","orgs":["independent"]},{"id":"2026-10-08-rank-3x3-matrix-multiplication-f2-is-23-lean-tomar","url":"https://postcutoff.com/e/2026-10-08-rank-3x3-matrix-multiplication-f2-is-23-lean-tomar/","date":"2026-10-08","date_precision":"day","short_title":"Laderman's 23 multiplications are optimal for 3×3 matrices over F_2 and over the integers","deck":"AI-found proof checked in Lean (Tomar)","takeaway":"Whether 3×3 matrices can be multiplied with fewer than Laderman's 23 multiplications has been open since 1976. A preprint claims the answer is no for coefficients in F_2 or the integers, ending a 2026 race in which the lower bound rose from 19 to 20, 21, 22 and now 23. The author says generative AI found and proved the result.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":8,"filed":"2026-10-12","updated":"2026-10-13","orgs":["independent"]},{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":5,"filed":"2026-09-30","updated":"2026-10-13","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[{"id":"fidji-simo","name":"Fidji Simo","url":"https://postcutoff.com/person/fidji-simo/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-13","type":"filed","text":"Created from a lead; abstract, introduction and AI statement read in the arXiv PDF (previously a key fact in the Smale entry)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-13","run":null,"sources_read":"A lead; abstract, introduction and AI statement read in the arXiv PDF (previously a key fact in the Smale entry)","updated":"2026-10-13","human_review":null,"version":{"date":"2026-10-13"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":151,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":90,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":181,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":120,"in_training_data":false}],"short_url":null}