{"schema":"postcutoff/event@1","as_of":"2026-10-12T22:30:00+02:00","url":"https://postcutoff.com/e/2026-10-09-smale-sixth-problem-five-bodies-lean-tomar/","md":"https://postcutoff.com/e/2026-10-09-smale-sixth-problem-five-bodies-lean-tomar/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"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":[{"n":1,"title":"arXiv 2610.13100: Finiteness of planar central configurations of five bodies for all positive masses","url":"https://arxiv.org/abs/2610.13100","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Zenodo archive: certificates, checkers and Lean project","url":"https://doi.org/10.5281/zenodo.23202861","type":"code","group":"primary","domain":"doi.org"},{"n":3,"title":"Albouy & Kaloshin, Finiteness of central configurations of five bodies in the plane (Annals of Mathematics, 2012)","url":"https://doi.org/10.4007/annals.2012.176.1.10","type":"paper","group":"primary","domain":"doi.org"}],"official":3,"filed":"2026-10-12","updated":"2026-10-12","orgs":["independent"],"title":"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)","summary":"On 9 Oct 2026 Tejasvi Singh Tomar (independent researcher) posted arXiv 2610.13100: for every choice of five positive masses the planar five-body problem has finitely many central configurations up to similarity. This is Smale's sixth problem for n = 5 in the plane; Albouy and Kaloshin (Annals of Mathematics, 2012) had proved it outside a codimension-two set of masses. The paper says the theorem \"is proved in Lean 4 with Mathlib\" and is accepted by two independent kernel implementations, and its \"Use of AI tools\" note reads: \"Generative AI tools were used to find and prove the results, to write the programs and the Lean formalization, and to draft the text.\" Unrefereed; no expert reaction found yet.","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","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"],"key_numbers":[],"tags":["math","celestial-mechanics","n-body-problem","smale-problems","central-configurations","lean","computer-assisted-proof","ai-generated-proof"],"science":{"field":"mathematics","subfield":"celestial mechanics / algebraic geometry","problem":"Smale's sixth problem: is the number of central configurations (relative equilibria) of the n-body problem finite for every choice of positive masses? Case n = 5 in the plane","result":"Claimed: finitely many planar central configurations for every five positive masses, removing the codimension-two exceptional set of Albouy–Kaloshin.","open_since":"1998","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":"peer-review","label":"Peer-reviewed","text":"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"},"status":"pending","shock":"A named problem from Smale's list, in a case specialists had left open since 2012, is claimed by an independent author with an AI-written formal proof.","short":"Smale's 6th problem, five bodies in the plane"},"body_md":"## What happened\n\nA central configuration is an arrangement of point masses in which the gravitational pull on each body points to the common centre\nof mass and is proportional to the distance from it. Such arrangements are the shapes that can rotate rigidly or collapse\nhomothetically. Whether there are only finitely many of them for given masses is the sixth problem on Steve Smale's 1998 list of\nproblems for the 21st century. Four bodies were settled in 2006. For five bodies in the plane, Alain Albouy and Vadim Kaloshin\nproved finiteness in 2012 except for masses in an explicit exceptional set.\n\nTomar's paper claims to remove that exceptional set. It keeps the Albouy–Kaloshin framework, adds expansions of the potential that\nrule out more limiting diagrams, and settles what remains by exact computer algebra with rational certificates. For two special\nfamilies of masses, where the complexified equations genuinely have infinitely many solutions, it counts the real configurations\ndirectly.\n\nThe author states that generative AI tools found and proved the results and wrote the Lean formalization. The Lean proof of the\nmain theorem is reported to pass the standard kernel and two independent re-implementations.\n\n## Why it matters\n\nIf it holds, it closes a case of a famous problem with a machine-checked proof produced largely by AI, by someone outside a\nuniversity. A Lean proof shifts the question from \"is the argument right\" to \"does the formal statement say what the theorem\nsays\". The paper points to the file with the formal statement so readers can compare, but nobody else is known to have done that\nyet.","disputed":[{"label":"Verification","text":"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","sources":[]}],"related":[{"id":"2026-10-09-geometrization-conjecture-lean-formalization","url":"https://postcutoff.com/e/2026-10-09-geometrization-conjecture-lean-formalization/","date":"2026-10-09","date_precision":"day","short_title":"Thurston's geometrization conjecture formalised in Lean","deck":"The Hamilton–Perelman proof machine-checked with GPT-6, Claude and DeepSeek agents (Chow's group)","takeaway":"Twelve days after the Poincaré conjecture, the same group machine-checked the full geometrization theorem, the result Perelman actually proved. The paper names the models: GPT-6 Astra reviews, and GPT-6 Luna, Sol, Astra, Claude Opus 5.5 and DeepSeek V4.1 Flash build.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-12","updated":"2026-10-12","orgs":["UC San Diego","Cornell University","Princeton University","Tsinghua University","Amazon AGI","NVIDIA Research","University of Oregon"]},{"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":5,"official":5,"filed":"2026-10-12","updated":"2026-10-12","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":9,"official":4,"filed":"2026-09-30","updated":"2026-10-12","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-12","type":"filed","text":"Created from the arXiv PDF (found by a manual arXiv scan)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-12","run":null,"sources_read":"The arXiv PDF (found by a manual arXiv scan)","updated":"2026-10-12","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":162,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":101,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":192,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":131,"in_training_data":false}],"short_url":"https://postcutoff.com/s/smale-sixth-problem-five"}