Infinitely many closed geodesics on every sphere, claimed and re-proved
Awaiting review
Importance: major (4 of 5)The takeaway
A manuscript in OpenAI’s 6 Oct 2026 math release (family 345, dated 24 Sept) claims that every smooth Riemannian metric on the n-sphere has infinitely many geometrically distinct closed geodesics, for every n ≥ 2. Only the 2-sphere was known (Bangert, Franks, Hingston, early 1990s). It also covers every closed 3-manifold. On 15 Oct Zhiqi Chen, Shaoqiang Deng and Hui Zhang posted a proof that does not use OpenAI’s lemmas, written with OpenAI Codex and checked by them, and extended it to the complex, quaternionic and Cayley projective spaces. They credit OpenAI’s paper as the source of the strategy.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 29 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 98 days before it.
Key facts
- Claim (abstract, verbatim): ‘We resolve the sphere case of the closed-geodesic infinitude problem: every smooth Riemannian metric on the standard sphere Sⁿ, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The two-sphere case is classical; the result in higher dimensions includes degenerate metrics. Using finite covers and theorems of Perelman and Rademacher–Taimanov, we also obtain the same conclusion for every smooth Riemannian metric on a nonempty closed smooth three-manifold.’
- Prior state (as Chen–Deng–Zhang recount it): Lyusternik–Fet give one closed geodesic on every closed manifold; Gromoll–Meyer and Vigué-Poirrier–Sullivan give infinitely many when the rational cohomology needs two or more generators; spheres and projective spaces are the exceptions; S² was done by Bangert with Franks or Hingston; Rademacher proved it for generic (strongly bumpy) metrics only
- Chen, Deng, Zhang, ‘Infinitely many closed geodesic images on simply connected compact rank-one symmetric spaces’ (arXiv 2610.21054, 15 Oct): Theorem 1.1 covers Sⁿ, CPᵐ, HPᵐ and OP². On OpenAI: ‘We cite that work explicitly as an antecedent for the sphere case and as a source of the finite-cut and doubling strategy. Accordingly, the sphere conclusion itself is not presented here as a new claim of priority.’ Independence: ‘Neither its sphere theorem nor its intermediate lemmas are assumed as mathematical inputs’
- AI disclosure of Chen–Deng–Zhang (verbatim): ‘OpenAI Codex was used in developing the draft arguments, some of the proof strategies and calculations, checking selected algebraic consequences. We take responsibility for the human verification and exposition.’
- Further variant: Kenta Kitamura (GitHub KitaKen1, draft of 9 Oct) claims the conclusion for smooth reversible Finsler metrics on Sⁿ, n ≥ 3, with a Lean proof; the n = 2 Finsler case is De Philippis–Marini–Mazzucchelli–Suhr (2022)
- Status: OpenAI manuscript has no Lean scope note; no revision or withdrawal in history.md; the general conjecture for all closed manifolds stays open outside these classes
What happened
Every closed Riemannian manifold has at least one closed geodesic (Lyusternik and Fet, 1951). Whether it always has infinitely many is a classical question of the calculus of variations. For most simply connected manifolds a topological argument settles it: their free loop spaces have too much homology for finitely many geodesics and their iterates to account for. Spheres and projective spaces are exactly the cases where this fails. On the 2-sphere the answer (yes) came from surface dynamics in the early 1990s. In higher dimensions it was known only for generic metrics.
OpenAI’s manuscript claims the answer is yes for every metric on every sphere, degenerate metrics included, using a length-filtered “finite-cut and doubling” argument in the loop space. Nine days after the release, Zhiqi Chen, Shaoqiang Deng and Hui Zhang posted a proof that rebuilds the argument without using OpenAI’s sphere theorem or lemmas, and that covers the complex, quaternionic and Cayley projective spaces too. They used OpenAI Codex to draft arguments and say they checked the proof themselves. They do not claim the idea: they name OpenAI’s paper as its source.
Why it matters
If correct, a long-open case of the closed-geodesic problem is settled for all spheres and all 3-manifolds. It is also an early example of the pattern the release invited: an AI-generated strategy, rewritten by human mathematicians into an independently checkable proof that goes further. Neither paper is refereed yet.
What is disputed or not yet verified
| Verification | Two unrefereed preprints; the second is a human-checked argument that does not rely on the first’s lemmas |
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Sources
4 sources from 2 sites. Numbers match the chips in the text.
4 sources: 3 primary, 1 reaction
Primary
- OpenAI: Infinitely many closed geodesic images on every Riemannian sphere (24 Sept 2026, PDF)github.com, paper
- Chen, Deng, Zhang: Infinitely many closed geodesic images on simply connected compact rank-one symmetric spaces (arXiv 2610.21054)arxiv.org, paper
- KitaKen1/reversible-finsler-spheres: Finsler variant with Lean proof (draft)github.com, code
Reactions
- oai-math-revised: community notes on the sphere manuscriptgithub.com, discussion
Changes
- Filed (families 251–378; OpenAI abstract, Chen–Deng–Zhang introduction and acknowledgements read, Kitamura README, oai-math-revised notes)