---
id: "2026-10-06-openai-closed-geodesics-spheres-infinitely-many"
url: "https://postcutoff.com/e/2026-10-06-openai-closed-geodesics-spheres-infinitely-many/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 4
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-06-openai-closed-geodesics-spheres-infinitely-many/

# Infinitely many closed geodesics on every sphere, claimed and re-proved

Full title: OpenAI's release claims every Riemannian metric on every sphere has infinitely many closed geodesics; Chen, Deng and Zhang give a logically independent, Codex-assisted proof and extend it to projective spaces

"Infinitely many closed geodesic images on every Riemannian sphere" (OpenAI, 24 Sept 2026, released 6 Oct, family 345) claims the sphere case of the closed-geodesic infinitude problem, including degenerate metrics, and, via finite covers, Perelman and Rademacher–Taimanov, every closed 3-manifold. Spheres and projective spaces were the hard cases: their loop spaces have bounded Betti numbers, so the Gromoll–Meyer criterion gives nothing. Chen, Deng and Zhang (arXiv 2610.21054, 15 Oct) prove the statement for all simply connected compact rank-one symmetric spaces with an argument they call logically independent of OpenAI's lemmas, while crediting OpenAI's "finite-cut and doubling strategy"; Codex helped draft arguments and "We take responsibility for the human verification and exposition."

## 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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI: Infinitely many closed geodesic images on every Riemannian sphere (24 Sept 2026, PDF)](https://github.com/openai/math/blob/main/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf) (github.com, paper)
2. [Chen, Deng, Zhang: Infinitely many closed geodesic images on simply connected compact rank-one symmetric spaces (arXiv 2610.21054)](https://arxiv.org/abs/2610.21054) (arxiv.org, paper)
3. [KitaKen1/reversible-finsler-spheres: Finsler variant with Lean proof (draft)](https://github.com/KitaKen1/reversible-finsler-spheres) (github.com, code)
4. [oai-math-revised: community notes on the sphere manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/AGENT_NOTES.md) (github.com, discussion)

## Changes

- 2026-10-29 (filed): Created (families 251–378; OpenAI abstract, Chen–Deng–Zhang introduction and acknowledgements read, Kitamura README, oai-math-revised notes)

## Related

- 2026-10-24: [oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)](https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/index.md)
- 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)