Infinitely many closed geodesics on every Riemannian sphere and every closed three-manifold
The closed-geodesic infinitude problem asks whether every closed Riemannian manifold of dimension at least two has infinitely many geometrically distinct (distinct image) closed geodesics. By Gromoll-Meyer and Vigue-Poirrier-Sullivan it holds when the rational cohomology needs at least two generators, which leaves spheres open. The two-sphere was settled by Bangert together with Franks or Hingston, and Rademacher proved infinitude for generic (strongly bumpy) metrics, but for , , only finite multiplicity was known for arbitrary metrics (two on , Long and Duan). Klingenberg's claimed general proof is regarded as erroneous. Does every smooth Riemannian metric on have infinitely many geometrically distinct closed geodesics?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Riemannian geometry; closed geodesics
- Posed by
- Classical closed-geodesic infinitude problem; the manuscript names no poser and traces it through Birkhoff (1917), Lyusternik-Fet (1951) and Klingenberg, Lectures on closed geodesics (1978)
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 55 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: every smooth Riemannian metric on , , has infinitely many prime closed geodesics with pairwise distinct images. Corollary 1.2: the same for every closed manifold finitely covered by a sphere (real projective spaces, spherical space forms). Corollary 1.3: the same for every closed three-manifold, orientable or not, using Rademacher-Taimanov and Perelman. It does NOT settle the problem for general closed manifolds in dimension four or more, and it gives no growth rate or length count for the geodesics.
What the AI did
The release README says the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result, out of roughly 4,000 problems posed; the output was aggregated into result families and manuscripts and kept if judged significant enough. This family has one manuscript, dated September 24, 2026. The manuscript is credited to 'OpenAI' alone and names no human author. The README's two exceptions to the fixed procedure (the Riemann zeta zero-free region work, whose Re(s) > 11/12 write-up was human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The README also cautions that unformalized results could have issues.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollaries 1.2 and 1.3 of the TeX source read against the infinitude problem as the introduction states it. The theorem covers every smooth metric on , , degenerate ones included; the two-sphere case is classical and the new content is . The three-manifold corollary is conditional on published inputs: Rademacher-Taimanov (infinite fundamental group) and Perelman's elliptization. The proof was not refereed. No Lean formalization exists for this family (no lean/docs/345.md and no matching Comparator challenge at the pinned commit). The general problem for all closed manifolds stays open; the paper disputes two steps of Charles's 2019 claimed general proof.