Not evaluated
Infinitely many closed geodesics on spheres and three-manifolds (family 345)
The gist
On October 6, 2026, OpenAI announced a manuscript dated September 24, 2026 claiming infinitely many closed geodesic images on every Riemannian sphere.
DiscoverIA status on October 12, 2026: not evaluated, meaning no third-party evaluation found.
AI draft, not reviewedWhat has not been reviewed?
Text written by an AI (DiscoverIA pipeline), on . The status and its sources come from the registry; only the explanatory text has not been reviewed. A reported error is corrected, then the text is reviewed.
Cite this entry
A single status snapshot since Oct 12, 2026: a citation always points to a dated snapshot.
Cite snapshot 1, Oct 12, 2026
APA citation with the dated link
BibTeX @misc, key discoveria-oai-fam-345-infinitely-many-closed-geodesics-spheres-r1
@misc{discoveria-oai-fam-345-infinitely-many-closed-geodesics-spheres-r1,
author = {{DiscoverIA}},
title = {Infinitely many closed geodesics on spheres and three-manifolds (family 345)},
howpublished = {DiscoverIA registry, entry oai-fam-345-infinitely-many-closed-geodesics-spheres},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-345-infinitely-many-closed-geodesics-spheres#releve-1},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
To understand ≈ 20 s reading time, estimated
What it changes
A closed geodesic is a curve that locally follows the shortest path and closes up on itself; a Riemannian sphere is a sphere equipped with a way of measuring lengths. If the manuscript held, its result would apply to every way of measuring, not to one particular sphere.
What remains unclear
No Lean (a language in which a computer checks every step of a proof) is supplied for this family, and no third-party assessment is read. The sources read do not say what the manuscript means by “images” of closed geodesics.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · README.md
Step 1 of 4 on the scale: not evaluated. Next step: under review.
- Announced
- Recordednot evaluated
- nextTo follow
What this status means
Source reading : Source read word for word (level 3 of 3)
The result is declared neither right nor wrong.
To move to “Under review”: a public review under way. To “Peer reviewed”: a journal publication or a review by named experts.
Sources
- Main sourcegithub.com · CONTENTS.mdlink only, reading not recorded
- Repositorygithub.com · Infinitely-many-closed…link only, reading not recorded
Method
Manuscript of September 24, 2026: “Infinitely many closed geodesic images on every Riemannian sphere”. The family is titled “Infinitely many closed geodesics on spheres and three-manifolds”; the manuscript's title mentions only spheres. Section “Differential geometry”; history.md records no revision. DiscoverIA status: not evaluated, meaning that no third-party evaluation has been found in the sources read. Set on October 12, 2026 from github.com (page read, quotation checked word for word: “Not all have accompanying Lean formalizations.”).
Limits
DiscoverIA has not checked this result itself.
Key facts
- Claimed by
- OpenAI (unreleased internal model)
announced on Oct 6, 2026 - Nature
- new result
- Field
- Mathematics (Differential geometry)
- In openai/math
- family 345, section “Differential geometry”, 1 manuscript dated Sep 24, 2026
- Lean proof
- none supplied
- Revisions
- none reported by the claimant
- Checked by
- no verifier identified
- Sources
- 2 : 2 links only
- This entry
- AI draft, not reviewed
Status history
Only one observation so far.
- 01
First observation:Not evaluatedCurrent status
Family with no Lean proof supplied. No third-party evaluation read.
Source: github.comread word for word
An observation is added; none is ever erased.
In the same set
openai/math: 719 manuscripts in 372 families from an unreleased internal model · entry 339 of 372, by family number
- Not evaluated : Donaldson's tamed-to-compatible conjecture (family 342)
- Not evaluated : Sharp symplectic ball-packing criteria in higher dimensions (family 343)
- Not evaluated : The metric Blaschke conjecture (family 344)
- Not evaluated : Infinitely many closed geodesics on spheres and three-manifolds (family 345)
- Not evaluated : Sharp singular-set bounds for stationary integral varifolds (family 346)
- Not evaluated : Counterexamples to strong forms of Arnold's fixed-point conjecture (family 347)
- Not evaluated : Nonnegatively curved Einstein four-manifolds and a topological gap (family 348)
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