# Closed geodesic research benchmark library

Family 345: Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Researchers can organize infinite distinct prime geodesic existence across spherical-cover and closed three-manifold cases.

## Applicability and commercial boundary

Existence of infinitely many paths supplies no efficient path enumerator, finite length budget or navigation improvement.

## Initial business decision

Research or conditional engineering only. A practical implementation, recurring buyer problem and willingness to pay have not been established. Mathematical usefulness alone is insufficient evidence for a standalone lucrative product.

## Next verification action

Extract a concrete computable metric and a limited geodesic-search reference.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and formal-scope comparison remain queued. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Infinitely many closed geodesic images on every Riemannian sphere](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf).