---
title: "Finite generation for the $K(n)$-local sphere"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 313; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Finite-generation-for-the-Kn-local-sphere-September-24-2026/Finite-generation-for-the-Kn-local-sphere-September-24-2026.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "topology"
  - "unformalized"
seed_rank: 2168
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Finite generation for the K(n)-local sphere"
    url: "https://github.com/openai/math/blob/main/preprints/Finite-generation-for-the-Kn-local-sphere-September-24-2026/Finite-generation-for-the-Kn-local-sphere-September-24-2026.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Finite generation for the $K(n)$-local sphere

## One-sentence takeaway

OpenAI's result family 313 (Topology) claims: Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the $K(n)$-local sphere is a finitely generated ℤp-module, for every prime, positive height and integer degree.

## Why it matters here

Topology results are background for knot, surface and configuration-space reasoning in geometry code. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): The same conclusion holds after $K(n)$-localizing any finite p-local spectrum.
- *Finite generation for the K(n)-local sphere*: We prove that the homotopy groups of the $K(n)$-local sphere are finitely generated ℤp-modules in every integer degree, for every prime p and positive height n. Equivalently, the same conclusion holds after $K(n)$-localizing any finite p-local spectrum. This answers the degreewise finiteness question of Hovey and Hovey–Strickland.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Finite generation for the K(n)-local sphere](https://github.com/openai/math/blob/main/preprints/Finite-generation-for-the-Kn-local-sphere-September-24-2026/Finite-generation-for-the-Kn-local-sphere-September-24-2026.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math