In OpenAI's preprint Finite generation for the K(n)-local sphere (PDF): 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

Yes, Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More on YouTube (29:20, chapter 19): https://www.youtube.com/watch?v=lckiipi69OY&t=377s

Finite generation for the K(n)-local sphere | OpenAI Math, Explained

Finite generation for the K(n)-local sphere

One of the 372 results in OpenAI's public mathematics catalogue, filed under topology: 1 preprint, the latest dated 24 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.

In one line: OpenAI reports a result titled ‘Finite generation for the K(n)-local sphere’ (topology) in a preprint dated 24 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More, is on YouTube.

What OpenAI 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. The same conclusion holds after K(n)-localizing any finite p-local spectrum.
OpenAI, in its public catalogue of maths results

Paper

Finite generation for the K(n)-local sphere
24 September 2026 · Preprint, not peer reviewed · PDF
OpenAI's abstract 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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.

Quick answers

Is it proven?

Not independently yet. OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.

What exactly does OpenAI claim?

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. The same conclusion holds after K(n)-localizing any finite p-local spectrum.

Where is the proof?

In the paper, Finite generation for the K(n)-local sphere (PDF).

Is there a video?

Yes, on YouTube (29:20, chapter 19).

Finite generation for the K(n)-local sphere

What OpenAI claims

Paper