Exactly n + 2?
An AI preprint claims the Hovey–Strickland and Chai conjectures, and a public question about where the proof came from
Covers result 311 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:07. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Chromatic homotopy theory studies the stable homotopy category one height at a time. At a prime p and a height n, the K(n)-local category has a natural family of dualizable objects. In their 1999 memoir, Hovey and Strickland found n + 2 thick tensor ideals among them, families closed under the natural operations, and conjectured that there are no others; equivalently, its Balmer spectrum would be a chain of n + 1 points. In 2020, Barthel, Heard and Naumann showed this would follow from a statement about the Lubin–Tate ring, which parametrizes the deformations of a formal group of height n: Chai’s hope that, even for open subgroups of the Morava stabilizer group, the only stable prime ideals are zero and the chain of ideals generated by p, u₁, …. They proved heights 1 and 2, and wrote that higher heights would need substantially new ideas.
Result 311 is one OpenAI paper, 34 pages, dated September 24, 2026, and made public October 6. It claims Chai’s statement at every prime and every positive height, for every open subgroup of the stabilizer. With Barthel, Heard and Naumann, this gives the Hovey–Strickland conjecture: exactly n + 2 thick tensor ideals, and a chain of n + 1 points. For Hovey–Strickland, the new cases are heights 3 and up.
What it doesn’t claim
Anything beyond the dualizable objects, ideals that are not radical, or residue fields other than the finite ones.
A question of provenance
The day after the release, Chenglong Ma, a doctoral student in Copenhagen, posted a proof of the same theorem on arXiv, using the Fargues–Fontaine curve, Cartier theory and Chai’s rigidity theorem; Ningyi Li shared another proof, by different methods, as a public PDF. Ma states that the OpenAI paper’s proof strategy and specific choices of notation are identical to a preliminary version of Ma’s paper that was uploaded to ChatGPT on September 8, 2026, sixteen days before the OpenAI paper’s date. That draft is not public, so we cannot check the claim. In the public versions, the overall strategy is closely parallel, down to some lemmas and a few symbols, but the technical machinery differs, and so do the basic names for the main objects. Both Ma and Li describe their own AI use: Ma used ChatGPT to check the height-3 case and then for auditing, and Li used AI to explore strategies and to audit. We found no response from OpenAI.
Where it stands
- Posted as a preprint
- OpenAI, 34 pages, dated September 24, 2026
- Peer reviewed
- Not yet
- Checked in Lean
- No formalization for this item
Last checked October 19, 2026. The paper has not been revised, and there is no Lean formalization for this item. OSVerify has not reviewed it. In the first twelve days we found no public specialist assessment of the mathematics and no reported error; the discussion has been about provenance.
How the proof goes
- Dense orbits. The core is a local orbit density theorem in characteristic p, built from Frobenius separation, a formal closure argument, and the geometry of the Fargues–Fontaine curve with perfectoid tools. Characteristic zero is handled through the Gross–Hopkins period map, an approach the paper credits to Chai, and the spectral half reassembles Barthel, Heard and Naumann’s argument, with credit.
The numbers in the video
- Ours: the lattice of n + 2 ideals and the chain of n + 1 points for heights up to 7; with one extra invariant piece at height 2 there would be 6 ideals, and the spectrum would no longer be a chain.
- Ours: in the p-series of the universal deformation, reduced by the h-th ideal for h below n, the leading term is u_h·x^(pʰ) in every case we tried; at p = 2, n = 2, [2](x) ≡ u₁x² + x⁴ + u₁²x⁶ + … mod 2.
Read the source
- Stabilizer Orbits and Thick Tensor Ideals of Dualizable K(n)-Local Spectra (PDF), 34 pages.
- Barthel, Heard and Naumann’s reduction (arXiv).
- Chenglong Ma’s paper and statement (arXiv).
- The collection’s catalog, where this is result 311.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
Chromatic homotopy theory studies the stable homotopy category one height at a time. At a prime p and a height n, the K(n)-local category has a natural family of dualizable objects. In their 1999 memoir, Hovey and Strickland found n + 2 thick tensor ideals among them, families closed under the natural operations, and conjectured that there are no others. Equivalently, its Balmer spectrum, a space of prime ideals, would be a chain of n + 1 points. Our computer builds that picture for heights up to 7, and shows what one extra invariant piece would do at height 2: 6 ideals, and a spectrum that is no longer a chain. In 2020, Barthel, Heard and Naumann showed it would follow from a statement about the Lubin–Tate ring, which parametrizes the deformations of a formal group of height n: Chai’s hope that, even for open subgroups of the Morava stabilizer group, the only stable prime ideals are zero and the chain of ideals generated by p, u₁, …. They proved heights 1 and 2, and wrote that higher heights would need substantially new ideas. Our computer checks the height ideals: in the p-series of the universal deformation, reduced by the h-th ideal for h below n, the leading term is u_h·x^(pʰ), in every case we tried.
1:23The claim
Result 311 is one OpenAI paper, 34 pages, dated September 24 and made public October 6, 2026. It claims Chai’s statement at every prime and every positive height, for every open subgroup of the stabilizer. With Barthel, Heard and Naumann, this gives the Hovey–Strickland conjecture: exactly n + 2 thick tensor ideals, and a chain of n + 1 points. For Hovey–Strickland, the new cases are heights 3 and up. Not claimed: anything beyond the dualizable objects, ideals that are not radical, or residue fields other than the finite ones.
2:04How the proof goes
The core is a local orbit density theorem in characteristic p, built from Frobenius separation, a formal closure argument, and the geometry of the Fargues–Fontaine curve with perfectoid tools. Characteristic zero is handled through the Gross–Hopkins period map, an approach the paper credits to Chai, and the spectral half reassembles Barthel, Heard and Naumann’s argument, with credit. There is also a question of provenance. The day after the release, Chenglong Ma, a doctoral student in Copenhagen, posted a proof of the same theorem on arXiv, using the Fargues–Fontaine curve, Cartier theory and Chai’s rigidity theorem; Ningyi Li shared another proof, by different methods, as a public PDF. Ma states that the OpenAI paper’s proof strategy and specific choices of notation are identical to a preliminary version of Ma’s paper that was uploaded to ChatGPT on September 8, 16 days before the OpenAI paper’s date. That draft is not public, so we cannot check the claim. In the public versions, the overall strategy is closely parallel, down to some lemmas and a few symbols, but the technical machinery differs, and so do the basic names for the main objects. Both Ma and Li describe their own AI use: Ma used ChatGPT to check the height-3 case and then for auditing, and Li used AI to explore strategies and to audit.
3:32Where it stands
The paper is unrefereed and unchanged since release, and there is no Lean formalization. In the first 12 days we found no public specialist assessment of the mathematics and no reported error; the discussion has been about provenance, and we found no response from OpenAI. If it holds, the dualizable K(n)-local category has exactly n + 2 thick tensor ideals at every height and every prime. How this proof came to be is a separate, open question.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.