Finitely generated?
An AI preprint claims every homotopy group of the K(n)-local sphere is finitely generated over the p-adic integers
Covers result 313 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
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Chapters
The claim, in plain words
At a prime p and a height n, the K(n)-local sphere is the part of the sphere spectrum that Morava K-theory sees. Its homotopy groups are modules over the p-adic integers. In their memoir, Hovey and Strickland asked whether each of them is finitely generated, noting that the evidence suggested it was true but subtle, and Hovey wrote on their own problem list that they believed we know almost nothing about these groups. Finitely generated, not finite: degrees 0 and −1 each contain a whole copy of the p-adic integers. Height 1 is essentially the image of J. Height 2 was known at primes 5 and up, from Hovey and Strickland’s analysis of Shimomura’s computations, and in 2024 Barthel, Schlank, Stapleton and Weinstein found the rational answer at every height: 2ⁿ rational dimensions spread over degrees 0 to −n². The torsion was the problem; experts expected the answer to follow from the chromatic splitting conjecture, whose strong form another paper in this release claims fails at height 3, for primes 5 and up.
Result 313 is one OpenAI paper, 103 pages, dated September 24, 2026, and made public October 6. It claims that every homotopy group of the K(n)-local sphere is finitely generated over the p-adic integers, at every prime, every positive height and every integer degree, and the same after K(n)-localizing any finite p-local spectrum. In other words, each group is finitely many copies of the p-adic integers plus a finite group, so the torsion in each degree is finite.
What it doesn’t claim
Bounds on the number of generators or the size of the torsion, new computations at height 2 or above, spectra that are not finite, or degrees other than integers. It does not assume the chromatic splitting conjecture.
Where it stands
- Posted as a preprint
- OpenAI, 103 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 comment, no reported error and no competing claim.
How the proof goes
- Reduce to the stabilizer group. Three pages reduce the theorem to one statement: the continuous cohomology of the Morava stabilizer group, with coefficients in Morava E-theory mod p, is finite in each degree. Descent turns that into finite homotopy mod p, and a compact form of Nakayama’s lemma then lifts finiteness mod p to finite generation.
- Eighty-eight pages for the hard part. The coefficient theorem takes about 88 pages of p-adic geometry and group cohomology: the Fargues–Fontaine curve, Banach–Colmez spaces, results of Anschütz and Le Bras, Kummer theory and Cartier transfers, ending with a Baire category argument that a countable compact Hausdorff group must be finite.
The numbers in the video
- Ours: at height 1 and odd p, a copy of the p-adic integers in degrees 0 and −1, and a cyclic group of order pᵏ⁺¹ in each degree 2(p − 1)pᵏm − 1 for nonzero m not divisible by p; it matches the image of J, computed from Bernoulli numbers, in 30 out of 30 stems for each prime tried.
- Ours: at height 4, degree −8 is the first with 2 rational dimensions.
- Ours: at height 1 and p = 3, working mod 3ᵃ, degree 0 grows like 3ᵃ while each torsion degree freezes.
Read the source
- Finite Generation for the K(n)-Local Sphere (PDF), 103 pages.
- The collection’s catalog, where this is result 313.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
At a prime p and a height n, the K(n)-local sphere is the part of the sphere spectrum that Morava K-theory sees. Its homotopy groups are modules over the p-adic integers. In their memoir, Hovey and Strickland asked whether each of them is finitely generated, noting that the evidence suggested it was true but subtle, and Hovey wrote on their own problem list that they believed we know almost nothing about these groups. Finitely generated, not finite: degrees 0 and −1 each contain a whole copy of the p-adic integers. Height 1 is essentially the image of J. Our computer rebuilds it at odd primes: a copy of the p-adic integers in degrees 0 and −1, and a cyclic group of order pᵏ⁺¹ in each degree 2(p − 1)pᵏm − 1, for nonzero m not divisible by p; it matches the image of J, computed from Bernoulli numbers, in 30 out of 30 stems for each prime tried. Height 2 was known at primes 5 and up, from Hovey and Strickland’s analysis of Shimomura’s computations, and in 2024 Barthel, Schlank, Stapleton and Weinstein found the rational answer at every height: 2ⁿ rational dimensions spread over degrees 0 to −n². Our computer tabulates them; at height 4, degree −8 is the first with 2. The torsion was the problem. Our computer lists modules over the p-adic integers that look tame but are not finitely generated, such as the p-adic numbers or an infinite product of copies of the field with p elements. Experts expected the answer to follow from the chromatic splitting conjecture, whose strong form another paper in this release claims fails at height 3, for primes 5 and up.
1:48The claim
Result 313 is one OpenAI paper, 103 pages, dated September 24 and made public October 6, 2026. It claims that every homotopy group of the K(n)-local sphere is finitely generated over the p-adic integers, at every prime, every positive height and every integer degree, and the same after K(n)-localizing any finite p-local spectrum. In other words, each group is finitely many copies of the p-adic integers plus a finite group, so the torsion in each degree is finite. Not claimed: bounds on the number of generators or the size of the torsion, new computations at height 2 or above, spectra that are not finite, or degrees other than integers. It does not assume the chromatic splitting conjecture.
2:40How the proof goes
Three pages reduce the theorem to one statement: the continuous cohomology of the Morava stabilizer group, with coefficients in Morava E-theory mod p, is finite in each degree. Descent turns that into finite homotopy mod p, and a compact form of Nakayama’s lemma then lifts finiteness mod p to finite generation. Our computer illustrates the stabilizer group: its standard open subgroups have index (pⁿ − 1)·p^(n(k − 1)), and as a p-adic Lie group it has dimension n². The coefficient theorem takes about 88 pages of p-adic geometry and group cohomology: the Fargues–Fontaine curve, Banach–Colmez spaces, results of Anschütz and Le Bras, Kummer theory and Cartier transfers, ending with a Baire category argument that a countable compact Hausdorff group must be finite. At height 1, our computer runs the descent by hand: degree zero grows like 3ᵃ as we work mod 3ᵃ, while each torsion degree freezes, which is exactly what finite generation looks like.
3:51Where 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 comment, no reported error and no competing claim. If it holds, every homotopy group of the K(n)-local sphere is a finitely generated module over the p-adic integers, at every height and every prime: the torsion in each degree is finite.
Spot a mistake?
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