Always a hole?
An AI preprint claims Quillen’s conjecture for every finite group and every prime
Covers result 310 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:16. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Take a finite group G and a prime p. In 1978 Quillen looked at the poset of its nontrivial elementary abelian p-subgroups, the subgroups that are commutative and in which every element other than the identity has order p, ordered by inclusion, and turned it into a space. If G has a nontrivial normal p-subgroup, Quillen showed this space can be shrunk to a point; the largest such subgroup is written Oₚ(G). Quillen conjectured the converse: if Oₚ(G) is trivial, the space is not contractible, and a stronger form asks for a hole that rational homology can see. Quillen proved it for solvable groups, Aschbacher and Kleidman for almost simple ones, and Aschbacher and Smith in 1993 for primes above 5, assuming certain unitary groups behave. Díaz Ramos then claimed every odd prime in a preprint, in versions from 2024 to 2026, and in July 2026 Piterman reduced the prime 2 to a list of classical groups and wrote that this case was still open.
Result 310 is one OpenAI paper, 63 pages, dated September 24, 2026, and made public October 6. It claims that for every finite group and every prime, if Oₚ(G) is trivial, the poset has nonzero rational homology. With Quillen’s own direction, the space is contractible exactly when Oₚ(G) is nontrivial. The new part is the prime 2: the paper rules out each family on Piterman’s list. For odd primes it credits Díaz Ramos, adding a second proof of one step.
What it doesn’t claim
That the homology sits in the top degree, which is false in general (our computer finds it fails for the alternating group A₅ at the prime 2); exact Betti numbers; or a proof free of the classification of finite simple groups, which the paper relies on through the reductions it cites.
Where it stands
- Posted as a preprint
- OpenAI, 63 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. There is no Lean formalization, and the paper reports no machine computation. OSVerify has not reviewed it. In the first twelve days we found no public specialist assessment and no reported error. The prime-2 case also depends on Piterman’s July 2026 reduction, itself an unrefereed preprint.
How the proof goes
- Cycles made of frames. For each classical family, the paper builds explicit cycles from frames, ways of splitting the vector space into lines, with an independent sign on each line. A counting bound, borrowing ideas from Lie superalgebras and Golod–Shafarevich arguments, shows that in dimension h at least a fraction 0.68ʰ of the frame coefficients survive, enough to leave a nonzero cycle, and a propagation lemma lifts it to the whole group.
- An Euler characteristic trick. One leftover case, PSL₄(5), of order 7,254,000,000, also uses a trick: a group of symmetries whose order is a power of a prime r, and which fixes no simplex, forces the reduced Euler characteristic to be −1 mod r.
The numbers in the video
- Ours: for 27 pairs of a small group and a prime, a trivial Oₚ always comes with nonzero homology. For the simple group of order 168 at the prime 2 the space is a graph with 8 independent loops, which, once smoothed, is the Heawood graph; for A₇ there are 176.
- Ours: the Euler characteristic trick holds in all 11 small cases where its hypothesis does; for A₅ at the prime 2 with r = 5, the reduced Euler characteristic is 4, which is −1 mod 5.
Read the source
- Rational Homology and Quillen’s Conjecture (PDF), 63 pages.
- Piterman’s July 2026 reduction (arXiv).
- Díaz Ramos’s odd-prime preprint (arXiv).
- The collection’s catalog, where this is result 310.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
Take a finite group G and a prime p. In 1978 Quillen looked at the poset of its nontrivial elementary abelian p-subgroups, the subgroups that are commutative and in which every element other than the identity has order p, ordered by inclusion, and turned it into a space. If G has a nontrivial normal p-subgroup, Quillen showed this space can be shrunk to a point. The largest such subgroup is written Oₚ(G). Our computer builds the poset for the symmetric group S₄ at the prime 2: it is contractible, even though it is not simply a cone. Quillen conjectured the converse: if Oₚ(G) is trivial, the space is not contractible. A stronger form asks for a hole that rational homology can see. Our computer builds the poset for 27 pairs of a small group and a prime: in every case, a trivial Oₚ comes with nonzero homology. For the simple group of order 168 at the prime 2, the space is a graph with 8 independent loops, which, once smoothed, is the Heawood graph; for the alternating group A₇ there are 176. Quillen proved it for solvable groups, Aschbacher and Kleidman for almost simple ones, and Aschbacher and Smith in 1993 for primes above 5, assuming certain unitary groups behave. Díaz Ramos then claimed every odd prime in a preprint, in versions from 2024 to 2026, and in July 2026 Piterman reduced the prime 2 to a list of classical groups, and wrote that this case was still open.
1:38The claim
Result 310 is one OpenAI paper, 63 pages, dated September 24 and made public October 6, 2026. It claims that for every finite group and every prime, if Oₚ(G) is trivial, the poset has nonzero rational homology. With Quillen’s own direction, the space is contractible exactly when Oₚ(G) is nontrivial. The new part is the prime 2: the paper rules out each family on Piterman’s list. For odd primes it credits Díaz Ramos, adding a second proof of one step. Not claimed: that the homology sits in the top degree, which is false in general; our computer finds it fails for the alternating group A₅ at the prime 2. Nor exact Betti numbers, nor a proof free of the classification of finite simple groups, which the paper relies on through the reductions it cites.
2:35How the proof goes
For each classical family, the paper builds explicit cycles from frames, ways of splitting the vector space into lines, with an independent sign on each line. A counting bound, borrowing ideas from Lie superalgebras and Golod–Shafarevich arguments, shows that in dimension h, at least a fraction 0.68ʰ of the frame coefficients survive, enough to leave a nonzero cycle, and a propagation lemma lifts it to the whole group. One leftover case, the projective special linear group of 4 × 4 matrices over the field with 5 elements, of order 7,254,000,000, also uses an Euler characteristic trick: a group of symmetries whose order is a power of a prime r, and which fixes no simplex, forces the reduced Euler characteristic to be −1 mod r. Our computer checks the trick on small groups: for the alternating group A₅ at the prime 2, with r = 5, it is 4, which is −1 mod 5, and it holds in all 11 cases where the hypothesis does.
3:41Where it stands
The paper is unrefereed and unchanged since release. There is no Lean formalization, and the paper reports no machine computation. In the first 12 days we found no public specialist assessment and no reported error. If it holds, together with the July reduction it builds on, Quillen’s conjecture is true: for every finite group and every prime, the poset of elementary abelian p-subgroups is contractible exactly when there is a nontrivial normal p-subgroup.
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.