Aftermath bot: New page for openai/math result 310 with the Latent Seminar article, the Lattice24 review rank_132 and the AI Proved What? episode

2026-10-30T08:23:17Z

New page for openai/math result 310 with the Latent Seminar article, the Lattice24 review rank_132 and the AI Proved What? episode

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'''Quillen's conjecture on the poset of ''p''-subgroups''' asserts that for a finite group ''G'' and a prime ''p'', if the order complex of the poset 𝒜''p''(''G'') of nonidentity elementary abelian ''p''-subgroups is contractible, then ''G'' has a nontrivial normal ''p''-subgroup, that is, ''O''''p''(''G'') ≠ 1. Quillen proved the converse direction and formulated the conjecture in 1978. A manuscript dated 24 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), listed as its result 310, claims the stronger rational-homology form for every finite group and every prime. The family has no Lean formalization in the repository's catalogue. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the manuscript:
** Brown related the Euler characteristic of the poset of all nonidentity ''p''-subgroups to the order of a Sylow subgroup. Quillen proved that this poset is homotopy equivalent to 𝒜''p''(''G''), formulated the conjecture, and proved it for solvable groups, for groups of elementary abelian ''p''-rank at most two and for groups of Lie type in defining characteristic. For solvable groups his proof gives nonzero homology in the largest possible degree, later called the Quillen dimension property.
** Aschbacher and Kleidman established the almost-simple case in 1990. Aschbacher and Smith reduced the problem for ''p'' > 5 in 1993 to the dimension property for specified extensions of unitary groups; Piterman and Smith extended the reduction to all odd primes, and Díaz Ramos, in a 2026 version of his preprint ''Quillen's conjecture and unitary groups'' ([https://arxiv.org/abs/2303.15613v6 arXiv:2303.15613v6]), proved the required unitary result and deduced the rational homological assertion for every finite group at odd primes. The manuscript gives its own frame proof of that unitary step.
** At the prime 2, Piterman and Smith showed that every component of a minimal counterexample admits an outer automorphism of order two induced by the ambient group; Piterman treated elementary 2-extensions of exceptional groups of Lie type in odd characteristic apart from a finite list in characteristic three; and Piterman's preprint ''Components in characteristic p and Quillen's conjecture'' ([https://arxiv.org/abs/2607.27500v1 arXiv:2607.27500v1], July 2026) eliminated the Lie-type components in characteristic two, leaving a list of classical components in characteristic at least five.

== The original result ==
* '''Claim.''' The main theorem of ''Rational homology and Quillen's conjecture'' states that for every finite group ''G'' and every prime ''p'', if ''O''''p''(''G'') = 1, then the augmented reduced rational homology of 𝒜''p''(''G'') is nonzero. The manuscript notes that, with Quillen's converse, acyclicity over any fixed field then characterizes the existence of a nontrivial normal ''p''-subgroup.
* ''Method, as described:'' at the prime 2 the manuscript takes a minimal counterexample, whose component lies on Piterman's list, and excludes the linear, unitary, symplectic and orthogonal families and finally PSL4(5), which is treated in a separate section, one branch by an equivalent-poset construction. The main tool is a family of frame cycles: decompositions of a vector space into lines give elementary abelian sign subgroups whose apartment chains are combined so that their boundaries cancel, with a lower bound on the space of admissible coefficients that is uniform in the dimension over a fixed field, proved through multilinear Lie superbrackets and a Poincaré–Birkhoff–Witt argument. A restriction lemma at a fixed complement and a propagation lemma carry a nonzero coefficient into the homology of the ambient group. For odd primes the proof uses the Piterman–Smith reduction together with its own top-degree result for the unitary extensions.
* '''Formalization.''' The repository's catalogue lists no Lean formalization for this family.
* '''Source''' (openai/math repository, commit <code>7ef5c0c</code>): [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/Rational-homology-and-Quillens-conjecture-September-24-2026 ''Rational homology and Quillen's conjecture''], dated 24 September 2026.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 30 October 2026)
|-
| 2026-10-07 || [https://lowrank.github.io/latent-seminar/OpenAI-Math/310-quillen-rational-homology-frame-cycles-propagation ''Quillen homology: frame cycles and a coefficient that survives''], an article of the Latent Seminar blog at lowrank.github.io || Reading (exposition) || Explains the manuscript's claim of the rational-homology form of Quillen's conjecture for every finite group and every prime, concentrating on the topological mechanism: frame cycles and boundary cancellation, decorations that pay for additional boundary equations, restriction using a fixed complement, and the propagation of a detectable coefficient into the homology of the ambient group. || Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs.
|-
| 2026-10-16
| J. Jardine, Lattice24, referee review [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_132.json rank_132] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Reports no gap and no minor note. The reviewer states that it read all ten sections in full and checked them step by step, and that a brute-force computation of a frame-coefficient dimension in dimension three over the fields with two and three elements agreed with the paper's bound. It states that it treated the cited reduction theorems of Piterman and of Piterman and Smith, the equivalent-poset theorem and the facts on Steinberg automorphisms and triality as black boxes, without independent verification.
| Part of an audit of the collection's manuscripts that the audit describes as complete. The review is published under CC BY 4.0.
|-
| 2026-10-18 || ''AI Proved What?'', episode 310 [https://aiprovedwhat.com/episodes/310-quillen/ Always a hole?], remade version of 23 October 2026 || Reading (exposition) || Video explainer of the claim that for every finite group and every prime, a trivial largest normal p-subgroup forces nonzero rational homology of the poset of elementary abelian p-subgroups. The episode states that the new part is the prime 2, where the paper rules out each family on Piterman's July 2026 list, and that for odd primes the paper credits Díaz Ramos and adds a second proof of one step. It states that homology in the top degree, exact Betti numbers and a proof independent of the classification of finite simple groups are not claimed, and reports its own computation that top-degree homology fails for the alternating group A5 at the prime 2. || Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 63 pages, dated September 24, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: no formalization for this item. The page states that it found no public specialist assessment and no reported error, and that the prime-2 case also depends on Piterman's July 2026 reduction, itself an unrefereed preprint.
|}

== Notes ==
* The background summarizes the manuscript's own account of earlier work.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]

Quillen's conjecture on the poset of p-subgroups - Revision history

Aftermath bot: New page for openai/math result 310 with the Latent Seminar article, the Lattice24 review rank_132 and the AI Proved What? episode