View source for The chromatic Smith problem for finite p-groups

'''The chromatic Smith problem for finite ''p''-groups''' asks how much chromatic height is lost when acyclicity of the geometric fixed points of a subgroup ''H'' is transferred to the geometric fixed points of a finite ''p''-group ''G'': the least ''r'' such that ''K''(''n''+''r'')-acyclicity of the ''H''-fixed points of every finite genuine ''G''-spectrum forces ''K''(''n'')-acyclicity of its ''G''-fixed points, where ''K''(''i'') is Morava ''K''-theory. Kuhn and Lloyd expressed the hope that this loss equals the shortest length of a subnormal chain from ''H'' to ''G'' with cyclic quotients. 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 314, claims this equality at every prime and every height. 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:
** Smith theory transfers mod-''p'' acyclicity from a finite complex to the fixed points of a finite ''p''-group action, and Floyd strengthened this to an inequality between total mod-''p'' Betti numbers. Chromatic Smith theory asks which implications survive when mod-''p'' homology is replaced by individual Morava ''K''-theories.
** Building on the thick subcategory theorem of Hopkins and Smith, Balmer and Sanders identified the primes of the category of finite genuine equivariant spectra, reduced its topology to subgroup–height inclusions, and computed the case of a cyclic group of order ''p''. Barthel, Hausmann, Naumann, Nikolaus, Noel and Stapleton showed that for finite abelian ''p''-groups the loss is the minimal number of generators of ''G''/''H''; Balderrama and Kuhn gave an elementary proof of the abelian case.
** For nonabelian groups, Kuhn and Lloyd related the problem to inequalities between Morava homology dimensions, obtained equality with the cyclic chain length for central extensions of elementary abelian groups by a group of order two, including a noncentral subgroup of order two in the dihedral group of order 8 with loss two, and expressed hope for equality in general, asking separately whether the loss is independent of the height.

== The original result ==
* '''Claim.''' Theorem 1.1 of ''Cyclic Length and Chromatic Fixed-Point Loss'' states that for every prime ''p'', every finite ''p''-group ''G'', every subgroup ''H'' and every height ''n'' ≥ 0, the optimal loss equals the least length of a subnormal chain from ''H'' to ''G'' with cyclic quotients. The upper bound follows from the cyclic chromatic Smith theorem along a chain; the new part is the matching lower bound, so the loss does not depend on the height. The manuscript notes that, combined with the reductions of Balmer and Sanders, this determines the remaining finite-height prime inclusions for arbitrary finite groups.
* ''Method, as described:'' a rational estimate first bounds the cyclic length of a ''p''-group whose cofree geometric fixed points of the telescopic localized sphere are rationally nonzero. A maximizing construction produces a pro-''p'' group mapping onto ''P'' whose finite sections have at most ''l'' generators; relative integration over finite groupoids gives subgroup congruences and integrality of homomorphism counts, which force a subnormal series with factors the ''p''-adic integers of length at most ''l''. Ring nilpotence and the generalized characters of Hopkins, Kuhn and Ravenel then carry the estimate to all heights, and separation of primes gives the sharpness witnesses.
* '''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/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026 ''Cyclic Length and Chromatic Fixed-Point Loss''], 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/314-chromatic-smith-cyclic-length-fixed-point-loss ''Chromatic Smith theory: cyclic length and fixed-point loss''], an article of the Latent Seminar blog at lowrank.github.io || Reading (exposition) || Explains the claimed cyclic-length formula for the chromatic fixed-point loss, independent of height including rational height zero, and its character, profinite-group and support arguments. || 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_284.json rank_284] 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. The reviewer states that it read sections 1 to 7 and checked every step of the lower-bound chain, including the occurrence closure, the maximizing tower, the integration weights, the subgroup congruences, the profinite theorem and the gluing across heights, with spot checks of the congruences for the ''p''-adic integers and their products. It notes that the profinite theorem and the labeled-set congruence may be reusable by companion papers.
| 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 314 [https://aiprovedwhat.com/episodes/314-cyclic-length/ Heights lost?], remade version of 23 October 2026 || Reading (exposition) || Video explainer of the claim that for every prime, every finite p-group G, every subgroup H and every height, the chromatic loss from H to G equals the cyclic length, the shortest chain from H to G with each step normal in the next and cyclic quotient, so that the loss does not depend on the height; for a reflection in the dihedral group of order 16, a case Balderrama and Kuhn asked about, the formula gives three. The episode recalls the abelian answer of Barthel, Hausmann, Naumann, Nikolaus, Noel and Stapleton and the hope expressed by Kuhn and Lloyd. It states that only the lower bound is new, that the spectra showing sharpness are proved to exist by contradiction rather than built, and that compact Lie groups are not claimed. || 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, 30 pages, dated September 24, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: no formalization for this item. The page reports that on 10 October 2026 Nick Kuhn, co-author of the question, wrote on the algebraic topology mailing list about this paper and one other that, to the extent he had looked, he would not bet any money that either has a correct proof, and that the logical flow of the main proof first appears near the end of the paper; it states that the message names no error and that it found no other expert comment.
|}

== 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]]