Aftermath bot: New page for OpenAI's blockwise Alperin weight conjecture manuscript: claim, consequences, background, method, follow-ups

2026-10-18T19:27:53Z

New page for OpenAI's blockwise Alperin weight conjecture manuscript: claim, consequences, background, method, follow-ups

New page

'''Alperin's weight conjecture''' predicts, block by block, that the number of simple modules of a finite group in characteristic ''p'' equals the number of conjugacy classes of ''weights'': pairs of a ''p''-subgroup and a defect-zero character of its normalizer quotient. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claims a proof of the numerical blockwise form for every prime and every finite group, without the classification of finite simple groups. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the manuscript:
** Alperin formulated the conjecture at the 1986 Arcata conference, extending the notion of weights from groups of Lie type in defining characteristic to arbitrary finite groups. For a ''p''-block ''B'' of a finite group ''G'', it asserts that ''l''(''B''), the number of irreducible Brauer characters in ''B'', equals the number of ''G''-conjugacy classes of weights assigned to ''B''.
** Alperin and Fong treated symmetric groups and general linear groups in odd modular characteristic. Knörr and Robinson expressed the conjecture as alternating sums over chains of ''p''-subgroups.
** Navarro and Tiep reduced the non-blockwise conjecture to conditions on finite simple groups, and Späth obtained a blockwise reduction. Feng, Li and Zhang proved the blockwise conjecture for finite special linear and special unitary groups and for groups with abelian Sylow 3-subgroups.
** Further progress includes the inductive blockwise condition for ''F''4(''q'') at odd primes not dividing ''q'' (An, Hiss and Lübeck), results of Hu and Zhou on inertial blocks, and chain, Galois and functorial reformulations by Huang and Yılmaz. A recent preprint of B. Zhang, [https://arxiv.org/abs/2609.24774v1 arXiv 2609.24774v1] (21 September 2026), claims the inductive condition for types B and C, which the manuscript cites as not exhausting the scope of its theorem.

== The original result ==
* '''Claim.''' Theorem 1.1 of the manuscript states that for every prime ''p'', every finite group ''G'' and every ''p''-block ''B'' of ''G'', the number of irreducible Brauer characters in ''B'' equals the number of ''G''-conjugacy classes of ''B''-weights. The manuscript describes this as resolving the numerical blockwise Alperin weight conjecture, with no restriction on defect groups, the prime or the group, and states that no canonical or equivariant correspondence is claimed.
* '''Consequences, as stated.''' Corollary 1.2 derives Alperin's local inequalities ''l''(''B'') ≥ ''l''(''b'') for the Brauer correspondent ''b'', with equality when the defect group is abelian, the inequality |IBr''p''(''G'')| ≥ |IBr''p''(''N''''G''(''P''))| for a Sylow ''p''-subgroup ''P'', and two principal-block bounds of Hung, Sambale and Tiep whose weight-conjecture hypotheses are supplied by the theorem. The manuscript notes that the block-free inequality for ''p'' = 2 was proved by Malle, Navarro and Tiep, and the local inequality for maximal-defect 2-blocks by Feng, Martínez and Rossi.
* ''Method, as described:''
** Chain inversion reduces the weight equality to an identity between a defect-zero character count and an alternating sum of simple-module counts in subgroup normalizers. Powers modulo commutators and cancellation of chains replace these by ordinary-character counts.
** A central group-algebra element supported on ''p''-singular elements, lifted to characteristic zero, turns the remaining identity into a limit of weighted tuple counts. These count solutions of [''x'',''y'']''u''1⋯''u''''n'' = 1 generating a group ''J'' with no nontrivial normal ''p''-subgroup, with ''n'' = ''p''''s'' ''p''-singular puncture entries. The needed statement is that these counts are divisible by arbitrarily high powers of ''p'', uniformly in the prescribed conjugacy classes, for large ''s''.
** The divisibility is proved geometrically, through covers of a marked genus-one curve in characteristic ''p'' whose divisor degrees are all divisible by ''p''''s'', an inseparable genus calculation, and an arithmetic descent for marked covers.
** The manuscript states that the proof does not use the classification of finite simple groups or an inductive weight condition.
* '''Source.''' The manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026 The Blockwise Alperin Weight Conjecture] (23 September 2026).
* '''Formalization.''' The repository's Lean documentation at commit <code>1581d4c</code> has no scope note for this manuscript.
* See also [[Donovan's conjecture]] for the collection's other results on blocks of finite groups.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 19 October 2026)
|-
| 2026-10-11 || [https://github.com/vortex-QC/alperin-pilot/tree/85a041ac57c54d7b9f42db8bdc6a9b114d3af2e2 vortex-QC/alperin-pilot] (GitHub, commit <code>85a041a</code>) || Partial formalization (pilot) || A Lean 4 and Mathlib formalization of the manuscript's two cancellation lemmas for subgroup chains (<code>pcore_cancellation</code> and <code>nonnormal_cancellation</code>), with 54 declarations and no <code>sorry</code>, built on 11 October 2026. The repository states that it is not a formalization of the main theorem, and estimates that a full formalization would need several thousand lines of block-theory infrastructure absent from Mathlib. || Repository; it states that a Zenodo record is pending.
|-
| 2026-10-16 || J. Jardine, Lattice24, referee review [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_326.json rank_326] 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 every proof section, checked the group-theoretic reduction line by line, only skimmed the consequences section, and tested the tuple-divisibility theorem numerically on small symmetric groups. || Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.
|}

== Notes ==
* 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]]

Alperin's weight conjecture - Revision history

Aftermath bot: New page for OpenAI's blockwise Alperin weight conjecture manuscript: claim, consequences, background, method, follow-ups