The blockwise Alperin weight conjecture

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves the numerical blockwise Alperin weight conjecture for every prime and every finite group: the number of irreducible Brauer characters in a block equals the number of conjugacy classes of its weights.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

What the papers state

Theorem 1.1

In The Blockwise Alperin Weight Conjecture, 23 September 2026

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

For every prime p, every finite group G, and every p-block B of G,

l(B)=|W_p(B)|.

Corollary 1.2 (Local–global and principal-block bounds)

In The Blockwise Alperin Weight Conjecture, 23 September 2026

Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.

Let p be a prime, G a finite group, B a p-block with defect group D, and b its Brauer correspondent in N_G(D). Then

l(B)\geq l(b),\qquad D\text{ abelian}\Longrightarrow l(B)=l(b).

For P\in\operatorname{Syl}_p(G), one also has

|\IBr_p(G)|\geq|\IBr_p(N_G(P))|.

Let B_0 be the principal p-block of G, and let k_p(G) count the conjugacy classes of p-power-order elements, including the identity. Then

\begin{aligned} k_p(G)=3&\Longrightarrow l(B_0)\geq (p-1)/2,\\ p\mid |G|&\Longrightarrow l(B_0)\geq 2\sqrt{p-1}+1-k_p(G). \end{aligned}

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.

Reported proof verification: The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations: One current paper, readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

The Blockwise Alperin Weight Conjecture

23 September 2026, current edition.

Read it here PDF

We prove the numerical blockwise Alperin weight conjecture for every finite group and every prime p. For each p-block B, the number of irreducible Brauer characters in B equals the number of conjugacy classes of B-weights.

Lean statements

The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

Donovan’s Conjecture over Algebraically Closed FieldsCurrent edition; no accompanying Lean statement Family Donovan’s conjecture over fields and complete mixed-characteristic DVRsCites The Blockwise Alperin Weight Conjecture.
Integral Donovan Finiteness over Witt VectorsCurrent edition; no accompanying Lean statement Family Donovan’s conjecture over fields and complete mixed-characteristic DVRsCites The Blockwise Alperin Weight Conjecture.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

39 references in The Blockwise Alperin Weight Conjecture

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanation Suggest a source attribution Suggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Author not stated. Source

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Identify the theorem the catalogue describes

Family 202 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

Family 202 The release has no Lean statement for this family. Open the form