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Donovan’s conjecture over fields and complete mixed-characteristic DVRs

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

Proves Donovan’s conjecture: over each fixed algebraically closed field of characteristic p, blocks of finite groups with bounded defect-group order have only finitely many Morita-equivalence classes, for every prime p. Also proves the integral form over each fixed complete mixed-characteristic discrete valuation ring with algebraically closed residue field.

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 (Donovan’s conjecture)

In Donovan’s Conjecture over Algebraically Closed Fields, 24 September 2026

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

Fix a prime p, an algebraically closed field K of characteristic p, and a positive integer M. As G ranges over finite groups and b over block idempotents of KG with defect groups of order at most M, only finitely many K-linear Morita equivalence classes of KGb occur. In particular, over this field K, Donovan’s conjecture holds for every finite p-group, including nonabelian groups.

Corollary 1.2 (Fixed complete coefficient rings)

In Integral Donovan Finiteness over Witt Vectors, 25 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, let \mathcal R be a fixed complete discrete valuation ring of characteristic zero whose residue field \ell is algebraically closed of characteristic p, and let M\geq1 be an integer. As G ranges over all finite groups and b ranges over primitive central idempotents of \mathcal R G for which the residue block \ell G\bar b has a defect group of order at most M, the algebras \mathcal R G b belong to only finitely many \mathcal R-linear Morita equivalence classes.

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.

One more possible match

  • Theorem 1.1 in Integral Donovan Finiteness over Witt Vectors, 25 September 2026.
Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Two current papers, each readable here as a page beside its PDF.

Papers

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Donovan’s Conjecture over Algebraically Closed Fields

24 September 2026, current edition.

Read it herePDF

We prove Donovan’s conjecture over every algebraically closed field K of characteristic p > 0. For each fixed K and bound on defect-group order, blocks of finite groups represent only finitely many K-linear Morita equivalence classes. The defect groups need not be abelian, and the result includes p = 2.

Integral Donovan Finiteness over Witt Vectors

25 September 2026, current edition.

Read it herePDF

We prove integral Donovan finiteness: for every prime p and positive integer M, the blocks of all finite groups with defect groups of order at most M have only finitely many Morita equivalence classes over \(W(\overline{\mathbb F}_p)\). The defect groups need not be abelian, and the result includes p = 2. The same bounded-defect finiteness holds over each fixed complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic p, including ramified rings.

Lean statements

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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

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Within this family

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References outside the release

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What people have said

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  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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Identify the theorem the catalogue describes

  • Family 203 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 203 The release has no Lean statement for this family. Open the form