---
title: "Donovan's conjecture over fields and complete mixed-characteristic DVRs"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 203; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026/main.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebra"
  - "unformalized"
seed_rank: 2058
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Donovan's Conjecture over Algebraically Closed Fields"
    url: "https://github.com/openai/math/blob/main/preprints/Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Integral Donovan Finiteness over Witt Vectors"
    url: "https://github.com/openai/math/blob/main/preprints/Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Donovan's conjecture over fields and complete mixed-characteristic DVRs

## One-sentence takeaway

OpenAI's result family 203 (Algebra) claims: 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.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): Also proves the integral form over each fixed complete mixed-characteristic discrete valuation ring with algebraically closed residue field.
- *Donovan's Conjecture over Algebraically Closed Fields*: 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*: 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.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Donovan's Conjecture over Algebraically Closed Fields](https://github.com/openai/math/blob/main/preprints/Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026/main.pdf)
- Manuscript: [Integral Donovan Finiteness over Witt Vectors](https://github.com/openai/math/blob/main/preprints/Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026/main.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math