---
title: "Integral counterexamples to Gersten’s conjecture"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 209; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebra"
  - "unformalized"
seed_rank: 2064
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "An integral counterexample to Gersten's conjecture"
    url: "https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "An integral degree-three Gersten counterexample"
    url: "https://github.com/openai/math/blob/main/preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Integral counterexamples to Gersten’s conjecture

## One-sentence takeaway

OpenAI's result family 209 (Algebra) claims: Disproves unrestricted integral Gersten injectivity in degrees 3 and 5.

## 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): Two explicit two-dimensional ramified regular local rings of mixed characteristic $(0,5)$ have nonzero integral K-theory classes that vanish over their fraction fields.
- *An integral counterexample to Gersten's conjecture*: We construct a two-dimensional ramified regular local ring A of mixed characteristic $(0,5)$ for which $K_5(A)\to K_5(\mathop{\mathrm{Frac}}\nolimits A)$ has a nonzero kernel. This disproves Gersten's conjecture in its unrestricted integral form.
- *An integral degree-three Gersten counterexample*: We construct a two-dimensional ramified regular local ring A in mixed characteristic $(0,5)$ for which the integral map $K_3(A)\to K_3(\mathop{\mathrm{Frac}}\nolimits A)$ has nonzero kernel. This gives a negative answer to the unrestricted integral Gersten conjecture.

## 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: [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf)
- Manuscript: [An integral degree-three Gersten counterexample](https://github.com/openai/math/blob/main/preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026/paper.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