---
title: "Counterexamples to Zariski’s multiplicity conjecture"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 059; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebraic-and-complex-geometry"
  - "unformalized"
seed_rank: 1918
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three"
    url: "https://github.com/openai/math/blob/main/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five"
    url: "https://github.com/openai/math/blob/main/preprints/Ambiently-homeomorphic-isolated-hypersurface-germs-in-C4-with-multiplicities-four-and-five-September-27-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Counterexamples to Zariski’s multiplicity conjecture

## One-sentence takeaway

OpenAI's result family 059 (Algebraic and complex geometry) claims: Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities.

## 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): The examples include hypersurfaces in ℂ4 with isolated critical points and multiplicities four and five.
- *Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three*: We give a negative answer to the embedded Zariski multiplicity conjecture. We construct two reduced hypersurface germs that are ambiently homeomorphic but have multiplicities two and three. Both have isolated singularities and lie in a common complex affine space of dimension divisible by eight.
- *Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five*: We give a negative answer to Zariski's multiplicity question in four complex variables. We construct two reduced convergent holomorphic function germs with isolated critical points and multiplicities four and five whose zero-set germs are ambiently homeomorphic.

## 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: [Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three](https://github.com/openai/math/blob/main/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf)
- Manuscript: [Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five](https://github.com/openai/math/blob/main/preprints/Ambiently-homeomorphic-isolated-hypersurface-germs-in-C4-with-multiplicities-four-and-five-September-27-2026/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