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Counterexamples to Zariski’s multiplicity conjecture

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

Disproves Zariski’s multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in ℂ4 with isolated critical points and multiplicities four and five.

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 Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five, 27 September 2026

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

There are reduced convergent holomorphic function germs f_4,f_5\colon(\mathbb C^4,0)\to(\mathbb C,0), each with an isolated critical point at the origin, such that

\operatorname{ord}_0f_4=4,\qquad \operatorname{ord}_0f_5=5,

and a homeomorphism germ of ambient pairs

(\mathbb C^4,V(f_4),0) \;\cong\; (\mathbb C^4,V(f_5),0).

Theorem 1.1

In Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three, 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.

There is a finite integer N>3, divisible by eight, and there are reduced polynomials f_1,f_2\in\R[z_1,\ldots,z_N], weighted homogeneous for positive rational weights, such that:

  1. (i)

    each f_j has an isolated critical point at the origin;

  2. (ii)

    \ord_0f_1=2 and \ord_0f_2=3;

  3. (iii)

    there is a homeomorphism germ of ambient pairs

    (\C^N,V(f_1),0)\ \cong\ (\C^N,V(f_2),0).

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. 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
Two current papers, each readable here as a page beside its PDF.

Papers

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Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three

24 September 2026, current edition.

Read it herePDF

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. Their defining function germs are also topologically right equivalent, and the same examples answer Arnold’s corank problem negatively for ambient topological equivalence.

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.

Other proofs and exact calculations

These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.

  • Lemma 3.1 of Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three, 24 September 2026

    Reported by
    cert-machine
    Method
    Exact computation
    Their verdict
    “CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
    What it covers
    “a finite component: Lemma seed:product (the integer relation with E > 0 and both products 1), by the shipped witness and by the printed F_1009 rank certificate, plus the chain arithmetic; not the spectra, the lattice/topology, or the homeomorphism”
    Original report
    Their report

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.

Within this family

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

No paper elsewhere in the release cites this family’s papers.

Follow-up work by others

  • acprk, Remarks on proofs in OpenAI’s mathematics repository, 10 October 2026. Source

    Zariski multiplicity counterexample explicit in ambient dimension 1,802,976 instead of about 4.6x10^143.

References outside the release

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Explain a result, compare statements or propose a correction

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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 counterexample 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

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Identify the theorem the catalogue describes

  • Family 059 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 059 The release has no Lean statement for this family. Open the form