Topological triviality of μ-constant surface singularities

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

Proves topological right-triviality for every holomorphic one-parameter family of isolated hypersurface singularities in ℂ³ with constant Milnor number, resolving the surface case of the μ-constant problem. After shrinking the parameter disk and representatives, ambient homeomorphisms vary jointly continuously, fix the origin section and preserve the defining functions.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

The paper states

Theorem 1.1

In Topological triviality of mu-constant families of surface singularities, 24 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.

Let F:(\mathbb C^3\times\Delta,\{0\}\times\Delta)\to(\mathbb C,0) be holomorphic. Suppose that, for every t\in\Delta, f_t(0)=0, f_t has an isolated critical point at 0, and \mu(f_t,0)=\mu(f_0,0). After shrinking the disk and the germ representatives, there is a homeomorphism germ

\Phi(x,t)=(\phi_t(x),t),\qquad \phi_t(0)=0,\quad \phi_0=\mathrm{id},\qquad f_t\bigl(\phi_t(x)\bigr)=f_0(x).

Both \Phi and \Phi^{-1} are jointly continuous on neighborhoods of the origin section over the original disk.

Reported proof verification: The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations: One current paper, readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Topological triviality of mu-constant families of surface singularities

24 September 2026, current edition.

Read it here PDF

We prove that every holomorphic one-parameter family of isolated hypersurface singularities in ℂ³ with constant Milnor number is topologically right-trivial. This gives a positive answer to the surface case of the μ-constant problem. The trivialization fixes the parameter and the origin section, and preserves the defining functions.

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.

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.

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

This family’s papers cite no paper elsewhere in the release.

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

Ambiently homeomorphic isolated hypersurfaces of multiplicities two and threeCurrent edition; no accompanying Lean statement Family Counterexamples to Zariski’s multiplicity conjectureCites Topological triviality of mu-constant families of surface singularities.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

34 references in Topological triviality of mu-constant families of surface singularities

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanation Suggest a source attribution Suggest a change

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 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

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Identify the theorem the catalogue describes

Family 064 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 064 The release has no Lean statement for this family. Open the form