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The Lê-Ramanujam mu-constant problem for surface singularities: are mu-constant families in three variables topologically trivial?

Let ft:(CN,0)→(C,0)f_t:(\mathbb C^N,0)\to(\mathbb C,0) be a holomorphic one-parameter family with isolated critical points at the origin, and μ(ft)=dim⁡COCN,0/(∂ft/∂xi)\mu(f_t)=\dim_{\mathbb C}\mathcal O_{\mathbb C^N,0}/(\partial f_t/\partial x_i) its Milnor number. Lê and Ramanujam (1976) proved that constant μ\mu implies constant embedded topological type whenever N≠3N\ne3, using the h-cobordism theorem, and Timourian and King obtained topological triviality of the family of functions there. The argument fails for complex surfaces, N=3N=3, where Briançon-Speder examples show that μ\mu-constant families need not be Whitney equisingular. This remaining case is the surface case of the μ\mu-constant problem. Is every μ\mu-constant holomorphic family of isolated hypersurface singularities in C3\mathbb C^3 topologically trivial?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Singularity theory; equisingularity
Posed by
Lê Dũng Tráng and C. P. Ramanujam (the case left open by their theorem)
Year posed
1976
Years open
50y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: every holomorphic one-parameter family ftf_t of isolated hypersurface singularities in C3\mathbb C^3 with constant μ(ft,0)\mu(f_t,0) is topologically right-trivial: after shrinking there is a homeomorphism germ Φ(x,t)=(ϕt(x),t)\Phi(x,t)=(\phi_t(x),t), with Φ\Phi and Φ−1\Phi^{-1} jointly continuous, ϕt(0)=0\phi_t(0)=0, ϕ0=id\phi_0=\mathrm{id} and ft(ϕt(x))=f0(x)f_t(\phi_t(x))=f_0(x). With Lê-Ramanujam and Timourian-King this gives μ\mu-constant topological triviality in every dimension. The key step shows that constant μ\mu forces constancy of Wahl's logarithmic invariant −P2-P^2 for non-log-canonical germs. Not shown: Whitney equisingularity (false in general), families over higher-dimensional parameter spaces, or Zariski's multiplicity question.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript, 'Topological triviality of mu-constant families of surface singularities' (September 24, 2026).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the problem as the manuscript frames it: for a holomorphic family FF on C3×Δ\mathbb C^3\times\Delta with isolated critical points and constant Milnor number there is, after shrinking, a homeomorphism germ Φ(x,t)=(ϕt(x),t)\Phi(x,t)=(\phi_t(x),t), jointly continuous with continuous inverse, with ϕt(0)=0\phi_t(0)=0 and ft∘ϕt=f0f_t\circ\phi_t=f_0. Topological right-triviality implies constancy of the embedded topological type, so this answers the surface case in full. There is no Lean formalization. The proof uses semistable reduction, the threefold log minimal model program, the Fernández de Bobadilla-Pełka equimultiplicity theorem, Okuma's simultaneous resolution and Langer's orbifold Bogomolov-Miyaoka-Yau inequality; it was not refereed.

Sources

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