The μ-constant problem for surface singularities

The μ-constant problem for surface singularities asks whether a holomorphic family of isolated hypersurface singularities in three complex variables whose Milnor number μ stays constant is topologically trivial. Lê and Ramanujam settled every hypersurface dimension other than two in 1976; the surface case remained open. A manuscript in OpenAI's public mathematics collection, dated 24 September 2026, claims a positive answer: every such one-parameter family is topologically right-trivial. The collection lists no Lean formalization of this result.

Background

• As recounted in the manuscript's introduction:
  • Lê–Ramanujam and its successors. Lê and Ramanujam proved that a μ-constant family of isolated hypersurface singularities has constant embedded topological type when the hypersurface dimension is not two, comparing Milnor fibers through a homologically trivial cobordism and the h-cobordism theorem. Timourian and King then established local triviality of the families of function germs outside complex dimension two.
  • Whitney regularity and multiplicity. Teissier proved that constancy of the full sequence of Milnor numbers of generic plane sections implies Whitney regularity, Briançon and Speder proved the converse and gave examples showing that topological triviality need not imply Whitney regularity, and Fernández de Bobadilla and Pełka proved that μ-constant families have constant multiplicity.
  • Vector fields and resolutions. The Lê–Saito–Teissier criterion relates constancy of μ to a gradient estimate, and Parusiński constructed trivializing vector fields for μ-constant linear families. For surfaces, Wahl introduced a logarithmic characteristic number, and Okuma showed that its constancy gives simultaneous resolutions after finite base change for non-log-canonical Gorenstein surface singularities. Recent work of Aldasoro Rosales and of Peters describes further structure of μ-constant degenerations.

The original result

• Claim, 24 September 2026. The main theorem of Topological triviality of mu-constant families of surface singularities states that for every holomorphic family f_t of germs on C³, over a disk, with an isolated critical point at the origin and constant Milnor number, there are, after shrinking, homeomorphism germs φ_t fixing the origin, with φ₀ the identity, jointly continuous in t together with their inverses, such that f_t ∘ φ_t = f₀. The manuscript states that this gives a positive answer to the surface case of the μ-constant problem, for arbitrary holomorphic parameter dependence and without assuming log canonical singularities.
• Scope, as stated: the result concerns one-parameter families in three variables and topological right-triviality; it makes no claim of Whitney regularity, which can fail in such families.
• Method, as described: for non-log-canonical germs the main step proves constancy of Wahl's logarithmic invariant from constancy of μ alone. After reduction to polynomial families over curves and a projective compactification, a semistable log minimal model gives additional surfaces over the singular point; a nearby Euler characteristic comparison, with nonnegative corrections from Greuel and Steenbrink, and Langer's orbifold Bogomolov–Miyaoka–Yau inequality force each of their logarithmic squares to vanish. Okuma's theorem then gives simultaneous resolutions after base change, an Euler characteristic argument excludes node smoothings, and lifts on the resolved zero family extend to the ambient function levels. The log canonical cases use constancy of the spectrum with Teissier's criterion, and the double-point case reduces to plane curves.
• Sources. Commit adc7f12 of the openai/math repository: Topological triviality of mu-constant families of surface singularities.
• Formalization. The collection lists no Lean scope note or Comparator challenge for this family.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 1 November 2026)
2026-10-13	AI Proved What?, episode 064 Same number, same shape?, remade version of 26 October 2026	Reading, exposition	Video explainer of the 37-page preprint. It recalls the theorems of Lê and Ramanujam and of Timourian and King and states that the surface case was the open μ-constant problem. It states that topological triviality is weaker than Whitney regularity, with the Briançon–Speder family as an example in which the Milnor number of a generic plane section drops, that the result concerns one-parameter families in three variables, and that constant multiplicity was already known from Fernández de Bobadilla and Pełka. It notes that one of the collection's papers on Zariski's multiplicity conjecture mentions this result for context.	Episode status, last checked 30 October 2026: posted as a preprint, released 6 October, not revised; under expert review, not yet; peer reviewed, not yet; checked in Lean, no; no public assessment by specialists found. The site states that it is made with Anthropic's Claude. Read on 1 November 2026.
2026-10-16	J. Jardine, Lattice24, referee review rank_241 in JJardine919/openai-math-audit	Review, AI-assisted audit	Reports no gap and lists no issues. It states that the reviewer read all main sections line by line, checked by hand among other steps the identity for Wahl's invariant, the compactification numbers, the nearby Euler vanishing, the sign cancellation that forces every extra logarithmic square to vanish and the node-count argument, and confirmed against Okuma's paper that his theorem gives exactly what is used. It lists the inputs taken on trust, among them Langer's inequality, the semistable minimal model program, Greuel–Steenbrink, Varchenko's spectrum constancy and the equimultiplicity theorem.	Part of the completed Lattice24 audit; the reviews are published under CC BY 4.0.

Notes

• The background summarizes the manuscript's own account of earlier work.
• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.