VibeMathedMath problems solved with AI
Sign in

The Budur-Fernandez de Bobadilla-Le-Nguyen conjecture on initial-form Milnor fibres of topologically equivalent germs

For a nonzero holomorphic germ gg vanishing at the origin, let g∗g_* be its lowest nonzero homogeneous component; the initial-form Milnor fibre is g∗−1(1)g_*^{-1}(1). Motivated by the relation between contact loci and monodromy, Budur, Fernandez de Bobadilla, Le and Nguyen conjectured (Cohomology of contact loci, Conjecture 1.7 in arXiv v3) that embedded topologically equivalent function germs have homotopy equivalent initial-form Milnor fibres; Sampaio (2025, Conjecture 4) studies the version for topological right equivalence. If f,g:(CN,0)→(C,0)f,g:(\mathbb{C}^N,0)\to(\mathbb{C},0) are topologically equivalent, are f∗−1(1)f_*^{-1}(1) and g∗−1(1)g_*^{-1}(1) homotopy equivalent?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Singularity theory; Milnor fibres and contact loci
Posed by
Nero Budur, Javier Fernandez de Bobadilla, Quy Thuong Le and Hong Duc Nguyen (J. Differential Geom. 120, 2022; Conjecture 1.7 of arXiv v3)
Year posed
2022
Years open
4y
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
15 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

With f1,f2f_1,f_2 the Zariski counterexample germs, f~j=fj(z)+t2\tilde f_j=f_j(z)+t^2 on CN+1\mathbb{C}^{N+1} are topologically right equivalent with isolated critical points, but H0((f~1)∗−1(1);Z)≅ZH_0((\tilde f_1)_*^{-1}(1);\mathbb{Z})\cong\mathbb{Z} while H0((f~2)∗−1(1);Z)≅Z2H_0((\tilde f_2)_*^{-1}(1);\mathbb{Z})\cong\mathbb{Z}^2: one initial-form Milnor fibre is connected and the other has two components. This disproves both the embedded and the right-equivalence formulations. It says nothing about weaker invariants the conjecture was meant to capture, such as cohomology of contact loci.

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). The manuscript is authored 'OpenAI' and names no human author. Two manuscripts in the family, September 24 and September 27, 2026, which the September 27 paper describes as independent arguments. The September 24 paper ships Python scripts (standard library only) that check its finite integer arithmetic; the README says they do not cover the lattice-theoretic or topological arguments.

Verification

No independent mathematician has checked this yet. Checked here: the corollary 'Initial-form Milnor-fibre counterexamples' in the September 24 manuscript was read against the conjecture as the manuscript states it. It follows from the Zariski counterexample by adding one common square t^2, which preserves topological right equivalence. Not refereed; no Lean formalization. The comparison of zeroth homology is elementary once the right equivalence is granted, so the weight rests on the main construction.

Sources

Changelog1 change

Discussion