Arnold's corank problem: is the Hessian corank of a hypersurface singularity an ambient topological invariant?
For a holomorphic germ with a critical point at the origin, let . Fernandes, Jelonek and Sampaio proved that the corank is preserved by ambient homeomorphisms in three complex variables, and Sampaio proved that its parity is an ambient topological invariant in every dimension. Arnold's question, as recorded by Sampaio (2026, Problem 1): if a homeomorphism germ of carries onto for reduced holomorphic germs with critical points at the origin, must ?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Singularity theory; topological invariants of hypersurface germs
- Posed by
- V. I. Arnold, as stated in J. E. Sampaio, 'On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems' (preprint, 2026), Problem 1
- Year posed
- —
- Years open
- —
- 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
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
The weighted homogeneous germs on (, divisible by eight) of the Zariski counterexample are ambiently homeomorphic, have isolated critical points and real coefficients, and have Hessian ranks and , so their coranks differ. Corank parity is preserved, as Sampaio's theorem requires. Nothing is said about corank in three variables (where it is invariant) or about a minimal dimension.
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 'Negative answer to Arnold's corank problem' in the September 24 manuscript was read against the question as the manuscript states it from Sampaio 2026. It is a corollary of the same pair of germs as the Zariski counterexample, whose Hessian ranks are 16E and 0 for an integer E fixed by the construction. Not refereed; no Lean formalization. The paper notes the result is consistent with Sampaio's corank-parity theorem, since both ranks are even, and with the three-variable invariance theorem, since N > 3.