The Planar Berenstein Conjecture
The unrestricted planar Berenstein conjecture holds that overdetermined Dirichlet-Neumann data characterize the disc. Disproved: a bounded simply connected domain with real-analytic Jordan boundary that is not a disc, carrying a nonzero real eigenfunction with zero Dirichlet data and constant nonzero Neumann data.
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Computation
- Field
- Overdetermined boundary problems
- Posed by
- Carlos A. Berenstein
- Year posed
- 1980
- Years open
- 46y
- Solved
- 2026-08-09
- Model
- ChatGPT 5.6
- Vendor
- OpenAI
- Collaborators
- Matthew J. Colbrook, Siavash Sadeghi, George Stepaniants
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 22 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
The domain has dihedral symmetry of order 26 and is neither a disc nor centrally symmetric, and its eigenfunction changes sign - which is why an additional sign assumption rescues the statement.
What the AI did
The authors declare the role and are careful about its limits: "We believe it is important to declare the use of AI in mathematical research, and in the present case its role is particularly noteworthy. ChatGPT 5.6 was given a substantial warm start consisting of an early draft and working code independently developed by MJC and GS" for the earlier Pompeiu-Schiffer paper, material that "already contained the central construction, conformal fixed-disc formulation, coefficient spaces, disk-polynomial algebra, tail strategy, and computational architecture on which the present paper rests. The system was used to explore the modification" to the Dirichlet endpoint.
Verification
A preprint days old. The existence claim is reduced by a Newton-Kantorovich argument to finitely many explicit inequalities certified in interval arithmetic, so the final step is machine-checkable, but nobody independent has rerun it.
Source
- PaperarXiv