VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
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.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
ChatGPT 5.6
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.
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.
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.
Known method families
computation (source-reported)
Source-reported tools: computation.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.