Schiffer's Conjecture
If a smooth bounded domain in admits a Neumann eigenfunction of the Laplacian that is constant on the boundary, must the domain be a ball? Cao-Labora and de Dios Pont construct infinitely many planar domains with large -fold symmetry that are not balls and admit such eigenfunctions, disproving the conjecture together with the closely related Pompeiu problem.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Spectral geometry
- Posed by
- M. M. Schiffer
- Year posed
- 1957
- Years open
- 69y
- Solved
- 2026-08-05
- Model
- GPT-5.6, Claude Opus 4.8, Claude Fable 5
- Vendor
- OpenAI, Anthropic
- Collaborators
- Gonzalo Cao-Labora, Jaume de Dios Pont
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 53 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
Models with coding harnesses were used in multiple parts of the research: numerically verifying the asymptotic estimates, producing first drafts of the proofs of the Bessel function estimates, and helping with exposition. The Lean4 verification of the proof was written by GPT 5.6 from an early draft of the paper. The novel construction strategy is the authors' own.
Verification
A day-old preprint. The paper states that a Lean4 verification of the proof was written by GPT 5.6, but no public artifact is linked yet, and no independent expert review has appeared.