Convex counterexamples to Schiffer and Pompeiu in dimensions 3, 4, 6, 8, 10 and 14, made with Claude Opus 5.5 and GPT-6 Astra/Sol
Awaiting review
The takeaway
On 28 Sep 2026 Jizhou Guo posted what he says are the first counterexamples to the Schiffer and Pompeiu conjectures in dimension three and higher.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 30 September 2026
Your AI and this story
- GPT-6 Astra151 days after its cutoff
- Claude Opus 5.590 days after its cutoff
- Gemini 4 Hydrogen181 days after its cutoff
- Grok 4.7120 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 90 days before it.
Key facts
- arXiv 2609.35419 (87 pages, v1 28 Sep 2026); supersedes the author’s Zenodo preprint on dimensions four and six (doi 10.5281/zenodo.22999505)
- Dimensions 4, 6, 8, 10 and 14: adjoint-invariant domains in the rank-two compact Lie algebras u(2), so(4), su(3), so(5) and g2. Harish-Chandra’s radial-part formula reduces each case to a planar Helmholtz problem, and Kostant’s theorem reduces convexity to the Cartan section
- Dimension 3: an axisymmetric convex domain, plus a separate strictly star-shaped non-convex example
- Existence by Newton–Kantorovich contractions in weighted coefficient spaces: finite blocks checked in Arb ball arithmetic, infinite tails bounded analytically; verification code and certificates are in the arXiv ancillary files and on GitHub
- Acknowledgement, verbatim: ‘This work was carried out with the assistance of AI models (Claude Opus 5.5, GPT-6 Astra and GPT-6 Sol). The author takes full responsibility for the content of this paper.’ The paper does not break down which parts each model did
- Builds on the August 2026 planar counterexamples of Colbrook–Stepaniants and Cao-Labora–de Dios Pont
- Status: single-author preprint, not peer-reviewed
What happened
Eight weeks after the planar disproofs, a single author extended the counterexamples to higher dimensions and to convex domains. He used Lie-theoretic symmetry to reduce each case to a planar problem and certified the solutions with interval arithmetic.
Why it matters
If it holds up, the Schiffer and Pompeiu conjectures fail even for convex domains, which was one of the natural fallback forms of the conjectures. It is also an example of a September 2026 paper that names the newest frontier models (Claude Opus 5.5 was released on 22 Sep) without saying what each one did.
What is disputed or not yet verified
| Verification | Computer-assisted proof with public certificates; preprint, not peer-reviewed |
|---|
Sources
3 sources from 2 sites. Numbers match the chips in the text.
3 sources: 3 primary
Primary
- arXiv 2609.35419: Convex counterexamples to the Schiffer and Pompeiu conjectures in dimensions three, four, six, eight, ten and fourteenarxiv.org, paper
- GitHub: aster2024/schiffer-pompeiu-counterexamples (paper and verification)github.com, code
- GitHub: aster2024/schiffer-pompeiu-r4 (earlier dimension-four version)github.com, code
Changes
- Filed