Disproved
Planar disk-rigidity conjecture refuted in the cited paper ↗
Simply connected non-disk with real-analytic boundary; this is not a multiply connected annulus construction.
Source dated Aug 3, 2026 · Observed Oct 13, 2026
Conjecture · version 1
Planar disk rigidity for overdetermined Helmholtz data
Statement
Every bounded simply connected planar domain Ω with C∞ boundary admitting a nonconstant smooth real u and λ > 0 with in Ω, u = 1 and on ∂Ω is a disk.
Assumptions
- Bounded simply connected planar domain with smooth boundary.
- A positive eigenvalue and a nonconstant smooth solution up to the boundary.
Scope
The positive rigidity assertion is refuted by the analytic and smooth constructions. The C² Lean construction has a separately stated regularity scope.
Evidence
Source reports
Connections
In this problem · publication v5 · Selected context
Schiffer and Pompeiu conjectures →
Primary formulation · Disproved
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v7
Schiffer and Pompeiu conjectures →
Primary formulation · Disproved
Relationship not independently reviewed. This concerns its use in this problem.
Sources
A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures →
- Main theorem, computational certificate appendix and AI declaration → · Original passage ↗
Recorded wording
Constructs a non-disk simply connected domain with real-analytic Jordan boundary and tenfold symmetry, using certified numerical estimates.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Matthew J. Colbrook · George Stepaniants · Preprint · Publication date Aug 3, 2026
Revision & attribution
Source revision: 2026-08-03. Observed Oct 13, 2026.
The paper credits GPT-5.5 Codex for debugging early computational code, checked line by line by the authors. The authors own the mathematical argument and certificate.
Counterexamples to Schiffer’s Conjecture →
- Theorem 1.1, Corollary 1.2 and AI/Lean acknowledgement → · Original passage ↗
Recorded wording
Constructs infinitely many symmetry orders with smooth star-shaped non-disk counterexamples through a uniform bifurcation argument.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Gonzalo Cao-Labora · Jaume de Dios Pont · Preprint · Publication date Aug 5, 2026
Revision & attribution
Source revision: 2026-08-05. Observed Oct 13, 2026.
The authors credit GPT-5.5/5.6 and Claude Opus 4.8/Fable 5 for exploration, checks and exposition; GPT-5.6 assisted the early-draft Lean verification. The authors retain responsibility.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Disproved
Planar disk-rigidity conjecture refuted in the cited paper ↗
Simply connected non-disk with real-analytic boundary; this is not a multiply connected annulus construction.
Source dated Aug 3, 2026 · Observed Oct 13, 2026 · Source passage →
Record details
- Claim ID
- claim-schiffer-planar-disk-rigidity
- Statement version
- 1
- SHA-256
- 4ae086a512098fe0472f3225370a92375674b63b5806e4ab99275e4448935d19