Theorem · version 1
For infinitely many integers N ≥ 2 there are a bounded simply connected star-shaped non-disk Ω with C∞ boundary and N-fold symmetry, λ > 0 and a nonconstant satisfying in Ω, u = 1 and on ∂Ω.
Cao-Labora–de Dios Pont’s uniform bifurcation theorem supplies an infinite family without a numerical certificate.
Proved
Infinitely many integers N ≥ 2; bounded simply connected star-shaped C∞ domains with N-fold symmetry.
Counterexamples to Schiffer’s Conjecture · Theorem 1.1, Corollary 1.2 and AI/Lean acknowledgement →
Source dated Aug 5, 2026 · Observed Oct 13, 2026
In this problem · publication v7
Counterexample · Proved
Relationship not independently reviewed. This concerns its use in this problem.
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
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.
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Infinitely many integers N ≥ 2; bounded simply connected star-shaped C∞ domains with N-fold symmetry.
Source dated Aug 5, 2026 · Observed Oct 13, 2026 · Source passage →