Counterexamples to the Schiffer and Pompeiu conjectures in dimensions four and six

Paper and computer-assisted verification (Arb interval arithmetic) of bounded convex domains in R^4 and R^6, not balls, with real-analytic boundaries diffeomorphic to spheres, carrying nonconstant Neumann eigenfunctions of the Laplacian that are constant on the boundary. This gives counterexamples to the Schiffer conjecture and to the Pompeiu conjecture in dimensions four and six. This work was carried out with the assistance of AI models (Claude Opus 5.5, GPT-6 Astra and GPT-6 Sol).