arXiv:2609.35419v2 Announce Type: replace-cross Abstract: We construct the first convex planar counterexample to the Schiffer and Pompeiu conjectures: a bounded strictly convex non-disc domain with real-analytic boundary carrying a nonconstant solution of $\Delta u+u=0$ with $u=1$ and $\partial_\nu u=0$ on the boundary. Together with the higher-dimensional constructions, this gives convex non-ball counterexamples in every dimension from two to eighteen, and in dimensions twenty and twenty-one, with real-analytic boundaries diffeomorphic to spheres and indicator Fourier transforms vanishing on the unit sphere. In dimensions $4$, $6$, $8$, $10$ and $14$, planar reductions through compact Lie algebras of rank two use Harish-Chandra's radial part formula and Kostant's convexity theorem. In dimensions $3$, $5$ and $7$, an axisymmetric formulation on the unit ball gives a quartic equation in weighted coefficient spaces. The exact two-sided inverse of the residual-corrected linearisation combines a Dirichlet Helmholtz inverse, a holomorphic boundary real-part problem and the material-derivative identity; it gives the planar Schiffer domain and, by one dimension-parametrised argument, the domains in dimensions $5$, $9$, $11$ to $13$, $15$ to $18$, $20$ and $21$. We also construct the first convex planar counterexample to Berenstein's conjecture: a strictly convex non-disc domain with real-analytic boundary carrying a sign-changing solution of $\Delta u+u=0$ with $u=0$ and $\partial_\nu u=1$ on the boundary. Planar existence for this problem is due to Colbrook, Sadeghi and Stepaniants. All existence proofs in this paper are computer-assisted.