A Smooth Counterexample to Second-Difference Stability on Graded Meshes

We give a smooth counterexample to the mesh-independent second-order difference-quotient estimate in Becker and Rannacher's 2005 Oberwolfach report, equation (5), represented by UnsolvedMath record OWR-1058-002. The fixed polynomial u(x,y)=xy(1-x^2)(1-y^2) on the unit square has homogeneous Dirichlet data. Its conforming piecewise affine Ritz projections are computed on uniformly shape-regular meshes whose maximum diameter tends to zero. They converge in the energy norm, but an interior normal-derivative jump, scaled by the edge length to represent a second-order difference, is unbounded. Exact local stiffness matrices yield a positive tridiagonal recurrence and an explicit geometric lower bound. A separate fixed-outer-mesh family is nested. The five-page English manuscript includes the mesh construction, complete proof, scope qualifications, a vector figure, and reproducible exact-arithmetic and numerical diagnostics. The separate averaging conjecture and specified adaptive refinement rules are not addressed. This is an unrefereed preprint, with no claim of formal verification or absolute priority.