Nonadditivity of the Mean Symmetric-Difference Risk for Kernel Level Sets

We give a smooth planar counterexample to simultaneous bandwidth-independent variance and bias limits for the mean symmetric-difference risk of kernel density level sets. For a standard bivariate normal density, a compactly supported radial C2 polynomial kernel, and critical bandwidths h_n(t) proportional to n^(-1/6), the normalized risk tends to C E|1+tZ|. Its strict convexity rules out universal constants A and B in sqrt(nh^2) I1 -> A and h^(-2) I2 -> B for any exact decomposition of the risk; three bandwidths suffice. An explicitly defined centered comparison gives unequal residual limits. A direct proof uses a triangular-array central limit theorem, an integrable boundary bound and an integrated-variance tail estimate. The general nonlinear risk law is earlier Doss-Weng/Qiao theory, which is fully credited. The result refutes the joint proposal in Chacon/Polonik's OWR32/2016 discussion while retaining its variance premise. The report does not define its I2 explicitly, so our centered comparison is not identified with that term and no whole UnsolvedMath OWR-14740-003 record closure is claimed. This five-page unrefereed preprint was prepared with AI assistance and self-audited; it has no independent referee or proof-assistant verification. Symbolic regression checks accompany the written proof.