Universal Polynomial Interpolation Need Not Be Hereditary

A finite family of affine algebraic sets may allow polynomial interpolation of every pointwise-compatible tuple although a subfamily does not. For every integer m >= 2, this note constructs an irredundant family of three proper, reduced, pure curves in C^3 with this property, while the sum of two vanishing ideals has quotient C[x]/(x^m). An explicit formula interpolates all compatible data on the whole family. The construction disproves the necessity direction of the radical-chain criterion in Theorem 2.5 of McKinley, Shekhtman and Tuesink, arXiv:2208.12406v1, even after correcting its indexing. The note gives the precise hereditary replacement and proves that the full-family criterion is restored when all pairwise intersections are finite. Classical two-set interpolation and Chinese remainder methods are credited. This is an unrefereed preprint, not a claim to a new solution of the full general interpolation program or an absolute priority claim.