Exchangeability and Trivial Tails Do Not Imply the Coordinate Intersection Property

For every prime p >= 5, we construct a minimal exchangeable sequence of trace-preserving embeddings of L(C_p) into a hyperfinite finite von Neumann algebra whose tail algebra is scalar but whose canonical filtration fails the intersection axiom for noncommutative factorizations. The intersection of the algebras on {0,1} and {1,2} has dimension p^2, while the algebra on {1} has dimension p. The example comes from an explicit class-three nilpotent group: a central coordinate is recovered by a triple commutator using any auxiliary index, but not from its single generator. We also identify the least enlargement of the coordinate algebras that does form a factorization. Thus the obstruction concerns retaining the given singleton ranges; it does not assert that the ambient algebra admits no factorization with enlarged coordinates. This is an unrefereed, AI-assisted preprint with a self-contained proof and exact reproducibility code. It makes no absolute priority claim.