Ambient Dependence of Extremal-Exchangeable Independence

We give explicit counterexamples answering the ambient-algebra question in Liebscher's unrestricted extremal-exchangeable free-product definition of independence. A state-preserving unital inclusion of the four-point algebra into M4(C) makes two distinct proper coordinate subalgebras with scalar intersection dependent downstairs and independent upstairs. Positivity forces every downstairs witness to be a nontrivial character mixture; a pure folding witness works upstairs. The three-page research note separately proves algebraic non-lifting via C[x] contained in C[x,x^-1], and explains extreme-state lifting for equivariant unital C*-inclusions. Pure folding and the extremality framework are established prior mathematics and are credited. The claim uses all-exchangeable extremality and unital algebraic/universal C*-free products, without a fixed-marginal convex slice, faithfulness, trace, or normal W*-requirement. The deposit includes source, exact rational verification, independent mathematical reviews and a bounded primary-source priority audit. Historical-first priority and a still-open-in-2026 assertion are not made. AI tools were used extensively in development, writing, reproduction and adversarial verification. This is an unrefereed preprint with internal AI adversarial reviews; it has not received human peer review.