A Three-Vertex Counterexample to Product-Order Lifting of TF Classes

Does every comparison in the facial semistable order admit coordinatewise ordered representatives in the corresponding TF-equivalence classes? We give a complete counterexample for the path algebra of 3 -> 2 -> 1. The classes C = {(-a,-b,a+b): a,b > 0} and D = {(-c,0,c): c > 0} satisfy a cover relation C < D, but no point of C is coordinatewise comparable with any point of D. Both classes lie on the hyperplane perpendicular to (1,1,1). We compute the strict and weak semistable torsion classes directly from the six interval modules and prove that even the transitive closure of the representative relation misses this cover. This answers the auxiliary lifting question in Hanson, A facial order for torsion classes, Remark 4.19. It does not resolve OWR-12732-005 or the general conjecture that g-vector cones determine the support tau-tilting order. The English manuscript includes an explicit proof, scope statement, source attribution and reproducible exact-arithmetic diagnostics. It is an unrefereed preprint with no claim of formal verification or absolute priority.