First-order separation of pro-orderable groups from a bi-orderable matrix group over Z[F₂]

Let \( O\) be the class of bi-orderable groups and let \( O^*\) consist of the groups in which every bi-invariant partial order extends to a bi-invariant total order. We construct a countable upper triangular matrix group \(H\) over \( Z[F_2]\) and a sentence \(σ\) in the pure group language such that \(H∈ O\), every member of \( O^*\) satisfies \(σ\), and \(H\) does not. Consequently, \(Mod(Th( O^*)) O\), giving a negative answer to Kourovka Problem 3.20. The construction makes two disjoint normal subgroups definable by commutators. Their disjointness follows from Fox derivatives, while their interaction with conjugation obstructs extension of a partial order.