The papers on orbit equivalence of free group actions and the additive arithmetic of linear orders both touch on automorphism groups of highly structured countable objects, and a natural gap appears at their intersection. Specifically, when one considers automorphism groups of countable linear orders built by transfinite addition, these groups can range from trivial to extremely rich, and it is unknown which of these automorphism groups support free ergodic probability-measure-preserving actions that are totally weakly mixing yet fail to be strongly ergodic. This question sits at the boundary of two separate research programs that have not yet been unified. The problem is hard because total weak mixing and strong ergodicity are separated by a subtle spectral gap condition, and the algebraic structure of automorphism groups of linear orders is highly sensitive to the arithmetic operations used to build those orders, making a uniform classification elusive. The automorphism groups in question need not be locally compact or finitely generated, so classical tools from representation theory and geometric group theory do not apply directly, and the infinite combinatorics of the underlying linear orders introduces obstructions that are not present for free groups or amenable groups. Resolving this would unlock a deeper understanding of how the combinatorial complexity of an ordered structure governs the ergodic-theoretic richness of its symmetry group, potentially providing new invariants for orbit equivalence classification and new examples or counterexamples relevant to the Connes embedding problem and cost theory for non-finitely-generated groups.