Stationary encoding in an aleph-one-like dense order ($L_S={\sum }_{\alpha <{\omega }_1}{B}_{\alpha }$)

For a subset $S$ of the nonzero countable limit ordinals, let $B_{\alpha }=1+\mathbb{Q}$ when $\alpha \in S$ and $B_{\alpha }=\mathbb{Q}$ otherwise. The order sum is an aleph-one-like linear order whose nonempty proper initial segments are $\mathbb{Q}$ or $\mathbb{Q}+1$. At the limit cut before block $\delta$, the complementary final segment has a least point exactly when $\delta \in S$. Club agreement of countable filtrations makes this condition invariant modulo a nonstationary difference. Disjoint stationary subsets of omega-one consequently supply $2^{{\aleph }_1}$ nonisomorphic orders.