 Comparison lower bound for interval disjointness

Deciding whether $n$ open real intervals are pairwise disjoint has comparison complexity $\Theta (n\log n)$. Sort by left endpoint and scan the maximum previous right endpoint for the upper bound. For the lower bound, map integer keys $z_i$ to $(2z_i,2z_i+1)$. Such intervals intersect exactly when the original integers agree. Each comparison between endpoints $2z_i+a$ and $2z_j+b$, with $a,b\in \{0,1\}$, is determined by a comparison of $z_i,z_j$ and their fixed offsets, so the reduction preserves the comparison model. Apply the element distinctness problem lower bound.