Example: A generated Lorentz cone without continuum events

Fix $m\ge 3$. Choose a family $(z_n{)}_{n\in \mathbb{Z}}$ dense in the open unit ball of ${\mathbb{R}}^{m-1}$, and a fixed spatial unit vector $e_1$. Take disjoint two-event chains $a_n◃b_n$, represented by

This graph is admissible. Its order is exactly the restriction of the Lorentz order. The event sample is a disjoint union of two-event chains; its generated cone is $L_m$, but no nonzero boundary displacement is realized (Figure fig:sparselorentz).