Proposition: Same-ray indivisibility is not horismos

In a faithful continuum realization of a cone order, every nonzero reachable displacement $v$ has reachable displacements $sv$ for every $0<s<1$. This includes boundary displacements.

Proof. Cone scaling gives $sv\in K\setminus \{0\}$, and continuum faithfulness supplies the corresponding events and protocols.