Point-free locale formulation of the causal completion

Relate the point-free causal boundary construction based on locales to the round-ideal completion of continuous posets, thereby extending the continuous-poset development from Scott-open frames on points to a locale-theoretic setting.

Background

The paper’s development is naturally expressed through the frame of Scott-open sets associated with a continuous poset. It suggests extending the theory one level of abstraction by working directly with locales rather than with points.

The proposed comparison concerns the point-free causal boundary introduced by van der Schaaf and the round-ideal completion developed in the paper. Establishing this relationship would clarify how the order-theoretic completion behaves in a point-free causal framework.