Determine whether Poisson sprinkling covers all faithfully embeddable causal sets

Ascertain whether a causal set (C,≺) that faithfully embeds into a given Lorentzian manifold (M,g) necessarily arises as a typical outcome of Poisson sprinkling of (M,g), i.e., establish whether faithful embeddability implies typicality under the Poisson process that samples points with uniform spacetime volume density and independent trials.

Background

Poisson sprinkling is widely used to generate causal sets from Lorentzian manifolds by random sampling with uniform density per spacetime volume and preserving causal relations among sampled points. This procedure underpins much of the empirical and mathematical evidence connecting discrete causal sets to continuum geometries.

Whether every causal set that is faithfully embeddable in (M,g) must be typical under Poisson sprinkling remains unknown. Resolving this would clarify the extent to which the probabilistic construction captures all continuum-approximating causal sets relevant to the recovery of Lorentzian geometry.