The theorem that inspires CST is due to Malament, and shows that causally isomorphic spacetimes must be entirely isomorphic up to some conformal factor. The causal relations uniquely determine a spacetime up to a volume. The idea of CST is that, if spacetime were composed of discrete parts, we could get the volume as well by “counting” the parts composing any region.

The Hauptvermutung: If we have two faithful or “uniform” embeddings of the same causal set into relativistic spacetimes, with a given density, then the two spacetimes are approximately isometric above the volume scale of 1/(the density). But what is it for an embedding to be “uniform”?

Normally this is defined in terms of the spacetime, but of course the causal sets are fundamental (not the spacetime), so uniformity must fundamentally mean something like “with high probability.”

But what does “high probability” mean? This is where observables can help. Uniformity will be an instrumentally useful standard relative to the observables we’re hoping to predict.

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I’d like to clarify the connection between “uniformity” and “arising with high probability.”

Causal set theorists typically insist on understanding embeddings probabilistically, so they take “uniform” to mean that the expectation value of the number of embedded points in a region is proportional to the volume of that region (as determined by the spacetime metric). In particular, they require it to be generated by a Poisson process,

But if one takes the causal set to be more fundamental, then it would be backwards to use understand them as *arising* from a uniform embedding. Rather, one must use the inverse, inferential process: the embedded causal set *could have arisen* “with high probability” from a Poisson process.