Thank you both for sharing your thoughts. I think I like the idea of a light cone that emerges fully only when taking the limit of a big causal graph embedded in a manifold.

Pavlo, interesting idea. I hadn't thought about that. However it may have some issues, because the causal graph could become rapidly so connected that there are no lightlike paths traversing it from top to bottom. For example, looking at the picture from Gorard's paper, there is only one lightlike path connecting the first event of the causal graph with the bottom events (the one I have drawn in violet)

In other words, only one of the bottom events can be reached in only one way. I suspect that doing some further step in the evolution, it may happen that no lightlike path traversing the whole graph exists anymore. This will definitely be the case if the system is confluent.

At the beginning of the Wolfram Physics Project, there was the following identification, corresponding to the assumption that spacetime is as Minkowski described it (a continuous manifold),

light cones: causal cones in the causal graph

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but later S. Wolfram developed a subtle distinction of these concepts in his theory of faster-than-light travel (this theory predicts that the interior of black holes may interact with the exterior environment via faster-than-light interactions across microscopic space tunnels... but this is another subject):

The “causal cone” of affected events is very well defined. But now the question is: how does this relate to what happens in space and time?

When one thinks about the propagation of effects in space and time one typically thinks of light cones. Given a light source somewhere in space and time, where in space and time can this affect?

And one might assume that the causal cone is exactly the light cone. But things are more subtle than that. The light cone is normally defined by the positions in space and time that it reaches. And that makes perfect sense if we’re dealing with a manifold representing continuous spacetime, on which we can, for example, set up numerical coordinates. But in our models, there’s not intrinsically anything like that. Yes, we can say what element in a hypergraph is affected after some sequence of events. But there’s no a priori way to say where that element is in space. That’s only defined in some limit, relative to everything else in the whole hypergraph.

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Therefore, if I am not mistaken, the notion of light-like separated events is extrinsic, depending on the embedding of the causal graph into a Minkowski space. Indeed, in Tommaso Bolognesi's paper Algorithmic Causal Sets for a Computational Spacetime, page 4, we can see an extrinsic definition of light-like separated events. This is the reason why faster-than-light interactions are possible in the Wolfram Model without violating special relativity, although they may be microscopic in practice (the detection of such interactions in a particle accelerator or a black hole would be evidence for the Wolfram Model).