There appears to be a mistake in the definition of causal invariance (Some Quantum Mechanical Properties of the Wolfram Model):
Definition 22. A multiway system is “causal invariant” if and only if the causal networks associated with
all paths through the multiway system are (eventually) isomorphic as directed, acyclic graphs.
Consider a pair of string substitution rules A->AB, A->AC with the starting string "A". At each step of the evolution we can choose whether we generate B or C, and the multiway graph for this rule looks like a binary tree. A causal graph for any update order will always look like a single chain, ,
and therefore, all paths are isomorphic as directed, acyclic graphs, and the system must be causally invariant according to the definition. On the other hand, the system is not confluent (Gorard says that confluence is a necessary condition for causal invariance in his presentations), and appears to contradict the whole intuition of
a causally invariant system, that was built before that. A definition, corresponding to the intent of causal invariance should look more like this:
"A multiway system is “causal invariant” if and only if the causal networks associated with
all paths through the multiway system are (eventually) identical."
It would help though to clarify what does it mean that two causal graphs are "identical" (isomorphism is necessary, but, clearly, not sufficient).
Does this look correct?