I realize that this quote may raise more doubts than it solves. In particular, how does one calculate the ratio of branchlike- to spacelike-separated events along two paths?

Do I have to compare every event in one path with every event in the other path, or do I first define a foliation of the multiway causal graph and then compare only events on the same slice? What should I do if two events are both spacelike and branchlike separated? This last question worries me particularly because Gorard writes:

spacelike-separation is, in general, a special case of branchlike-separation, since the application of a pair of purely spacelike-separated updating events will usually yield a branch pair in the multiway graph

Does this mean that every spacelike-separated event is also branchlike-separated?

More importantly, I don't see how to apply this reasoning to the system in your question. The rule ABA->AAB, ABA->ABA, ABA->BAA with initial condition AAAABAAAA can only generate pure branchlike separations. With this I mean that there is no way to do two updates "simultaneously", because all the rules apply only to the part with the B, the right hand side of the rules always "overlap". If no event is spacelike-separated, the ratio becomes infinite, and the phase difference between any two given paths is always $\pi$. This conclusion, apart from giving the wrong physical result, is also logically impossible.

I am sorry, I know I should answer your question, not pose other questions. My hope is that someone will eventually come and clarify all these doubts that we are exposing in these days.

Thank you for the second reference. It was helpful, but unfortunately it left me disappointed. The first two examples in the reference are just regular constructions of the double slit experiment in Lagrangian and Hamiltonian formalisms and have nothing to do with Wolfram model. The third example is a Wolfram model, but it is more "construction of two negatively interfering paths", rather than "a double slit experiment". Each transition in the model was constructed by hand, making it not scalable (and I am still not sure how exactly we assign phases after the Knuth–Bendix completion).

Thank you for the explanation, I now understand why you were talking about a probabilistic cellular automaton. You are absolutely right when you say that if all the paths are summed with the same phase the only thing you can obtain is diffusion and not quantum mechanics. However, I suspect that the random walk we were discussing is only a special case, where by incident or construction every path has the same phase.

The quote where Gorard mentions the ArcTan seems to imply that in general every path will have a different phase, therefore allowing quantum interference. It would be nice if someone could explain how to calculate in practice these phase difference with a real example.

Thank you for clarification. I am not sure it resolves my concerns though.

The Wolfram Model is 100% deterministic.

Every probabilistic model is 100% deterministic with respect to the probability distribution function (including probabilistic cellular automatons). The question one should be asking is "can evolution of the weights of the states in a Wolfram model be represented by the evolution of probabilities in a probabilistic cellular automaton?". And if the answer "Yes", then WM will have big problems modeling any interacting quantum fields.

Now concerning Gorard's response

By allowing for the existence of causal connections not only between updating events on the same branch of evolutionary history, but also between updating events on distinct branches of evolution history

Is he talking about Knuth–Bendix completion algorithm here, or the regular update rules? Because the update rules in a Wolfram model are fairly local, by which I mean that there exists a space in which they are local.

Concerning superdeterminism:

In my case, I prefer to explore the option of superdeterminism as it was developed by Gerard 't Hooft

Superdeterminism is interesting, but not particularly useful for developing working theories. In specific cases, the results of Bell's theorem can be reformulated without the requirement of the free will, but with an additional equality imposed onto the three-variable correlation function (see chapter 3.6.1. The mouse dropping function in Hooft's book, Eq. 3.26). The specific form of the correlation function is a necessary and sufficient condition for a local deterministic classical theory to satisfy Bell's theorem (it follows from the way Eq. 3.26 was derived). Unfortunately, The requirement is by itself is extremely constraining, because any noise in the model usually disrupts correlations between variables over time (assuming the variables are associated with distant non-communicating structures). I am not saying it is impossible to create a theory which would satisfy a particular form of the three variable correlation equation, but it is extremely hard (no one succeeded in making one so far, and I doubt that anyone ever will).

The reference is actually great. It clarifies my questions about the way one assigns weights and treats different nodes (the question was from the other thread, but I didn't get a clear answer there). From what I can see, the evolution of the system ("oX"->"Xo", "oX"->"Xo") is essentially a probabilistic cellular automaton, where each possible substitution is assigned the same probability (or weight). This immediately rises a number of concerns about Gorard's results and Wolfram model in general:

1. The first concern is very specific. The substitution system ("oX"->"Xo", "oX"->"Xo") resembles classical random walk very much. The only difference from a classical random walk is that a state "oooXXooo" can bounce only into the state "ooXooXoo", while in the classical random walk it can also bounce into "oooXXooo". This is a minor difference, and it cannot create "waves" in the probability distribution of the position of the letter X. The wavy interference pattern, that Gorard gets, is the result of the string sorting algorithm, which sets the order in which the strings are presented. This point was discussed in detail by Matt Kelly in the comments to the bulletin. To be honest, I am puzzled by Gorard being seemingly unaware about the interpretation of the rules as a modified random walk, then not being surprised after obtaining interference in such system, and then not addressing the issue of the ambiguity of the sorting algorithm, which is the source of the interference pattern.

2. My second concern is about Wolfram model in general. If my understanding is correct, and a Wolfram model can be interpreted as a probabilistic cellular automaton, it makes it very difficult to make anything resembling quantum fields. The reason is the following: probabilistic cellular automatons are observationally equivalent to deterministic automatons with hidden variables. It is problematic to reproduce quantum mechanics by a local theory of hidden variables due to Bell's theorem (there are some loopholes in the theorem, but no one managed to actually use them, so... may be the loopholes are just an illusion). This result is very concerning to the prospects of a Wolfram model being able to model the real world.