From what I can see, ZX calculus is a particular way of doing quantum computing. I am not sure though, what are the conclusions of the ZX paper about the mapping between ZX calculus and Wolfram model. The paper says in the conclusion:

[We] Demonstrated that the diagrammatic rewriting formalism of the ZX-calculus can indeed be embedded and realized within the more general formalism of Wolfram model multiway operator systems, using a novel reformulation of the Wolfram model in terms of double-pushout rewriting systems and adhesive categories.

I do not understand this. ZX calculus is "a graphical language for reasoning about linear maps between qubits", while Wolfram Model is a class of algorithms for updating graphs/strings/etc (with some physical meaning attached to the procedure). What does it even mean, that a language (ZX calculus) is imbedded into the model? Does it mean that "some "words" of the language are realized sometimes in some Wolfram models"? Or does it mean that we can "translate" ZX-calculus language into something made of Wolfram notations? To me, saying "ZX-calculus can be embedded and realized within the more general formalism of Wolfram model", is like saying "calculus can be realized and embedded within a more general formalism of electromagnetism" (in short, nonsense).

I am not sure I see how the paper really helps with the questions in the original post. Could you elaborate, please (I am not particularly knowledgeable in ZX calculus or categorical quantum information theory, so would be nice if you could make your point more or less self-contained)?

Thank you for the link. It was quite helpful in the sense that it clarified that Wolfram considers a single link in the multiway system to contribute about $10^{-116}$ to the phase of the path integral. We could assume this is the fundamental constant of "the" Wolfram model, which represents our universe.

Concerning the clarification of the paragraph:

But now we can make a potential identification with standard quantum formalism: we suppose that the Lagrangian density ℒ corresponds to the total flux in all directions (or, in other words, the divergence) of causal edges at each point in multiway space.

Complemented by:

By the way, it’s worth mentioning what a “flux of causal edges.” corresponds to. Each causal edge represents a causal connection between events that are, in a sense “carried” by some element in the underlying hypergraph (the “spatial hypergraph”). So a “flux of causal edges” is in effect the communication of activity (i.e. events), either in time (i.e. through spacelike hypersurfaces) or in space. (i.e. through timelike hypersurfaces). And at least in some approximation we can then say that energy is associated with activity in the hypergraph that propagates information through time, while momentum is associated with the activity that propagates information in space.

Reference: Finally We May Have a Path to the Fundamental Theory of Physics

Notice that S. Wolfram is proposing a more fundamental framework than the usual framework of energy and momentum. The idea is not to compute energy, momentum and the Lagrangians from the Wolfram Model and then go back to the mainstream physics, but to use the Wolfram Model to explain physical phenomena without using the notions from the previous paradigm: energy, momentum, Lagrangian, etc. The association between energy and activity in the hypergraph that propagates information through time is a hint for substituting energy in the explanations by the role of the activity in the hypergraph that propagates information through time.

For example, imagine an explanation of some property of a black hole in terms of energy. This explanation should be easy to translate to an explanation in the Wolfram Model using the activity in the hypergraph that propagates information through time without any need to compute the energy for the explanation to be consistent. Of course, at some point, the activity in the hypergraph that propagates information through time should explain the results of the measurements of energy.

The obvious question is: What is the problem with energy, momentum, etc? Well, these are real numbers and maybe real numbers are not the best way to describe nature at its most fundamental level. Maybe a real number is too reductive. To use discrete relations rather than real numbers is one of the starting points of the Wolfram Model.

Here is a reflection about the role of explanations in physics due to David Deutsch.