"people from Earth call analytic function what we call static electric fields in a region of the plane containing no electric charge"

This sounds like they are just using different words for the same thing. They would still need to write partial differential equation for the field. Fields, functions and differential equations are more general things than electric fields, and Maxwell equations. You can explain second through the first, but not the other way around.

Anyway, I don't know what we are arguing about (and I don't want to argue about it). Have you read the paper? Can you explain, what Gorard does there? Does it help with the questions in this post? You are sending references with 100 pages in each, without explaining the context or how the references are related to the questions in this thread, or where I should even look. It doesn't help. I asked specific questions and I need specific directions. If you don't know the answer to the questions, just say it (or don't say anything at all). If you know there are no answers yet, just say the questions were not worked out yet. If you know the information is contained in a specific source, and it answers my questions, then tell where I can find it. If you have a source, which you suspect may be helpful, but you have no idea if it actually is, then say directly that you have no idea if the source is of help because you haven't read it yet. Please, don't waste my and your time. Just give the useful information, suggestions, clarifications. I asked questions to clarify technical details of Wolfram model. Do you have anything specific to say about the questions?

The ZX paper is meant to answer the requirement

it is imperative to make sure that Wolfram model reproduces quantum mechanics to a satisfactory degree

ZX calculus is an important part of quantum mechanics. The adjective categorical is not about the physics, but about the mathematical framework in which quantum mechanics is expressed. Of course, ZX calculus is not the end of the story, it is just the beginning.

What is "framework"? By "translating into Wolfram's notation", I meant just enriching words and notations of Wolfram Models to include definitions from ZX calculus.

I prefer to use the word framework, rather than notation, because a framework is conceptual (interaction among several concepts), whereas a notation is just a way to write something.

What do you mean by this? (Me: I agree that differential and integral calculus can be realized and embedded in the general formalism of electromagnetism)

Imagine an intelligent extraterrestrial civilization where the formalism of electromagnetism was discovered before than differential and integral calculus because these people are more related to electromagnetism than to mechanics for some historical reason. Then they visit Earth and they find a textbook of differential and integral calculus. In order to understand this textbook, they will express calculus using their formalism of electromagnetism. For example, in order to understand analytic functions of a complex variable they may say: "people from Earth call analytic function what we call static electric fields in a region of the plane containing no electric charge".

In this sense, calculus can be developed from the formalism of electromagnetism, but for historical reasons, it is done the other way around: electromagnetism is developed using calculus. This is the reason why Bob Coecke used the phrase When worlds collide...in a good way! (his world of categorical quantum mechanics and the world of the Wolfram Model). In the example above, the worlds are the formalism of electromagnetism and the world of calculus.

Here is a reference to the reproduction of a part of categorical quantum mechanics in the Wolfram Model: the ZX calculus.

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.

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.