This approach could be useful for running efficient simulations of particle physics in a curved spacetime. However, this is not useful for uncovering the more fundamental hypergraph rewriting rules. Let me explain why, and how to improve on this.

In this Feynman Checkers, the Lorentz symmetry emerges despite the underlying graph clearly not being Lorentz symmetric. A simple square lattice is known not to be a viable model of spacetime since it obviously degenerates under Lorentz transformation, as the causal set theorists have long known ( https://www.slideserve.com/altessa/an-introduction-to-causal-sets ). It follows that the structure of the underlying graph must already be Lorentz symmetric itself. So how can you uncover the correct underlying structure? You would have to do it by completely abandoning approaches that will yield Lorentz symmetry on top of graphs that are not Lorentz symmetric, i.e., don't use use Feynman checkers. The most straightforward alternative would be to use the longest path metric for graphs, a simple discrete graph distance between vertices, which is the maximal number of steps across edges to get from one vertex to another. Obviously, such a simple measure will not yield Lorentz symmetry on a simple square grid lattice, which is good. Now the actually interesting problem is how to construct a graph from simple rewriting rules alone, such that even this simple distance metric will then approximate Lorentz symmetry. This had actually first been achieved by Tommaso Bolognesi (Title: Towards algorithmic causal sets with special-relativistic properties). However, his approach has remained confined to 1+1 dimensions only. I have then taken a different approach and have solved the analogous problem for the 2D Euclidean plane perfectly. I then generalized this approach to 3+1-dimensional Minkowski spacetime, where it worked less perfectly but still remarkably well. In the various illustrations, you can see that this extremely simple approach leads to the emergence of richly patterned graphical structures (The paper: https://www.researchgate.net/profile/Gabriel-Leuenberger/publication/358898864_Emergence_of_Minkowski_Spacetime_by_Simple_Deterministic_Graph_Rewriting/links/621c4e3e2542ea3cacb70ea7/Emergence-of-Minkowski-Spacetime-by-Simple-Deterministic-Graph-Rewriting.pdf )

So, this is approximately how it is really supposed to be done, i.e., first figuring out what the true microstructure of spacetime must look like and this can then allow you to derive a better understanding of particle physics from it afterward, since particles are 'made of space' as Wolfram would say.

And there is one more problem with this Feynman checkers, which is that the time scale of the lattice is supposed to be Planck time while the time required for one oscillation of an electron is orders of magnitude greater. So to make these scales compatible would reqiure the model to use continuous complex numbers, which is not very compatible pure (hyper-) graph rewriting. However, luckily in quantum gravity, one oscillation is actually supposed to be of the same order of magnitude as Planck time. This means that one should first try to completely solve quantum gravity first before moving to QED, at least when it comes to making progress at the most fundamental level.