First of all, I'm very unsurprised that you can get special and general relativity out of something like this. Relativity fundamentally just isn't that complicated and you can see what are basically relativistic phenomenon pop out of all sorts of natural setups where you have some sort of space with an emergent distance metric.
The real question is how this approach handles quantum mechanics. The fact that causal graph updates produce branching structure that's consistent with quantum mechanics is nice—and certainly suggestive that graphs could form a nice underlying substrate for quantum field theory (which isn't really new; I would have told you that before reading this)—but it's not a solution in and of itself. And again what the article calls “branchial space” does look vaguely like what you want out of Hilbert space on top of an underlying graph substrate. And it's certainly nice that it connects entanglement to distance, but again that was already theorized to be true in ER = EPR. Beyond that, though, it doesn't seem to really have all that much additional content—the best steelman I can give is that it's saying “hey, graphs could be a really good underlying substrate for QFT,” which I agree with, but isn't really all that new, and leaves the bulk of the work still undone.
That being said—credit where credit is due—I think this is in fact working on what is imo the “right problem” to be working on if you want to find an actual theory of everything. And it's certainly nice to have more of the math worked out for quantum mechanics on top of graphs. But beyond that I don't think this really amounts to much yet other than being pointed in the right direction (which does make it promising in terms of potentially producing real results eventually, even if doesn't have them right now).
TL;DR: This looks fairly pointed in the right direction to me but not really all that novel.
EDIT 1: If you're interested in some of the existing work on quantum mechanics on top of graphs, Sean Carroll wrote up a pretty accessible explanation of how that could work in this 2016 post (which also does a good job of summarizing what is basically my view on the subject).
EDIT 2: It looks like Scott Aaronson has a proof that a previous version of Wolfram's graph stuff is incompatible with quantum mechanics—if you really want to figure out how legit this stuff is I'd probably recommend taking a look at that and determining whether it still applies to this version.
I think Wolfram's "theory" is complete gibberish. Reading through "some relativistic and gravitational properties of the Wolfram model" I haven't encountered a single claim that was simultaneously novel, correct and non-trivial.
Using a set of rules for hypergraph evolution they construct a directed graph. Then they decide to embed it into a lattice that they equip with the Minkowski metric. This embedding is completely ad hoc. It establishes as much connection between their formalism and relativity, as writing the two formalisms next to each other on the same page would. Their "proof" of Lorentz covariance consists of observing that they can apply a Lorentz transformation (but there is nothing non-trivial it preserves). At some point they mention the concept of "discrete Lorentzian metric" without giving the definition. As far as I know it is a completely non-standard notion and I have no idea what it means. Later they talk about discrete analogues of concepts in Riemannian geometry and completely ignore the Lorentzian signature. Then they claim to derive Einstein's equation by assuming that the "dimensionality" of their causal graph converges, which is supposed to imply that something they call "global dimension anomaly" goes to zero. They claim that this global dimension anomaly corresponds to the Einstein-Hilbert action in the continuum limit. Only, instead of concluding the action converges to zero, they inexplicably conclude the variation of the action converges to zero, which is equivalent to the Einstein equation.
Alas, no theory of everything there.
The property such universes have in common is that they are computable on a hypothetical classical computer with unlimited capacity. (Potentially very inefficiently, like maybe computing one Planck unit of time in a tiny part of the simulated universe would require greater computing capacity than our universe could provide during its entire existence. These are mathematical abstractions unrelated to the real world.)
That implies a few things, for example that in none of these universe you could solve a halting problem. (That there would be certain potential... (read more)