The central claim in particular is not proven because a physical theory P need not be able to express statements like "there exists a number G, which, when interpreted as the text of a theory T, essentially states that the theory T itself is unprovable in the broader physical theory P" as an empirical physical fact.
> Arithmetic expressiveness; LQG can internally model the natural numbers with their basic operations. This is important as quantum gravity should reproduce calculations used for amplitudes, curvature scalars, entropy, etc in appropriate limits. Both string theory [34, 37] and LQG [35, 38] satisfy this by reproducing GR and QM in appropriate limits
Here the citations are four entire books. How am I supposed to very that LQG can model N with that?
Sure, I'm assuming here that nothing Gödel's brain did is fundamentally non-computable, but that's a pretty easy lift I think. Math is hard but it's not that hard.
Or just start simulating QM on a limited basis, just inside their brains. You might need to run evolution for another few million years until they start taking advantage of whatever Penrosian effects there are.
Godel then latches onto that to create an alphabet of the symbols which then are mapped to numbers; thus formulas are even bigger numbers, and derivations are even bigger bigger numbers. So for any statements there should be a derivation that prove the statement is true or a derivation that proves the statement is false. Of course most statements will be false, but even then there will be a derivation showing so.
Then Godel does some clever manipulation to show that there will be some statements for which there can be no such derivation in either way. But that does not need the physics theory to express things about itself. It only requires to be mathematically complex enough (it'd be weird if a theory of everything was simpler than Robinson Arithmetic) and that it has rules of derivation of its statements (ie, that mechanical math can be applied to deduce the truth of the matter from the first principles of the theory).
Of course, the actual undecidable godel number and the associated physical proposition would be immensely complex. But that is only cause nobody has tried to improve on Godel's methodology of assigning numbers to propositions. He used what was simpler, prime factorization, cause it was easy to reason about, but results in astronomical numbers. But there is no reason a better, less explosive way of encoding propositions could be found that made an undecidible Godel number to be translated into something comprehensible.
But this is largely unnecessary; Godel proof forces the mathematical system to speak about itself and then abuses this reflection to create a contradiction. It means the system is not complete, that there are statements in the system that cannot be proven from its first principles and derivation rules; the fact that the one Godel showed to exist is self referential does not mean all the undecidable propositions _are_ self referential. There well could be other, non self referential undecidable propositions, that could very well have a comprehensible physical interpretation.
And, regardless of the universe being a simulation or not, the physical theory will ultimately need to deal with this incompleteness.
Note Godel's proof is mechanically exactly analogous to Turing's proof of the undecidability of the halting problem, because ultimately it's the same thing (Curry-Howard, Prolog, and all that). So you can bypass arithmetic, but you can't really bypass self-reference; just like programming languages need some looping or recursion (or equivalent expressiveness) to be Turing-complete, mathematical theories need universal quantification to be subject to Godel's Incompleteness Theorem.
Of course, you can have a physical theory that _is_ Turing-complete, say the Newtonian billiard ball model (and, y'know, we can build computers); but that doesn't mean the theory will necessarily tell you, as a static, measureable physical fact, whether a particular physical process (say, an n-body system) will ever halt or loop, or go on forever with ever-increasing complexity; so you could (in principle, in Newtonian mechanics) build some (mechanical!) physical system that simulates the Goldbach conjecture, or looks for solutions to an arbitrary Diophantine equation, but if there are no integer solutions you'll never actually be able to show it; the theory is incomplete in the mathematical sense, but just as complete a description of reality's rules.
EDIT: I should also mention the idea that reality can tell us if a statement about a theory is true, given that the theory is an accurate description of reality. So if there’s an accurate Turing complete theory of reality, and we see some process that’s supposed to encode a decision on an undecidable statement being resolved (I guess in a non-probabilistic way as well), then we can conclude that reality is deciding undecidable statements in some nontrivial way.
Mostly as an abstraction on top of a continuous wavefunction/quantum field
> Spacetime ends up being discretizable
As far as I know this is speculative and usually assumed by physicists to be false; it's definitely not a required feature of quantum mechanics per se, and as far as I know not of any other well-accepted theory.
> Quantum mechanics dictates that certain properties, like energy and angular momentum, are quantized, meaning they can only exist in discrete packets or "quanta".
This was from a cursory google search.