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.

The paper itself [1] seems quite compact and extremely high level, so I'm sure some heavy hitters would try to reformulate it. Would be the most unintuitive thing to happen since Bell's theorem [2].

[1] https://jhap.du.ac.ir/article_488.html

[2] https://en.wikipedia.org/wiki/Bell%27s_theorem

It wasn't stated why all truths need to be provable though. Perhaps the paper goes into this detail that I'd like explained.

Seems at best they may have proved you can't simulate the universe on hardware that exists within this universe, which is a bit of a no-duh kinda thing.

Imagine running a simulation in our universe and using a hardware random generator. And AI mathematicians inside your simulation proclaiming confidently that it would be impossible for them to be in a simulation because all randomness must be algorithmic and thus impossible to generate such randomness.

Doesn’t mean the universe isn’t a simulation.

Everything you perceive is through the brain. Brain could be in a jar receiving the same neuron signals, it wouldn’t be able to know if it is in a simulation or not.

There is no way for a program to know if it’s inside a virtual machine or not.