science

Any viable F_QG must meet four intertwined criteria:

This statement is simply defining the fundamental structure of how a full theory of everything would be composed. A consistent and complete theory must meet all four criteria.

Also the core concept of F_QG is defined in a very hand-wavy way. I'd like to see a concrete example of an existing theory formalized in the way they proposed in the paper.

The above four criteria are how F_QG is defined. The author, in presenting these four criteria, provides two very specific, concrete examples of theories (String Theory and Loop Quantum Gravity) while introducing the premise of his argument. He clearly affirms that these theories do meet three of these four criteria but fail on the fourth. If there were an example of a theory that meets all four criteria than that theory would be the theory of everything and the whole issue would be resolved.

It's unclear to me how mathematical derivability from the formal system correspond to how laws of physics apply. Specifically mathematical logic is a discrete process, yet the world described by physics is generally contiguous.

The rest of the paper explains exactly this. Mainly that the only way to satisfy all four criteria is to include non-algorithmic components that bridge the discreteness of math with the observable continuity of physics. The author goes on to describe several examples where this process can apply in modern physics theory.

I do agree that the author is making a dramatic and bold statement regarding a proof of a theory of everything (that being that the theory of everything can never be computational) which requires heavy scrutiny. However, I am in no way an expert in these fields and so I have accept that the journal that published the proof can provide that scrutiny. It is easy to check on the reliability of that journal as a lay person, and in doing so doesn't seem to raise any flags about the validity of the arguments the author is presenting.