I agree that Godel is suggesting that the mathematics itself has creativity, another word for solutions, that goes beyond the axioms of the mathematical system. This I would think is your best argument that the universe is primarily mathematics.

All fields and particles including life forms can be thought of as mathematical simulations. They all follow basic axioms or laws and through evolution or just random interaction (ala Chaitin) they arrive at solutions, or forms, really they create forms that are not inherent in the basic laws or axioms. The new forms then imply the existense of new axioms, the ones that are needed to derive the new forms.

I suspect that you would agree with this much.

Where we probable depart is my suspician that the godel mathematics underly communication between highly complex systems. This is of course an intuitive belief. But the metaphor, or simulation, is that an overall system of higher complexity allows communication between subset systems having partial sets of axioms. One such subset system is for example human life forms. I can only speculate as to what other subset systems might be. I am tempted to suggest angels as another subset, and god as the overall system. But it's really all mathematics. And if such things exist, they are just simulations that happen to simulate intelligence and consciousness in addition to spatial form. Right?

BTW I like what's happening on this forum. The thinking is really opening up to new ideas.