I’d say if your axioms don’t hold you wouldn’t go far in your quest for truth.

Axioms can be demonstrated. They don't have to be purely theoretical.

Mass and Energy are axiomatic to the study of physics, for instance. The periodic table is axiomatic to understanding chemistry. You can establish something as self-evident that's also demonstrably true.

One could argue that mathematics is less a physical thing than a language to describe a thing. But once you have that shared language, you can factually guarantee certain fundamental ideas. The idea of an empty set is demonstrable, for instance. You can even demonstrate the idea of infinity, assuming you're not existing in a closed system.

You can posit axioms that don't fit reality, too. And you can build up features of this hypothetical space that diverge from our own. But then you can demonstrate why those axioms can't apply to this space and agree as such with whomever you're trying to convey ideas.

When we talk about "absolute truth", we're talking about a point of universal rational consensus. Mathematics is a language that helps us extend subjective observation into objective conclusion. That's what makes it a useful tool in scientific inquiry.

You could say it's an absolute truth that absolute truths do not exist.

What about my Sith friend?

"This line on the map is perpendicular to this other line on the map" is not a statement about the territory.

I don't think that's a settled debate IIRC

That just means we can't know everything about the system. Not that it is not true.

That's computer science alongside with Church/Turing. Maths could have tried to claim it but they doubled down on formalism so they don't deserve it.

That said though incompleteness follows from nothing but logical implication itself so it's more fundamental than physics (try to imagine a physics without cause and effect that doesn't get you cancelled because Boltzmann) and philosophy (find me a philosopher who wasn't asleep during their logic lectures).