I'm hoping for more specificity about where a generalization might break down.

My comment was more about horizontal exploration (before it most comments were physics-related) than about elaborating any details.

No one that I've asked seems to know much about how mathematicians choose axioms-- there's got to be some process of choosing axioms which are likely to generate interesting mathematics.

That is one aspect of my first item "Mathematics is explained by reduction of propositions to axioms". The axioms as "premise so evident as to be accepted as true without controversy" (Wikipedia) still possess or require some structure - albeit a non-mathematical implied and/or often 'soft' one. I once had a discussion with a mathematician about this and also a longer web dialog about how vague notions crystalize into concrete structures but couldn't convice anyone that this is a real problem instead of a wishy washy relativization of the truth of math.

I will address your other items with separate comments.