I disagree. The definitions may have some gray areas, but at least looking back in history there are some breakthroughs that would fall firmly within the "a result found within the convex hull of ideas that existed at that time" type (e.g. Einstein's special theory of relativity), but also some truly novel, "out of the blue" breakthroughs that people agree are fully outside the convex hull of ideas from that time (e.g. Einstein's general theory of relativity).
Based on that, mathematicians and scientists should be able to provide some falsifiable hypotheses about the theorems and kinds of proof methods that would be truly novel.
Most importantly, you don't need to define it 100%. You just need to come up with a bunch of specific examples where one would say "the proof of this would be outside the convex hull" and get wide consensus as to why, and then you'd have your falsifiable hypothesis.