> we don't know the physical curvature goes to infinity--the math says it does More precisely, the math of the theory that works everywhere we can test it, whe

> we don't know the physical curvature goes to infinity--the math says it does More precisely, the math of the theory that works everywhere we can test it, when extrapolated to r = 0 inside a black hole, where we cannot directly test it, says the curvature increases without bound as r = 0 is approached. The mathematical process of extrapolating a well tested theory into a domain where we cannot directly test it is used all the time in science; there's nothing inherently questionable or unreliable about it. The issue in this particular case is that the extrapolation gives us an answer that has a problem: it doesn't seem physically reasonable to have curvature increasing without bound in a finite time, along a finite arc length on the worldline of some observer, at which point spacetime just ends and the observer is destroyed. So most physicists believe that GR, the theory, will actually break down somewhere before this point is reached, and we will need a new theory, which most physicists believe will be a theory of quantum gravity, to tell us what actually happens in this regime. But you're correct that we won't know for sure unless and until we have some way of testing what happens in this regime.