Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?

it doesn't overturn much. For example, here is a post from 2004

https://www.math.columbia.edu/~woit/wordpress/?p=105

it is about a purported (though incorrect) positive proof of the Jacobian conjecture in 2 dimemnsions. It is true in 1 dimension. The Fable proof is that it is false in >= 3 dimensions. 2 dimensions is still open.

Anyway, in that post it says

> It now seems that a proof has been found by Carolyn Dean of the University of Michigan, for the case of polynomials in two complex variables *(for more variables, many people believe it is not even true)*

so the resolution of this is a "surprise" in that it is a very long open with many failed proof attempts. But the direction it resolved was not surprising.

Not much.

But it does give credible plausibility to the concept that we might be mistaken about the exact boundaries of hardness for adjacent (but not equivalent) polynomial systems. Most (all?) of which have also stood up to a whole lot of undeniably sharp people poking at them for about as long.

No. This is about polynomials. The assumption that the Jacobian is nowhere zero is what is doing so much of the work. This means the Jacobian must in fact be constant. But obviously there are many mappings whose Jacobians are not constant.