The Jacobian Conjecture, a long-standing mathematical problem, questioned whether certain polynomial functions must always be invertible. A recent disproof, stemming from work by Levent Alpöge and Fable, provides a concrete counterexample.

Essentially, the conjecture requires a function to smoothly deform a space without pinching, creasing, tearing, or collapsing volumes. This is analogous to deforming an infinitely stretchy rubber block. The disproof uses a polynomial function in three-dimensional space that meets these smooth deformation requirements (its Jacobian determinant is constant and nonzero). However, this function maps multiple distinct inputs to the same output, meaning the deformed rubber block would overlap itself, proving the conjecture false.

This counterexample, understandable with basic algebra and calculus, demonstrates that even if local regions of the deformation are well-behaved, distant parts of the space can still occupy the same location. The disproof was found by a human working with AI, solving a problem mathematicians had pondered for nearly a century. The two-dimensional version of the conjecture remains an open question.