A local inverse that fails globally
The Jacobian conjecture says that a polynomial map from complex n-space to itself with a nonzero constant Jacobian determinant must have a polynomial inverse. The determinant condition gives local invertibility, so the question is whether that local condition forces a global inverse. Tao's exposition writes down an explicit map from C³ to C³ whose determinant is identically minus two, then gives three distinct inputs that all map to the same output. That collision rules out global injectivity and therefore supplies the counterexample. [1]
The displayed map is not a numerical search result that depends on trusting a benchmark. Its coefficients, constant determinant and repeated output can be checked directly. Tao reconstructs it through multiplication of a linear and a quadratic binary polynomial, a resultant normalization and a three-dimensional affine slice, showing why the construction is locally invertible while retaining three-to-one behavior. [1]
Which conjecture fell
A counterexample in three variables extends to every higher dimension by adding untouched coordinates. Tao therefore describes the general-dimensional conjecture as false in dimension three and higher, while stating that the two-dimensional problem remains open and the one-dimensional case is easy. The distinction is material: this event does not settle the planar Jacobian conjecture. [1]
A later self-contained paper dates the first three-dimensional counterexample to July 19, attributes it to Alpöge, and develops the geometric mechanism behind it. That paper constructs further examples in every dimension greater than two, including maps with larger generic fiber sizes, and says its displayed identities and fiber calculations were checked in exact rational arithmetic. [2]
What the Claude attribution does and does not show
Tao reports that the three-dimensional result was shown using Anthropic's Fable AI, and the base record credits Levent Alpöge working with Claude Fable 5. The public mathematical account is centered on the explicit map and its verification rather than on a transcript, a prompt, a training-data audit or a controlled evaluation of the model. [1]
Atlas interpretation: That makes this a stronger historical claim about a checkable mathematical artifact than about autonomous research performance. The public sources support that Claude was part of the work and that the resulting map can be independently checked; they do not establish a general success rate, isolate the human contribution, or show that a model independently produced a peer-reviewed proof. Keeping those claims separate preserves the significance of the counterexample without turning its provenance into a broader capability conclusion. [1][2]
Sources
- A digestion of the Jacobian conjecture counterexample
Terence Tao · Jul 21, 2026
- Counterexamples to the Jacobian conjecture in dimensions greater than two
arXiv · Jul 31, 2026