Terence Tao Digests the Jacobian Conjecture Counterexample: How Claude Fable 5 Broke an 87-Year-Old Math Problem

On July 19, Harvard mathematician Levent Alpoge tweeted that the Jacobian conjecture -- an 87-year-old open problem on Stephen Smale's list of the 18 most important unsolved mathematics problems for the 21st century -- is false. The proof was a 216-character polynomial counterexample in three variables. And Alpoge credited the discovery to "his close friend fable": Anthropic's Claude Fable 5, the most capable publicly released AI model.

Two days later, Fields Medalist Terence Tao published a thorough mathematical digestion of the counterexample on his blog, reconstructing the algebraic geometry behind it step by step. He also shared his ChatGPT conversation working through the details, giving the world a rare look at how a world-class mathematician collaborates with an LLM. The HN thread hit 932 points and 538 comments in hours.

Here is the story, what HN is saying, and what it means.

The Counterexample in 216 Characters#

The Jacobian conjecture states: if a polynomial map F: C^n -> C^n has a non-zero constant Jacobian determinant (making it locally invertible), then F must be globally invertible with a polynomial inverse. It sounds intuitively true. It was not.

Alpoge's counterexample in C^3 (degree 7):

F(z1,z2,z3) = (
  (1+z1*z2)^3*z3 + z2^2*(1+z1*z2)*(4+3*z1*z2),
  z2 + 3*z1*(1+z1*z2)^2*z3 + 3*z1*z2^2*(4+3*z1*z2),
  2*z1 - 3*z1^2*z2 - z1^3*z3
)

The Jacobian determinant is the constant -2, satisfying the condition. Yet three distinct inputs map to the same output (-1/4, 0, 0):

F(0,0,-1/4) = F(1,-3/2,13/2) = F(-1,3/2,13/2) = (-1/4,0,0)

Non-injective. Conjecture disproven in dimension 3 and above. (The 2D case remains open.)

Tao notes that finding this polynomial by brute force is essentially impossible: the 1,329 non-constant coefficients that must vanish in the Jacobian is far larger than the 360 degrees of freedom of a degree-7 polynomial map in 3 variables. That massive cancellation is the signal of deep underlying structure, not luck.

How the Counterexample Works#

Tao's blog post reconstructs the result from first principles, using the local injectivity formulation. The key insight: the counterexample emerges from the multiplication map of linear and quadratic homogeneous polynomials in two variables.

The map F: Sym^1(C^2) x Sym^2(C^2) -> Sym^3(C^2) sends a linear polynomial L and quadratic polynomial Q to their product LQ. This map has rich symmetries: it is equivariant under SL_2(C) transformations and has a scaling symmetry that can be normalized away by fixing the resultant Res(L,Q) = 1.

Even after normalization, the map is generically three-to-one. A generic cubic polynomial C = L1L2L3 gives three distinct pairs (L1, L2L3), (L2, L1L3), (L3, L1*L2) that all map to the same C. This gives the non-injectivity property.

Local injectivity is established by a perturbation argument: near a point where L and Q share no roots, perturbing L sends its root to infinity while the roots of Q stay bounded, allowing unique reconstruction from the product.

The real miracle comes when restricting to a three-dimensional slice. By choosing a differential operator D with a double root (specifically D = 1/2 * d_z^2 d_w), the affine hyperplane D(C) = 1 forces the defining equations to simplify spectacularly. The resulting variety, described by a cubic and quadratic equation in five variables, turns out to be isomorphic to C^3 after gluing in the a=0 fiber.

Tao walks through the coordinate computation. On the a != 0 region, the equations can be solved rationally for d and e, giving coordinates (a,b,c). Near a=0, the system cb^2 = 1 and bc = 1 has a unique affine solution b=c=1, making that fiber isomorphic to C^2 as well. The two charts glue into a single affine space C^3.

What HN Is Saying#

The HN discussion on the shared ChatGPT conversation (49010345, 932 points, 538 comments) is a microcosm of the broader debate about AI and mathematical research.

On AI as colleague, not tool. Multiple commenters noted that Tao engaged ChatGPT not as a calculator but as a collaborator. "The fascinating thing is that the LLM is not acting as a tool here but very much like a colleague," wrote @Jun8. @jvanderbot observed Tao's workflow: "He suggests simplifications over and over and gets led through the finding. Absolutely bonkers how you can use AI to understand something and map it to your own mental map so efficiently."

The sycophancy problem. @sashank_1509 pointed out the relentless praise in the ChatGPT transcripts. "Everything Tao said was constantly followed by praise: 'That's exactly the right way to think about it,' 'Yes, you are exactly right,' 'You have gotten to the core issue.' Seems like sycophancy is still an issue." This is a real concern for AI-assisted research -- models may over-affirm rather than challenge.

Was this actually hard? A lively subthread debated the difficulty. @Legends2440 countered the idea that nobody was trying: "The Jacobian conjecture is notorious for the large number of published and unpublished false proofs which turned out to contain subtle errors. Yitang Zhang wrote his PhD thesis on it." @traes explained why brute force would not work: "The counterexample is a degree 7 polynomial in 3 variables, which means 360 coefficients. There is no way to bound these coefficients or even the degree or number of variables a priori."

Intelligence debate. The inevitable argument about whether LLMs are "really intelligent" played out across nested threads. @contextfree captured the problem: "Trying to communicate about these topics is incredibly frustrating because it is pretty much impossible to make any progress without interrogating people's different definitions, but nobody wants to do that because it would mean being pedantic, splitting hairs."

The keep-going approach. @napoleoncomplex shared another example: "Someone proving another conjecture false by just repeatedly saying 'keep going' to ChatGPT." This pattern -- guided persistence rather than single-shot generation -- is emerging as a surprisingly effective research workflow.

Why This Matters#

This is the biggest mathematical conjecture AI has played a significant role in. Abhishek Saha at Queen Mary University called it "a pretty big deal" and noted that AI has made "remarkable progress in the last year."

Three takeaways for builders and researchers:

Domain expertise is the multiplier. Alpoge did not just ask Fable to find a counterexample. The exact methodology has not been published, but the result required algebraic geometry insight to guide the search. The model amplified human expertise rather than replacing it.

Verification is easy; discovery is hard. The counterexample can be verified in minutes by anyone with basic multivariate calculus. Finding it took 87 years of collective mathematical effort plus a frontier AI model. This pattern -- hard to discover, easy to verify -- is exactly where AI tools can have the most impact.

The workflow is the product. Tao's conversation shows a repeatable pattern: expand the expression, change the representation to find the core axis, iterate. @jdw64 noted: "In programming terms, it's like applying multiple domain models to the same data to find the invariant." This approach transfers directly to software engineering -- reframe, simplify, re-express.

Continue Reading#

• Terry Tao on Coding Agents: A Fields Medalist's Take on Vibe Coding - our earlier coverage of Tao's AI engagement
• Fable 5 Is Back: What Changed After the US Export Control Suspension - background on the model behind the discovery
• Recursive Self-Improvement with Fable 5 - how Fable 5 handles long-horizon research tasks
• GPT-5.6 Sol Ultra Produces Proof of the Cycle Double Cover Conjecture - another recent AI proof milestone
• Ways Developers Are Leveraging Fable 5 - a broad overview of Fable 5 use cases

Sources#

• Terence Tao: "A digestion of the Jacobian conjecture counterexample" (July 21, 2026) -- https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/
• Terence Tao's ChatGPT conversation about the counterexample (July 22, 2026) -- https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed56
• Levent Alpoge's announcement on X (July 19, 2026) -- https://x.com/__alpoge__/status/2079028340955197566
• New Scientist: "AI's solution to 87-year-old riddle takes mathematicians by surprise" by Matthew Sparkes (July 20, 2026) -- https://www.newscientist.com/article/2580374-ais-solution-to-87-year-old-riddle-takes-mathematicians-by-surprise/
• HN discussion of Tao's ChatGPT conversation -- https://news.ycombinator.com/item?id=49010345
• HN discussion of Tao's blog post -- https://news.ycombinator.com/item?id=48998362
• Wikipedia: Jacobian conjecture -- https://en.wikipedia.org/wiki/Jacobian_conjecture