explainx / blog
Terence Tao published a ChatGPT conversation on the Jacobian conjecture counterexample. It shows how a world-class expert actually prompts AI.
Jul 21, 2026
Excited that AI helped disprove the Jacobian conjecture, but not sure what the conjecture says? This beginner-friendly guide builds from derivatives and local invertibility to the explicit three-variable counterexample, its independent verification, and the still-open plane case.
Jul 20, 2026
Around 2:19 AM UTC on July 20, 2026, Levent Alpoge posted that Claude Fable 5 helped produce a polynomial map C³→C³ with Jacobian determinant −2 that is not invertible. Mathematicians and a verification preprint have since checked the arithmetic. explainx.ai covers the announcement thread, J-lens confusion, and links the beginner explainer.
Jun 27, 2026
AI now disproves Erdős conjectures, formalizes Fields Medal proofs in Lean, and scores IMO gold. IEEE asked top mathematicians whether humans become “priests to oracles” or partners in “Big Mathematics.” The honest answer depends on which future you choose.
TL;DR: Terence Tao, one of the world's most accomplished living mathematicians, published a shared ChatGPT conversation in which he worked through an alternate construction related to the recently discovered counterexample to the Jacobian conjecture — and it became one of the most discussed AI-and-expertise stories of 2026, with 665+ points on Hacker News. The math is genuinely dense. The real story explainx.ai's audience should care about is how an expert directs a model compared to how most people prompt — and what that implies for anyone trying to get real value out of AI in their own domain.
The Jacobian conjecture — a long-standing open problem about polynomial maps — was recently settled by a counterexample, found separately from the conversation covered here. Explainx.ai has a full beginner-friendly explainer of the conjecture and the original counterexample if you want the mathematical background.
What made news this week wasn't the original discovery — it was Tao publishing his own shared ChatGPT session, in which he explored whether an alternate approach, given in an attached PDF, could "retroactively help explain the construction" of the original counterexample. Over several exchanges, Tao guided the model through:
Tao later published a blog post digesting the exchange at terrytao.wordpress.com, treating the shared conversation as a companion artifact to that writeup.
| Question | Answer |
|---|---|
| Who? | Terence Tao, Fields Medalist |
| What model? | Reportedly GPT-5.6 Sol on a "Pro" tier, per session context |
| What was verified? | An alternate construction of the (already-found) Jacobian conjecture counterexample |
| Did AI discover the counterexample? | No — it predates this conversation |
| HN reception? | 665+ points, one of the most discussed threads of the week |
| What's the real story? | How an expert prompts and directs a model vs. typical usage |
| Companion resource? | Explainx.ai's Jacobian conjecture explainer |
The detail that generated the most discussion wasn't the mathematics itself — it was how Tao interacted with the model. Commenters repeatedly noted that his questions were short, jargon-dense, and strategically sequenced, closer to how a mathematician would direct a capable research assistant than how a typical user prompts a chatbot:
"I'm also surprised to see that even Terence Tao seems to use it in a way that resembles, in progression, how I use llms in my area of expertise... short pointed questions that goes all in on the jargon and machinery of the field and steers the llm hard (eg no softballs)." — @lukebuehler
This connects to a broader, recurring observation about AI usefulness: the value you get out of a model scales with how well you can direct and verify its output — which in turn depends heavily on your own domain expertise.
"It reinforces how to 'learn AI' is to first master the problem domain. I can use AI for coding after decades of coding. I can't use it for theoretical physics because I can't evaluate the responses." — @michaelchisari
This is a genuinely important framing for anyone building or evaluating AI tools: a model's raw capability is only half the story. The other half is whether the user asking the questions can tell a good answer from a confidently wrong one — and that's a property of the user, not the model.
A separate, related example circulated alongside the Tao story: someone reportedly disproved a different conjecture by repeatedly replying "keep going" to ChatGPT rather than accepting an incomplete or hedged answer. One commenter's generalized version of the pattern:
"REPEAT THIS PROCESS UNTIL CONVERGENCE AND YOU ARE OUT OF OPTIMIZATION IDEAS. You have permission to keep iterating." — @minimaxir
The underlying mechanism is straightforward: LLMs don't inherently know a problem is supposed to be hard or unsolved — they'll keep attempting a task if given permission to continue, rather than concluding "this is famously open, I should stop." That can surface genuine progress a model might otherwise have prematurely abandoned. It's also a double-edged pattern — the same persistence that helps a model push through a genuinely tractable subproblem can just as easily produce confident-sounding hallucinated progress if there's no domain expert checking each step, which is exactly why this technique reads very differently coming from Tao (who verifies every step) than from a non-expert applying the same "keep going" pressure to a problem they can't independently check.
Not every reaction treated the conversation as pure evidence of AI capability. Some commenters pointed out the conversation is Tao steering a highly capable calculator, not the model doing independent mathematical discovery:
"I don't think chatgpt could have come to this on its own without the amount of steering he did, which just validates the idea that AI is not a replacement for human expertise but an amplifier." — @seamossfet
Others noted this specific conversation wasn't the one that found the original counterexample — it's a follow-up exploration of an alternate approach, and no equivalent "how was the original counterexample actually found" transcript has been published for comparison:
"You are badly informed the counter-example was found shortly before. Terence just tried to replicate how it was found." — @singularity2001
Both readings can be true simultaneously: the shared conversation is a legitimate, technically substantive verification exercise, and it's not evidence that an unguided model could have found the counterexample from scratch without an expert directing every step.
For explainx.ai's audience — builders, engineers, and AI practitioners rather than research mathematicians — the transferable lesson isn't about the Jacobian conjecture specifically. It's a data point in an ongoing argument about what "using AI well" actually looks like:
Primary sources: chatgpt.com/share — Tao's published conversation · terrytao.wordpress.com · Hacker News discussion, 665+ points
Details reflect the publicly shared ChatGPT conversation and associated Hacker News and blog discussion as of publication. Mathematical claims in the transcript are Tao's own verification work; readers seeking the underlying conjecture and original counterexample should consult explainx.ai's companion explainer and Tao's own blog post.
