Why couldn't mathematicians prove this classic conjecture for 90 years — and what if it was actually false all along?
Why did the Jacobian conjecture resist proof for over a century?
This explores why the Jacobian conjecture stayed unresolved for so long, and what the corpus says about how it was finally settled: according to the corpus, it was disproved rather than proved.
This explores why the Jacobian conjecture stayed open so long. Most of the corpus answer lies in the question's assumption. The conjecture says that if a polynomial map has a constant, nonzero Jacobian (meaning it never locally squashes or folds space), then it must have a polynomial inverse that undoes it exactly. Mathematicians spent decades trying to prove this. According to the corpus, they couldn't because it isn't true in three dimensions or more: Levent Alpöge used Fable 5 to find a three-dimensional polynomial counterexample Can AI search find what human proof cannot?. One small correction to the premise: Ott-Heinrich Keller posed the problem in 1939, so it was open for nearly 90 years, not more than a century. Part of the answer, then, is that people were looking for a proof of something false.
The second half of the answer is about where the counterexample was hiding. The note's main point is that the AI's contribution was search, not argument. It moved through a huge space of candidate polynomials that no human could check by hand, and it found one that broke the rule. A counterexample like this can be extremely specific, and there's no reason intuition or elegant theory should lead to it. This fits a pattern elsewhere in the corpus. OpenAI's counterexample to Erdős's unit distance conjecture was built from well-known number-theory tools. The new step was letting one parameter grow without limit, which is a configuration people hadn't thought to try What made OpenAI's unit distance counterexample succeed?. In both cases, the conjecture survived because it held up across every case people tended to check.
Terence Tao's view explains why search can be trusted even when the searcher is a black box. An opaque model's suggestions are fine as long as a reliable checker confirms the result Can opaque machine learning models help prove new mathematics?. Counterexamples suit this setup well: once you have the candidate polynomial, anyone can compute its Jacobian and confirm that it fails to have a polynomial inverse. This is also why the result doesn't contradict the corpus's view that LLM theorem provers still behave like solvers rather than research agents Can LLM theorem provers tackle genuinely open-ended research problems?. Finding a counterexample is a hunt for a single object. It doesn't require sustained, open-ended proof construction.
A few cautions from the same corpus. Claims of AI cracking open problems have sometimes collapsed under scrutiny. In one case, GPT-5's supposed solutions to Erdős problems turned out to be retrievals of existing papers Did GPT-5 really solve previously unsolved math problems?. In another, a claim of 'very little human input' on a Navier-Stokes result didn't hold up How much human input did OpenAI's Navier-Stokes proof actually require?. Who gets credit is also disputed: the Leiden Declaration holds that responsibility for a result belongs to its human authors Can AI-generated proofs ever replace human mathematical understanding?. Be clear about what the corpus doesn't cover. It holds nothing on the conjecture's history, such as the many failed proofs or why the two-dimensional case is different, so a deeper 'why was it hard' answer would need sources outside this collection. One naming trap: the 'Jacobian lens' used to read a language model's internal states Can we read a language model's unspoken thoughts? uses the same calculus tool but has nothing to do with this conjecture.
Sources 8 notes
Levent Alpöge used Fable 5 to find a three-dimensional polynomial counterexample to the Jacobian conjecture, a century-old open problem. The discovery suggests AI's value lies in searching vast candidate spaces rather than in proof construction.
OpenAI's counterexample to Erdős's unit distance conjecture builds on classical Golod–Shafarevich towers and number-field methods, but achieves its breakthrough by letting the degree [K : Q] → ∞. This allows a fixed split prime to suppress the class number and discriminant, enabling the construction of planar point sets with superlinear unit distances.
Tao argues ML tools' opacity matters less than pairing them with reliable validators like proof assistants or numerical methods. He cites finite-time blowup for Boussinesq equations, where a neural network suggested solutions later verified through perturbation arguments.
Current systems excel at isolated, well-defined proofs but cannot address truly open problems like Millennium Prize Problems. Many claimed successes rediscover existing results, and formal verification does not guarantee the proof addresses the intended mathematical claim.
OpenAI's announcement conflated 'open to one maintainer' with 'unsolved in mathematics.' Thomas Bloom confirmed the problems had existing solutions GPT-5 surfaced; the actual contribution was literature retrieval, not proof discovery.
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Buckmaster documented that OpenAI's public framing of "very little human input" contradicted details he learned directly: an entire team worked on it, tested simpler problems first, even the prompt was AI-generated, and timing suggests work accelerated after learning of competing results.
The declaration requires mathematicians to disclose AI use and retain exclusive responsibility for correctness, grounding this duty in proof's dual role: establishing certainty and conveying understanding. Formal verification alone cannot secure both goods.
The Jacobian lens identifies representations a model is poised to verbalize that exhibit functional signatures of global workspace theory: coherent content in intermediate layers, capacity for tens of concepts, and wider broadcasting. This enables cheap alignment auditing by revealing strategic reasoning even when hidden from output.
Papers this line draws on 8
The research behind the notes this line reads — ranked by how closely each paper relates.
- From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier
- The crisis of AI-generated mathematics
- Verification abundance, adjudication scarcity: what happens to mathematical knowledge when proof checking becomes free
- Machine-Assisted Proof
- Mathematicians are developing rules for AI use — other fields should follow
- 'hello there the jacobian conjecture is false thanx': why a tiny social media post has mathematicians rethinking AI
- Remarks on the disproof of the unit distance conjecture
- Mathematical methods and human thought in the age of AI