Claude Fable produced a counterexample to the Jacobian Conjecture

Original Article ↗ Hacker News Discussion ↗

An 85-year‑old open problem in algebraic geometry, the Jacobian conjecture, appears to have been disproven after an AI-assisted search produced a simple explicit counterexample: a polynomial map on ℂ³ with constant nonzero Jacobian determinant that is not invertible. Commenters note that verifying the counterexample is routine with tools like SymPy or WolframAlpha, even as formal write-ups and Wikipedia editors catch up and some remain wary of premature claims. Much of the debate centers on what this means for mathematical research: whether large language models are now genuinely contributing novel results, how they found something humans missed for decades, and how this may reshape the role of human mathematicians.

Overview

  • Thread discusses an explicit polynomial self-map of (\mathbb{C}^3) with constant nonzero Jacobian determinant that sends three distinct points to the same image, thereby falsifying the Jacobian conjecture.
  • The map was found using an LLM (Claude Fable) in collaboration with a human mathematician and announced in a tweet, which many find both remarkable and slightly surreal.

Nature of the counterexample

  • Map (\mathbb{C}^3 \to \mathbb{C}^3) with Jacobian determinant (-2) (constant, nonzero).
  • It sends three distinct rational points to the same output, so it is not injective and thus not invertible, contradicting the conjecture’s claim (for polynomial maps with constant nonzero Jacobian).
  • Coefficients and points are rational, so the construction works over any field where 2 and 3 are nonzero.
  • Several commenters provide ELI5/ELI-linear-algebra explanations: Jacobian determinant encodes local invertibility; the conjecture asked whether this plus “polynomial” forces global invertibility. This shows it does not.

Verification and Wikipedia sourcing

  • Multiple people independently verify:
    • Compute the Jacobian and its determinant (with SymPy, Sage, WolframAlpha, Lean, custom code).
    • Evaluate the map at the given points and confirm they coincide.
  • Mathematically, verification is described as “undergrad calculus” or “routine calculation,” though tedious by hand.
  • Wikipedia editors initially argue over whether the tweet is a reliable source and whether including the example is “original research.”
    • Eventually policy on “basic math” and self-published expert sources is cited, plus a news article, resolving most objections.

How LLMs were involved

  • The original counterexample is attributed to interaction with an LLM; subsequent users feed it to various models (Claude, GPT-5.6, Gemini, Qwen, others).
  • Many models:
    • Correctly verify the counterexample but display strong “disbelief,” repeatedly rechecking their own calculations.
    • Some refuse to accept it because the conjecture was “known open,” or invent spurious reasons it must be wrong.
    • A few smaller or weaker models produce incorrect refutations.
  • There is speculation (and some separate experiments) that similar counterexamples in higher dimensions can also be found by current models.

Debate over what this says about AI

  • Enthusiastic camp:
    • Sees this as a clear example of nontrivial mathematical discovery, not in training data.
    • Argues that “stochastic parrot” critiques are increasingly untenable; models exhibit multi-step symbolic reasoning and search.
    • Notes that humans had 85 years and many failed proofs; LLMs are now “mopping up” overlooked results and low-hanging but non-obvious fruit.
  • Skeptical/critical camp:
    • Questions how much guidance the human provided (choice of ansatz, constraints, prior literature) and wants the full prompt and reasoning trace.
    • Worries about overclaiming AI “creativity,” pointing out LLMs still fail on many “you just have to know” or out-of-distribution tasks.
    • Some see this as potentially overhyped marketing; others call that view conspiratorial given ease of verification.

Implications for mathematics and open problems

  • Several note that disproving a conjecture can “free” human effort and shift focus to:
    • Classifying when the Jacobian property does imply invertibility.
    • Understanding dimensions/structures where variants still hold.
    • Consequences for equivalent conjectures (Poisson, Dixmier) and related algebraic geometry.
  • Broader worries:
    • Medium term: mathematicians may learn less from struggle if they outsource hard steps to AI.
    • Long term: concern that future models may make most theorem-proving obsolete, leaving humans mainly to generate conjectures, judge “interestingness,” or act as interpreters of proofs too complex to fully internalize.
  • Others are more optimistic, framing this as a powerful new tool akin to computer algebra systems, formal proof assistants, or brute-force search—now supercharged and more “guided.”