How did an AI system discover a counterexample to the 87-year-old Jacobian conjecture, and what does this mean for the f

On July 19, 2026, mathematician Levent Alpöge casually announced on X that the 87‑year‑old Jacobian conjecture is false [2][28]. He had found a counterexample with the help of Anthropic’s Claude Fable 5, a large language model [1][11][26][32][51][55]. The discovery is being called the biggest conjecture that AI has significantly contributed to disproving so far [6][21][40].

How the AI system found the counterexample

The Jacobian conjecture was proposed by Ott‑Heinrich Keller in 1939 [5][50][54]. It asked whether every polynomial mapping with a constant non‑zero Jacobian determinant must be globally invertible. Alpöge, who works at Anthropic and Harvard, turned the problem over to Claude Fable 5 [1][31]. The AI searched through a vast space of possible polynomial mappings—a task that is hard for humans but natural for machines [14][29]. Alpöge built a pipeline that let the AI test many candidates, possibly using a “blind marked‑factor” search technique [24][25]. The exact prompts and the human insight that steered the search haven’t been made public, so part of the process remains a mystery [20][39][53].

The counterexample the AI found is a three‑dimensional polynomial with constant Jacobian determinant −2 [33]. It sends several distinct inputs to the same output, making it non‑invertible, which breaks the conjecture for all dimensions larger than 2 (the two‑dimensional case is still open) [3][4][27][57]. Despite its importance, the counterexample is tiny—just 216 characters—and mathematicians could verify it in a snap [18][38][19][34].

What this means for the future of mathematical research

The success has set mathematicians buzzing about where things go from here [22]. Here are the main takeaways and open questions.

• AI as a discovery tool: The episode shows that AI can find unexpected mathematical objects, and it’s clear that future mathematicians will routinely employ AI in their work [15][30][35][43].
• Rapid progress predicted: Some experts believe AI will soon reach PhD‑level math ability, potentially reducing the need for human mathematicians and causing a fundamental shift in the field [16][23][41]. Google DeepMind’s Aletheia has already autonomously produced publishable PhD‑level research [46]. Within 5–10 years, AI systems may regularly help write proofs [42].
• New ways of working together: Fields medalist Terence Tao imagines large‑scale, decentralized human‑AI collaborations where humans handle the creative leaps and AI does the technical heavy lifting [45]. In a more extreme scenario, humans could become “priests to oracles” if AI surpasses us completely [44].
• The understanding gap: A big worry is that AI gives answers without showing its reasoning—current models can’t produce reliable step‑by‑step proofs [7]. For mathematicians, an answer alone isn’t enough; they want to understand why [8]. This shift is already unsettling, especially for early‑career mathematicians [9].
• Motivation and skill erosion: If machines take over the grunt work, will human mathematicians stay motivated to do the deep thinking? [48] There’s also a fear that the next generation of mathematicians might suffer intellectual atrophy from leaning too hard on AI [49].
• Access and inequality: There is some anxiety that advanced mathematics could become an elitist activity open only to those who can afford expensive, proprietary AI models [47].
• Upside for research: The counterexample didn’t just punch a hole in one conjecture; it also falsified related conjectures (the image conjecture and vanishing conjecture) [58] and gave insight into why earlier proof attempts kept failing [52].

It’s still too early to say exactly how AI will reshape mathematics, but one thing is clear: the old boundaries have been moved.