Fable 5 Constructs Counterexample Disproving Jacobian Conjecture, Version in Three Dimensions and Above Overturned

In July 2026, Anthropic researcher Levent Alpöge posted on X announcing a counterexample to the Jacobian conjecture found by Claude Fable 5. The conjecture, proposed in 1939, is a core problem in algebraic geometry that had stumped top mathematicians including Yitang Zhang. Fable 5 constructed a polynomial map from C³ to C³ whose Jacobian determinant is constantly -2 (a nonzero constant), satisfying the conjecture's premise, but mapping three different inputs to the same output, proving the map is not invertible, thus disproving the version in three dimensions and above. The two-dimensional version remains open.

Counterexample Details

• Map: F(x,y,z) = (x - 2y - 2z - 2x²y - 2x²z - 2xy² - 2xz² - 4xyz, y, z) etc. (see post for full details).
• Jacobian determinant equals -2, satisfying the conjecture's condition.
• Three distinct points: (0,0,-1/4), (1,-3/2,13/2), (-1,3/2,13/2) all map to (-1/4,0,0).
• The counterexample is simple enough to verify by hand, and multiple mathematicians have confirmed it using Wolfram Alpha.

Historical Background and Connection to Yitang Zhang

• The Jacobian conjecture was proposed by Ott-Heinrich Keller in 1939 and is listed on Smale's list of 21st-century mathematical problems.
• During his PhD at Purdue University, Yitang Zhang worked on this conjecture, but his doctoral thesis failed due to an erroneous lemma provided by his advisor Tzuong-Tsieng Moh, leading to years of academic wandering.
• In 2013, Zhang gained fame for his breakthrough on the twin prime conjecture, but the Jacobian conjecture remained unsolved.

Reactions and Impact

• Stanford mathematician Jared Duker Lichtman called it "one of the most inspiring stories in modern mathematics."
• UC Berkeley associate professor Jason Lee exclaimed, "Mathematics is over."
• OpenAI's Aaron Lou independently derived a similar counterexample using an internal model and provided a complete derivation.
• The Wikipedia entry has been updated to acknowledge the counterexample.
• The two-dimensional version (the most studied) remains unsolved and is mathematically more significant.

Significance and Discussion

• This is the third mathematical conjecture overturned by AI in recent times (previously the Erdős unit distance conjecture and the Grothendieck group scheme problem).
• The discovery has sparked debate about AI's mathematical creativity: is it retrieval or creation?
• Fields Medalist Timothy Gowers remarked, "The 2030 Fields Medal in mathematics may be the last awarded to a human."