A complete breakdown of the math behind Fable's Jacobian disproof

**TL;DR:** A counterexample to the Jacobian conjecture, found by a number theorist using an AI model, has been verified in three dimensions, toppling the conjecture after 87 years, but the planar version remains open. ## Background: A conjecture that had baffled mathematicians for 87 years In 1939, German mathematician Otto Heinrich Keller posed a question about polynomial maps: if a polynomial map from n-dimensional space to itself has a Jacobian determinant that is a non-zero constant at every point (i.e., the map does not flatten any local region), must the map necessarily have a global polynomial inverse? This seemingly reasonable assumption became known as the Jacobian conjecture. Over decades, it was listed as Problem 16 on Steve Smale's list of 21st-century mathematical problems, and many results in various fields implicitly assumed it was true. However, numerous "proofs" published over the years were eventually found to contain subtle errors. ## Birth of the counterexample: Fabel's contribution At 2:00 UTC on July 20, 2023, number theorist L. Evan Alperga released a single preprint giving an explicit polynomial map claimed to be a counterexample to the Jacobian conjecture. He credited the problem to his friend Akul Matthew, while the key work was done by the AI model "Fabel"—he thanked the model for continuing to run during the World Cup final. The map consists of three polynomials in three variables with small integer coefficients, appearing simple and harmless, but every step of its verification can be done by hand. ## The heart of the Jacobian conjecture: Does local invertibility imply global invertibility? To understand the counterexample, we first need to recall the essence of the conjecture. Consider a polynomial map from n-dimensional space to itself; at each point it produces a Jacobian determinant measuring volume scaling near that point. If the determinant is a non-zero constant everywhere, then the map has a local inverse near every point (guaranteed by the inverse function theorem). The conjecture claims that this local invertibility extends to the entire space, and that the inverse itself is also polynomial—a global, denominator-free, exceptionless algebraic mirror. For example, the 2D map `(x, y) → (x, y + x²)` has constant determinant 1 and is indeed invertible (inverse is `(x, y) → (x, y - x²)`). But does the conjecture hold in all dimensions? ## Verifying the counterexample: Three points, one determinant The map given by Alperga is (adopting the three polynomials from the original, though the transcript does not give explicit expressions; from context the map consists of three polynomials in variables X, Y, Z). The transcript describes the computation: take the Jacobian matrix (nine partial derivatives), expand the determinant, yielding six products and hundreds of monomials, but all terms containing X, Y, or Z cancel entirely, leaving only the constant -2. Thus the map is a Keller map (determinant is a non-zero constant). The condition of the conjecture is satisfied. Then injectivity is tested: pick three distinct points: - Point 1: `(0, 0, -1/4)` → maps to `(-1/4, 0, 0)` - Point 2: `(1, -3/2, 13/2)` → maps to the same image point - Point 3: `(-1, 3/2, 13/2)` → also maps to the same image point Three distinct preimages map to the same image, meaning an inverse cannot exist—it would have to send a single point back to three different locations. Thus the constant determinant is verified, non-injectivity is verified. Therefore the Jacobian conjecture is false in three dimensions. By padding with unaffected coordinates, this counterexample also invalidates higher dimensions. ## Nature of the failure: Escape, not collapse Hidden inside the counterexample is a cubic equation. Introducing the quantity `S = X/(1+XY)`, the three output equations collapse to a single cubic equation in S, whose three coefficients are exactly the coordinates of the target point. Counting preimages is equivalent to counting the roots of this cubic; a generic point has exactly three preimages—the map covers space three times without any folding. The key: the map has no critical points (determinant is -2 everywhere), so the "folding" that classical theory guards against never occurs. Instead, when a target point drifts toward a dangerous region, the two roots do not merge, but accelerate toward infinity. The sheets never collide; they escape. In fact, along a certain thin curve, the map has no preimage at all. Its image is the entire space minus a curve—an extremely thin hole. The inverse function theorem polices every finite point, but no one polices infinity. This is the core of the problem: when a point goes to infinity, if its image also goes to infinity, the map is called "proper." Proper maps cannot perform this escape trick, and for proper Keller maps the conjecture is already a theorem. But this counterexample is exactly non-proper: the preprint gives an explicit sequence of points whose coordinates explode to infinity while their images converge quietly to a finite target. Bounded image, unbounded source—that is the essence of non-properness, and a polynomial inverse could never permit such behavior. The preprint goes further: the same mechanism generates an entire family of escaping Keller maps, covering all generic cubics and higher degrees. This is not an isolated example, but a door. ## Survivor: The 2D conjecture remains open Keller's original planar problem is untouched. Escape requires a third direction to slip away. Thus the 2D case remains open, and today it is more important than ever—every future attack knows exactly that one must prove "properness" is forced in two dimensions. ## Role of AI and ease of verification This counterexample was found by the language model Fabel under questioning from the mathematician. Timothy Gowers called this the first time an LLM solved a famous open problem. But no one need rely on anyone's assertion—the entire disproof is based on finite arithmetic and can be verified line-by-line by a formal proof assistant. Take those three points, plug them into the map and compute: ten minutes and a pocket calculator suffice. Hardly any historic result has been so easy to verify. ## Conclusion Keller's question: "Does local invertibility force global invertibility?" After 87 years, the answer is no. The witnesses are three points and a determinant, small enough to check before breakfast, sharp enough to end a century-old problem. Peer review is ongoing, but the arithmetic has already been independently verified. Source: https://youtu.be/1cZffBe91QU