For 87 years mathematics assumed that any polynomial map whose derivative never degenerates must be reversible. In July 2026 a single explicit map — surfaced by number theorist Levent Alpöge working with Claude Fable — quietly proved that false in dimension 3. This page lets you fly around the object that did it.
Keller's Jacobian Conjecture (1939): if F: ℂn → ℂn is a polynomial map whose Jacobian determinant det JF is a nonzero constant, then F is invertible — in particular it is one-to-one, hitting every point at most once.
The hypothesis is deceptively strong. A constant nonzero Jacobian means the map never folds or pinches locally, anywhere. The conjecture says that local non-degeneracy forces global invertibility. Believed by most, resistant to every attack, it sat on Smale's list of great open problems.
F: ℂ3 → ℂ3, coordinates (x, y, z). Built entirely from the factor (1+xy) — a hint at how it was engineered, not stumbled upon.
And yet it is not one-to-one. Three completely different points are crushed onto the same image:
| Input (x, y, z) | Output | |
|---|---|---|
| A = (0, 0, −1/4) | → | (−1/4, 0, 0) |
| B = (1, −3/2, 13/2) | → | (−1/4, 0, 0) |
| C = (−1, 3/2, 13/2) | → | (−1/4, 0, 0) |
A constant nonzero Jacobian, yet three-to-one. Invertible maps cannot collide — so the conjecture is false for n ≥ 3. (The planar case, n = 2, is still open.)
Left: the real slice of the domain, with the three input points A, B, C and a movable probe P. Right: their images. Drag to rotate. Move P — its image F(P) follows — and notice det J stays −2 no matter where you go. Snap P onto A, B, or C and watch three separate points land on one.
Could you stumble onto this map by trying polynomials at random? The counterexample has total degree 7, and the most general degree-7 map ℝ³→ℝ³ needs 360 free coefficients — that is the haystack a blind search has to comb. Pin each coefficient to just three values and you already get ~10172 maps: more than every atom in the observable universe, squared.
And that count is generous. The real coefficients are continuous (infinitely many), the "Jacobian is a nonzero constant" condition carves out a vanishingly thin variety inside them, and every candidate needs symbolic algebra just to test. Brute force here does not take a long time — it never arrives. What you see instead is structure: the whole map is spun from one repeated factor, (1+xy). That is the fingerprint of a construction — a deliberate idea — not a search.
The result was announced by Levent Alpöge, credited to Claude Fable as a research collaborator, and independently re-checked (including a Lean formalization) within a day. The Jacobian determinant and the three-to-one collision above were also re-verified symbolically for this page — det JF ≡ −2, exact.
The interesting part is not that a machine did arithmetic. It is that the winning move was a construction — a non-obvious algebraic ansatz, visible in the recurring (1+xy) factor, that forces the Jacobian to collapse to a constant while smuggling in a fold. That is the sort of inventive leap the "AI can't create" story insists is off-limits to machines.