Interactive note / 01
The base map G below is Alpöge’s counterexample to the Jacobian conjecture, announced in July 2026 and rescaled here so that det JG = 1 exactly, at every point of ℂ³ — even though G is not injective. Add polynomial shears, compose another copy of G, or rescale one output. The chain rule keeps an exact determinant ledger, the plot shows a live two-dimensional slice, and the three-point collision below survives anything you build.
(x, y) grid at fixed zt (e.g. 1/2), pick a power m, and
press either shear button — u₁ ← u₁ + t·u₂ᵐ or u₂ ← u₂ + t·u₃ᵐ.
Shears are triangular, so their ledger factor is exactly 1. Pressing Enter in the
t field applies the first shear.
k and press Rescale output 1 (or Enter): the first output is
multiplied by k / det J_F, so the ledger lands on exactly k.
k = 0 is allowed as a deliberately singular contrast.
F ← G ∘ F. The determinant is unchanged — and the
collision can only deepen. Three extra copies are the cap that keeps the live plot responsive.
det J_F in exact rational
arithmetic; the degree stat is an upper bound on total degree; the plot shows the
image of an (x, y) grid at the slider’s z, switching to an asinh-compressed view when values
spread over many orders of magnitude — drag the plot with the mouse, or use the
view α/β sliders, to orbit the image in output space (display only; rotation never
enters the pipeline or the ledger); and the collision panel re-evaluates the three
witness points exactly — their images stay identical no matter what you build.
These three points share one image under G. Every operation above is applied after G, so no composition you build here can pull them apart — and for every nonzero rescale target, the ledger keeps the determinant a nonzero constant. A Keller map that merges points is exactly what the Jacobian conjecture said could not exist. The images update live, in exact arithmetic, as you edit the pipeline.
(-1/16, 0, 0)
(-1/16, 0, 0)
(-1/16, 0, 0)
All three images agree while det J_F = 1 everywhere: a nonzero constant determinant without injectivity.
Three states rendered by this page’s own code, so you know what to expect before touching anything. Reproduce them with Reset, Load a worked example, and Rescale output 1 with k = 0.
F = G
The counterexample itself, untouched: determinant exactly 1 at every point, yet the three witness points already share one image.
Exported from the live canvas · z = 0 · u₁ 0…1.069 · u₂ -1.674…1.674 · 32 of 1044 beyond view
F = D₁[6] ∘ S₂₃[-1,3] ∘ G ∘ S₁₂[1/2,2] ∘ G
The built-in four-stage pipeline: shear, compose another G, shear, rescale. The ledger telescopes to exactly 6, the collision survives every stage, and the slice needs asinh compression because the composed degrees reach the bound.
Exported from the live canvas · z = 0 · asinh-compressed view · u₁ -2.588…1.10e+4 · u₂ -6367…1.30e+4 · 34 of 1044 beyond view
F = D₁[0] ∘ G
Rescaling the first output to zero is the one move that breaks the Keller condition: the determinant becomes 0, the first coordinate dies, every witness lands on the same trivial point, and the slice collapses onto a line.
Exported from the live canvas · z = 0 · u₁ = 0 · u₂ -1.674…1.674 · 32 of 1044 beyond view
det J_F = 1
G₁(x,y,z) = (1 + 2xy)³z + y²(1 + 2xy)(4 + 6xy)
G₂(x,y,z) = y + 6x(1 + 2xy)²z + 6xy²(4 + 6xy)
G₃(x,y,z) = −x + 3x²y + 2x³z
G is the map F announced by Levent Alpöge in July 2026 [1] — with the question posed by Akhil Mathew and the search carried out with Claude (Fable) — in the determinant-one form due to N8Programs [2]: G = diag(¼, ½, −½) ∘ F ∘ diag(1, 2, 4) moves det J from −2 to exactly 1 while changing nothing essential. Every fiber of G has three, one, or zero points, and the image misses only a smooth curve. G is étale but not proper: preimages can escape to infinity while their images converge, and properness is precisely the loophole — a proper Keller map is a polynomial automorphism. The construction falsifies the conjecture in every dimension n ≥ 3; dimension two remains open.
The displayed composition stays in factored pipeline form because expanding repeated copies of G grows very quickly. Every stage is still a polynomial map, and its determinant follows exactly from the stage factors shown above. See the verification note [3] for the fiber-by-fiber geometry.