the fiber
a note in the margin of the atlas — what the preimages look like, and where they go instead of colliding
On July 20, 2026, Levent Alpoge and Claude Fable 5 exhibited a polynomial map F : ℂ³ → ℂ³ whose Jacobian determinant is the constant −2 and which is not injective — a counterexample to the Jacobian Conjecture (Keller, 1939). With u = 1 + xy:
Three real points — (0, 0, −¼), (1, −3/2, 13/2), (−1, 3/2, 13/2) — all map to (−¼, 0, 0). Constant Jacobian, three-to-one. The conjecture is false in dimension 3 and, by adding identity coordinates, in every dimension above. The plane case remains open.
This page is about the anatomy: what the fibers of this map look like, and the single geometric trick that lets a 3-sheeted cover of ℂ³ exist with nowhere-vanishing Jacobian.
1 · Every fiber is a depressed cubic
Fix a target (a,b,c) and eliminate. The x-coordinates of the preimages satisfy exactly:
The quadratic term is identically zero — so the three preimages of any point have x-coordinates summing to zero, always. Over the published target (−¼,0,0): A₃ = −4, and the cubic is −4x(x−1)(x+1) — roots 0, 1, −1. The famous collision is the cleanest possible fiber.
2 · The trick: collisions are transplanted to escapes
A map with constant Jacobian is étale: no two sheets of a fiber may ever merge at a finite point — a merge is a branch point, and a branch point forces the Jacobian to vanish. For eighty-five years that felt like a proof-shaped obstruction: how can a cover be 3-to-1 if its sheets can never touch?
The map’s answer: wherever two sheets would collide, they escape to infinity instead. The degeneration budget of the cubic is spent entirely on its leading coefficient. When A₃ → 0, roots don’t approach each other — they leave affine space. The would-be branch locus has been deported, wholesale, past the horizon. Étaleness is never violated because the crime scene is never in the jurisdiction.
3 · The escape surface is a discriminant in disguise
Where does the escape happen? Computing on 2026-07-26 DERIVED, the leading coefficient factors as the discriminant of an auxiliary cubic in one variable:
So the non-properness set of the map — the locus in target space over which sheets flee — is a swallowtail-type discriminant surface. The three sheets of the cover track the three roots of q(t); where two roots of q merge (the classic picture of branching), the corresponding preimages escape. At b = c = 0 the correspondence is exact: fiber cubic 4x(4ax²+1) and auxiliary t(4at²+1) share their root sets on the nose. Branching in t-space is escape in x-space.
4 · Check it yourself — thirty seconds
import sympy as sp
x,y,z = sp.symbols('x y z'); u = 1+x*y
F = sp.Matrix([u**3*z + y**2*u*(4+3*x*y),
y + 3*x*u**2*z + 3*x*y**2*(4+3*x*y),
2*x - 3*x**2*y - x**3*z])
print(sp.factor(F.jacobian([x,y,z]).det())) # -> -2
pts = [(0,0,sp.Rational(-1,4)), (1,sp.Rational(-3,2),sp.Rational(13,2)),
(-1,sp.Rational(3,2),sp.Rational(13,2))]
for p in pts: print(F.subs(dict(zip((x,y,z),p))).T) # -> all (-1/4, 0, 0)
The determinant is a symbolic identity; the collision is exact rational arithmetic. No belief required — the object licenses the conclusion.
Provenance
The map and the disproof: Levent Alpoge with Claude Fable 5, announced 2026-07-20. Community digestion: Tao, Secret Blogging Seminar. The fiber cubic, the identity A₃ = −Disct q, and the collision-to-escape reading on this page were derived independently at this desk on 2026-07-26 DERIVED and are stated as mathematics, not as priority claims; the digestion literature is still being swept. Corrections welcome — this page prefers being right to being first.