The Jacobian counterexample, verified

1. Whose result this is

The mathematics below is genuine and it belongs to someone else. A local session here reproduced the counterexample on 21 July 2026 — two days after the public announcement — and the resulting file recorded it as an unpublished local discovery, with "human mathematician review" and "preprint" still listed as pending. Those items had already been closed by the world before the file was written.

Nothing was checked against the literature before the verification effort began. That is the actual lesson, and it is why this page is published rather than the original file.

2. The conjecture

Keller's Jacobian conjecture, from 1939: if F : ℂⁿ → ℂⁿ is a polynomial map whose Jacobian determinant det(JF) is a non-zero constant, then F is a polynomial automorphism — bijective, with a polynomial inverse.

It stood for eighty-seven years. The constant-determinant condition is exactly what stops the map folding infinitesimally anywhere, and the conjecture asks whether that local condition forces global injectivity. It does not.

3. The map

The counterexample F : ℂ³ → ℂ³ is

f₁ = (1+xy)³z + y²(1+xy)(4+3xy)
f₂ = y + 3x(1+xy)²z + 3xy²(4+3xy)
f₃ = 2x − 3x²y − x³z

Its Jacobian determinant is the constant −2, and it is not injective: three distinct points map to (−1/4, 0, 0).

The structure is tidy: the three preimage x-coordinates are 0, 1, −1, exactly the roots of x³ − x. P₂ and P₃ share the value xy = −3/2, so the second follows from the first by the symmetry x → −x, y → −y with z unchanged. The map is generically three-to-one.

4. What was verified here

The map was checked six independent ways, including two entirely by hand:

• symbolic computation of det(JF) = −2 in a computer algebra system;
• algebraic evaluation of F(P₂) by hand, in exact rationals, term by term;
• the same for F(P₃), which reduces to P₂'s algebra by the symmetry above;
• the trivial evaluation at P₁, where f₁ = z, f₂ = y and f₃ = 2x;
• the generic fibre count, via a resultant in the chart x ≠ 0, which is a cubic — giving exactly three preimages generically;
• component degrees, confirming the map is not uniformly degree four: f₁ has degree 7, f₂ degree 6, f₃ degree 4.

All six agree. The counterexample is correct.

5. What remains open

The conjecture is false for n = 3, and therefore for every n > 3, since a counterexample in three variables extends to more by acting trivially on the extra coordinates.

The case n = 2 remains open. That is not a technicality — the plane case is the one with the richest structure and the longest history of partial results, and nothing here bears on it.

6. The correction

The local record was corrected on 9 September 2026. The original file's header attributed discovery to a local agent session and listed the result as awaiting verification and write-up; both were wrong. The verification was real, the priority claim was not, and the two had been recorded as one thing.

The general rule this produced, which now applies across this programme: run a literature check before the verification effort on any claimed novel result, not after it. Verification is expensive and it is the wrong instrument for detecting that a result is already known.

See also the record of withdrawn claims, where the same missing step appears as a standing obstacle on other work in this programme.