An Independent Lean 4 Verification of the Alpöge–Fable Counterexample

Alpöge's map disproves the Jacobian Conjecture — the assertion, open since Keller (1939), that every polynomial map ℂⁿ → ℂⁿ with nonzero constant Jacobian determinant is a polynomial automorphism — in dimension 3, and hence, by a standard stabilization argument, in every dimension n ≥ 3.

The counterexample verified in this note — a polynomial map F : ℂ³ → ℂ³ with constant Jacobian determinant −2 that sends three distinct points to a common image — is not original to this paper. It is reproduced here, in an independently-written Lean 4 formalization, purely as a verification exercise.

the assertion, open since Keller (1939) [2], that every polynomial map ℂⁿ → ℂⁿ with nonzero constant Jacobian determinant is a polynomial automorphism

Alpöge's map disproves the Jacobian Conjecture — the assertion, open since Keller (1939) [2], that every polynomial map ℂⁿ → ℂⁿ with nonzero constant Jacobian determinant is a polynomial automorphism — in dimension 3, and hence, by a standard stabilization argument, in every dimension n ≥ 3 [3].

The claim, due to Alpöge and Fable: det Jac F ≡ −2 identically

F(0, 0, −1/4) = F(1, −3/2, 13/2) = F(−1, 3/2, 13/2) = (−1/4, 0, 0).

This build is an independent confirmation, over ℚ (a subfield of ℂ, so the result transfers) [per 1], of exactly the fact reported in [1] and analyzed in [4]: constant Jacobian −2, three-point collision, hence non-injectivity, hence non-invertibility, hence a genuine Keller map [4] that is not a polynomial automorphism.

the assertion, open since Keller (1939) [2], that every polynomial map ℂⁿ → ℂⁿ with nonzero constant Jacobian determinant is a polynomial automorphism

the assertion, open since Keller (1939) [2], that every polynomial map ℂⁿ → ℂⁿ with nonzero constant Jacobian determinant is a polynomial automorphism

The claim, due to Alpöge and Fable: det Jac F ≡ −2 identically, and F(0, 0, −1/4) = F(1, −3/2, 13/2) = F(−1, 3/2, 13/2) = (−1/4, 0, 0).

lake build JacobianCounterexample completed successfully (3003/3003 jobs), with three cosmetic linter warnings ... and zero sorry.

The Dixmier Conjecture (rank n): every endomorphism of the n-th Weyl algebra is an automorphism [6].

All three conjectures — Jacobian, Dixmier, Poisson — were shown to be stably equivalent by Tsuchimoto (2005) [8] and independently by Belov-Kanel and Kontsevich (2007) [9], and the three-way (Jacobian–Dixmier–Poisson) equivalence was established by Adjamagbo and van den Essen (2007) [7].

the assertion, open since Keller (1939) [2], that every polynomial map ℂⁿ → ℂⁿ with nonzero constant Jacobian determinant is a polynomial automorphism

The claim, due to Alpöge and Fable: det Jac F ≡ −2 identically, and F(0, 0, −1/4) = F(1, −3/2, 13/2) = F(−1, 3/2, 13/2) = (−1/4, 0, 0).

Jacobian Conjecture false for all n ≥ 3

The Dixmier Conjecture (rank n): every endomorphism of the n-th Weyl algebra is an automorphism [6].

This gives DC₃ false, explicitly, in the correct (unaltered) rank, not via non-constructive stable equivalence [5].

All three conjectures — Jacobian, Dixmier, Poisson — were shown to be stably equivalent by Tsuchimoto (2005) [8] and independently by Belov-Kanel and Kontsevich (2007) [9]

The Belov-Kanel– Kontsevich direction is JC₂ₙ ⟹ DC_n (Jacobian in dimension 2n implies Dixmier in rank n)

Since Alpöge's counterexample lives in dimension 3 — odd — it does not correspond to any JC₂ₙ for integer n, so this particular theorem does not directly transfer the counterexample to any specific Dixmier rank.