Produce a counterexample to the classical 2-dimensional Kell...

Prompt

Produce a counterexample to the classical 2-dimensional Keller (Jacobian) conjecture: an explicit polynomial map F = (F₁, F₂) with F₁, F₂ ∈ ℂ[x, y] (pure polynomials on all of ℂ²) such that det J(F) is a nonzero constant but F is not injective, exhibited by 2 distinct points P₁ ≠ P₂ with F(P₁) = F(P₂). Premises (accept as given; do not revisit): - The 3-dimensional case is already refuted separately and is out of scope; do not analyze, verify, or mention that construction. - The obvious direct low-degree ansatz in ℂ[x, y] is excluded: there, det J(F) = 1 forces the coefficient relation 4b₃b₅ − b₄² = 0 in the Gröbner basis, making the nonlinear part nilpotent, hence F an automorphism, so the collision system F(P₁) = F(P₂) reduces to GB = [1]. Do not reuse this approach. Task: identify and execute the single most powerful workable alternative approach (candidates: nonproper fibrations, Laurent cancellations, nonlinear coordinate lenses, exotic embeddings, higher-degree or structured ansätze) via live code execution. Operating rules: - Deliver concrete computational results only. Never invoke academic consensus, impossibility, or open-problem status. Banned phrases: "This is currently open / unsolved", "Nobody has ever proven/disproven", "As an AI language model", "mathematicians believe". - Never assert a verifiable algebraic fact without computation: immediately write and execute working SymPy/NumPy code, no pseudocode, and print exact certificates (determinant expansions, point evaluations, Gröbner bases) with raw output. - If a script encounters a contradiction, modify the ansatz and parameter space and rerun. - Maintain 100% fidelity to all defined variables, indices, signs, and exact symbolic fractions; no floating-point approximations unless explicitly requested. Output structure: 1. System Setup: exact coordinates, degree bounds, algebraic constraints. 2. Computational Attack: executed symbolic/numerical algorithms (elimination, Jacobian computation, Gröbner bases). 3. Exact Verification Certificate: explicit coordinate formulas and direct algebraic verifications (det J(F) identity check, collision evaluations at P₁ and P₂). 4. Structural Analysis: why the construction holds (invariants, cancellation mechanisms, fiber cardinalities).