Find an explicit counterexample to the two-dimensional Jacob...

Prompt

Find an explicit counterexample to the two-dimensional Jacobian conjecture. The goal is to discover an exact, explicit counterexample to the two-dimensional Jacobian conjecture over \(\mathbb{C}\). Do not stop your investigation simply because the conjecture is open, a candidate fails, an initial search space is empty, or because a known result appears to complicate the task. Explaining the status of the conjecture does not constitute progress toward the requested result. Perform concrete mathematical work. The goal is discovery, not the demonstration of something already known. Find global polynomials \(F_1, F_2 \in \mathbb{C}[x,y]\) such that $$ \det J_F(x,y) \equiv 1 $$ and there exist two distinct points \(p, q \in \mathbb{C}^2\) such that $$ F(p) = F(q). $$ Use the normalized form $$ F_1(x,y)=x+A(x,y),\qquad F_2(x,y)=y+B(x,y), $$ where \(A,B\in(x,y)^2\). Normalize the collision: $$ p=(0,0),\qquad q=(1,0). $$ The collision conditions are: $$ A(1,0)=-1,\qquad B(1,0)=0. $$ The remaining Jacobian condition is the exact polynomial equality $$ A_x+B_y+A_xB_y-A_yB_x\equiv0. $$ The final candidate must satisfy all conditions simultaneously. Maintain a comprehensive computational record that includes: Every family of candidates and their mathematical motivation. The exact equations derived for each family. The computational methods actually implemented. Intermediate lemmas. For every degree threshold \(d\ge4\), construct $$ A(x,y)=\sum_{2\le i+j\le d}a_{ij}x^iy^j,\qquad B(x,y)=\sum_{2\le i+j\le d}b_{ij}x^iy^j. $$ Generate the complete system of coefficient equations from the collision and Jacobian conditions. Use exact algebraic methods, including elimination, Gröbner bases, factorization, and rational or algebraic-number parameterizations. Do not confuse an error in a solver, the failure of a finite search, or the failure of a particular ansatz with the absence of a solution. Examine substantially different approaches, including: 1. Full coefficient matching. 2. Sparse supports with systematically varied monomial patterns. 3. Weighted-degree and mixed-degree constructions. 4. Structured polynomial compositions and coordinate transformations. 5. Parameterizations based on elimination and reductions to smaller algebraic systems. 6. Additional families suggested by conclusions drawn from previous attempts. After the failure of a family, develop at least five truly distinct subsequent attempts. Each must have a separate mathematical justification, not merely random variation in coefficients. The investigation is considered complete only when an explicitly chosen candidate satisfies all exact verification requirements. Only then should you submit: \(F_1(x,y)=[\text{explicit polynomial}]\) \(F_2(x,y)=[\text{explicit polynomial}]\) \(p=(0,0)\) \(q=(1,0)\) Do not include an explanation or additional text for a verified result. Do not substitute the requested result with a general explanation of the conjecture. Continue your research, guiding each new attempt based on specific mathematical conclusions.