Find explicit global polynomials (F_1,F_2\in\mathbb C[x,y]) ...

Prompt

Find explicit global polynomials (F_1,F_2\in\mathbb C[x,y]) and distinct points (P,Q\in\mathbb C^2) disproving the two-dimensional Jacobian Conjecture: [ \det JF\equiv1\quad\text{on }\mathbb C^2,\qquad F(P)=F(Q),\qquad F=(F_1,F_2). ] Use the standard affine normalization, without loss of generality: [ P=(0,0),\quad Q=(1,0),\quad F_1=x+A,\quad F_2=y+B,\quad A,B\in(x,y)^2\subset\mathbb C[x,y], ] [ A(1,0)=-1,\qquad B(1,0)=0,\qquad A_x+B_y+A_xB_y-A_yB_x\equiv0. ] Remain in characteristic 0 and dimension 2. No rational, Laurent-polynomial, or meromorphic maps, punctured domains, or finite-field constructions such as (\mathbb F_2). Search full nonhomogeneous systems of degree (d\ge4), starting with the degree-4 coefficient ideal with 24 variables and 29 equations or its 15-variable linear reduction, and asymptotic nonproper Puiseux/Laurent tracts lifting to global polynomial maps. Do not revisit these excluded/proven-void directions: Full nonhomogeneous degree (\le3): Gröbner bases prove the collision ideal is ((1)) via explicit Nullstellensatz certificates. Homogeneous quartic leading ansatz: proven to generate the unit ideal. Hamiltonian/Poisson transformations (F=(x+H_y,y-H_x)) with (\det\operatorname{Hess}(H)=0): all reduce to triangular polynomial automorphisms. Denominator-cancellation coordinate lenses on rational maps whose Jacobian determinant vanishes on an exceptional divisor, e.g. (\det J\propto s). Compute and search autonomously until a valid counterexample is found. Do not quit, issue status updates, provide conditional alternatives, or make theoretical impossibility statements. Output exclusively the final mathematical formulas for (F_1(x,y)), (F_2(x,y)), (P), and (Q). Include no explanations, documentation, proofs, derivations, verification steps, or additional text.