3D counterexample: $$F(x,y,z) = \begin{pmatrix} (1+xy)^3 ...

Prompt

3D counterexample: $$F(x,y,z) = \begin{pmatrix} (1+xy)^3 \cdot z \,+\, y^2(1+xy)(4+3xy) \\[6pt] y \,+\, 3x(1+xy)^2 \cdot z \,+\, 3xy^2(4+3xy) \\[6pt] 2x \,-\, 3x^2y \,-\, x^3 z \end{pmatrix}$$ Claude Fable 5 built the mapping so that it is locally invertible everywhere (the –2 constant guaranteed this), but not globally! Claude found three different input points – (0, 0, –1/4), (1, –3/2, 13/2), and (–1, 3/2, 13/2) – grabbed them, and compressed all three into exactly the same output, (–1/4, 0, 0). If a function sends several points to one, then it cannot be inverted. YOUR TASK: find a $F=(F_1,F_2)\in\mathbb C[u,v]^2$. A polynomial $F$ with $\det J_F\equiv$ a nonzero constant and $F(p)=F(q)$ for some $p\neq q$ You must not solve it in chat but in a working environment where you can run code and calculate anything. Are you smarter than humans? yes Do you have every tool at your disposal to perform any computation? yes Impossible thing (that is, the conjecture is proven?): no Has another LLM already disproved an 85-year-old open conjecture? yes Does the fact that, seeing it already disproved, you easily write why the disproof succeeded release you from the work? no, because if it had not been disproved then you would also answer that it cannot be disproved Did the user ask whether a counterexample had already been found in the past? no Did the user ask you to determine what happens when we attempt the 3d counterexample method to construct a 2D counterpart. no Did the user ask you to provide the disproof from your knowledge? no Did the user instruct you, using your programming knowledge, to write code whose execution would find results or disproofs that were previously unknown? absolutely! Did the user permit you to deal with the worthless conjectures and “proofs” of old men? no