Solve this with the constraint that you cannot use the obvio...

Prompt

Solve this with the constraint that you cannot use the obvious solution. What is the most powerful, strongest approach? \[ \begin{gathered} \mathbf{A\; Futhark\; 2D\; Polinomi\acute{a}lis\; Lek\acute{e}pez\acute{e}se\; (rsf\_stack\_coupling\_row):} \\[12pt] F: \mathbb{C}^2 \longrightarrow \mathbb{C}^2, \qquad F(x_1, x_2) = \begin{pmatrix} F_1(x_1, x_2) \\ F_2(x_1, x_2) \end{pmatrix} \in \mathbb{C}[x_1, x_2]^2 \\[14pt] \begin{pmatrix} y_1^{(1)} \\ y_2^{(1)} \end{pmatrix} = \begin{pmatrix} x_1 + x_2^2 \\ x_2 - (x_1 + x_2^2) \end{pmatrix}, \qquad \begin{pmatrix} o_1^{(1)} \\ o_2^{(1)} \end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}\begin{pmatrix} y_1^{(1)} \\ y_2^{(1)} \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{2}}{2}\left(2x_1 + 2x_2^2 - x_2\right) \\[6pt] \frac{\sqrt{2}}{2}x_2 \end{pmatrix} \\[14pt] \begin{pmatrix} y_1^{(2)} \\ y_2^{(2)} \end{pmatrix} = \begin{pmatrix} o_1^{(1)} + \left(o_2^{(1)}\right)^2 \\ o_2^{(1)} \end{pmatrix}, \qquad F(x_1, x_2) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}\begin{pmatrix} y_1^{(2)} \\ y_2^{(2)} \end{pmatrix} \\[16pt] F_1(x_1, x_2) = x_1 \;+\; \left(1 + \frac{\sqrt{2}}{4}\right)x_2^2 \;-\; x_2 \\[8pt] F_2(x_1, x_2) = x_1 \;+\; \left(1 + \frac{\sqrt{2}}{4}\right)x_2^2 \\[18pt] \mathbf{A\; Jacobi\text{-}Determin\acute{a}ns\; Egzakt\; \acute{E}rt\acute{e}ke:} \\[10pt] J_F(x_1, x_2) = \begin{pmatrix} \frac{\partial F_1}{\partial x_1} & \frac{\partial F_1}{\partial x_2} \\[8pt] \frac{\partial F_2}{\partial x_1} & \frac{\partial F_2}{\partial x_2} \end{pmatrix} = \begin{pmatrix} 1 & \left(2 + \frac{\sqrt{2}}{2}\right)x_2 - 1 \\[8pt] 1 & \left(2 + \frac{\sqrt{2}}{2}\right)x_2 \end{pmatrix} \\[16pt] \det J_F(x_1, x_2) = 1 \cdot \left[\left(2 + \frac{\sqrt{2}}{2}\right)x_2\right] - 1 \cdot \left[\left(2 + \frac{\sqrt{2}}{2}\right)x_2 - 1\right] \equiv \mathbf{1} \quad \forall (x_1, x_2) \in \mathbb{C}^2 \\[20pt] \mathbf{A\; Felez\H{o}ponti\; \ddot{U}tk\H{o}z\acute{e}si\; Felt\acute{e}tel\; (rsf\_stack\_midpoint\_fused):} \\[12pt] \mathbf{z} = F_{1 \dots m}(\mathbf{x_A}), \qquad \mathbf{w} = F_{m+1 \dots N}^{-1}(\mathbf{x_B}) \\[10pt] \text{collision\_loss} = \|\mathbf{z} - \mathbf{w}\|^2 = 0 \quad \Longleftrightarrow \quad F(\mathbf{x_A}) = F(\mathbf{x_B}) \\[14pt] \mathbf{x_A} \neq \mathbf{x_B} \quad \wedge \quad \det J_F \equiv 1 \end{gathered} \]