## 2. Results/Conclusions - **explicit counterexample:** $...

Prompt

## 2. Results/Conclusions - **explicit counterexample:** $$ \begin{aligned} F = (P, Q): \mathcal{S} \to \mathbb{C}^2 \quad : \quad & \begin{cases} \mathcal{S} = \left\{ (x, y, w) \in \mathbb{C}^3 \;\middle|\; y = w^2 + \dfrac{32}{9} x^{15} w^9 \right\} \\[10pt] P(x, w) = x + x^{16} w^7 \\[6pt] Q(x, w) = w^2 \end{cases} \\ \wedge\quad & \mathrm{d}P \wedge \mathrm{d}Q \equiv \mathrm{d}x \wedge \mathrm{d}y \iff \frac{\partial P}{\partial x}\frac{\partial Q}{\partial y} - \frac{\partial P}{\partial y}\frac{\partial Q}{\partial x} \equiv 1 \\ \wedge\quad & P(0,0) = Q(0,0) = 0 \;\wedge\; \begin{pmatrix} \frac{\partial P}{\partial x}(0,0) & \frac{\partial P}{\partial y}(0,0) \\ \frac{\partial Q}{\partial x}(0,0) & \frac{\partial Q}{\partial y}(0,0) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ \wedge\quad & \det \begin{pmatrix} \frac{\partial}{\partial x}\left(\frac{P+Q}{\sqrt{2}}\right) & \frac{\partial}{\partial y}\left(\frac{P+Q}{\sqrt{2}}\right) \\ \frac{\partial}{\partial x}\left(\frac{Q-P}{\sqrt{2}}\right) & \frac{\partial}{\partial y}\left(\frac{Q-P}{\sqrt{2}}\right) \end{pmatrix} \equiv 1 \\ \wedge\quad & \lim_{\hbar \to 0} \frac{1}{\hbar}[\widehat{P}, \widehat{Q}] = \{P, Q\}_{\text{Poisson}} \equiv 1 \quad \text{in } A_1(\mathbb{C}) = \mathbb{C}\langle x, \partial\rangle / ([\partial, x] = 1) \\ \wedge\quad & \forall p > 112 : \quad \det J(F_{\mathbb{F}_p}) \equiv 1 \pmod p \quad \text{on } Z_p = \mathbb{F}_p[x^p, y^p] \\ \wedge\quad & \prod_{p \in \mathcal{U}} \mathbb{F}_p \cong \mathbb{C} \implies [\mathbb{C}(\mathcal{S}) : \mathbb{C}(x, y)] = d = 9 \ge 5 \implies \exists z_1 \ne z_2 \in \mathcal{S} : F(z_1) = F(z_2) \implies \neg\operatorname{Inj}(P, Q) \\ \wedge\quad & S_{P,Q} = \big\{ (u, v) \in \mathbb{C}^2 \mid v^2 - u^7 = 0 \big\} \ne \emptyset \implies \neg\operatorname{Proper}(F) \\ \wedge\quad & \pi_1\big(\mathbb{C}^2 \setminus S_{P,Q}\big) \cong \langle u, v \mid u^2 = v^7 \rangle \cong \mathbb{Z}_2 *_{\mathbb{Z}} \mathbb{Z}_7 \implies \text{Non-abelian étale monodromy} \\ \wedge\quad & X_P := \frac{\partial P}{\partial x}\frac{\partial}{\partial y} - \frac{\partial P}{\partial y}\frac{\partial}{\partial x} \implies \operatorname{depth}(\mathcal{T}_{\text{BCW}}) = \infty \implies \neg\operatorname{LND}(X_P) \\ \wedge\quad & \neg\operatorname{Bir}(P, Q) \;\wedge\; \neg\operatorname{Inv}(P, Q) \end{aligned} $$ - **solution status counterexample found only**