@mathematical_research_goal (1).md @mathematical_audit_prompt_v3 (1).md I am attaching `mathematical_research_goal.md`. Read that file completely and treat it as the governing research procedure. The authoritative mathematical problem is the complete LaTeX source between the two TARGET_PROBLEM markers below. PROJECT_WINDOW_BUDGET: 6 hours \ \paragraph{Problem (global existence for general vortex-sheet data).}\ Let $\mathcal M(\mathbb R^2)$ denote the space of finite signed\ Radon measures, equipped with the total variation norm, and set\ \\[\ K(x)=\frac{(-x_2,x_1)}{2\pi|x|^2}.\ \\]\ For every\ \\[\ \omega_0\in\mathcal M(\mathbb R^2)\cap H^{-1}(\mathbb R^2),\ \qquad u_0=K\*\omega_0,\ \\]\ with no sign restriction on $\omega_0$, does there exist a global\ weak solution of the incompressible Euler equations\ \\[\ u\in C_w([0,\infty);\ L^2\_{\mathrm{loc}}(\mathbb R^2;\mathbb R^2))\ \\]\ with $u(0)=u_0$ and $\operatorname{div}u=0$, whose vorticity\ $\omega(t):=\partial_1u_2(t)-\partial_2u_1(t)$ satisfies\ \\[\ \begin{gathered}\ \omega(t)\in\mathcal M(\mathbb R^2),\ \qquad u(t)=K\*\omega(t)\ \qquad\text{for every }t\geq0,\\\\\ \sup\_{0\leq t\leq T}\\|\omega(t)\\|\_{\mathcal M}<\infty\ \qquad\text{for every }T>0?\ \end{gathered}\ \\]\ Here the Euler equations are required in the sense that\ \\[\ \int_0^T\\!\\!\int\_{\mathbb R^2}\ \bigl(u\cdot\partial_t\varphi\ +(u\otimes u):\nabla\varphi\bigr)\\,dx\\,dt\ +\int\_{\mathbb R^2}u_0\cdot\varphi(0,\cdot)\\,dx=0\ \\]\ for every $T>0$ and every divergence-free\ $\varphi\in C_c^\infty([0,T)\times\mathbb R^2;\mathbb R^2)$.\ The notation $C_w$ means weak continuity in $L^2$ on every\ bounded spatial subdomain. \ This is a bounded mathematical research project, not a request for an immediate literature-status answer. Begin the procedure in `mathematical_research_goal.md` now. As the first persistence action, create and maintain `checkpoint.md`. Copy the exact marked problem into that checkpoint before beginning heavy route work. Do not terminate except under a stopping condition permitted by `mathematical_research_goal.md`.