Incompressible Euler Blowup at the C^{1,frac{1}{3}} Threshold

We prove finite-time Type-I blowup for the three-dimensional incompressible Euler equations in the axisymmetric no-swirl class, with initial velocity in $C^{1,\alpha}(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$, odd symmetry in $z$, and $0<\alpha<\tfrac13$, for an explicit class of finite-energy initial data. The singularity forms at a stagnation point on the symmetry axis. The on-axis axial strain and the global vorticity norm blow up at the Type-I rates $-\partial_z u_z(0,0,t)\sim (T^*-t)^{-1}$ and $\|\omega(\cdot,t)\|_{L^\infty}\sim (T^*-t)^{-1}$, while the meridional Jacobian collapses according to $J(t)\sim (T^*-t)^{1/(1-3\alpha)}$. The proof introduces a Lagrangian clock-and-strain framework that replaces the Eulerian self-similar ansatz used in prior work with a Lagrangian flow decomposition. The collapse dynamics are governed by a Riccati law for the on-axis axial strain, coupled to a clock ODE for the meridional Jacobian. The decisive step is a non-perturbative strain-pressure comparison showing that the pressure Hessian cannot cancel the quadratic compressive strain responsible for collapse. This gives a dynamical explanation of the threshold $\alpha=\tfrac13$. The blowup mechanism is structurally stable and persists for an open set of admissible angular profiles in a weighted H\"older topology.