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 $0<\alpha <\frac{1}{3}$, 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 $-{\mathrm{\partial }}_z{u}_z(0,0,t)\sim (T^{\ast }-t{)}^{-1}$ and $\mathrm{\parallel }\omega (\cdot ,t){\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\sim (T^{\ast }-t{)}^{-1}$, while the meridional Jacobian collapses according to $J(t)\sim (T^{\ast }-t{)}^{1\mathrm{/}(1-3\alpha )}$. 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 $\alpha =\frac{1}{3}$. The blowup mechanism is structurally stable and persists for an open set of admissible angular profiles in a weighted Hölder topology.