A conditional Lagrangian clock barrier at the $C^{1,\frac{1}{3}}$ threshold for axisymmetric Euler without swirl

We consider axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations, with initial velocity in $C^{1,\alpha }\cap L^2$, where $\alpha \in [\frac{1}{3},1)$. Motivated by Shkoller’s Lagrangian clock-and-driver framework for finite-time blow-up below the $C^{1,\frac{1}{3}}$ threshold, we introduce coherent Lagrangian classes of initial data and conditional solutions for which the same clock mechanism yields a supercritical/critical barrier when $\alpha \ge \frac{1}{3}$, with a genuinely depleted barrier for $\alpha >\frac{1}{3}$ and an exponential bound at the critical endpoint $\alpha =\frac{1}{3}$. In the general case, we formulate a matrix-clock criterion in terms of the smallest singular value of the deformation gradient and show that, under cusp-tail, Dini coherence, near-field compatibility, and bounded transverse-distortion hypotheses, this singular value cannot collapse in finite time. In the on-axis case, the criterion reduces to the scalar clock inequality ${\displaystyle \dot{J}(t)\gtrsim -B(t)J(t)-CJ(t{)}^{3\alpha }}$, which rules out Shkoller-type clock collapse for $\alpha \ge \frac{1}{3}$. These results do not enlarge the known Lorentz-space global regularity classes. Rather, they in particular identify the supercritical Lagrangian obstruction dual to Shkoller’s subcritical blow-up mechanism in the case $\alpha >\frac{1}{3}$.