Euler Singularities I: Boundary Blow-Up for Smooth Exact-Odd Axisymmetric Euler with Swirl

We construct smooth axisymmetric-with-swirl initial data in a periodic cylinder for which the three-dimensional incompressible Euler evolution develops a finite-time boundary singularity. The construction is carried out in the dynamically invariant exact-odd class $$\mathrm{\Gamma }(r,-z,t)=-\mathrm{\Gamma }(r,z,t),G(r,-z,t)=-G(r,z,t),$$ where $\mathrm{\Gamma }=r{u}^{\theta }$ and $G={\omega }^{\theta }\mathrm{/}r$. At the side-wall point $(r,z)=(1,0)$, exact oddness gives the pointwise identities $${\mathrm{\partial }}_t{\mathrm{\partial }}_{z}G(1,0,t)=\sigma (t){\mathrm{\partial }}_{z}G(1,0,t)+2({\mathrm{\partial }}_z\mathrm{\Gamma }(1,0,t){)}^2,{\mathrm{\partial }}_t{\mathrm{\partial }}_z\mathrm{\Gamma }(1,0,t)=\sigma (t){\mathrm{\partial }}_z\mathrm{\Gamma }(1,0,t),$$ with $\sigma (t)=-{\mathrm{\partial }}_z{u}^z(1,0,t)$. The proof is based on a side-wall Dirichlet parametrix for the five-dimensional lifted recovery equation $-{\mathrm{\Delta }}_5\phi =G$. Near the wall, the effective compression kernel has leading term $$K_0(x,y)=C_0\frac{xy}{(x^2+y^2{)}^2},C_0>0,$$ with controlled remainders, parity-based shear cancellation, and strain-variation bounds on narrow diagonal cones. These estimates are combined with an over-compressed dyadic angular cluster functional. The cluster functional absorbs same-scale angular fragmentation, growing dyadic windows, dynamically separated far tails, and fixed-distance exterior fields into an integrably small affine Campanato defect. The resulting invariant cluster contains a uniformly coherent component with amplitudes $A_{\ast }(t)$ and $B_{\ast }(t)$ satisfying the Dini comparison system $$D^{+}{A}_{\ast }(t)\ge c{B}_{\ast }(t{)}^2,D^{+}{B}_{\ast }(t)\ge c{A}_{\ast }(t)B_{\ast }(t)\mathrm{.}$$