Stable Finite-Time Singularity Formation for 3D Navier--Stokes via 5D-Lifted Axisymmetric Reductions

We present a 5D-lifted analytic-profile program for finite-time singularity formation in the 3D incompressible Navier–Stokes equations on the periodic torus ${\text{T}}^3$. The core of the construction is a stationary rescaled profile $\text{Ombar}$ satisfying a nonlinear elliptic fixed-point equation in an analytically weighted Hilbert space $\text{Xspace}$, together with a computer-assisted Newton–Kantorovich validation based on interval arithmetic. The profile is reconstructed into a nearly self-similar singular evolution and then transferred to ${\text{T}}^3$ by periodic extension and exact Leray projection. The manuscript is organized in the style of a computer-assisted proof paper, with theorem statements, proof packages, and explicit validation constants for the residual, inverse stability, and Lipschitz bounds.