Large-Data Global Regularity for Three-Dimensional Navier--Stokes I: A Direct First-Threshold Continuation Proof for the Axisymmetric Swirl Class

This is the first paper in a two-part direct-threshold series on large-data global regularity for the three-dimensional Navier–Stokes equations. We prove a direct first-threshold continuation theorem for the axisymmetric class with swirl. The proof is written entirely in the lifted variables $$\mathrm{\Gamma }=r{u}_{\theta },G={\omega }_{\theta }\mathrm{/}r,d{\mu }_5=r^{3}drdz,$$ and uses the five-dimensional full-Dirichlet visibility ${\mathcal{V}}_{\chi }$ as the local coercive quantity. The argument is organized by a finite first-threshold stopping time. We define a critical axis score envelope, follow it to a first possible threshold time, and prove that the corresponding normalized packet cannot exist. The proof has three quantitative ingredients. First, a small-envelope continuation theorem converts bounded score and regularized source size into smooth continuation. Second, a finite-overlap descendant-extraction theorem shows that every large collar leakage, exterior tail, low-frequency residue, source concentration, or fragmentation channel either produces a smaller descendant packet or is perturbative. Third, in the remaining coherent case, the strict full-Dirichlet bridge $$\mathrm{\mid }{\mathcal{T}}_{G,\chi }[G]\mathrm{\mid }\le \theta {\mathcal{V}}_{\chi }[G]+C{\mathfrak{E}}_{\mathrm{d}\mathrm{i}\mathrm{r}}[G],0<\theta <1,$$ and a coefficient-calibrated local balance contract the selected packet. Consequently no first threshold occurs, the critical envelope stays