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 $$\Gamma=ru_\theta,\qquad G=\omega_\theta/r,\qquad d\mu_5=r^3\,dr\,dz,$$ 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 $$|\mathcal T_{G,\chi}[G]| \le \theta\mathcal V_\chi[G]+C\mathfrak E_{\rm dir}[G], \qquad 0<\theta<1,$$ and a coefficient-calibrated local balance contract the selected packet. Consequently no first threshold occurs, the critical envelope stays