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

Abstract

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 \[ Γ=ru_θ,\qquad G=ω_θ/r,\qquad dμ_5=r^3\,dr\,dz, \] and uses the five-dimensional full-Dirichlet visibility \(\mathcal V_χ\) 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 \[ |\mathcal T_{G,χ}[G]| \le θ\mathcal V_χ[G]+C\mathfrak E_{\rm dir}[G], \qquad 0<θ<1, \] and a coefficient-calibrated local balance contract the selected packet. Consequently no first threshold occurs, the critical envelope stays