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We prove a direct first-threshold continuation theorem for the axisymmetric class with swirl. The proof is written entirely in the lifted variables \\[\n  \\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 ","authors_text":"Rishad Shahmurov","cross_cats":[],"headline":"Axisymmetric Navier-Stokes solutions with swirl have no first threshold and remain smooth for all time.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-03T13:38:14Z","title":"Large-Data Global Regularity for Three-Dimensional Navier--Stokes I: A Direct First-Threshold Continuation Proof for the Axisymmetric Swirl Class"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.01875","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-09T16:47:05.195381Z","id":"4aa4c121-a583-4319-b5ec-d1c1c6d33685","model_set":{"reader":"grok-4.3"},"one_line_summary":"Axisymmetric Navier-Stokes solutions with swirl remain globally smooth because no first threshold occurs in the defined critical axis score envelope.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Axisymmetric Navier-Stokes solutions with swirl have no first threshold and remain smooth for all time.","strongest_claim":"We prove a direct first-threshold continuation theorem for the axisymmetric class with swirl. 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