{"paper":{"title":"Large-Data Global Regularity for Three-Dimensional Navier--Stokes I: A Direct First-Threshold Continuation Proof for the Axisymmetric Swirl Class","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Axisymmetric Navier-Stokes solutions with swirl have no first threshold and remain smooth for all time.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Rishad Shahmurov","submitted_at":"2026-05-03T13:38:14Z","abstract_excerpt":"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 \\[\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 "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove a direct first-threshold continuation theorem for the axisymmetric class with swirl. [...] Consequently no first threshold occurs, the critical envelope stays bounded, and the solution remains smooth for all time.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The strict full-Dirichlet bridge inequality |T_{G,χ}[G]| ≤ θ V_χ[G] + C E_dir[G] with 0<θ<1 holds and, together with the coefficient-calibrated local balance, contracts the selected packet; additionally the finite-overlap descendant-extraction theorem covers every possible leakage, tail, residue, concentration or fragmentation channel.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Axisymmetric Navier-Stokes solutions with swirl remain globally smooth because no first threshold occurs in the defined critical axis score envelope.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Axisymmetric Navier-Stokes solutions with swirl have no first threshold and remain smooth for all time.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7657dcde4d1cc3e8c3f8adc383653554e7cc8c49f7c4189fc6dfe6575efab54f"},"source":{"id":"2605.01875","kind":"arxiv","version":3},"verdict":{"id":"4aa4c121-a583-4319-b5ec-d1c1c6d33685","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T16:47:05.195381Z","strongest_claim":"We prove a direct first-threshold continuation theorem for the axisymmetric class with swirl. [...] Consequently no first threshold occurs, the critical envelope stays bounded, and the solution remains smooth for all time.","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","weakest_assumption":"The strict full-Dirichlet bridge inequality |T_{G,χ}[G]| ≤ θ V_χ[G] + C E_dir[G] with 0<θ<1 holds and, together with the coefficient-calibrated local balance, contracts the selected packet; additionally the finite-overlap descendant-extraction theorem covers every possible leakage, tail, residue, concentration or fragmentation channel.","pith_extraction_headline":"Axisymmetric Navier-Stokes solutions with swirl have no first threshold and remain smooth for all time."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.01875/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T16:55:53.229900Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"46b18e6621e7eeac150f62b47dde5f425238740fc567e2454f81c037574c17a6"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}