{"paper":{"title":"Finite-Chain CKN-Bad Scale Counting for Navier-Stokes: Standard PDE Closure and Canonical Detector Realization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Runlong Yu","submitted_at":"2026-06-19T22:11:40Z","abstract_excerpt":"This paper proves a finite-chain counting theorem for Caffarelli--Kohn--Nirenberg bad scales of suitable weak solutions to the three-dimensional incompressible Navier--Stokes equations. The main standard-PDE result bounds the weighted size of a finite set of CKN-bad scales by nonnegative channel costs consisting of vertical one-component concentration, annular leakage, pressure-tail terms, and pressure--flux--energy residuals. The proved closing mechanism is qualitative one-component compactness under a full local critical bound: small vertical component forces CKN smallness at a smaller radiu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.21783","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.21783/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}