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The first part is a local resolution lemma for the scale-critical quantity: for every spatial filter length ell > 0, Psi(r) <= 4 Psi^ell(r) + 4 Omega^ell(r), where Psi^ell(r) is the corresponding coarse-grained velocity-pressure quantity and Omega^ell(r) is the explicitly defined subfilter residual. Thus a CKN-bad scale is either visible at the resolved level or is carried by unresolved velocity-pressure oscillation"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.25322","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-24T02:38:56Z","cross_cats_sorted":[],"title_canon_sha256":"a2e8edd2c885fc3617f4359136c9f7364aac1b8f0b5f37aef922216116690fcc","abstract_canon_sha256":"ef1a11bfc3109181a7e5ce0a37f410831d78a6efdc276d4a7af562b551bc6461"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-25T01:18:02.027142Z","signature_b64":"2Sr1uUgn6okegdOwzusQa/vxQR6lAC8mTFOq3yVuJn5pSm+iGNsnGV4Dj8lQDJL/Rpj/TPqdeNPWz3B5dqEmDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2e77afdc4f2b8d4f46cbe702398c1cc3bf5102a0c67a63185af0cd773881962","last_reissued_at":"2026-06-25T01:18:02.026723Z","signature_status":"signed_v1","first_computed_at":"2026-06-25T01:18:02.026723Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Coarse-Grained Resolution and Pressure-Flux Work Depletion for Navier-Stokes CKN Badness","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Runlong Yu","submitted_at":"2026-06-24T02:38:56Z","abstract_excerpt":"We prove a finite-scale coarse-grained decomposition for the three-dimensional incompressible Navier-Stokes equations near the Caffarelli-Kohn-Nirenberg local regularity framework. The first part is a local resolution lemma for the scale-critical quantity: for every spatial filter length ell > 0, Psi(r) <= 4 Psi^ell(r) + 4 Omega^ell(r), where Psi^ell(r) is the corresponding coarse-grained velocity-pressure quantity and Omega^ell(r) is the explicitly defined subfilter residual. 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