{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:DJAEQ6AOISYABJOGI7PIG3DBUS","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"fa2a436ce8f552170357ff7242bbcd554e33180095e2890be3865f8996ecb207","cross_cats_sorted":["cs.NA","math.AP"],"license":"","primary_cat":"math.NA","submitted_at":"2006-07-07T07:02:19Z","title_canon_sha256":"579907244a5d8969f131d73f08dc37eec6666a27adcdca4fb7b5b42678b7b5df"},"schema_version":"1.0","source":{"id":"math/0607181","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0607181","created_at":"2026-06-03T22:06:20Z"},{"alias_kind":"arxiv_version","alias_value":"math/0607181v2","created_at":"2026-06-03T22:06:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0607181","created_at":"2026-06-03T22:06:20Z"},{"alias_kind":"pith_short_12","alias_value":"DJAEQ6AOISYA","created_at":"2026-06-03T22:06:20Z"},{"alias_kind":"pith_short_16","alias_value":"DJAEQ6AOISYABJOG","created_at":"2026-06-03T22:06:20Z"},{"alias_kind":"pith_short_8","alias_value":"DJAEQ6AO","created_at":"2026-06-03T22:06:20Z"}],"graph_snapshots":[{"event_id":"sha256:10ba03ccf2f6c6f60823460cd85dfb9c05e66d02ea672bc0d319a491bb0f6a13","target":"graph","created_at":"2026-06-03T22:06:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0607181/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we consider the r\\^ole that numerical computations -- in particular Galerkin approximations -- can play in problems modelled by the 3d Navier-Stokes equations, for which no rigorous proof of the existence of unique solutions is currently available. We prove a robustness theorem for strong solutions, from which we derive an {\\it a posteriori} check that can be applied to a numerical solution to guarantee the existence of a strong solution of the corresponding exact problem.\n  We then consider Galerkin approximations, and show that {\\it if} a strong solution exists the Galerkin app","authors_text":"Edriss S. Titi, James C. Robinson, Peter Constantin, Sergei I. Chernyshenko","cross_cats":["cs.NA","math.AP"],"headline":"","license":"","primary_cat":"math.NA","submitted_at":"2006-07-07T07:02:19Z","title":"A Posteriori Regularity of the Three-dimensional Navier-Stokes Equations from Numerical Computations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607181","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2dfabb4301ff87932202067c1467a5397ae18ccc236ac459f4830d5f1ea8de82","target":"record","created_at":"2026-06-03T22:06:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"fa2a436ce8f552170357ff7242bbcd554e33180095e2890be3865f8996ecb207","cross_cats_sorted":["cs.NA","math.AP"],"license":"","primary_cat":"math.NA","submitted_at":"2006-07-07T07:02:19Z","title_canon_sha256":"579907244a5d8969f131d73f08dc37eec6666a27adcdca4fb7b5b42678b7b5df"},"schema_version":"1.0","source":{"id":"math/0607181","kind":"arxiv","version":2}},"canonical_sha256":"1a4048780e44b000a5c647de836c61a487ea368dc70547a7c4eee9746b356782","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1a4048780e44b000a5c647de836c61a487ea368dc70547a7c4eee9746b356782","first_computed_at":"2026-06-03T22:06:20.029771Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T22:06:20.029771Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hXTkz6vk/O/629lUtxz3hQMQXIU3YapKOBE3+bDk93M0GbKNw8FuVE3h49TvaGdHoVFZUXVtLOZnjRS1flnmDA==","signature_status":"signed_v1","signed_at":"2026-06-03T22:06:20.030181Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0607181","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2dfabb4301ff87932202067c1467a5397ae18ccc236ac459f4830d5f1ea8de82","sha256:10ba03ccf2f6c6f60823460cd85dfb9c05e66d02ea672bc0d319a491bb0f6a13"],"state_sha256":"5c2c1e928dbc7fd20c2c47cc831e0c65edffa963b8d725e1923cb426ae5eb00b"}