{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:CRGQIW44ZBJPVZE2DPE6VXP374","short_pith_number":"pith:CRGQIW44","canonical_record":{"source":{"id":"1407.7329","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-28T05:54:49Z","cross_cats_sorted":[],"title_canon_sha256":"e8ed36183c79835ca5e64730968efdfef2746cc77370a3fa73ca84e4a181b59d","abstract_canon_sha256":"14684b3a5f4e51c3639b69f23ffbb1d7a6940df534e8a20c8b193144a88f8299"},"schema_version":"1.0"},"canonical_sha256":"144d045b9cc852fae49a1bc9eaddfbff38e10eccb0f3dec3cc9da0916a01345a","source":{"kind":"arxiv","id":"1407.7329","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.7329","created_at":"2026-05-18T02:19:16Z"},{"alias_kind":"arxiv_version","alias_value":"1407.7329v2","created_at":"2026-05-18T02:19:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.7329","created_at":"2026-05-18T02:19:16Z"},{"alias_kind":"pith_short_12","alias_value":"CRGQIW44ZBJP","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"CRGQIW44ZBJPVZE2","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"CRGQIW44","created_at":"2026-05-18T12:28:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:CRGQIW44ZBJPVZE2DPE6VXP374","target":"record","payload":{"canonical_record":{"source":{"id":"1407.7329","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-28T05:54:49Z","cross_cats_sorted":[],"title_canon_sha256":"e8ed36183c79835ca5e64730968efdfef2746cc77370a3fa73ca84e4a181b59d","abstract_canon_sha256":"14684b3a5f4e51c3639b69f23ffbb1d7a6940df534e8a20c8b193144a88f8299"},"schema_version":"1.0"},"canonical_sha256":"144d045b9cc852fae49a1bc9eaddfbff38e10eccb0f3dec3cc9da0916a01345a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:16.725172Z","signature_b64":"7oOxQLwBOfOWevEQCH3Kida0fZZlNCDSWg+KYZlVSaUkh4rxvoITzLLLWfyrvwPuChxMcfufHdqrHMga5ZhJCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"144d045b9cc852fae49a1bc9eaddfbff38e10eccb0f3dec3cc9da0916a01345a","last_reissued_at":"2026-05-18T02:19:16.724507Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:16.724507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1407.7329","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:19:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V2eYHNU8CrfDqoP/xA+MgBEVy1R3pcMxX9VLH5qbTgyTjDPGttretNn8h9D8YFugTu9R5LDbvcQ0CDf3PecEBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T14:17:21.179067Z"},"content_sha256":"45e0bb171ddcdbe7e079a8a3510bf2a0a29cea6db99b4b7fa79c1b02dcb4a97d","schema_version":"1.0","event_id":"sha256:45e0bb171ddcdbe7e079a8a3510bf2a0a29cea6db99b4b7fa79c1b02dcb4a97d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:CRGQIW44ZBJPVZE2DPE6VXP374","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Regularity criterion of the 4D Navier-Stokes equations involving two velocity field components","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kazuo Yamazaki","submitted_at":"2014-07-28T05:54:49Z","abstract_excerpt":"We study the Serrin-type regularity criteria for the solutions to the four-dimensional Navier-Stokes equations and magnetohydrodynamics system. We show that the sufficient condition for the solution to the four-dimensional Navier-Stokes equations to preserve its initial regularity for all time may be reduced from a bound on the four-dimensional velocity vector field to any two of its four components, from a bound on the gradient of the velocity vector field to the gradient of any two of its four components, from a gradient of the pressure scalar field to any two of its partial derivatives. Res"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.7329","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:19:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XwVPTNMbI4xgYSI0kOAM5HPQ+e0X3joW3Mmo9Eg960a4V6C1Vy2tWLXEZ9Kcfp/wHpmu1y3a3d2dbzo7geNeDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T14:17:21.179732Z"},"content_sha256":"c966f8cc36d153668c088afe998157ca2d5e73b573d02a5f5a62aa61f994db94","schema_version":"1.0","event_id":"sha256:c966f8cc36d153668c088afe998157ca2d5e73b573d02a5f5a62aa61f994db94"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CRGQIW44ZBJPVZE2DPE6VXP374/bundle.json","state_url":"https://pith.science/pith/CRGQIW44ZBJPVZE2DPE6VXP374/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CRGQIW44ZBJPVZE2DPE6VXP374/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-07T14:17:21Z","links":{"resolver":"https://pith.science/pith/CRGQIW44ZBJPVZE2DPE6VXP374","bundle":"https://pith.science/pith/CRGQIW44ZBJPVZE2DPE6VXP374/bundle.json","state":"https://pith.science/pith/CRGQIW44ZBJPVZE2DPE6VXP374/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CRGQIW44ZBJPVZE2DPE6VXP374/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:CRGQIW44ZBJPVZE2DPE6VXP374","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":"14684b3a5f4e51c3639b69f23ffbb1d7a6940df534e8a20c8b193144a88f8299","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-28T05:54:49Z","title_canon_sha256":"e8ed36183c79835ca5e64730968efdfef2746cc77370a3fa73ca84e4a181b59d"},"schema_version":"1.0","source":{"id":"1407.7329","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.7329","created_at":"2026-05-18T02:19:16Z"},{"alias_kind":"arxiv_version","alias_value":"1407.7329v2","created_at":"2026-05-18T02:19:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.7329","created_at":"2026-05-18T02:19:16Z"},{"alias_kind":"pith_short_12","alias_value":"CRGQIW44ZBJP","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"CRGQIW44ZBJPVZE2","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"CRGQIW44","created_at":"2026-05-18T12:28:22Z"}],"graph_snapshots":[{"event_id":"sha256:c966f8cc36d153668c088afe998157ca2d5e73b573d02a5f5a62aa61f994db94","target":"graph","created_at":"2026-05-18T02:19:16Z","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"},"paper":{"abstract_excerpt":"We study the Serrin-type regularity criteria for the solutions to the four-dimensional Navier-Stokes equations and magnetohydrodynamics system. We show that the sufficient condition for the solution to the four-dimensional Navier-Stokes equations to preserve its initial regularity for all time may be reduced from a bound on the four-dimensional velocity vector field to any two of its four components, from a bound on the gradient of the velocity vector field to the gradient of any two of its four components, from a gradient of the pressure scalar field to any two of its partial derivatives. Res","authors_text":"Kazuo Yamazaki","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-28T05:54:49Z","title":"Regularity criterion of the 4D Navier-Stokes equations involving two velocity field components"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.7329","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":"sha256:45e0bb171ddcdbe7e079a8a3510bf2a0a29cea6db99b4b7fa79c1b02dcb4a97d","target":"record","created_at":"2026-05-18T02:19:16Z","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":"14684b3a5f4e51c3639b69f23ffbb1d7a6940df534e8a20c8b193144a88f8299","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-28T05:54:49Z","title_canon_sha256":"e8ed36183c79835ca5e64730968efdfef2746cc77370a3fa73ca84e4a181b59d"},"schema_version":"1.0","source":{"id":"1407.7329","kind":"arxiv","version":2}},"canonical_sha256":"144d045b9cc852fae49a1bc9eaddfbff38e10eccb0f3dec3cc9da0916a01345a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"144d045b9cc852fae49a1bc9eaddfbff38e10eccb0f3dec3cc9da0916a01345a","first_computed_at":"2026-05-18T02:19:16.724507Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:19:16.724507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7oOxQLwBOfOWevEQCH3Kida0fZZlNCDSWg+KYZlVSaUkh4rxvoITzLLLWfyrvwPuChxMcfufHdqrHMga5ZhJCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:19:16.725172Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.7329","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:45e0bb171ddcdbe7e079a8a3510bf2a0a29cea6db99b4b7fa79c1b02dcb4a97d","sha256:c966f8cc36d153668c088afe998157ca2d5e73b573d02a5f5a62aa61f994db94"],"state_sha256":"0446c8d3ff652042d4aea642f2ec1b4f6fd052160588ca1e5810d5f268f1d196"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xFX/wCuxs/JNUqMlt/mTk3IYTtWzSEOpDrX7WNerWbqvzzfM80QdCwfX7kaLtZHN5wSBSVYqzdA3zjLQDvNYCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T14:17:21.183254Z","bundle_sha256":"9192ecb3e386e4dc3f14ba0223645674fc2c82aae338ec70b3283cd648457484"}}