{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:UALXZYWPWBV6YI37ZHDOSTUMPA","short_pith_number":"pith:UALXZYWP","canonical_record":{"source":{"id":"2411.13896","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-21T07:16:41Z","cross_cats_sorted":[],"title_canon_sha256":"3308ab0142107008dee5e17e78c71c4ebaa8870fb45d762d5b9894723b209bce","abstract_canon_sha256":"57583a383f3402e0dda7110b0cc61418197b316194cb8ea68e3fe0f9d24ab8ce"},"schema_version":"1.0"},"canonical_sha256":"a0177ce2cfb06bec237fc9c6e94e8c7816c4c5902bc7d7b335330a75af939c23","source":{"kind":"arxiv","id":"2411.13896","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.13896","created_at":"2026-07-05T09:55:14Z"},{"alias_kind":"arxiv_version","alias_value":"2411.13896v3","created_at":"2026-07-05T09:55:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.13896","created_at":"2026-07-05T09:55:14Z"},{"alias_kind":"pith_short_12","alias_value":"UALXZYWPWBV6","created_at":"2026-07-05T09:55:14Z"},{"alias_kind":"pith_short_16","alias_value":"UALXZYWPWBV6YI37","created_at":"2026-07-05T09:55:14Z"},{"alias_kind":"pith_short_8","alias_value":"UALXZYWP","created_at":"2026-07-05T09:55:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:UALXZYWPWBV6YI37ZHDOSTUMPA","target":"record","payload":{"canonical_record":{"source":{"id":"2411.13896","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-21T07:16:41Z","cross_cats_sorted":[],"title_canon_sha256":"3308ab0142107008dee5e17e78c71c4ebaa8870fb45d762d5b9894723b209bce","abstract_canon_sha256":"57583a383f3402e0dda7110b0cc61418197b316194cb8ea68e3fe0f9d24ab8ce"},"schema_version":"1.0"},"canonical_sha256":"a0177ce2cfb06bec237fc9c6e94e8c7816c4c5902bc7d7b335330a75af939c23","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:55:14.384857Z","signature_b64":"vMyInwojT1efDBKl/WsEEfod710Fgs/OvajN0kj/B4V0GOjgN14TPGxPvpCNQDY3xu1YLywdYwDGx8bEN+m3BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a0177ce2cfb06bec237fc9c6e94e8c7816c4c5902bc7d7b335330a75af939c23","last_reissued_at":"2026-07-05T09:55:14.384425Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:55:14.384425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.13896","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:55:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HO+UvsPUz/oRVzTDCJPV2lsvilqDyPf5W0cYDQI+WKc+8D7vKwwVCYF757I3r2pGbD5IH2qstx5v27KOXUvPBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T02:28:50.011114Z"},"content_sha256":"f0f9bcfb45db4b19a3a075fc13709d5d24ee25fb2f284807d86b3d4c3b6ea797","schema_version":"1.0","event_id":"sha256:f0f9bcfb45db4b19a3a075fc13709d5d24ee25fb2f284807d86b3d4c3b6ea797"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:UALXZYWPWBV6YI37ZHDOSTUMPA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A blow up solution of the Navier-Stokes equations with a critical force","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Qi S. Zhang","submitted_at":"2024-11-21T07:16:41Z","abstract_excerpt":"A forced solution $v$ of the Navier-Stokes equation in any open domain with no slip\n  boundary condition is constructed. The scaling factor of the forcing term is the\n  critical order $-2$. The velocity, which is smooth until its final blow up moment,\n  is in the energy space through out. Since most physical forces from a point source\n  in nature are regarded as order $-2$, such as Coulomb force, Yukawa force, this\n  result indicates possible singularity formation under these kind of forces. The result\n  even holds for some log subcritical forces or some forces in the standard critical space $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.13896","kind":"arxiv","version":3},"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/2411.13896/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:55:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZoAUyxISXAGnjPFMsyNikosPnpdpv/l7ddLqNbW1FltYDgFVxpqTS4JrckbIckOW85r92cLk7MRq/K3HnCm4Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T02:28:50.011523Z"},"content_sha256":"d39c88e0b7dea621826726c42eba2eab1071043c40858aa68a7c2d39d0993d9f","schema_version":"1.0","event_id":"sha256:d39c88e0b7dea621826726c42eba2eab1071043c40858aa68a7c2d39d0993d9f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UALXZYWPWBV6YI37ZHDOSTUMPA/bundle.json","state_url":"https://pith.science/pith/UALXZYWPWBV6YI37ZHDOSTUMPA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UALXZYWPWBV6YI37ZHDOSTUMPA/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":"2026-08-13T02:28:50Z","links":{"resolver":"https://pith.science/pith/UALXZYWPWBV6YI37ZHDOSTUMPA","bundle":"https://pith.science/pith/UALXZYWPWBV6YI37ZHDOSTUMPA/bundle.json","state":"https://pith.science/pith/UALXZYWPWBV6YI37ZHDOSTUMPA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UALXZYWPWBV6YI37ZHDOSTUMPA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:UALXZYWPWBV6YI37ZHDOSTUMPA","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":"57583a383f3402e0dda7110b0cc61418197b316194cb8ea68e3fe0f9d24ab8ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-21T07:16:41Z","title_canon_sha256":"3308ab0142107008dee5e17e78c71c4ebaa8870fb45d762d5b9894723b209bce"},"schema_version":"1.0","source":{"id":"2411.13896","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.13896","created_at":"2026-07-05T09:55:14Z"},{"alias_kind":"arxiv_version","alias_value":"2411.13896v3","created_at":"2026-07-05T09:55:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.13896","created_at":"2026-07-05T09:55:14Z"},{"alias_kind":"pith_short_12","alias_value":"UALXZYWPWBV6","created_at":"2026-07-05T09:55:14Z"},{"alias_kind":"pith_short_16","alias_value":"UALXZYWPWBV6YI37","created_at":"2026-07-05T09:55:14Z"},{"alias_kind":"pith_short_8","alias_value":"UALXZYWP","created_at":"2026-07-05T09:55:14Z"}],"graph_snapshots":[{"event_id":"sha256:d39c88e0b7dea621826726c42eba2eab1071043c40858aa68a7c2d39d0993d9f","target":"graph","created_at":"2026-07-05T09:55:14Z","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/2411.13896/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A forced solution $v$ of the Navier-Stokes equation in any open domain with no slip\n  boundary condition is constructed. The scaling factor of the forcing term is the\n  critical order $-2$. The velocity, which is smooth until its final blow up moment,\n  is in the energy space through out. Since most physical forces from a point source\n  in nature are regarded as order $-2$, such as Coulomb force, Yukawa force, this\n  result indicates possible singularity formation under these kind of forces. The result\n  even holds for some log subcritical forces or some forces in the standard critical space $","authors_text":"Qi S. Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-21T07:16:41Z","title":"A blow up solution of the Navier-Stokes equations with a critical force"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.13896","kind":"arxiv","version":3},"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:f0f9bcfb45db4b19a3a075fc13709d5d24ee25fb2f284807d86b3d4c3b6ea797","target":"record","created_at":"2026-07-05T09:55:14Z","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":"57583a383f3402e0dda7110b0cc61418197b316194cb8ea68e3fe0f9d24ab8ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-21T07:16:41Z","title_canon_sha256":"3308ab0142107008dee5e17e78c71c4ebaa8870fb45d762d5b9894723b209bce"},"schema_version":"1.0","source":{"id":"2411.13896","kind":"arxiv","version":3}},"canonical_sha256":"a0177ce2cfb06bec237fc9c6e94e8c7816c4c5902bc7d7b335330a75af939c23","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a0177ce2cfb06bec237fc9c6e94e8c7816c4c5902bc7d7b335330a75af939c23","first_computed_at":"2026-07-05T09:55:14.384425Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:55:14.384425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vMyInwojT1efDBKl/WsEEfod710Fgs/OvajN0kj/B4V0GOjgN14TPGxPvpCNQDY3xu1YLywdYwDGx8bEN+m3BQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:55:14.384857Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.13896","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f0f9bcfb45db4b19a3a075fc13709d5d24ee25fb2f284807d86b3d4c3b6ea797","sha256:d39c88e0b7dea621826726c42eba2eab1071043c40858aa68a7c2d39d0993d9f"],"state_sha256":"e1d13d1bcba35f3353b88fdcebfca1ed6e2b050a23e51543f384830615d32c9c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IAB3Fq2IHfRJGXMth66MzrpACtO+AUM02te0Znx+ZYl7aP4uGEmaMWNTONLcZS645oe5PsjFCT79PHinRf3ZCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T02:28:50.013809Z","bundle_sha256":"229dfb6726a8fef69ce661a2aae79009b364e822c60abbd68564b4199141a84f"}}