{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:MKAWC2MDBAC4LRH6GV4YOS5BKX","short_pith_number":"pith:MKAWC2MD","canonical_record":{"source":{"id":"2606.19508","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-06-17T18:46:49Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"69c610585118f9bb39c89e25c02bbc503fc978ab73397557b0ce80f1c8ee5fba","abstract_canon_sha256":"1c5ce17fbfec39624022d3f6fbdd8e8e6f06b37bf38783516b33ecc43f81c16f"},"schema_version":"1.0"},"canonical_sha256":"62816169830805c5c4fe3579874ba155c671985ec93b7a5cc9dc00c57442aa1d","source":{"kind":"arxiv","id":"2606.19508","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.19508","created_at":"2026-06-19T16:12:27Z"},{"alias_kind":"arxiv_version","alias_value":"2606.19508v1","created_at":"2026-06-19T16:12:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.19508","created_at":"2026-06-19T16:12:27Z"},{"alias_kind":"pith_short_12","alias_value":"MKAWC2MDBAC4","created_at":"2026-06-19T16:12:27Z"},{"alias_kind":"pith_short_16","alias_value":"MKAWC2MDBAC4LRH6","created_at":"2026-06-19T16:12:27Z"},{"alias_kind":"pith_short_8","alias_value":"MKAWC2MD","created_at":"2026-06-19T16:12:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:MKAWC2MDBAC4LRH6GV4YOS5BKX","target":"record","payload":{"canonical_record":{"source":{"id":"2606.19508","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-06-17T18:46:49Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"69c610585118f9bb39c89e25c02bbc503fc978ab73397557b0ce80f1c8ee5fba","abstract_canon_sha256":"1c5ce17fbfec39624022d3f6fbdd8e8e6f06b37bf38783516b33ecc43f81c16f"},"schema_version":"1.0"},"canonical_sha256":"62816169830805c5c4fe3579874ba155c671985ec93b7a5cc9dc00c57442aa1d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:12:27.474286Z","signature_b64":"Ga4N8sTQ64wSfl7m76y4rS2upYKYyKP8yLjjJBLn6lO1Oibh4P6Ws+4QYhHHa36iRUZ4MOVnpXkF9lB/8J22CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62816169830805c5c4fe3579874ba155c671985ec93b7a5cc9dc00c57442aa1d","last_reissued_at":"2026-06-19T16:12:27.473938Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:12:27.473938Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.19508","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-19T16:12:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TBXGKqLnl6v2V7GiOVswc1JuemHUnk7cFeqBhRnCf+GbZ8eldZqlGN7qgvJe12EuKyNOb77lsNgArlD+/K51BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T11:38:46.386146Z"},"content_sha256":"4d3085798b46651317729d91a4fe426d17ca8cf9eaaf8fb569f6693942b1c04e","schema_version":"1.0","event_id":"sha256:4d3085798b46651317729d91a4fe426d17ca8cf9eaaf8fb569f6693942b1c04e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:MKAWC2MDBAC4LRH6GV4YOS5BKX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Higher Accuracy Modular Data Assimilation for the Navier-Stokes Equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Troy Yang","submitted_at":"2026-06-17T18:46:49Z","abstract_excerpt":"This paper develops an accurate and effective combination of second order backward differentiation time discretization (BDF2) with modular, 2-step nudging-based data assimilation \\begin{align} \\text{Forecast step: } \\quad &\\frac{3\\widetilde{v}^{n+2}-4v^{n+1}+v^n}{2\\Delta t}+\\widetilde{v}^{n+2} \\cdot \\nabla \\widetilde{v}^{n+2} - \\nu \\Delta \\widetilde{v}^{n+2} + \\nabla q^{n+2}=f(x) \\notag \\\\ &\\nabla \\cdot \\widetilde{v}^{n+2} = 0 \\notag \\\\ \\text{Analysis step: } \\quad &\\frac{3v^{n+2}-3\\widetilde{v}^{n+2}}{2\\Delta t}-\\chi I_H(u(t^{n+2})-v^{n+2})=0. \\notag \\end{align} If $I_H=I_H^2$, the analysis s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.19508","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/2606.19508/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":"2026-06-19T16:12:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UUy3+HAFq0FEkbJg0aYQaqFhV9x7a30YVSR1hp6Ka0QGdKSYcNABAKObbG59lys0yKnaN5EIKDoxhNzSPu+LDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T11:38:46.387273Z"},"content_sha256":"3ce9e96d12f4c316e9287e2c792a32aa0bd6b7f88ebdea7cba642c5be11902ad","schema_version":"1.0","event_id":"sha256:3ce9e96d12f4c316e9287e2c792a32aa0bd6b7f88ebdea7cba642c5be11902ad"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MKAWC2MDBAC4LRH6GV4YOS5BKX/bundle.json","state_url":"https://pith.science/pith/MKAWC2MDBAC4LRH6GV4YOS5BKX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MKAWC2MDBAC4LRH6GV4YOS5BKX/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":"2026-08-11T11:38:46Z","links":{"resolver":"https://pith.science/pith/MKAWC2MDBAC4LRH6GV4YOS5BKX","bundle":"https://pith.science/pith/MKAWC2MDBAC4LRH6GV4YOS5BKX/bundle.json","state":"https://pith.science/pith/MKAWC2MDBAC4LRH6GV4YOS5BKX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MKAWC2MDBAC4LRH6GV4YOS5BKX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:MKAWC2MDBAC4LRH6GV4YOS5BKX","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":"1c5ce17fbfec39624022d3f6fbdd8e8e6f06b37bf38783516b33ecc43f81c16f","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-06-17T18:46:49Z","title_canon_sha256":"69c610585118f9bb39c89e25c02bbc503fc978ab73397557b0ce80f1c8ee5fba"},"schema_version":"1.0","source":{"id":"2606.19508","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.19508","created_at":"2026-06-19T16:12:27Z"},{"alias_kind":"arxiv_version","alias_value":"2606.19508v1","created_at":"2026-06-19T16:12:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.19508","created_at":"2026-06-19T16:12:27Z"},{"alias_kind":"pith_short_12","alias_value":"MKAWC2MDBAC4","created_at":"2026-06-19T16:12:27Z"},{"alias_kind":"pith_short_16","alias_value":"MKAWC2MDBAC4LRH6","created_at":"2026-06-19T16:12:27Z"},{"alias_kind":"pith_short_8","alias_value":"MKAWC2MD","created_at":"2026-06-19T16:12:27Z"}],"graph_snapshots":[{"event_id":"sha256:3ce9e96d12f4c316e9287e2c792a32aa0bd6b7f88ebdea7cba642c5be11902ad","target":"graph","created_at":"2026-06-19T16:12:27Z","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/2606.19508/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper develops an accurate and effective combination of second order backward differentiation time discretization (BDF2) with modular, 2-step nudging-based data assimilation \\begin{align} \\text{Forecast step: } \\quad &\\frac{3\\widetilde{v}^{n+2}-4v^{n+1}+v^n}{2\\Delta t}+\\widetilde{v}^{n+2} \\cdot \\nabla \\widetilde{v}^{n+2} - \\nu \\Delta \\widetilde{v}^{n+2} + \\nabla q^{n+2}=f(x) \\notag \\\\ &\\nabla \\cdot \\widetilde{v}^{n+2} = 0 \\notag \\\\ \\text{Analysis step: } \\quad &\\frac{3v^{n+2}-3\\widetilde{v}^{n+2}}{2\\Delta t}-\\chi I_H(u(t^{n+2})-v^{n+2})=0. \\notag \\end{align} If $I_H=I_H^2$, the analysis s","authors_text":"Troy Yang","cross_cats":["cs.NA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-06-17T18:46:49Z","title":"Higher Accuracy Modular Data Assimilation for the Navier-Stokes Equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.19508","kind":"arxiv","version":1},"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":"sha256:4d3085798b46651317729d91a4fe426d17ca8cf9eaaf8fb569f6693942b1c04e","target":"record","created_at":"2026-06-19T16:12:27Z","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":"1c5ce17fbfec39624022d3f6fbdd8e8e6f06b37bf38783516b33ecc43f81c16f","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-06-17T18:46:49Z","title_canon_sha256":"69c610585118f9bb39c89e25c02bbc503fc978ab73397557b0ce80f1c8ee5fba"},"schema_version":"1.0","source":{"id":"2606.19508","kind":"arxiv","version":1}},"canonical_sha256":"62816169830805c5c4fe3579874ba155c671985ec93b7a5cc9dc00c57442aa1d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62816169830805c5c4fe3579874ba155c671985ec93b7a5cc9dc00c57442aa1d","first_computed_at":"2026-06-19T16:12:27.473938Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:12:27.473938Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ga4N8sTQ64wSfl7m76y4rS2upYKYyKP8yLjjJBLn6lO1Oibh4P6Ws+4QYhHHa36iRUZ4MOVnpXkF9lB/8J22CA==","signature_status":"signed_v1","signed_at":"2026-06-19T16:12:27.474286Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.19508","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4d3085798b46651317729d91a4fe426d17ca8cf9eaaf8fb569f6693942b1c04e","sha256:3ce9e96d12f4c316e9287e2c792a32aa0bd6b7f88ebdea7cba642c5be11902ad"],"state_sha256":"b10724a6b84206c83c4468f198945e04f18e1fb3b00f1ef0511d488798bd39bd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MtniQX5OxpXgqsrsOQoSDAajJRm3qU2itwWQnIcQfIme6LSAzBXDkA0LgDXlLI8DHLAGV+Kb8yxVOOGGoNG9AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T11:38:46.399108Z","bundle_sha256":"f5ade5fbcf50775d52dd2fa90590cd53facc90be75122a37bdff39a3d280d335"}}