{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:PIVTBKPHDOZA2JKFQZ3FSMEFIF","merge_version":"pith-open-graph-merge-v1","event_count":3,"valid_event_count":3,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8f36588b2cfcfdd7837aff5437fef2f318da98a27a3597eac22779bc2c3207d8","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-06T16:56:38Z","title_canon_sha256":"cd2d2c49e047ee9d804a9257a709d30eccae1b876b3598b59554af527b512671"},"schema_version":"1.0","source":{"id":"2605.05129","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.05129","created_at":"2026-07-01T01:17:51Z"},{"alias_kind":"arxiv_version","alias_value":"2605.05129v2","created_at":"2026-07-01T01:17:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.05129","created_at":"2026-07-01T01:17:51Z"},{"alias_kind":"pith_short_12","alias_value":"PIVTBKPHDOZA","created_at":"2026-07-01T01:17:51Z"},{"alias_kind":"pith_short_16","alias_value":"PIVTBKPHDOZA2JKF","created_at":"2026-07-01T01:17:51Z"},{"alias_kind":"pith_short_8","alias_value":"PIVTBKPH","created_at":"2026-07-01T01:17:51Z"}],"graph_snapshots":[{"event_id":"sha256:fe0a80c15a230dc1f974782f8a981a940513fcf7909305ee34b257bfe960c41a","target":"graph","created_at":"2026-07-01T01:17:51Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"In combination with our previous work, this establishes the first higher-order-in-time and linear integrator that converges both to weak and strong solutions of LLG."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"Sufficient regularity assumptions on the exact solution and the external field are required for the optimal-order a-priori error estimates to hold."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"A linear BDF2 finite-element integrator for the LLG equation achieves first-order spatial and second-order temporal convergence rates and converges to both weak and strong solutions."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"A linear BDF2 scheme paired with finite elements achieves optimal-order convergence for the Landau-Lifshitz-Gilbert equation."}],"snapshot_sha256":"ea6927d81aefc45c65ddf31bbc120bc003b7a7512242a0e5eb8d7de5b3808493"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":false,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-20T10:36:56.562986Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_title_agreement","ran_at":"2026-05-19T21:31:19.510990Z","status":"completed","version":"1.0.0"},{"findings_count":1,"name":"doi_compliance","ran_at":"2026-05-19T13:49:15.581701Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.05129/integrity.json","findings":[{"audited_at":"2026-05-19T13:49:15.581701Z","detected_arxiv_id":null,"detected_doi":"10.1007/978-3-540-85054-","detector":"doi_compliance","finding_type":"broken_identifier","note":"DOI '10.1007/978-3-540-85054-' as printed in the bibliography is syntactically invalid and cannot resolve.","ref_index":10,"severity":"critical","verdict_class":"incontrovertible"}],"snapshot_sha256":"b32ec979e5b9b80f829871589825cfc1141bc382dee1d2feee48e93df55bfaef","summary":{"advisory":0,"by_detector":{"doi_compliance":{"advisory":0,"critical":1,"informational":0,"total":1}},"critical":1,"informational":0}},"paper":{"abstract_excerpt":"We consider the Landau-Lifshitz-Gilbert equation (LLG), which models time-dependent micromagnetic phenomena. We analyze a fully discrete scheme that combines first-order finite elements in space with a BDF2 method in time. The method requires the solution of only one linear system of equations per time step and does not enforce the pointwise unit-length constraint of the magnetization. While unconditional weak convergence has been analyzed in an earlier work, we now prove optimal-order convergence rates under sufficient regularity assumptions on the exact solution and the external field. In co","authors_text":"Dirk Praetorius, Michael Feischl, Michele Ald\\'e","cross_cats":["cs.NA"],"headline":"A linear BDF2 scheme paired with finite elements achieves optimal-order convergence for the Landau-Lifshitz-Gilbert equation.","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-06T16:56:38Z","title":"BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: a-priori error estimates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.05129","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-08T15:49:24.570215Z","id":"0fe8a8cd-ec5e-46db-a672-562513c1d3fc","model_set":{"reader":"grok-4.3"},"one_line_summary":"A linear BDF2 finite-element integrator for the LLG equation achieves first-order spatial and second-order temporal convergence rates and converges to both weak and strong solutions.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"A linear BDF2 scheme paired with finite elements achieves optimal-order convergence for the Landau-Lifshitz-Gilbert equation.","strongest_claim":"In combination with our previous work, this establishes the first higher-order-in-time and linear integrator that converges both to weak and strong solutions of LLG.","weakest_assumption":"Sufficient regularity assumptions on the exact solution and the external field are required for the optimal-order a-priori error estimates to hold."}},"verdict_id":"0fe8a8cd-ec5e-46db-a672-562513c1d3fc"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8cae580d06172c088e9ff5c3723f5a6f1a972da38e89989e0670d33745e42ec5","target":"record","created_at":"2026-07-01T01:17:51Z","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":"8f36588b2cfcfdd7837aff5437fef2f318da98a27a3597eac22779bc2c3207d8","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-06T16:56:38Z","title_canon_sha256":"cd2d2c49e047ee9d804a9257a709d30eccae1b876b3598b59554af527b512671"},"schema_version":"1.0","source":{"id":"2605.05129","kind":"arxiv","version":2}},"canonical_sha256":"7a2b30a9e71bb20d254586765930854173c321fdc4d371943cfe1b5582b089d0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7a2b30a9e71bb20d254586765930854173c321fdc4d371943cfe1b5582b089d0","first_computed_at":"2026-07-01T01:17:51.573391Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-01T01:17:51.573391Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rqB+HWYkMV21pKl78FtpYlY53aLLchZ/z8oUTWc2LBRut9U56Ud/U0ULNU2hacyf6kF91eXB6MuQJUrj4rnoBg==","signature_status":"signed_v1","signed_at":"2026-07-01T01:17:51.573816Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.05129","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b74d4883356522d0a800b9a146eca1a0545dfdbcd422b10f84a8e774fc3d5286","sha256:8cae580d06172c088e9ff5c3723f5a6f1a972da38e89989e0670d33745e42ec5","sha256:fe0a80c15a230dc1f974782f8a981a940513fcf7909305ee34b257bfe960c41a"],"state_sha256":"8f7282916a1a9fb76dacc73afafdad4405dd546b8fbc5abfd5eec50c57fee103"}