{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:KP6QKW6N4N3LTYTX2JGYXR6UTD","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":"e1367c5c5d396fac435d3631f2a54d815ec0ed718737d5d19bf81d65856320f9","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-13T22:16:17Z","title_canon_sha256":"ba1afdede9dcf2a2d7ab28eb5e12396a78be0e80d326bc4c8fa1fde9b392410c"},"schema_version":"1.0","source":{"id":"2605.14157","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14157","created_at":"2026-05-17T23:39:11Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14157v1","created_at":"2026-05-17T23:39:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14157","created_at":"2026-05-17T23:39:11Z"},{"alias_kind":"pith_short_12","alias_value":"KP6QKW6N4N3L","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"KP6QKW6N4N3LTYTX","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"KP6QKW6N","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:73412fe0f1f7902784f372a4c9fe83152a77dba886f8a7b4d6d9b304a5ade2d5","target":"graph","created_at":"2026-05-17T23:39:11Z","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":"At the core of the paper is the division of the associated matrices into ``block-arrow'' and ``block-tridiagonal'' forms."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The systems under consideration are symmetric double saddle-point systems whose matrices admit the described block structures."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"A classification of symmetric double saddle-point systems into block-arrow and block-tridiagonal matrix forms is presented, including invertibility conditions, spectral properties, and block preconditioners."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Symmetric double saddle-point systems divide into block-arrow and block-tridiagonal matrix forms that control their invertibility, spectra, and preconditioners."}],"snapshot_sha256":"e696e0bd602f0aefcaf4b1dc8b512830a2c12fc47664ff57d95092cb3c96d6ed"},"formal_canon":{"evidence_count":1,"snapshot_sha256":"276b63fd0f60f4171ea86f367984e14c1726c7b454c199c0e2f80dea77c60c0b"},"paper":{"abstract_excerpt":"We offer a classification of a broad and practically relevant class of symmetric double saddle-point system. At the core of the paper is the division of the associated matrices into ``block-arrow'' and ``block-tridiagonal'' forms. We describe relevant applications, invertibility conditions, spectral properties, and block preconditioners. Our discussion is kept within a general framework rather than tailored to specific applications.","authors_text":"Chen Greif, Susanne Bradley","cross_cats":["cs.NA"],"headline":"Symmetric double saddle-point systems divide into block-arrow and block-tridiagonal matrix forms that control their invertibility, spectra, and preconditioners.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-13T22:16:17Z","title":"Classification of Double Saddle-Point Systems"},"references":{"count":86,"internal_anchors":0,"resolved_work":86,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"James H. Adler, Thomas R. Benson, Eric C. Cyr, Patrick E. Farrell, Scott P. MacLachlan, and Ray S. Tuminaro. Monolithic multigrid meth- ods for magnetohydrodynamics.SIAM Journal on Scientific Computin","work_id":"3430b725-83d6-4d64-9900-b021864df9fe","year":2021},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"Safique Ahmad and Pinki Khatun","work_id":"718bd984-c237-40a8-8e32-cea11e15667f","year":2025},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Safique Ahmad and Pinki Khatun","work_id":"68125046-f111-4625-b5f0-ad685b9efecc","year":2026},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Ali Beik and M","work_id":"ae6663ab-b47e-4149-a311-92ba63e76083","year":2018},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Antonietti, Jacopo De Ponti, Luca Formaggia, and Anna Scotti","work_id":"3a85891a-fedb-48be-9157-1fcf904998d7","year":2020}],"snapshot_sha256":"1b0ae7ee4bd71538732448ed7bbe972be777d9e8064f77de3ec39b32a8b45fd1"},"source":{"id":"2605.14157","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-15T01:52:54.804514Z","id":"babc917f-d273-4c1f-b880-0d29f1104304","model_set":{"reader":"grok-4.3"},"one_line_summary":"A classification of symmetric double saddle-point systems into block-arrow and block-tridiagonal matrix forms is presented, including invertibility conditions, spectral properties, and block preconditioners.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Symmetric double saddle-point systems divide into block-arrow and block-tridiagonal matrix forms that control their invertibility, spectra, and preconditioners.","strongest_claim":"At the core of the paper is the division of the associated matrices into ``block-arrow'' and ``block-tridiagonal'' forms.","weakest_assumption":"The systems under consideration are symmetric double saddle-point systems whose matrices admit the described block structures."}},"verdict_id":"babc917f-d273-4c1f-b880-0d29f1104304"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:382a302896294dbd47c1777fe08c5746644e78524c33fc5ff19eea30e1f99f60","target":"record","created_at":"2026-05-17T23:39:11Z","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":"e1367c5c5d396fac435d3631f2a54d815ec0ed718737d5d19bf81d65856320f9","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-13T22:16:17Z","title_canon_sha256":"ba1afdede9dcf2a2d7ab28eb5e12396a78be0e80d326bc4c8fa1fde9b392410c"},"schema_version":"1.0","source":{"id":"2605.14157","kind":"arxiv","version":1}},"canonical_sha256":"53fd055bcde376b9e277d24d8bc7d498c3110f79f8381a1a33a2111ba3ed2b14","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"53fd055bcde376b9e277d24d8bc7d498c3110f79f8381a1a33a2111ba3ed2b14","first_computed_at":"2026-05-17T23:39:11.522867Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:11.522867Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"w+RX/2zIkhT3FfWHjMw4InTvl6U++x3GWTJC0dj3Xk4O5IfqDxdyvS4P41OyuTytJ/if6htQodB0uiA9hjjFAQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:11.523449Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.14157","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:382a302896294dbd47c1777fe08c5746644e78524c33fc5ff19eea30e1f99f60","sha256:73412fe0f1f7902784f372a4c9fe83152a77dba886f8a7b4d6d9b304a5ade2d5"],"state_sha256":"a641f7b4715749eeda2cc6a75c66219d95d01a204c7ec29028ff6931593c3f2d"}