{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:KP6QKW6N4N3LTYTX2JGYXR6UTD","short_pith_number":"pith:KP6QKW6N","canonical_record":{"source":{"id":"2605.14157","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-13T22:16:17Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"ba1afdede9dcf2a2d7ab28eb5e12396a78be0e80d326bc4c8fa1fde9b392410c","abstract_canon_sha256":"e1367c5c5d396fac435d3631f2a54d815ec0ed718737d5d19bf81d65856320f9"},"schema_version":"1.0"},"canonical_sha256":"53fd055bcde376b9e277d24d8bc7d498c3110f79f8381a1a33a2111ba3ed2b14","source":{"kind":"arxiv","id":"2605.14157","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"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:KP6QKW6N4N3LTYTX2JGYXR6UTD","target":"record","payload":{"canonical_record":{"source":{"id":"2605.14157","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-13T22:16:17Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"ba1afdede9dcf2a2d7ab28eb5e12396a78be0e80d326bc4c8fa1fde9b392410c","abstract_canon_sha256":"e1367c5c5d396fac435d3631f2a54d815ec0ed718737d5d19bf81d65856320f9"},"schema_version":"1.0"},"canonical_sha256":"53fd055bcde376b9e277d24d8bc7d498c3110f79f8381a1a33a2111ba3ed2b14","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:11.523449Z","signature_b64":"w+RX/2zIkhT3FfWHjMw4InTvl6U++x3GWTJC0dj3Xk4O5IfqDxdyvS4P41OyuTytJ/if6htQodB0uiA9hjjFAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53fd055bcde376b9e277d24d8bc7d498c3110f79f8381a1a33a2111ba3ed2b14","last_reissued_at":"2026-05-17T23:39:11.522867Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:11.522867Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.14157","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":"2026-05-17T23:39:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cXxPu/txTe1dOW/mv8Gw4wKkHvJZUfVOeKaQLNPVoJtBNqCZRTcq5v5JZUX+S9/iNNHiTUVq4QHHcuHCGENsDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T12:52:24.812300Z"},"content_sha256":"382a302896294dbd47c1777fe08c5746644e78524c33fc5ff19eea30e1f99f60","schema_version":"1.0","event_id":"sha256:382a302896294dbd47c1777fe08c5746644e78524c33fc5ff19eea30e1f99f60"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:KP6QKW6N4N3LTYTX2JGYXR6UTD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Classification of Double Saddle-Point Systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Symmetric double saddle-point systems divide into block-arrow and block-tridiagonal matrix forms that control their invertibility, spectra, and preconditioners.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Chen Greif, Susanne Bradley","submitted_at":"2026-05-13T22:16:17Z","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."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"At the core of the paper is the division of the associated matrices into ``block-arrow'' and ``block-tridiagonal'' forms.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The systems under consideration are symmetric double saddle-point systems whose matrices admit the described block structures.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Symmetric double saddle-point systems divide into block-arrow and block-tridiagonal matrix forms that control their invertibility, spectra, and preconditioners.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e696e0bd602f0aefcaf4b1dc8b512830a2c12fc47664ff57d95092cb3c96d6ed"},"source":{"id":"2605.14157","kind":"arxiv","version":1},"verdict":{"id":"babc917f-d273-4c1f-b880-0d29f1104304","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:52:54.804514Z","strongest_claim":"At the core of the paper is the division of the associated matrices into ``block-arrow'' and ``block-tridiagonal'' forms.","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","weakest_assumption":"The systems under consideration are symmetric double saddle-point systems whose matrices admit the described block structures.","pith_extraction_headline":"Symmetric double saddle-point systems divide into block-arrow and block-tridiagonal matrix forms that control their invertibility, spectra, and preconditioners."},"references":{"count":86,"sample":[{"doi":"","year":2021,"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","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Safique Ahmad and Pinki Khatun","work_id":"718bd984-c237-40a8-8e32-cea11e15667f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"Safique Ahmad and Pinki Khatun","work_id":"68125046-f111-4625-b5f0-ad685b9efecc","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"Ali Beik and M","work_id":"ae6663ab-b47e-4149-a311-92ba63e76083","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Antonietti, Jacopo De Ponti, Luca Formaggia, and Anna Scotti","work_id":"3a85891a-fedb-48be-9157-1fcf904998d7","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":86,"snapshot_sha256":"1b0ae7ee4bd71538732448ed7bbe972be777d9e8064f77de3ec39b32a8b45fd1","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"276b63fd0f60f4171ea86f367984e14c1726c7b454c199c0e2f80dea77c60c0b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"babc917f-d273-4c1f-b880-0d29f1104304"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:39:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8epGCcrqkBdCzFrgfvJmDgGbeIm5X2hCNMH3G2IwXMnrkgqYc4r5c6X4jngBIQyXbbUILUTU9+mY1c8E9dJ0AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T12:52:24.812828Z"},"content_sha256":"73412fe0f1f7902784f372a4c9fe83152a77dba886f8a7b4d6d9b304a5ade2d5","schema_version":"1.0","event_id":"sha256:73412fe0f1f7902784f372a4c9fe83152a77dba886f8a7b4d6d9b304a5ade2d5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KP6QKW6N4N3LTYTX2JGYXR6UTD/bundle.json","state_url":"https://pith.science/pith/KP6QKW6N4N3LTYTX2JGYXR6UTD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KP6QKW6N4N3LTYTX2JGYXR6UTD/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":"2026-05-21T12:52:24Z","links":{"resolver":"https://pith.science/pith/KP6QKW6N4N3LTYTX2JGYXR6UTD","bundle":"https://pith.science/pith/KP6QKW6N4N3LTYTX2JGYXR6UTD/bundle.json","state":"https://pith.science/pith/KP6QKW6N4N3LTYTX2JGYXR6UTD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KP6QKW6N4N3LTYTX2JGYXR6UTD/bundle.json"},"state":{"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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e5PSwZfm7tXzmt4IJzOkCfNSuj+RDEzpBz1FfW9mzTVKn0RXaOiNgdtKl/97iC9bPAlu8M1lbzwO7Tq34CoMAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-21T12:52:24.815494Z","bundle_sha256":"676d86d5f23917c350b631015bb34bdab4bd32f67f7e64dd0adc2dc74c74fbfc"}}