{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:UIEA7ZSF6KUDWM4ECWLNQNP52C","short_pith_number":"pith:UIEA7ZSF","schema_version":"1.0","canonical_sha256":"a2080fe645f2a83b33841596d835fdd08f0eecff33a253f4fdb34317c60569a0","source":{"kind":"arxiv","id":"2605.11056","version":2},"attestation_state":"computed","paper":{"title":"Diagonal parity and loop toggling for symmetric matrices over $\\mathbb F_2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Symmetric matrices over F₂ always have their diagonal vector in the column space, with solutions obeying a rank parity rule.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mohsen Aliabadi","submitted_at":"2026-05-11T15:57:52Z","abstract_excerpt":"Let $M$ be a symmetric matrix over $\\mathbb F_2$, and let $\\diag(M)$ be its diagonal vector. It is known that \\[\n  \\diag(M)\\in \\Img(M). \\] Thus the affine system $Mx=\\diag(M)$ is always solvable. We strengthen this existence statement to a parity rigidity theorem: every solution satisfies \\[\n  \\diag(M)^T x\\equiv \\rank(M)\\pmod 2 . \\] For graph matrices this gives a common extension of Sutner's odd-domination theorem and Batal's parity theorem from closed-neighborhood matrices $A(G)+I$ to arbitrary partially looped graph matrices $A(G)+D$.\n  We also study how rank and nullity change when loops a"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.11056","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-11T15:57:52Z","cross_cats_sorted":[],"title_canon_sha256":"31fe717a7445042025def55fa0de6c6c24fb303f1c2ae0a51176a2b350a84a0e","abstract_canon_sha256":"2f690fa798dc88731a17aa58ad0ea1ad4688f3679bb0618dcab62235fe4902db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T01:03:33.345063Z","signature_b64":"5wbjVTEE6moyPqtFJhehsWdo0zZsSwVkuc0smv1effaGPuHDGcbDGB0VvhwjHec60fCWUERf9rGAme99MseeAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2080fe645f2a83b33841596d835fdd08f0eecff33a253f4fdb34317c60569a0","last_reissued_at":"2026-05-26T01:03:33.344211Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T01:03:33.344211Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Diagonal parity and loop toggling for symmetric matrices over $\\mathbb F_2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Symmetric matrices over F₂ always have their diagonal vector in the column space, with solutions obeying a rank parity rule.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mohsen Aliabadi","submitted_at":"2026-05-11T15:57:52Z","abstract_excerpt":"Let $M$ be a symmetric matrix over $\\mathbb F_2$, and let $\\diag(M)$ be its diagonal vector. It is known that \\[\n  \\diag(M)\\in \\Img(M). \\] Thus the affine system $Mx=\\diag(M)$ is always solvable. We strengthen this existence statement to a parity rigidity theorem: every solution satisfies \\[\n  \\diag(M)^T x\\equiv \\rank(M)\\pmod 2 . \\] For graph matrices this gives a common extension of Sutner's odd-domination theorem and Batal's parity theorem from closed-neighborhood matrices $A(G)+I$ to arbitrary partially looped graph matrices $A(G)+D$.\n  We also study how rank and nullity change when loops a"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We include a self-contained proof that diag(M) ∈ Img(M), and we prove that every solution of Mx=diag(M) satisfies diag(M)^T x ≡ rank(M) mod 2. We also give a complete rank and nullity formula for rank-one diagonal perturbations.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The matrix M is symmetric over F_2; without symmetry the inclusion diag(M) ∈ Img(M) need not hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For symmetric matrices M over F_2, diag(M) is always in the image of M and solutions satisfy a rank-parity relation, with explicit formulas for diagonal perturbations and tree-structured systems.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Symmetric matrices over F₂ always have their diagonal vector in the column space, with solutions obeying a rank parity rule.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"90a9a590d7853af5ab63a8849078b96eff0677ea07398cc3f94c187e18db79e4"},"source":{"id":"2605.11056","kind":"arxiv","version":2},"verdict":{"id":"2c16aefe-1862-44f5-b475-a7028d5a575b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T01:36:05.832952Z","strongest_claim":"We include a self-contained proof that diag(M) ∈ Img(M), and we prove that every solution of Mx=diag(M) satisfies diag(M)^T x ≡ rank(M) mod 2. We also give a complete rank and nullity formula for rank-one diagonal perturbations.","one_line_summary":"For symmetric matrices M over F_2, diag(M) is always in the image of M and solutions satisfy a rank-parity relation, with explicit formulas for diagonal perturbations and tree-structured systems.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The matrix M is symmetric over F_2; without symmetry the inclusion diag(M) ∈ Img(M) need not hold.","pith_extraction_headline":"Symmetric matrices over F₂ always have their diagonal vector in the column space, with solutions obeying a rank parity rule."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.11056/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T05:22:00.402014Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T14:36:18.189004Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T11:01:16.767204Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T09:00:04.103088Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"cacb21052649e017e51a6af998ef8e3e5e1387ce0b6c788af400b42c00e0bcef"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.11056","created_at":"2026-05-26T01:03:33.344371+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.11056v2","created_at":"2026-05-26T01:03:33.344371+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.11056","created_at":"2026-05-26T01:03:33.344371+00:00"},{"alias_kind":"pith_short_12","alias_value":"UIEA7ZSF6KUD","created_at":"2026-05-26T01:03:33.344371+00:00"},{"alias_kind":"pith_short_16","alias_value":"UIEA7ZSF6KUDWM4E","created_at":"2026-05-26T01:03:33.344371+00:00"},{"alias_kind":"pith_short_8","alias_value":"UIEA7ZSF","created_at":"2026-05-26T01:03:33.344371+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UIEA7ZSF6KUDWM4ECWLNQNP52C","json":"https://pith.science/pith/UIEA7ZSF6KUDWM4ECWLNQNP52C.json","graph_json":"https://pith.science/api/pith-number/UIEA7ZSF6KUDWM4ECWLNQNP52C/graph.json","events_json":"https://pith.science/api/pith-number/UIEA7ZSF6KUDWM4ECWLNQNP52C/events.json","paper":"https://pith.science/paper/UIEA7ZSF"},"agent_actions":{"view_html":"https://pith.science/pith/UIEA7ZSF6KUDWM4ECWLNQNP52C","download_json":"https://pith.science/pith/UIEA7ZSF6KUDWM4ECWLNQNP52C.json","view_paper":"https://pith.science/paper/UIEA7ZSF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.11056&json=true","fetch_graph":"https://pith.science/api/pith-number/UIEA7ZSF6KUDWM4ECWLNQNP52C/graph.json","fetch_events":"https://pith.science/api/pith-number/UIEA7ZSF6KUDWM4ECWLNQNP52C/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UIEA7ZSF6KUDWM4ECWLNQNP52C/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UIEA7ZSF6KUDWM4ECWLNQNP52C/action/storage_attestation","attest_author":"https://pith.science/pith/UIEA7ZSF6KUDWM4ECWLNQNP52C/action/author_attestation","sign_citation":"https://pith.science/pith/UIEA7ZSF6KUDWM4ECWLNQNP52C/action/citation_signature","submit_replication":"https://pith.science/pith/UIEA7ZSF6KUDWM4ECWLNQNP52C/action/replication_record"}},"created_at":"2026-05-26T01:03:33.344371+00:00","updated_at":"2026-05-26T01:03:33.344371+00:00"}