{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:XJOP3OTCNT4OZRP27KHK3CBTRC","short_pith_number":"pith:XJOP3OTC","canonical_record":{"source":{"id":"1010.4497","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-21T15:27:33Z","cross_cats_sorted":[],"title_canon_sha256":"48a5ec10a1d218abeb8f45942f4758bdada826835d9f51d86b90373349630d5a","abstract_canon_sha256":"ee9cac54b4bdb14645e1f091d5d1b5927d6399b40ebd6f7f5d2c912bffb1fdb3"},"schema_version":"1.0"},"canonical_sha256":"ba5cfdba626cf8ecc5fafa8ead883388b18837fa85a6acd96bc5cc2d81598b81","source":{"kind":"arxiv","id":"1010.4497","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.4497","created_at":"2026-05-18T02:55:39Z"},{"alias_kind":"arxiv_version","alias_value":"1010.4497v5","created_at":"2026-05-18T02:55:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.4497","created_at":"2026-05-18T02:55:39Z"},{"alias_kind":"pith_short_12","alias_value":"XJOP3OTCNT4O","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XJOP3OTCNT4OZRP2","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XJOP3OTC","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:XJOP3OTCNT4OZRP27KHK3CBTRC","target":"record","payload":{"canonical_record":{"source":{"id":"1010.4497","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-21T15:27:33Z","cross_cats_sorted":[],"title_canon_sha256":"48a5ec10a1d218abeb8f45942f4758bdada826835d9f51d86b90373349630d5a","abstract_canon_sha256":"ee9cac54b4bdb14645e1f091d5d1b5927d6399b40ebd6f7f5d2c912bffb1fdb3"},"schema_version":"1.0"},"canonical_sha256":"ba5cfdba626cf8ecc5fafa8ead883388b18837fa85a6acd96bc5cc2d81598b81","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:39.583130Z","signature_b64":"dBOszsI6IBZeHSU9JA67EFsS0onOguijkEfGqkjYgK7LR8XctkSxEUG6c5wEIPt8DmX+seZ5cha73EaStyTiDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba5cfdba626cf8ecc5fafa8ead883388b18837fa85a6acd96bc5cc2d81598b81","last_reissued_at":"2026-05-18T02:55:39.582489Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:39.582489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1010.4497","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:55:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uApp8jjkHfWmGa2YRDyY0iTMjUNThbajc0qyGTzmGkWiLgOOBeH2d/+juLhoKohfiBa1lxgc+EiJQ2J6e/2SBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T00:44:42.518739Z"},"content_sha256":"3e529a7c4c903d4a61fd1f8906b4085c00894b5624ec88f36c9d2a6b07414b00","schema_version":"1.0","event_id":"sha256:3e529a7c4c903d4a61fd1f8906b4085c00894b5624ec88f36c9d2a6b07414b00"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:XJOP3OTCNT4OZRP27KHK3CBTRC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nullity and Loop Complementation for Delta-Matroids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hendrik Jan Hoogeboom, Robert Brijder","submitted_at":"2010-10-21T15:27:33Z","abstract_excerpt":"We show that the symmetric difference distance measure for set systems, and more specifically for delta-matroids, corresponds to the notion of nullity for symmetric and skew-symmetric matrices. In particular, as graphs (i.e., symmetric matrices over GF(2)) may be seen as a special class of delta-matroids, this distance measure generalizes the notion of nullity in this case. We characterize delta-matroids in terms of equicardinality of minimal sets with respect to inclusion (in addition we obtain similar characterizations for matroids). In this way, we find that, e.g., the delta-matroids obtain"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.4497","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T02:55:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kuTuQya6uYsP1x9WZgyWGYDokp9y8M2mSBv8sCne34YuYM09eopK6kDRz0g9ZmwS9Awlk0PyQ9AfOrhOyTVsDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T00:44:42.519083Z"},"content_sha256":"061d7f482929e1d114e3beebc4bdbdecebb9b2d5486a0094821fdda407f87beb","schema_version":"1.0","event_id":"sha256:061d7f482929e1d114e3beebc4bdbdecebb9b2d5486a0094821fdda407f87beb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XJOP3OTCNT4OZRP27KHK3CBTRC/bundle.json","state_url":"https://pith.science/pith/XJOP3OTCNT4OZRP27KHK3CBTRC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XJOP3OTCNT4OZRP27KHK3CBTRC/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":"2026-06-21T00:44:42Z","links":{"resolver":"https://pith.science/pith/XJOP3OTCNT4OZRP27KHK3CBTRC","bundle":"https://pith.science/pith/XJOP3OTCNT4OZRP27KHK3CBTRC/bundle.json","state":"https://pith.science/pith/XJOP3OTCNT4OZRP27KHK3CBTRC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XJOP3OTCNT4OZRP27KHK3CBTRC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:XJOP3OTCNT4OZRP27KHK3CBTRC","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":"ee9cac54b4bdb14645e1f091d5d1b5927d6399b40ebd6f7f5d2c912bffb1fdb3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-21T15:27:33Z","title_canon_sha256":"48a5ec10a1d218abeb8f45942f4758bdada826835d9f51d86b90373349630d5a"},"schema_version":"1.0","source":{"id":"1010.4497","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.4497","created_at":"2026-05-18T02:55:39Z"},{"alias_kind":"arxiv_version","alias_value":"1010.4497v5","created_at":"2026-05-18T02:55:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.4497","created_at":"2026-05-18T02:55:39Z"},{"alias_kind":"pith_short_12","alias_value":"XJOP3OTCNT4O","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XJOP3OTCNT4OZRP2","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XJOP3OTC","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:061d7f482929e1d114e3beebc4bdbdecebb9b2d5486a0094821fdda407f87beb","target":"graph","created_at":"2026-05-18T02:55:39Z","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"},"paper":{"abstract_excerpt":"We show that the symmetric difference distance measure for set systems, and more specifically for delta-matroids, corresponds to the notion of nullity for symmetric and skew-symmetric matrices. In particular, as graphs (i.e., symmetric matrices over GF(2)) may be seen as a special class of delta-matroids, this distance measure generalizes the notion of nullity in this case. We characterize delta-matroids in terms of equicardinality of minimal sets with respect to inclusion (in addition we obtain similar characterizations for matroids). In this way, we find that, e.g., the delta-matroids obtain","authors_text":"Hendrik Jan Hoogeboom, Robert Brijder","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-21T15:27:33Z","title":"Nullity and Loop Complementation for Delta-Matroids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.4497","kind":"arxiv","version":5},"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":"sha256:3e529a7c4c903d4a61fd1f8906b4085c00894b5624ec88f36c9d2a6b07414b00","target":"record","created_at":"2026-05-18T02:55:39Z","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":"ee9cac54b4bdb14645e1f091d5d1b5927d6399b40ebd6f7f5d2c912bffb1fdb3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-21T15:27:33Z","title_canon_sha256":"48a5ec10a1d218abeb8f45942f4758bdada826835d9f51d86b90373349630d5a"},"schema_version":"1.0","source":{"id":"1010.4497","kind":"arxiv","version":5}},"canonical_sha256":"ba5cfdba626cf8ecc5fafa8ead883388b18837fa85a6acd96bc5cc2d81598b81","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ba5cfdba626cf8ecc5fafa8ead883388b18837fa85a6acd96bc5cc2d81598b81","first_computed_at":"2026-05-18T02:55:39.582489Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:55:39.582489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dBOszsI6IBZeHSU9JA67EFsS0onOguijkEfGqkjYgK7LR8XctkSxEUG6c5wEIPt8DmX+seZ5cha73EaStyTiDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:55:39.583130Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.4497","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e529a7c4c903d4a61fd1f8906b4085c00894b5624ec88f36c9d2a6b07414b00","sha256:061d7f482929e1d114e3beebc4bdbdecebb9b2d5486a0094821fdda407f87beb"],"state_sha256":"5172a204b6ca4b3db9b063bf34084e37f36cfc89f1b8e85bf0c862ba1147792d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l5ir70y8ZYCexiKVRWwfu5jD1Wvze07M5VDvcWD0OWob/nkQquVU3ZvNrvXhFg4/ohLXe1is2HI+hKxUkLraBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T00:44:42.521041Z","bundle_sha256":"bc4376c7e47e41875ffbca7a39d460d8814350c7b69237e9607ef3197d9ebd74"}}