{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:A2KYCGC3M3OCHS3M55R2TJRW6X","short_pith_number":"pith:A2KYCGC3","canonical_record":{"source":{"id":"1110.4824","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-21T15:27:27Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"675b6b9f4029e45e08f4bc7bab721da1647c9ee90acff08b11c48e61cd689214","abstract_canon_sha256":"90cc194b91bfa456581fd5c7bed4061b2dc9c98448bbffd91630a27a9b99b4f1"},"schema_version":"1.0"},"canonical_sha256":"069581185b66dc23cb6cef63a9a636f5cc1738cb5094ef0b3b22f93fccbf6bb3","source":{"kind":"arxiv","id":"1110.4824","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.4824","created_at":"2026-05-18T03:50:21Z"},{"alias_kind":"arxiv_version","alias_value":"1110.4824v1","created_at":"2026-05-18T03:50:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.4824","created_at":"2026-05-18T03:50:21Z"},{"alias_kind":"pith_short_12","alias_value":"A2KYCGC3M3OC","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"A2KYCGC3M3OCHS3M","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"A2KYCGC3","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:A2KYCGC3M3OCHS3M55R2TJRW6X","target":"record","payload":{"canonical_record":{"source":{"id":"1110.4824","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-21T15:27:27Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"675b6b9f4029e45e08f4bc7bab721da1647c9ee90acff08b11c48e61cd689214","abstract_canon_sha256":"90cc194b91bfa456581fd5c7bed4061b2dc9c98448bbffd91630a27a9b99b4f1"},"schema_version":"1.0"},"canonical_sha256":"069581185b66dc23cb6cef63a9a636f5cc1738cb5094ef0b3b22f93fccbf6bb3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:21.991653Z","signature_b64":"511s4Lq76G2+U05kYX/j/UoIjaBUWqKcD0hkZ2B+uHWO9Y2IJBMhGPX9/pY/sddI5mmhpQzNmI6yQGKXurupDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"069581185b66dc23cb6cef63a9a636f5cc1738cb5094ef0b3b22f93fccbf6bb3","last_reissued_at":"2026-05-18T03:50:21.990822Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:21.990822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.4824","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":"2026-05-18T03:50:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w+6bGuwua7a2Cg3TTrepOa12lGu26jdsA65qHkKQx5wVBxnYUp6lGDVlYcxfcFGDbn6qO83fgMNuPd5dlXsmDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T18:52:55.852687Z"},"content_sha256":"49e2977cc88c6aad786f08ab0885c74721682bbe7d6f0c203f66c09a446e5e41","schema_version":"1.0","event_id":"sha256:49e2977cc88c6aad786f08ab0885c74721682bbe7d6f0c203f66c09a446e5e41"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:A2KYCGC3M3OCHS3M55R2TJRW6X","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Improved lower bounds for the 2-page crossing numbers of K_{m,n} and K_n via semidefinite programming","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.CO","authors_text":"Dmitrii V. Pasechnik, Etienne de Klerk","submitted_at":"2011-10-21T15:27:27Z","abstract_excerpt":"It has been long conjectured that the crossing numbers of the complete bipartite graph K_{m,n} and of the complete graph K_n equal Z(m,n) (the value conjectured by Zarankiewicz, who came up with a drawing reaching this value) and Z(n) :=Z(n,n-2)/4, respectively.\n  In a 2-page drawing of a graph, the vertices are drawn on a straight line (the spine), and each edge is contained in one of the half-planes of the spine. The 2-page crossing number v_2(G) of a graph G is the minimum number of crossings in a 2-page drawing of G. Somewhat surprisingly, there are 2-page drawings of K_{m,n} (respectively"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.4824","kind":"arxiv","version":1},"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":"2026-05-18T03:50:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"taIFQX6joFsF93P/1UK+0gQlJ5eDQvhHEot6DZrkA9ira4ZPY8zj3tbBFjPh0pi4kvn1fchNeUb1YjlJFdvzDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T18:52:55.853031Z"},"content_sha256":"c83366e33d7e48bb48dac7dd7567f6803e54b1ded324fc3786887a697da5713d","schema_version":"1.0","event_id":"sha256:c83366e33d7e48bb48dac7dd7567f6803e54b1ded324fc3786887a697da5713d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A2KYCGC3M3OCHS3M55R2TJRW6X/bundle.json","state_url":"https://pith.science/pith/A2KYCGC3M3OCHS3M55R2TJRW6X/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A2KYCGC3M3OCHS3M55R2TJRW6X/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":"2026-06-25T18:52:55Z","links":{"resolver":"https://pith.science/pith/A2KYCGC3M3OCHS3M55R2TJRW6X","bundle":"https://pith.science/pith/A2KYCGC3M3OCHS3M55R2TJRW6X/bundle.json","state":"https://pith.science/pith/A2KYCGC3M3OCHS3M55R2TJRW6X/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A2KYCGC3M3OCHS3M55R2TJRW6X/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:A2KYCGC3M3OCHS3M55R2TJRW6X","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":"90cc194b91bfa456581fd5c7bed4061b2dc9c98448bbffd91630a27a9b99b4f1","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-21T15:27:27Z","title_canon_sha256":"675b6b9f4029e45e08f4bc7bab721da1647c9ee90acff08b11c48e61cd689214"},"schema_version":"1.0","source":{"id":"1110.4824","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.4824","created_at":"2026-05-18T03:50:21Z"},{"alias_kind":"arxiv_version","alias_value":"1110.4824v1","created_at":"2026-05-18T03:50:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.4824","created_at":"2026-05-18T03:50:21Z"},{"alias_kind":"pith_short_12","alias_value":"A2KYCGC3M3OC","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"A2KYCGC3M3OCHS3M","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"A2KYCGC3","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:c83366e33d7e48bb48dac7dd7567f6803e54b1ded324fc3786887a697da5713d","target":"graph","created_at":"2026-05-18T03:50:21Z","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":"It has been long conjectured that the crossing numbers of the complete bipartite graph K_{m,n} and of the complete graph K_n equal Z(m,n) (the value conjectured by Zarankiewicz, who came up with a drawing reaching this value) and Z(n) :=Z(n,n-2)/4, respectively.\n  In a 2-page drawing of a graph, the vertices are drawn on a straight line (the spine), and each edge is contained in one of the half-planes of the spine. The 2-page crossing number v_2(G) of a graph G is the minimum number of crossings in a 2-page drawing of G. Somewhat surprisingly, there are 2-page drawings of K_{m,n} (respectively","authors_text":"Dmitrii V. Pasechnik, Etienne de Klerk","cross_cats":["math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-21T15:27:27Z","title":"Improved lower bounds for the 2-page crossing numbers of K_{m,n} and K_n via semidefinite programming"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.4824","kind":"arxiv","version":1},"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":"sha256:49e2977cc88c6aad786f08ab0885c74721682bbe7d6f0c203f66c09a446e5e41","target":"record","created_at":"2026-05-18T03:50:21Z","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":"90cc194b91bfa456581fd5c7bed4061b2dc9c98448bbffd91630a27a9b99b4f1","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-21T15:27:27Z","title_canon_sha256":"675b6b9f4029e45e08f4bc7bab721da1647c9ee90acff08b11c48e61cd689214"},"schema_version":"1.0","source":{"id":"1110.4824","kind":"arxiv","version":1}},"canonical_sha256":"069581185b66dc23cb6cef63a9a636f5cc1738cb5094ef0b3b22f93fccbf6bb3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"069581185b66dc23cb6cef63a9a636f5cc1738cb5094ef0b3b22f93fccbf6bb3","first_computed_at":"2026-05-18T03:50:21.990822Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:50:21.990822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"511s4Lq76G2+U05kYX/j/UoIjaBUWqKcD0hkZ2B+uHWO9Y2IJBMhGPX9/pY/sddI5mmhpQzNmI6yQGKXurupDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:50:21.991653Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.4824","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:49e2977cc88c6aad786f08ab0885c74721682bbe7d6f0c203f66c09a446e5e41","sha256:c83366e33d7e48bb48dac7dd7567f6803e54b1ded324fc3786887a697da5713d"],"state_sha256":"1b0954fc13f733d02670cee52ccbed75453f4caadc4745b482b1ac16a66ba684"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xDG7Op+tYQqiC5hepez78xWC5DRwG1pMnEpWt+ZqneW6g9RmELL12PiwWZRkUEjm/LT/GDdvM7BoLJfcO5J3Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T18:52:55.854904Z","bundle_sha256":"06829d9ecd7e30f273030648d332522a21e85ec8bf0836b12ed3a4e4430e15ad"}}