{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:R4XD5XZ2EZ3MRU3ZXYCE6IMMIW","short_pith_number":"pith:R4XD5XZ2","canonical_record":{"source":{"id":"2502.02301","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-04T13:15:17Z","cross_cats_sorted":[],"title_canon_sha256":"4be1563135f2aa9bb1b0cdd466046493e24961f3e8803b6a39403dc5b9c976ba","abstract_canon_sha256":"2c922cb6a627939d4b653bf3f08afe43ab4e7c8089dd2ad770f846c58a333580"},"schema_version":"1.0"},"canonical_sha256":"8f2e3edf3a2676c8d379be044f218c45bcd16e6600c79b82c8d35f8055e62564","source":{"kind":"arxiv","id":"2502.02301","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.02301","created_at":"2026-07-05T10:09:26Z"},{"alias_kind":"arxiv_version","alias_value":"2502.02301v1","created_at":"2026-07-05T10:09:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.02301","created_at":"2026-07-05T10:09:26Z"},{"alias_kind":"pith_short_12","alias_value":"R4XD5XZ2EZ3M","created_at":"2026-07-05T10:09:26Z"},{"alias_kind":"pith_short_16","alias_value":"R4XD5XZ2EZ3MRU3Z","created_at":"2026-07-05T10:09:26Z"},{"alias_kind":"pith_short_8","alias_value":"R4XD5XZ2","created_at":"2026-07-05T10:09:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:R4XD5XZ2EZ3MRU3ZXYCE6IMMIW","target":"record","payload":{"canonical_record":{"source":{"id":"2502.02301","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-04T13:15:17Z","cross_cats_sorted":[],"title_canon_sha256":"4be1563135f2aa9bb1b0cdd466046493e24961f3e8803b6a39403dc5b9c976ba","abstract_canon_sha256":"2c922cb6a627939d4b653bf3f08afe43ab4e7c8089dd2ad770f846c58a333580"},"schema_version":"1.0"},"canonical_sha256":"8f2e3edf3a2676c8d379be044f218c45bcd16e6600c79b82c8d35f8055e62564","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:09:26.707162Z","signature_b64":"I23WHBdVpnIJPvyFERN9k9YVm6Tye9xXaCy5EN738lrkmpcpY0dp4ziVnVLOUpfMO4UIsezbsBHQB1O4WZCmAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8f2e3edf3a2676c8d379be044f218c45bcd16e6600c79b82c8d35f8055e62564","last_reissued_at":"2026-07-05T10:09:26.706728Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:09:26.706728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2502.02301","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":"2026-07-05T10:09:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"omZnFu1FdlxejLve2OzXeRnD/t+XJpLeXWmbl2WKw+WG7XjZka/naxIpBt0MluWKg4uhROwrugEO07xyWby4DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T16:42:28.719562Z"},"content_sha256":"43f0e13bc149305b3d8ff390554d36ae7d792520cf6e88f9d43519a6511741d0","schema_version":"1.0","event_id":"sha256:43f0e13bc149305b3d8ff390554d36ae7d792520cf6e88f9d43519a6511741d0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:R4XD5XZ2EZ3MRU3ZXYCE6IMMIW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a conjecture of Pach-Spencer-T\\'oth for graph crossing numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Ma, Kaizhe Chen","submitted_at":"2025-02-04T13:15:17Z","abstract_excerpt":"The crossing number of a graph $G$ denotes the minimum number of crossings in any planar drawing of $G$. In this short note, we confirm a long-standing conjecture posed by Pach, Spencer, and T\\'oth over 25 years ago, establishing an optimal lower bound on the crossing number of graphs that satisfy some monotone properties. Furthermore, we address a related open problem introduced by Pach and T\\'oth in 2000, which explores the interplay between the crossing number of a graph, its degree sequence, and its bisection width."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.02301","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.02301/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T10:09:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FKJqGC0afHIyhKDMA24kP0GqV8WHPsmPewoyMZj4B4+krO1wAQNMTySNpw31I9yOmPEHHKXtXXwMlv3aTfrtBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T16:42:28.720068Z"},"content_sha256":"cefe2d32ba15efd9467737c0f771f3a9d12bc97013cdea0ff0df15eb76185f19","schema_version":"1.0","event_id":"sha256:cefe2d32ba15efd9467737c0f771f3a9d12bc97013cdea0ff0df15eb76185f19"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/R4XD5XZ2EZ3MRU3ZXYCE6IMMIW/bundle.json","state_url":"https://pith.science/pith/R4XD5XZ2EZ3MRU3ZXYCE6IMMIW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/R4XD5XZ2EZ3MRU3ZXYCE6IMMIW/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":"2026-08-03T16:42:28Z","links":{"resolver":"https://pith.science/pith/R4XD5XZ2EZ3MRU3ZXYCE6IMMIW","bundle":"https://pith.science/pith/R4XD5XZ2EZ3MRU3ZXYCE6IMMIW/bundle.json","state":"https://pith.science/pith/R4XD5XZ2EZ3MRU3ZXYCE6IMMIW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/R4XD5XZ2EZ3MRU3ZXYCE6IMMIW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:R4XD5XZ2EZ3MRU3ZXYCE6IMMIW","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":"2c922cb6a627939d4b653bf3f08afe43ab4e7c8089dd2ad770f846c58a333580","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-04T13:15:17Z","title_canon_sha256":"4be1563135f2aa9bb1b0cdd466046493e24961f3e8803b6a39403dc5b9c976ba"},"schema_version":"1.0","source":{"id":"2502.02301","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.02301","created_at":"2026-07-05T10:09:26Z"},{"alias_kind":"arxiv_version","alias_value":"2502.02301v1","created_at":"2026-07-05T10:09:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.02301","created_at":"2026-07-05T10:09:26Z"},{"alias_kind":"pith_short_12","alias_value":"R4XD5XZ2EZ3M","created_at":"2026-07-05T10:09:26Z"},{"alias_kind":"pith_short_16","alias_value":"R4XD5XZ2EZ3MRU3Z","created_at":"2026-07-05T10:09:26Z"},{"alias_kind":"pith_short_8","alias_value":"R4XD5XZ2","created_at":"2026-07-05T10:09:26Z"}],"graph_snapshots":[{"event_id":"sha256:cefe2d32ba15efd9467737c0f771f3a9d12bc97013cdea0ff0df15eb76185f19","target":"graph","created_at":"2026-07-05T10:09:26Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2502.02301/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The crossing number of a graph $G$ denotes the minimum number of crossings in any planar drawing of $G$. In this short note, we confirm a long-standing conjecture posed by Pach, Spencer, and T\\'oth over 25 years ago, establishing an optimal lower bound on the crossing number of graphs that satisfy some monotone properties. Furthermore, we address a related open problem introduced by Pach and T\\'oth in 2000, which explores the interplay between the crossing number of a graph, its degree sequence, and its bisection width.","authors_text":"Jie Ma, Kaizhe Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-04T13:15:17Z","title":"On a conjecture of Pach-Spencer-T\\'oth for graph crossing numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.02301","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":"sha256:43f0e13bc149305b3d8ff390554d36ae7d792520cf6e88f9d43519a6511741d0","target":"record","created_at":"2026-07-05T10:09:26Z","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":"2c922cb6a627939d4b653bf3f08afe43ab4e7c8089dd2ad770f846c58a333580","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-04T13:15:17Z","title_canon_sha256":"4be1563135f2aa9bb1b0cdd466046493e24961f3e8803b6a39403dc5b9c976ba"},"schema_version":"1.0","source":{"id":"2502.02301","kind":"arxiv","version":1}},"canonical_sha256":"8f2e3edf3a2676c8d379be044f218c45bcd16e6600c79b82c8d35f8055e62564","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8f2e3edf3a2676c8d379be044f218c45bcd16e6600c79b82c8d35f8055e62564","first_computed_at":"2026-07-05T10:09:26.706728Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:09:26.706728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"I23WHBdVpnIJPvyFERN9k9YVm6Tye9xXaCy5EN738lrkmpcpY0dp4ziVnVLOUpfMO4UIsezbsBHQB1O4WZCmAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:09:26.707162Z","signed_message":"canonical_sha256_bytes"},"source_id":"2502.02301","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:43f0e13bc149305b3d8ff390554d36ae7d792520cf6e88f9d43519a6511741d0","sha256:cefe2d32ba15efd9467737c0f771f3a9d12bc97013cdea0ff0df15eb76185f19"],"state_sha256":"d93df33ec563e05d148a518d898076c4f4ab0df69b2c06a3963870f68b75a684"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d1lvYQ5A1P43dUwv0we1HNLo58PWbmKREdETmrbQ/cECGTt9E9ZMVIFSKx2jMXh8e8TTVbSnI4CGp7Pl8Vk0DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T16:42:28.724686Z","bundle_sha256":"01cb7ed6d454b611bbc6ef98994e0d4f0913f543eee64a80ef966e01260f1253"}}