{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:7AUKQOLKEHCQK2OPNSAOANGFDV","short_pith_number":"pith:7AUKQOLK","canonical_record":{"source":{"id":"2409.04821","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-09-07T13:26:48Z","cross_cats_sorted":["cs.DM","cs.DS"],"title_canon_sha256":"35e2b57768d81a64f265c2acc6fc0e2c5e252568a896f068ee1fa87a610e73aa","abstract_canon_sha256":"526dfc995c4f11e6f5267dedf2811e8813093227046bce540b526078e336fd35"},"schema_version":"1.0"},"canonical_sha256":"f828a8396a21c50569cf6c80e034c51d7430bbfc321a65d002952ba7e3f78966","source":{"kind":"arxiv","id":"2409.04821","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.04821","created_at":"2026-07-05T09:04:18Z"},{"alias_kind":"arxiv_version","alias_value":"2409.04821v1","created_at":"2026-07-05T09:04:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.04821","created_at":"2026-07-05T09:04:18Z"},{"alias_kind":"pith_short_12","alias_value":"7AUKQOLKEHCQ","created_at":"2026-07-05T09:04:18Z"},{"alias_kind":"pith_short_16","alias_value":"7AUKQOLKEHCQK2OP","created_at":"2026-07-05T09:04:18Z"},{"alias_kind":"pith_short_8","alias_value":"7AUKQOLK","created_at":"2026-07-05T09:04:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:7AUKQOLKEHCQK2OPNSAOANGFDV","target":"record","payload":{"canonical_record":{"source":{"id":"2409.04821","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-09-07T13:26:48Z","cross_cats_sorted":["cs.DM","cs.DS"],"title_canon_sha256":"35e2b57768d81a64f265c2acc6fc0e2c5e252568a896f068ee1fa87a610e73aa","abstract_canon_sha256":"526dfc995c4f11e6f5267dedf2811e8813093227046bce540b526078e336fd35"},"schema_version":"1.0"},"canonical_sha256":"f828a8396a21c50569cf6c80e034c51d7430bbfc321a65d002952ba7e3f78966","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:04:18.474244Z","signature_b64":"ec12OtAYK0qp1KkmQKUakxmbPs7E1IcymwfPXkclhEW3CVFtnQ91s2zUqL7gaK30F0iK8M90W0UnuNw/78mGAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f828a8396a21c50569cf6c80e034c51d7430bbfc321a65d002952ba7e3f78966","last_reissued_at":"2026-07-05T09:04:18.473897Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:04:18.473897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2409.04821","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":"2026-07-05T09:04:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AKfR4J+Nyl6wPGWskpWRQkeesmwtBmsi/tl8llXqD09nVqy8mZuexK1pMKAsl1zp7evnm/RBsk7CmSe4faSJBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T17:07:52.227578Z"},"content_sha256":"07ede18ef9f64e0afafa7712f7a5ee88c060b2db96a72dbb9c9d1faeec1ea0eb","schema_version":"1.0","event_id":"sha256:07ede18ef9f64e0afafa7712f7a5ee88c060b2db96a72dbb9c9d1faeec1ea0eb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:7AUKQOLKEHCQK2OPNSAOANGFDV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Adjacency Labeling Schemes for Small Classes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","cs.DS"],"primary_cat":"math.CO","authors_text":"\\'Edouard Bonnet, John Sylvester, Julien Duron, Viktor Zamaraev","submitted_at":"2024-09-07T13:26:48Z","abstract_excerpt":"A graph class admits an implicit representation if, for every positive integer $n$, its $n$-vertex graphs have a $O(\\log n)$-bit (adjacency) labeling scheme, i.e., their vertices can be labeled by binary strings of length $O(\\log n)$ such that the presence of an edge between any pair of vertices can be deduced solely from their labels. The famous Implicit Graph Conjecture posited that every hereditary (i.e., closed under taking induced subgraphs) factorial (i.e., containing $2^{O(n \\log n)}$ $n$-vertex graphs) class admits an implicit representation. The conjecture was recently refuted [Hatami"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.04821","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/2409.04821/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":"2026-07-05T09:04:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OUufhSCnGixcXXVz0v0fqW0gBHeRTGhjYdgfSP0ZUHgo2D/ryvXjZnp7oj2/mrT/DR0pyhP8OWMwSPRIOVV0BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T17:07:52.228076Z"},"content_sha256":"73646722ed93a7646cb0877dc317716e6d551c2f901103e0a8c5817a5248b482","schema_version":"1.0","event_id":"sha256:73646722ed93a7646cb0877dc317716e6d551c2f901103e0a8c5817a5248b482"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7AUKQOLKEHCQK2OPNSAOANGFDV/bundle.json","state_url":"https://pith.science/pith/7AUKQOLKEHCQK2OPNSAOANGFDV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7AUKQOLKEHCQK2OPNSAOANGFDV/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":"2026-08-12T17:07:52Z","links":{"resolver":"https://pith.science/pith/7AUKQOLKEHCQK2OPNSAOANGFDV","bundle":"https://pith.science/pith/7AUKQOLKEHCQK2OPNSAOANGFDV/bundle.json","state":"https://pith.science/pith/7AUKQOLKEHCQK2OPNSAOANGFDV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7AUKQOLKEHCQK2OPNSAOANGFDV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:7AUKQOLKEHCQK2OPNSAOANGFDV","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":"526dfc995c4f11e6f5267dedf2811e8813093227046bce540b526078e336fd35","cross_cats_sorted":["cs.DM","cs.DS"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-09-07T13:26:48Z","title_canon_sha256":"35e2b57768d81a64f265c2acc6fc0e2c5e252568a896f068ee1fa87a610e73aa"},"schema_version":"1.0","source":{"id":"2409.04821","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.04821","created_at":"2026-07-05T09:04:18Z"},{"alias_kind":"arxiv_version","alias_value":"2409.04821v1","created_at":"2026-07-05T09:04:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.04821","created_at":"2026-07-05T09:04:18Z"},{"alias_kind":"pith_short_12","alias_value":"7AUKQOLKEHCQ","created_at":"2026-07-05T09:04:18Z"},{"alias_kind":"pith_short_16","alias_value":"7AUKQOLKEHCQK2OP","created_at":"2026-07-05T09:04:18Z"},{"alias_kind":"pith_short_8","alias_value":"7AUKQOLK","created_at":"2026-07-05T09:04:18Z"}],"graph_snapshots":[{"event_id":"sha256:73646722ed93a7646cb0877dc317716e6d551c2f901103e0a8c5817a5248b482","target":"graph","created_at":"2026-07-05T09:04:18Z","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/2409.04821/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A graph class admits an implicit representation if, for every positive integer $n$, its $n$-vertex graphs have a $O(\\log n)$-bit (adjacency) labeling scheme, i.e., their vertices can be labeled by binary strings of length $O(\\log n)$ such that the presence of an edge between any pair of vertices can be deduced solely from their labels. The famous Implicit Graph Conjecture posited that every hereditary (i.e., closed under taking induced subgraphs) factorial (i.e., containing $2^{O(n \\log n)}$ $n$-vertex graphs) class admits an implicit representation. The conjecture was recently refuted [Hatami","authors_text":"\\'Edouard Bonnet, John Sylvester, Julien Duron, Viktor Zamaraev","cross_cats":["cs.DM","cs.DS"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-09-07T13:26:48Z","title":"Adjacency Labeling Schemes for Small Classes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.04821","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":"sha256:07ede18ef9f64e0afafa7712f7a5ee88c060b2db96a72dbb9c9d1faeec1ea0eb","target":"record","created_at":"2026-07-05T09:04:18Z","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":"526dfc995c4f11e6f5267dedf2811e8813093227046bce540b526078e336fd35","cross_cats_sorted":["cs.DM","cs.DS"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-09-07T13:26:48Z","title_canon_sha256":"35e2b57768d81a64f265c2acc6fc0e2c5e252568a896f068ee1fa87a610e73aa"},"schema_version":"1.0","source":{"id":"2409.04821","kind":"arxiv","version":1}},"canonical_sha256":"f828a8396a21c50569cf6c80e034c51d7430bbfc321a65d002952ba7e3f78966","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f828a8396a21c50569cf6c80e034c51d7430bbfc321a65d002952ba7e3f78966","first_computed_at":"2026-07-05T09:04:18.473897Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:04:18.473897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ec12OtAYK0qp1KkmQKUakxmbPs7E1IcymwfPXkclhEW3CVFtnQ91s2zUqL7gaK30F0iK8M90W0UnuNw/78mGAA==","signature_status":"signed_v1","signed_at":"2026-07-05T09:04:18.474244Z","signed_message":"canonical_sha256_bytes"},"source_id":"2409.04821","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:07ede18ef9f64e0afafa7712f7a5ee88c060b2db96a72dbb9c9d1faeec1ea0eb","sha256:73646722ed93a7646cb0877dc317716e6d551c2f901103e0a8c5817a5248b482"],"state_sha256":"0e8edc7bc6fe4a4de0fba39a4b25374e4101c1aafaaf5b7c9dcda51427ee1e81"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1lfGey/cCunkpAk4G4j5/oHi+QR6lKJUbRfrmZ7wIHbnGstxrxkkZrQYbNiIQ69xPDZkWMtnN2MhnsJVAW4hDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T17:07:52.232350Z","bundle_sha256":"07806c6e2544f4819ea64073177f2db4c99c8c52e9ead84fd4c8b84fc70bd4bd"}}