{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:2S6YDZ3U5Z4WSXOARB4QY2UEJF","short_pith_number":"pith:2S6YDZ3U","canonical_record":{"source":{"id":"1001.4888","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2010-01-27T10:03:48Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"a429a48489c44ebc1d1155d0500eed9ab7ed8ab93abe3a7b8b6077d76141733d","abstract_canon_sha256":"af493105e78c1cf74cc4aa9c530648e2129adaaec852c8f493242eb07e8002bd"},"schema_version":"1.0"},"canonical_sha256":"d4bd81e774ee79695dc088790c6a8449770bb849c3eff3daa77e19a5a47b9411","source":{"kind":"arxiv","id":"1001.4888","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.4888","created_at":"2026-05-18T02:24:39Z"},{"alias_kind":"arxiv_version","alias_value":"1001.4888v2","created_at":"2026-05-18T02:24:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.4888","created_at":"2026-05-18T02:24:39Z"},{"alias_kind":"pith_short_12","alias_value":"2S6YDZ3U5Z4W","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"2S6YDZ3U5Z4WSXOA","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"2S6YDZ3U","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:2S6YDZ3U5Z4WSXOARB4QY2UEJF","target":"record","payload":{"canonical_record":{"source":{"id":"1001.4888","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2010-01-27T10:03:48Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"a429a48489c44ebc1d1155d0500eed9ab7ed8ab93abe3a7b8b6077d76141733d","abstract_canon_sha256":"af493105e78c1cf74cc4aa9c530648e2129adaaec852c8f493242eb07e8002bd"},"schema_version":"1.0"},"canonical_sha256":"d4bd81e774ee79695dc088790c6a8449770bb849c3eff3daa77e19a5a47b9411","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:39.416328Z","signature_b64":"XChu1x7HfyDOswVhzigqj6QqWwnMcE1n6AYTjtuZmLKjLVe7MHeYy436OIrdVDA4iPcRvdmwGJpN7l4kTxgZBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4bd81e774ee79695dc088790c6a8449770bb849c3eff3daa77e19a5a47b9411","last_reissued_at":"2026-05-18T02:24:39.415872Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:39.415872Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1001.4888","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:24:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/Pghczcrga/dQuiMK74H/e4mnRDS3CkVnoEbnWf2BuMG4+NUwXGrH0secPhRv27cqUWTaiBPEAmBR77ldX13AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T20:28:52.661059Z"},"content_sha256":"db83cba4a533a796ce216c11918e8c48cd331f350fd00b2487016e745f04da59","schema_version":"1.0","event_id":"sha256:db83cba4a533a796ce216c11918e8c48cd331f350fd00b2487016e745f04da59"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:2S6YDZ3U5Z4WSXOARB4QY2UEJF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multiple gaps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.LO","authors_text":"Antonio Avil\\'es, Stevo Todorcevic","submitted_at":"2010-01-27T10:03:48Z","abstract_excerpt":"We consider the notion of multiple gap as a finite set of ideals that cannot be separated. We study the different types of such objects that can be found in the Boolean algebra of subsets of the natural numbers modulo finite sets."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.4888","kind":"arxiv","version":2},"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":"2026-05-18T02:24:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2wPL0P75+i5Szn+5YI12QB/YPyAtqPuRJCeR9DMf36bIJFttNMOxednsVkK6GouprxO3QYMHWUtBDfJl0w81Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T20:28:52.661516Z"},"content_sha256":"cf32a672df769469928cedb2fa255ea50d75142773d3d4b36b61cc729bebf77a","schema_version":"1.0","event_id":"sha256:cf32a672df769469928cedb2fa255ea50d75142773d3d4b36b61cc729bebf77a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2S6YDZ3U5Z4WSXOARB4QY2UEJF/bundle.json","state_url":"https://pith.science/pith/2S6YDZ3U5Z4WSXOARB4QY2UEJF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2S6YDZ3U5Z4WSXOARB4QY2UEJF/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":"2026-05-28T20:28:52Z","links":{"resolver":"https://pith.science/pith/2S6YDZ3U5Z4WSXOARB4QY2UEJF","bundle":"https://pith.science/pith/2S6YDZ3U5Z4WSXOARB4QY2UEJF/bundle.json","state":"https://pith.science/pith/2S6YDZ3U5Z4WSXOARB4QY2UEJF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2S6YDZ3U5Z4WSXOARB4QY2UEJF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:2S6YDZ3U5Z4WSXOARB4QY2UEJF","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":"af493105e78c1cf74cc4aa9c530648e2129adaaec852c8f493242eb07e8002bd","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2010-01-27T10:03:48Z","title_canon_sha256":"a429a48489c44ebc1d1155d0500eed9ab7ed8ab93abe3a7b8b6077d76141733d"},"schema_version":"1.0","source":{"id":"1001.4888","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.4888","created_at":"2026-05-18T02:24:39Z"},{"alias_kind":"arxiv_version","alias_value":"1001.4888v2","created_at":"2026-05-18T02:24:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.4888","created_at":"2026-05-18T02:24:39Z"},{"alias_kind":"pith_short_12","alias_value":"2S6YDZ3U5Z4W","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"2S6YDZ3U5Z4WSXOA","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"2S6YDZ3U","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:cf32a672df769469928cedb2fa255ea50d75142773d3d4b36b61cc729bebf77a","target":"graph","created_at":"2026-05-18T02:24:39Z","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 consider the notion of multiple gap as a finite set of ideals that cannot be separated. We study the different types of such objects that can be found in the Boolean algebra of subsets of the natural numbers modulo finite sets.","authors_text":"Antonio Avil\\'es, Stevo Todorcevic","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2010-01-27T10:03:48Z","title":"Multiple gaps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.4888","kind":"arxiv","version":2},"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:db83cba4a533a796ce216c11918e8c48cd331f350fd00b2487016e745f04da59","target":"record","created_at":"2026-05-18T02:24:39Z","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":"af493105e78c1cf74cc4aa9c530648e2129adaaec852c8f493242eb07e8002bd","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2010-01-27T10:03:48Z","title_canon_sha256":"a429a48489c44ebc1d1155d0500eed9ab7ed8ab93abe3a7b8b6077d76141733d"},"schema_version":"1.0","source":{"id":"1001.4888","kind":"arxiv","version":2}},"canonical_sha256":"d4bd81e774ee79695dc088790c6a8449770bb849c3eff3daa77e19a5a47b9411","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d4bd81e774ee79695dc088790c6a8449770bb849c3eff3daa77e19a5a47b9411","first_computed_at":"2026-05-18T02:24:39.415872Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:24:39.415872Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XChu1x7HfyDOswVhzigqj6QqWwnMcE1n6AYTjtuZmLKjLVe7MHeYy436OIrdVDA4iPcRvdmwGJpN7l4kTxgZBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:24:39.416328Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.4888","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:db83cba4a533a796ce216c11918e8c48cd331f350fd00b2487016e745f04da59","sha256:cf32a672df769469928cedb2fa255ea50d75142773d3d4b36b61cc729bebf77a"],"state_sha256":"ffa8a4eb57c20859bbcead88c9155fd0bbc130391f36bacd1841d2660069c76c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ner3RrpDuX/G3/f8j7OoiNcvFwfTbLsc0cmIT93hyNNQ9N4kAWA1DIAxsY2YcG44gMNPeydklEE1tL4MrSTaDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T20:28:52.665649Z","bundle_sha256":"4f8a99c7d34398ff67f195f17cab263f916a01eebcc6c8f31fed8a48cc4ec534"}}