{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:3EWNLTM3L6DLLWSL6KFS7O67NM","short_pith_number":"pith:3EWNLTM3","canonical_record":{"source":{"id":"0901.0240","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-01-02T14:42:03Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a7d6e63d8f73916de82fb063281f2783beb91a022cddca1106eaece069b23385","abstract_canon_sha256":"cec8069c1cdee9acbc6462b84990fe9759bba771f61a643f4a157804af6c23d9"},"schema_version":"1.0"},"canonical_sha256":"d92cd5cd9b5f86b5da4bf28b2fbbdf6b07a70820b36d0567a33ad08519f2486a","source":{"kind":"arxiv","id":"0901.0240","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0901.0240","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"arxiv_version","alias_value":"0901.0240v3","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.0240","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"pith_short_12","alias_value":"3EWNLTM3L6DL","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"3EWNLTM3L6DLLWSL","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"3EWNLTM3","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:3EWNLTM3L6DLLWSL6KFS7O67NM","target":"record","payload":{"canonical_record":{"source":{"id":"0901.0240","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-01-02T14:42:03Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a7d6e63d8f73916de82fb063281f2783beb91a022cddca1106eaece069b23385","abstract_canon_sha256":"cec8069c1cdee9acbc6462b84990fe9759bba771f61a643f4a157804af6c23d9"},"schema_version":"1.0"},"canonical_sha256":"d92cd5cd9b5f86b5da4bf28b2fbbdf6b07a70820b36d0567a33ad08519f2486a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:43.206332Z","signature_b64":"6Hi5XCf1VDaLOwKQEFMmHPaR6j0CY3krfs2W7+1og/VLhEwcQrScoi6aHl++HwKPcm5x3wl9ZJJEacaOgVRvDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d92cd5cd9b5f86b5da4bf28b2fbbdf6b07a70820b36d0567a33ad08519f2486a","last_reissued_at":"2026-05-18T04:12:43.205698Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:43.205698Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0901.0240","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:12:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9s3gY8Z+Zw4SsrFoR+or/zq0DTTKTdj3GdL4hjOEpLj0SiLI06ZMTXSHTQ+ZRwCy1gGbXQRNYDcCGKvbbJ6zBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T13:13:23.850445Z"},"content_sha256":"63092d6ef0be50f29798f20628e2dca18990aaef6c3dab9d59f6e87c02d06546","schema_version":"1.0","event_id":"sha256:63092d6ef0be50f29798f20628e2dca18990aaef6c3dab9d59f6e87c02d06546"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:3EWNLTM3L6DLLWSL6KFS7O67NM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Order-invariant measures on causal sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Graham Brightwell, Malwina Luczak","submitted_at":"2009-01-02T14:42:03Z","abstract_excerpt":"A causal set is a partially ordered set on a countably infinite ground-set such that each element is above finitely many others. A natural extension of a causal set is an enumeration of its elements which respects the order. We bring together two different classes of random processes. In one class, we are given a fixed causal set, and we consider random natural extensions of this causal set: we think of the random enumeration as being generated one point at a time. In the other class of processes, we generate a random causal set, working from the bottom up, adding one new maximal element at ea"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.0240","kind":"arxiv","version":3},"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":"2026-05-18T04:12:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"URh+ZmvW75vmI9glepywnDWvrrX3z9UbAYvRODT8gxPFhEN7ezOaCnduMBP6J+2t6ejJ2BRhX3eONZRf/sOFAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T13:13:23.850786Z"},"content_sha256":"6de60484beb0809c2214b1876ee14b87554515c134dfbc9bd18e2050f81a6433","schema_version":"1.0","event_id":"sha256:6de60484beb0809c2214b1876ee14b87554515c134dfbc9bd18e2050f81a6433"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3EWNLTM3L6DLLWSL6KFS7O67NM/bundle.json","state_url":"https://pith.science/pith/3EWNLTM3L6DLLWSL6KFS7O67NM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3EWNLTM3L6DLLWSL6KFS7O67NM/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":"2026-07-03T13:13:23Z","links":{"resolver":"https://pith.science/pith/3EWNLTM3L6DLLWSL6KFS7O67NM","bundle":"https://pith.science/pith/3EWNLTM3L6DLLWSL6KFS7O67NM/bundle.json","state":"https://pith.science/pith/3EWNLTM3L6DLLWSL6KFS7O67NM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3EWNLTM3L6DLLWSL6KFS7O67NM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:3EWNLTM3L6DLLWSL6KFS7O67NM","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":"cec8069c1cdee9acbc6462b84990fe9759bba771f61a643f4a157804af6c23d9","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-01-02T14:42:03Z","title_canon_sha256":"a7d6e63d8f73916de82fb063281f2783beb91a022cddca1106eaece069b23385"},"schema_version":"1.0","source":{"id":"0901.0240","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0901.0240","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"arxiv_version","alias_value":"0901.0240v3","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.0240","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"pith_short_12","alias_value":"3EWNLTM3L6DL","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"3EWNLTM3L6DLLWSL","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"3EWNLTM3","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:6de60484beb0809c2214b1876ee14b87554515c134dfbc9bd18e2050f81a6433","target":"graph","created_at":"2026-05-18T04:12:43Z","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":"A causal set is a partially ordered set on a countably infinite ground-set such that each element is above finitely many others. A natural extension of a causal set is an enumeration of its elements which respects the order. We bring together two different classes of random processes. In one class, we are given a fixed causal set, and we consider random natural extensions of this causal set: we think of the random enumeration as being generated one point at a time. In the other class of processes, we generate a random causal set, working from the bottom up, adding one new maximal element at ea","authors_text":"Graham Brightwell, Malwina Luczak","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-01-02T14:42:03Z","title":"Order-invariant measures on causal sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.0240","kind":"arxiv","version":3},"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":"sha256:63092d6ef0be50f29798f20628e2dca18990aaef6c3dab9d59f6e87c02d06546","target":"record","created_at":"2026-05-18T04:12:43Z","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":"cec8069c1cdee9acbc6462b84990fe9759bba771f61a643f4a157804af6c23d9","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-01-02T14:42:03Z","title_canon_sha256":"a7d6e63d8f73916de82fb063281f2783beb91a022cddca1106eaece069b23385"},"schema_version":"1.0","source":{"id":"0901.0240","kind":"arxiv","version":3}},"canonical_sha256":"d92cd5cd9b5f86b5da4bf28b2fbbdf6b07a70820b36d0567a33ad08519f2486a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d92cd5cd9b5f86b5da4bf28b2fbbdf6b07a70820b36d0567a33ad08519f2486a","first_computed_at":"2026-05-18T04:12:43.205698Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:43.205698Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6Hi5XCf1VDaLOwKQEFMmHPaR6j0CY3krfs2W7+1og/VLhEwcQrScoi6aHl++HwKPcm5x3wl9ZJJEacaOgVRvDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:43.206332Z","signed_message":"canonical_sha256_bytes"},"source_id":"0901.0240","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:63092d6ef0be50f29798f20628e2dca18990aaef6c3dab9d59f6e87c02d06546","sha256:6de60484beb0809c2214b1876ee14b87554515c134dfbc9bd18e2050f81a6433"],"state_sha256":"345c3a6a3b970451e46c03c8c25c7c5ff95ea36340cafabc4ba0053819988be7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Jgr6vSb8lRk7bSHwrGjedki2D+5v4ju5E+LDOX9FEEz3xFgggGgl2x+TM4icxvEPjpUtwJtfk1x1f7YE3/iKBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T13:13:23.852688Z","bundle_sha256":"cf551bd8edbb1bf3ee0ebf0f89f9ff202532137ee24e8263b8d807aa163b401d"}}