{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:2E3NFEUX7Y5AT3BLMLPXTBV5WZ","short_pith_number":"pith:2E3NFEUX","canonical_record":{"source":{"id":"1603.04972","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2016-03-16T05:58:20Z","cross_cats_sorted":[],"title_canon_sha256":"08432f6143d9729c2fc0f94aa067b523f58d615868b2e9aec94c59aab4fde7c0","abstract_canon_sha256":"67af15ce35b56ced5ece8ce83e6eb092862ff9ff9ff4be15454524a00a223b4f"},"schema_version":"1.0"},"canonical_sha256":"d136d29297fe3a09ec2b62df7986bdb64f3e9460f0ed1458e61c55657f69e202","source":{"kind":"arxiv","id":"1603.04972","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.04972","created_at":"2026-05-18T01:07:20Z"},{"alias_kind":"arxiv_version","alias_value":"1603.04972v2","created_at":"2026-05-18T01:07:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.04972","created_at":"2026-05-18T01:07:20Z"},{"alias_kind":"pith_short_12","alias_value":"2E3NFEUX7Y5A","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"2E3NFEUX7Y5AT3BL","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"2E3NFEUX","created_at":"2026-05-18T12:29:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:2E3NFEUX7Y5AT3BLMLPXTBV5WZ","target":"record","payload":{"canonical_record":{"source":{"id":"1603.04972","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2016-03-16T05:58:20Z","cross_cats_sorted":[],"title_canon_sha256":"08432f6143d9729c2fc0f94aa067b523f58d615868b2e9aec94c59aab4fde7c0","abstract_canon_sha256":"67af15ce35b56ced5ece8ce83e6eb092862ff9ff9ff4be15454524a00a223b4f"},"schema_version":"1.0"},"canonical_sha256":"d136d29297fe3a09ec2b62df7986bdb64f3e9460f0ed1458e61c55657f69e202","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:20.779802Z","signature_b64":"PyA6opMvbwLPkJVOMGRSN0s0dkvR8uuhW6EQNLcSAgHn+P4oiRZZR0CGvsNBAeKjxlsyTd7cnCbzi1SFbvJoDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d136d29297fe3a09ec2b62df7986bdb64f3e9460f0ed1458e61c55657f69e202","last_reissued_at":"2026-05-18T01:07:20.779197Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:20.779197Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.04972","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":"2026-05-18T01:07:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"54J34+EhPI2vCgz602GuzCYuvXk0u/8kiC8uxkZXzc7sz2dyjZ1CcWkrfFFuokTf1LkjzhHYzCU4lMuYzVFECQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T19:07:05.517784Z"},"content_sha256":"63689e3f2bb466443f4182045faa229cdd4ed677c67f8d0b6be0605f9fb99a80","schema_version":"1.0","event_id":"sha256:63689e3f2bb466443f4182045faa229cdd4ed677c67f8d0b6be0605f9fb99a80"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:2E3NFEUX7Y5AT3BLMLPXTBV5WZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Representable posets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Rob Egrot","submitted_at":"2016-03-16T05:58:20Z","abstract_excerpt":"A poset is representable if it can be embedded in a field of sets in such a way that existing finite meets and joins become intersections and unions respectively (we say finite meets and joins are preserved). More generally, for cardinals $\\alpha$ and $\\beta$ a poset is said to be $(\\alpha,\\beta)$-representable if an embedding into a field of sets exists that preserves meets of sets smaller than $\\alpha$ and joins of sets smaller than $\\beta$. We show using an ultraproduct/ultraroot argument that when $2\\leq\\alpha,\\beta\\leq \\omega$ the class of $(\\alpha,\\beta)$-representable posets is elementa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.04972","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":"2026-05-18T01:07:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KYRoht1MLessI/g4mGPEiV1UPvwj7SQ+zFuhQ7QCO6kN+ye/Ulk00T0cDXedmSOJRQ5/q2Lktuywvl4k1/KCAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T19:07:05.518462Z"},"content_sha256":"1e71bc2f28868024663d74cd4f24a729dbe1379d52eabdfefae4354e41289530","schema_version":"1.0","event_id":"sha256:1e71bc2f28868024663d74cd4f24a729dbe1379d52eabdfefae4354e41289530"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2E3NFEUX7Y5AT3BLMLPXTBV5WZ/bundle.json","state_url":"https://pith.science/pith/2E3NFEUX7Y5AT3BLMLPXTBV5WZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2E3NFEUX7Y5AT3BLMLPXTBV5WZ/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":"2026-06-07T19:07:05Z","links":{"resolver":"https://pith.science/pith/2E3NFEUX7Y5AT3BLMLPXTBV5WZ","bundle":"https://pith.science/pith/2E3NFEUX7Y5AT3BLMLPXTBV5WZ/bundle.json","state":"https://pith.science/pith/2E3NFEUX7Y5AT3BLMLPXTBV5WZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2E3NFEUX7Y5AT3BLMLPXTBV5WZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:2E3NFEUX7Y5AT3BLMLPXTBV5WZ","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":"67af15ce35b56ced5ece8ce83e6eb092862ff9ff9ff4be15454524a00a223b4f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2016-03-16T05:58:20Z","title_canon_sha256":"08432f6143d9729c2fc0f94aa067b523f58d615868b2e9aec94c59aab4fde7c0"},"schema_version":"1.0","source":{"id":"1603.04972","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.04972","created_at":"2026-05-18T01:07:20Z"},{"alias_kind":"arxiv_version","alias_value":"1603.04972v2","created_at":"2026-05-18T01:07:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.04972","created_at":"2026-05-18T01:07:20Z"},{"alias_kind":"pith_short_12","alias_value":"2E3NFEUX7Y5A","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"2E3NFEUX7Y5AT3BL","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"2E3NFEUX","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:1e71bc2f28868024663d74cd4f24a729dbe1379d52eabdfefae4354e41289530","target":"graph","created_at":"2026-05-18T01:07:20Z","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 poset is representable if it can be embedded in a field of sets in such a way that existing finite meets and joins become intersections and unions respectively (we say finite meets and joins are preserved). More generally, for cardinals $\\alpha$ and $\\beta$ a poset is said to be $(\\alpha,\\beta)$-representable if an embedding into a field of sets exists that preserves meets of sets smaller than $\\alpha$ and joins of sets smaller than $\\beta$. We show using an ultraproduct/ultraroot argument that when $2\\leq\\alpha,\\beta\\leq \\omega$ the class of $(\\alpha,\\beta)$-representable posets is elementa","authors_text":"Rob Egrot","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2016-03-16T05:58:20Z","title":"Representable posets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.04972","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":"sha256:63689e3f2bb466443f4182045faa229cdd4ed677c67f8d0b6be0605f9fb99a80","target":"record","created_at":"2026-05-18T01:07:20Z","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":"67af15ce35b56ced5ece8ce83e6eb092862ff9ff9ff4be15454524a00a223b4f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2016-03-16T05:58:20Z","title_canon_sha256":"08432f6143d9729c2fc0f94aa067b523f58d615868b2e9aec94c59aab4fde7c0"},"schema_version":"1.0","source":{"id":"1603.04972","kind":"arxiv","version":2}},"canonical_sha256":"d136d29297fe3a09ec2b62df7986bdb64f3e9460f0ed1458e61c55657f69e202","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d136d29297fe3a09ec2b62df7986bdb64f3e9460f0ed1458e61c55657f69e202","first_computed_at":"2026-05-18T01:07:20.779197Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:07:20.779197Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PyA6opMvbwLPkJVOMGRSN0s0dkvR8uuhW6EQNLcSAgHn+P4oiRZZR0CGvsNBAeKjxlsyTd7cnCbzi1SFbvJoDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:07:20.779802Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.04972","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:63689e3f2bb466443f4182045faa229cdd4ed677c67f8d0b6be0605f9fb99a80","sha256:1e71bc2f28868024663d74cd4f24a729dbe1379d52eabdfefae4354e41289530"],"state_sha256":"18f55aa4befdd8bc424e28a9feb17551c0a5ee6f5a0a06034bacaec9cc70f635"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nUt4gAnbyxldMyJVjFo7aWt2QiPBc+KbK0Mrcga6jANKDnQLtm21BBBU8qefHshFaX19iEgmKGX2JOGuZ3VnAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T19:07:05.522241Z","bundle_sha256":"80492b9429cdde16f9c1e485300aa1c6928ebb7bcc941e48078b613d29e381f5"}}