{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:H3ZXLNUMTKEASOJT7QDEH7CIGM","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":"ddab48f83d02d0aa305224099f5ec927dd1d8822b14b2a1a319412d1d8087e18","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2008-02-28T12:27:26Z","title_canon_sha256":"ee1b2dff865687743679c04d1084e7da48cb8835661765993c33c6d99f107bec"},"schema_version":"1.0","source":{"id":"0802.4185","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0802.4185","created_at":"2026-05-18T03:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"0802.4185v1","created_at":"2026-05-18T03:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0802.4185","created_at":"2026-05-18T03:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"H3ZXLNUMTKEA","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"H3ZXLNUMTKEASOJT","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"H3ZXLNUM","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:e7c152e5b3dab3f44e4054b0caefff49b5d623fee92c964a6abcd350b6bf0462","target":"graph","created_at":"2026-05-18T03:46:32Z","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 prove a Theorem about the relationship between the Depth of the ultraproduct of Boolean algebras, divided by an ultrafilter, and the products of the depths of each component. This answers (partly) an open problem of Monk.","authors_text":"Saharon Shelah, Shimon Garti","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2008-02-28T12:27:26Z","title":"Depth of Boolean algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0802.4185","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":"sha256:72a4b03b962e3218368e078096505f4e3a9f8df9aaf6cfbaa92a955a901ac81f","target":"record","created_at":"2026-05-18T03:46:32Z","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":"ddab48f83d02d0aa305224099f5ec927dd1d8822b14b2a1a319412d1d8087e18","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2008-02-28T12:27:26Z","title_canon_sha256":"ee1b2dff865687743679c04d1084e7da48cb8835661765993c33c6d99f107bec"},"schema_version":"1.0","source":{"id":"0802.4185","kind":"arxiv","version":1}},"canonical_sha256":"3ef375b68c9a88093933fc0643fc48330569e7169cc8418731a45049a6dc0b77","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3ef375b68c9a88093933fc0643fc48330569e7169cc8418731a45049a6dc0b77","first_computed_at":"2026-05-18T03:46:32.267698Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:46:32.267698Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kJdy3apGE+K+1MKUXMEjFd0Ca4WAf6SLVRoM8ZVvkS1sP2Y+vLvKiQ4FE/P9i/e6WAycQn24hzw2SSOXHyWoCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:46:32.268512Z","signed_message":"canonical_sha256_bytes"},"source_id":"0802.4185","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:72a4b03b962e3218368e078096505f4e3a9f8df9aaf6cfbaa92a955a901ac81f","sha256:e7c152e5b3dab3f44e4054b0caefff49b5d623fee92c964a6abcd350b6bf0462"],"state_sha256":"125363bda9193c09a337c5edf22efb9386eb70ae372b6ffec8326620496ed747"}