{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1994:N6S66Y6IRDNTE6WCJ3IKXEARPC","short_pith_number":"pith:N6S66Y6I","canonical_record":{"source":{"id":"math/9412208","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"1994-12-03T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"9e2bda60bd5237625a312e4c489fd1f0717fa238bbb0c0d8c9c33f7cb4ecd63c","abstract_canon_sha256":"d06d70cd5498f251a1ab7b23ae80bc34f125ba1a425cc78f9bbde1e9bb0b319a"},"schema_version":"1.0"},"canonical_sha256":"6fa5ef63c888db327ac24ed0ab901178ad2e186b0a896fbd4fbb872be4408c41","source":{"kind":"arxiv","id":"math/9412208","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9412208","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"arxiv_version","alias_value":"math/9412208v1","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9412208","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"pith_short_12","alias_value":"N6S66Y6IRDNT","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"N6S66Y6IRDNTE6WC","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"N6S66Y6I","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1994:N6S66Y6IRDNTE6WCJ3IKXEARPC","target":"record","payload":{"canonical_record":{"source":{"id":"math/9412208","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"1994-12-03T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"9e2bda60bd5237625a312e4c489fd1f0717fa238bbb0c0d8c9c33f7cb4ecd63c","abstract_canon_sha256":"d06d70cd5498f251a1ab7b23ae80bc34f125ba1a425cc78f9bbde1e9bb0b319a"},"schema_version":"1.0"},"canonical_sha256":"6fa5ef63c888db327ac24ed0ab901178ad2e186b0a896fbd4fbb872be4408c41","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:50.775827Z","signature_b64":"z5bzjWYN7wnqfkjX5pt/ryQB1IAVhcdFF8F61Oxl+mt6KwRUH/Wb9dXP/aZTK2hW/aDoFDJTmBfeI8CNnxyzBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6fa5ef63c888db327ac24ed0ab901178ad2e186b0a896fbd4fbb872be4408c41","last_reissued_at":"2026-05-18T01:05:50.775250Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:50.775250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9412208","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":"2026-05-18T01:05:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ukIx0WL26RmKouaEFjeKQBEF/xF4V25UtnYrKZq7XUfaX/gaPogXdgNTnN0Wd+VFHtGJU+uk/9MMRLBv6lCGBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T18:02:52.198303Z"},"content_sha256":"4b5e211b0cf8a8e56775ce50bd6be8c5454b31ead7533783dc568a62dac57a24","schema_version":"1.0","event_id":"sha256:4b5e211b0cf8a8e56775ce50bd6be8c5454b31ead7533783dc568a62dac57a24"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1994:N6S66Y6IRDNTE6WCJ3IKXEARPC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Possible pcf algebras","license":"","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Saharon Shelah, Thomas Jech","submitted_at":"1994-12-03T00:00:00Z","abstract_excerpt":"There exists a family $\\{B_{\\alpha}\\}_{\\alpha<\\omega_1}$ of sets of countable ordinals such that\n o $\\max B_{\\alpha}=\\alpha$,\n o if $\\alpha\\in B_{\\beta}$ then $B_{\\alpha}\\subseteq B_{\\beta}$,\n o if $\\lambda\\leq \\alpha$ and $\\lambda$ is a limit ordinal then $B_{\\alpha}\\cap\\lambda$ is not in the ideal generated by the $B_{\\beta}$, $\\beta<\\alpha$, and by the bounded subsets of $\\lambda$,\n o there is a partition $\\{A_n\\}_{n=0}^{\\infty}$ of $\\omega_1$ such that for every $\\alpha$ and every $n,$ $B_{\\alpha}\\cap A_n$ is finite."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9412208","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":""},"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":"2026-05-18T01:05:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"obigbB0DX4sq7Y+6Lhr9uXnNY6Cx81HIvFmMFFIVh29m50AhHXrx3fiUl3ur+mXt8CqSi9o1uWEmD7dzVY76BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T18:02:52.198662Z"},"content_sha256":"1bdb29d10b54a7ee86e3d6378da5b224c3955b27a2230f1f5ab0796e5f360fac","schema_version":"1.0","event_id":"sha256:1bdb29d10b54a7ee86e3d6378da5b224c3955b27a2230f1f5ab0796e5f360fac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N6S66Y6IRDNTE6WCJ3IKXEARPC/bundle.json","state_url":"https://pith.science/pith/N6S66Y6IRDNTE6WCJ3IKXEARPC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N6S66Y6IRDNTE6WCJ3IKXEARPC/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":"2026-06-23T18:02:52Z","links":{"resolver":"https://pith.science/pith/N6S66Y6IRDNTE6WCJ3IKXEARPC","bundle":"https://pith.science/pith/N6S66Y6IRDNTE6WCJ3IKXEARPC/bundle.json","state":"https://pith.science/pith/N6S66Y6IRDNTE6WCJ3IKXEARPC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N6S66Y6IRDNTE6WCJ3IKXEARPC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1994:N6S66Y6IRDNTE6WCJ3IKXEARPC","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":"d06d70cd5498f251a1ab7b23ae80bc34f125ba1a425cc78f9bbde1e9bb0b319a","cross_cats_sorted":[],"license":"","primary_cat":"math.LO","submitted_at":"1994-12-03T00:00:00Z","title_canon_sha256":"9e2bda60bd5237625a312e4c489fd1f0717fa238bbb0c0d8c9c33f7cb4ecd63c"},"schema_version":"1.0","source":{"id":"math/9412208","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9412208","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"arxiv_version","alias_value":"math/9412208v1","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9412208","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"pith_short_12","alias_value":"N6S66Y6IRDNT","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"N6S66Y6IRDNTE6WC","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"N6S66Y6I","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:1bdb29d10b54a7ee86e3d6378da5b224c3955b27a2230f1f5ab0796e5f360fac","target":"graph","created_at":"2026-05-18T01:05:50Z","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":"There exists a family $\\{B_{\\alpha}\\}_{\\alpha<\\omega_1}$ of sets of countable ordinals such that\n o $\\max B_{\\alpha}=\\alpha$,\n o if $\\alpha\\in B_{\\beta}$ then $B_{\\alpha}\\subseteq B_{\\beta}$,\n o if $\\lambda\\leq \\alpha$ and $\\lambda$ is a limit ordinal then $B_{\\alpha}\\cap\\lambda$ is not in the ideal generated by the $B_{\\beta}$, $\\beta<\\alpha$, and by the bounded subsets of $\\lambda$,\n o there is a partition $\\{A_n\\}_{n=0}^{\\infty}$ of $\\omega_1$ such that for every $\\alpha$ and every $n,$ $B_{\\alpha}\\cap A_n$ is finite.","authors_text":"Saharon Shelah, Thomas Jech","cross_cats":[],"headline":"","license":"","primary_cat":"math.LO","submitted_at":"1994-12-03T00:00:00Z","title":"Possible pcf algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9412208","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":"sha256:4b5e211b0cf8a8e56775ce50bd6be8c5454b31ead7533783dc568a62dac57a24","target":"record","created_at":"2026-05-18T01:05:50Z","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":"d06d70cd5498f251a1ab7b23ae80bc34f125ba1a425cc78f9bbde1e9bb0b319a","cross_cats_sorted":[],"license":"","primary_cat":"math.LO","submitted_at":"1994-12-03T00:00:00Z","title_canon_sha256":"9e2bda60bd5237625a312e4c489fd1f0717fa238bbb0c0d8c9c33f7cb4ecd63c"},"schema_version":"1.0","source":{"id":"math/9412208","kind":"arxiv","version":1}},"canonical_sha256":"6fa5ef63c888db327ac24ed0ab901178ad2e186b0a896fbd4fbb872be4408c41","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6fa5ef63c888db327ac24ed0ab901178ad2e186b0a896fbd4fbb872be4408c41","first_computed_at":"2026-05-18T01:05:50.775250Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:50.775250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z5bzjWYN7wnqfkjX5pt/ryQB1IAVhcdFF8F61Oxl+mt6KwRUH/Wb9dXP/aZTK2hW/aDoFDJTmBfeI8CNnxyzBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:50.775827Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9412208","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4b5e211b0cf8a8e56775ce50bd6be8c5454b31ead7533783dc568a62dac57a24","sha256:1bdb29d10b54a7ee86e3d6378da5b224c3955b27a2230f1f5ab0796e5f360fac"],"state_sha256":"6a5f0a097e4d81f59db9469bc978622055e77578d2e557b187d1ff8f308b2e82"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t+mGNCP1MjNBcE2EABtL1jVL17Jpt8eJMaueOlE+1E7HHXd/KNuvX2oHd1QhGn04bUn+jj3Sp11oGdzJjxdCAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T18:02:52.200611Z","bundle_sha256":"3986ea7f6640ba9a288a7aec27a52f8006fd7ed8de646254edfc4bfc640d9259"}}