{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1994:DO7GAA4TW364X7NJOQJZIF372B","short_pith_number":"pith:DO7GAA4T","canonical_record":{"source":{"id":"math/9409207","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"1994-09-17T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"ecf82c3badc5b53acd853d66ea0e9f54a1539e87e598aeb478c4120f984698f7","abstract_canon_sha256":"c98313c049dc69fc118ce05dc93d0155580c9db3b1c4779bb63e60e9948d9a28"},"schema_version":"1.0"},"canonical_sha256":"1bbe600393b6fdcbfda9741394177fd04bde5817d589b359dcb95e6b55a63d09","source":{"kind":"arxiv","id":"math/9409207","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9409207","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/9409207v1","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9409207","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"pith_short_12","alias_value":"DO7GAA4TW364","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"DO7GAA4TW364X7NJ","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"DO7GAA4T","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1994:DO7GAA4TW364X7NJOQJZIF372B","target":"record","payload":{"canonical_record":{"source":{"id":"math/9409207","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"1994-09-17T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"ecf82c3badc5b53acd853d66ea0e9f54a1539e87e598aeb478c4120f984698f7","abstract_canon_sha256":"c98313c049dc69fc118ce05dc93d0155580c9db3b1c4779bb63e60e9948d9a28"},"schema_version":"1.0"},"canonical_sha256":"1bbe600393b6fdcbfda9741394177fd04bde5817d589b359dcb95e6b55a63d09","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:51.147551Z","signature_b64":"EAld5j0qFpyUzqFPMQPg+UaD1Q40u/QT4dK7/vjdGiq4wrLCTk2wfAfgoIGid/a1v71O1sQnpj3+w+za7Tm8AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1bbe600393b6fdcbfda9741394177fd04bde5817d589b359dcb95e6b55a63d09","last_reissued_at":"2026-05-18T01:05:51.146833Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:51.146833Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9409207","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":"2026-05-18T01:05:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zHlu2DcX1Za0msZ3dPaU1ERgN0uVPTqtoaA4i04/chRAe+UUxxiqo4Fvwur5VvecaQ6XGxmCyCfy157OISUqDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T09:33:58.159355Z"},"content_sha256":"d93a96e3d226baa3b8d16b17acc67c9d0ca4ad34b46b12532d005c0d732b9108","schema_version":"1.0","event_id":"sha256:d93a96e3d226baa3b8d16b17acc67c9d0ca4ad34b46b12532d005c0d732b9108"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1994:DO7GAA4TW364X7NJOQJZIF372B","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Universal abelian groups","license":"","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Menachem Kojman, Saharon Shelah","submitted_at":"1994-09-17T00:00:00Z","abstract_excerpt":"We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example:\n Theorem: For $n\\ge 2$, there is a purely universal separable $p$-group in $\\aleph_n$ if, and only if, $\\cont\\le \\aleph_n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9409207","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":"2026-05-18T01:05:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HDj+yMDxblXxBefCO7zK5KxYK4bnezRirrF5MomXAieP8Fy+Xzs2/eKksDjxEZmFYitnxP1vGpJ59ONmryPwAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T09:33:58.159711Z"},"content_sha256":"663346f57ffc3f8057c3602fa422948ca379dbcc7e4ffb6c697726123cae237c","schema_version":"1.0","event_id":"sha256:663346f57ffc3f8057c3602fa422948ca379dbcc7e4ffb6c697726123cae237c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DO7GAA4TW364X7NJOQJZIF372B/bundle.json","state_url":"https://pith.science/pith/DO7GAA4TW364X7NJOQJZIF372B/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DO7GAA4TW364X7NJOQJZIF372B/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":"2026-06-12T09:33:58Z","links":{"resolver":"https://pith.science/pith/DO7GAA4TW364X7NJOQJZIF372B","bundle":"https://pith.science/pith/DO7GAA4TW364X7NJOQJZIF372B/bundle.json","state":"https://pith.science/pith/DO7GAA4TW364X7NJOQJZIF372B/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DO7GAA4TW364X7NJOQJZIF372B/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1994:DO7GAA4TW364X7NJOQJZIF372B","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":"c98313c049dc69fc118ce05dc93d0155580c9db3b1c4779bb63e60e9948d9a28","cross_cats_sorted":[],"license":"","primary_cat":"math.LO","submitted_at":"1994-09-17T00:00:00Z","title_canon_sha256":"ecf82c3badc5b53acd853d66ea0e9f54a1539e87e598aeb478c4120f984698f7"},"schema_version":"1.0","source":{"id":"math/9409207","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9409207","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/9409207v1","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9409207","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"pith_short_12","alias_value":"DO7GAA4TW364","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"DO7GAA4TW364X7NJ","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"DO7GAA4T","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:663346f57ffc3f8057c3602fa422948ca379dbcc7e4ffb6c697726123cae237c","target":"graph","created_at":"2026-05-18T01:05:51Z","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 examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example:\n Theorem: For $n\\ge 2$, there is a purely universal separable $p$-group in $\\aleph_n$ if, and only if, $\\cont\\le \\aleph_n$.","authors_text":"Menachem Kojman, Saharon Shelah","cross_cats":[],"headline":"","license":"","primary_cat":"math.LO","submitted_at":"1994-09-17T00:00:00Z","title":"Universal abelian groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9409207","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:d93a96e3d226baa3b8d16b17acc67c9d0ca4ad34b46b12532d005c0d732b9108","target":"record","created_at":"2026-05-18T01:05:51Z","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":"c98313c049dc69fc118ce05dc93d0155580c9db3b1c4779bb63e60e9948d9a28","cross_cats_sorted":[],"license":"","primary_cat":"math.LO","submitted_at":"1994-09-17T00:00:00Z","title_canon_sha256":"ecf82c3badc5b53acd853d66ea0e9f54a1539e87e598aeb478c4120f984698f7"},"schema_version":"1.0","source":{"id":"math/9409207","kind":"arxiv","version":1}},"canonical_sha256":"1bbe600393b6fdcbfda9741394177fd04bde5817d589b359dcb95e6b55a63d09","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1bbe600393b6fdcbfda9741394177fd04bde5817d589b359dcb95e6b55a63d09","first_computed_at":"2026-05-18T01:05:51.146833Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:51.146833Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EAld5j0qFpyUzqFPMQPg+UaD1Q40u/QT4dK7/vjdGiq4wrLCTk2wfAfgoIGid/a1v71O1sQnpj3+w+za7Tm8AA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:51.147551Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9409207","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d93a96e3d226baa3b8d16b17acc67c9d0ca4ad34b46b12532d005c0d732b9108","sha256:663346f57ffc3f8057c3602fa422948ca379dbcc7e4ffb6c697726123cae237c"],"state_sha256":"29220e60c7798f50943898609575f77ca245853c45e806f3ab39a874addae20f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bRv9/7lzzYk/VTQkPt7KyA8Su4Gumox5CvWQ77jA8bocbw3vfu+RxbMT0o9ZfoRelWAHYYfpTf9nKqk81gKVAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T09:33:58.161999Z","bundle_sha256":"b136367de1ae92875c8a470fc0b5cfc8a3a9baecb378b6ddd53994a955bc44c2"}}