{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1993:ENTUPLHC6SNUH4MIZEBPMK753E","short_pith_number":"pith:ENTUPLHC","canonical_record":{"source":{"id":"math/9301205","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"1993-01-15T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"070f7d94a670b5918b300a7f5eff234943098de8d91362b52d7ecf6cc11e8330","abstract_canon_sha256":"ea758f280c40e0355bbf95ec05e1ea4291293cbdef8b9cc9bc04ac1d74735bde"},"schema_version":"1.0"},"canonical_sha256":"236747ace2f49b43f188c902f62bfdd908e2ab1a5fa157cb9bfca077bd330b07","source":{"kind":"arxiv","id":"math/9301205","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9301205","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"arxiv_version","alias_value":"math/9301205v1","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9301205","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"pith_short_12","alias_value":"ENTUPLHC6SNU","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"ENTUPLHC6SNUH4MI","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"ENTUPLHC","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1993:ENTUPLHC6SNUH4MIZEBPMK753E","target":"record","payload":{"canonical_record":{"source":{"id":"math/9301205","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"1993-01-15T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"070f7d94a670b5918b300a7f5eff234943098de8d91362b52d7ecf6cc11e8330","abstract_canon_sha256":"ea758f280c40e0355bbf95ec05e1ea4291293cbdef8b9cc9bc04ac1d74735bde"},"schema_version":"1.0"},"canonical_sha256":"236747ace2f49b43f188c902f62bfdd908e2ab1a5fa157cb9bfca077bd330b07","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:52.392242Z","signature_b64":"BiSVfG5czN9ueHv3qja3NENEI7ccyfVlSIV+PqE7rLnBFgpLNAuJBy8vPtQVeE7syiypCHaZ2p9HKAjdEEHaAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"236747ace2f49b43f188c902f62bfdd908e2ab1a5fa157cb9bfca077bd330b07","last_reissued_at":"2026-05-18T01:05:52.391800Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:52.391800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9301205","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":"2026-05-18T01:05:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O5lK4JfDCFdot1+5DjkrFQOU0zZUKK7Yj9X8nrjTphImby/3v54GFQIbOgbs+0YXnRNH7Fcxg/jFkr/BWvrEDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T12:08:19.659681Z"},"content_sha256":"f1871d4ea4b94601ec4a95ee941ea814157f85d2ae6f0323968efd51a183254f","schema_version":"1.0","event_id":"sha256:f1871d4ea4b94601ec4a95ee941ea814157f85d2ae6f0323968efd51a183254f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1993:ENTUPLHC6SNUH4MIZEBPMK753E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the number of automorphism of uncontable models","license":"","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Heikki Tuuri, Jouko V\\\"a\\\"an\\\"anen, Saharon Shelah","submitted_at":"1993-01-15T00:00:00Z","abstract_excerpt":"Let s(A) denote the number of automorphisms of a model A of power omega_1. We derive a necessary and sufficient condition in terms of trees for the existence of an A with omega_1 < s(A) < 2^{omega_1}. We study the sufficiency of some conditions for s(A)=2^{omega_1}. These conditions are analogous to conditions studied by D.Kueker in connection with countable models."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9301205","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":"2026-05-18T01:05:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7IrK4BEveMCeqlwSOSnBxI7DfobWYeAerHnseAryA+NjP9hEn2FLh0tjm5CD3oUixpxVakEg6+R3pdGPg5mIDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T12:08:19.660018Z"},"content_sha256":"b5142c28cf0d0a4a364b57c7f56c1d2463974623d4cc1e1e5dad7de7c436ba2f","schema_version":"1.0","event_id":"sha256:b5142c28cf0d0a4a364b57c7f56c1d2463974623d4cc1e1e5dad7de7c436ba2f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ENTUPLHC6SNUH4MIZEBPMK753E/bundle.json","state_url":"https://pith.science/pith/ENTUPLHC6SNUH4MIZEBPMK753E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ENTUPLHC6SNUH4MIZEBPMK753E/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":"2026-06-01T12:08:19Z","links":{"resolver":"https://pith.science/pith/ENTUPLHC6SNUH4MIZEBPMK753E","bundle":"https://pith.science/pith/ENTUPLHC6SNUH4MIZEBPMK753E/bundle.json","state":"https://pith.science/pith/ENTUPLHC6SNUH4MIZEBPMK753E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ENTUPLHC6SNUH4MIZEBPMK753E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1993:ENTUPLHC6SNUH4MIZEBPMK753E","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":"ea758f280c40e0355bbf95ec05e1ea4291293cbdef8b9cc9bc04ac1d74735bde","cross_cats_sorted":[],"license":"","primary_cat":"math.LO","submitted_at":"1993-01-15T00:00:00Z","title_canon_sha256":"070f7d94a670b5918b300a7f5eff234943098de8d91362b52d7ecf6cc11e8330"},"schema_version":"1.0","source":{"id":"math/9301205","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9301205","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"arxiv_version","alias_value":"math/9301205v1","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9301205","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"pith_short_12","alias_value":"ENTUPLHC6SNU","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"ENTUPLHC6SNUH4MI","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"ENTUPLHC","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:b5142c28cf0d0a4a364b57c7f56c1d2463974623d4cc1e1e5dad7de7c436ba2f","target":"graph","created_at":"2026-05-18T01:05:52Z","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":"Let s(A) denote the number of automorphisms of a model A of power omega_1. We derive a necessary and sufficient condition in terms of trees for the existence of an A with omega_1 < s(A) < 2^{omega_1}. We study the sufficiency of some conditions for s(A)=2^{omega_1}. These conditions are analogous to conditions studied by D.Kueker in connection with countable models.","authors_text":"Heikki Tuuri, Jouko V\\\"a\\\"an\\\"anen, Saharon Shelah","cross_cats":[],"headline":"","license":"","primary_cat":"math.LO","submitted_at":"1993-01-15T00:00:00Z","title":"On the number of automorphism of uncontable models"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9301205","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:f1871d4ea4b94601ec4a95ee941ea814157f85d2ae6f0323968efd51a183254f","target":"record","created_at":"2026-05-18T01:05:52Z","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":"ea758f280c40e0355bbf95ec05e1ea4291293cbdef8b9cc9bc04ac1d74735bde","cross_cats_sorted":[],"license":"","primary_cat":"math.LO","submitted_at":"1993-01-15T00:00:00Z","title_canon_sha256":"070f7d94a670b5918b300a7f5eff234943098de8d91362b52d7ecf6cc11e8330"},"schema_version":"1.0","source":{"id":"math/9301205","kind":"arxiv","version":1}},"canonical_sha256":"236747ace2f49b43f188c902f62bfdd908e2ab1a5fa157cb9bfca077bd330b07","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"236747ace2f49b43f188c902f62bfdd908e2ab1a5fa157cb9bfca077bd330b07","first_computed_at":"2026-05-18T01:05:52.391800Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:52.391800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BiSVfG5czN9ueHv3qja3NENEI7ccyfVlSIV+PqE7rLnBFgpLNAuJBy8vPtQVeE7syiypCHaZ2p9HKAjdEEHaAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:52.392242Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9301205","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f1871d4ea4b94601ec4a95ee941ea814157f85d2ae6f0323968efd51a183254f","sha256:b5142c28cf0d0a4a364b57c7f56c1d2463974623d4cc1e1e5dad7de7c436ba2f"],"state_sha256":"6cda7cf1085d88f97ef7eaeea73af11b1df46e6e70d39a1279c8b098f91261f0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LWpuXCAIo15SOC05cBQeIQ5BkePt+6n7Bo05zDPOeO60XoJdZ4q7GkaNKKlLuT39jcyUKkyu1o3XRLtdtxRyCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T12:08:19.662173Z","bundle_sha256":"60d0e06f80db4c84ffbcc221316f34db6d2578dd017e604b45e74c49266b206a"}}