{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:N6EJXYEYCW3KQADFCITW3U6454","short_pith_number":"pith:N6EJXYEY","canonical_record":{"source":{"id":"1503.08781","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-03-30T18:34:55Z","cross_cats_sorted":[],"title_canon_sha256":"bdc01b115d59b7397da0105c127a6ff9c01cda992c1f31f0fc834ced04391d9b","abstract_canon_sha256":"83ff6ba5732f7fb0afb206ec0e81141212d2cb84626b00b0a1faaa9236b91ebd"},"schema_version":"1.0"},"canonical_sha256":"6f889be09815b6a8006512276dd3dcef3ee0db62ee4daf23838193bdc78531b2","source":{"kind":"arxiv","id":"1503.08781","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.08781","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"1503.08781v4","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.08781","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"N6EJXYEYCW3K","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"N6EJXYEYCW3KQADF","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"N6EJXYEY","created_at":"2026-05-18T12:29:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:N6EJXYEYCW3KQADFCITW3U6454","target":"record","payload":{"canonical_record":{"source":{"id":"1503.08781","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-03-30T18:34:55Z","cross_cats_sorted":[],"title_canon_sha256":"bdc01b115d59b7397da0105c127a6ff9c01cda992c1f31f0fc834ced04391d9b","abstract_canon_sha256":"83ff6ba5732f7fb0afb206ec0e81141212d2cb84626b00b0a1faaa9236b91ebd"},"schema_version":"1.0"},"canonical_sha256":"6f889be09815b6a8006512276dd3dcef3ee0db62ee4daf23838193bdc78531b2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:32.218591Z","signature_b64":"PWVeyNUa23Os/Jpy/HHUSpwImboLsyglKH6JRKQLfO4xG/O7zhJgXNoYDrsfHpMuhY5UlTyHteZCs8c2stbVCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f889be09815b6a8006512276dd3dcef3ee0db62ee4daf23838193bdc78531b2","last_reissued_at":"2026-05-18T00:46:32.217876Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:32.217876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.08781","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:46:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oMo37eKuiP++YjqL/IAtxBZSN8m8+3XGi/mFUsGWFe5Zacek54LmMO9fyXtimB3x5U1oBTphOhnFWekQdDdgAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T17:44:16.874941Z"},"content_sha256":"654229e17fc41a304a0971690c3f296e6ea00616416c8170c7f7865ca9021abe","schema_version":"1.0","event_id":"sha256:654229e17fc41a304a0971690c3f296e6ea00616416c8170c7f7865ca9021abe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:N6EJXYEYCW3KQADFCITW3U6454","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Chains of saturated models in AECs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Sebastien Vasey, Will Boney","submitted_at":"2015-03-30T18:34:55Z","abstract_excerpt":"We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove:\n  $\\mathbf{Theorem}$\n  If $K$ is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough $\\lambda$:\n  * The union of an increasing chain of $\\lambda$-saturated models is $\\lambda$-saturated.\n  * There exists a type-full good $\\lambda$-frame with underlying class the saturated models of size $\\lambda$.\n  * There exists a unique limit model of siz"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08781","kind":"arxiv","version":4},"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":"2026-05-18T00:46:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/5FnBB8lAFUPhr6L7P0Wn0afeoHS4kDPcWvc7pbviAYZX/WTo2vw4kZrsBU1JGqEVCz7PfBaiQEPAG+QnHk6BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T17:44:16.875659Z"},"content_sha256":"6978fde1ef17cd57d9fea9abf5908e4442b2d4c6450cd9be49906d5e9d197f8a","schema_version":"1.0","event_id":"sha256:6978fde1ef17cd57d9fea9abf5908e4442b2d4c6450cd9be49906d5e9d197f8a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N6EJXYEYCW3KQADFCITW3U6454/bundle.json","state_url":"https://pith.science/pith/N6EJXYEYCW3KQADFCITW3U6454/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N6EJXYEYCW3KQADFCITW3U6454/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":"2026-05-25T17:44:16Z","links":{"resolver":"https://pith.science/pith/N6EJXYEYCW3KQADFCITW3U6454","bundle":"https://pith.science/pith/N6EJXYEYCW3KQADFCITW3U6454/bundle.json","state":"https://pith.science/pith/N6EJXYEYCW3KQADFCITW3U6454/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N6EJXYEYCW3KQADFCITW3U6454/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:N6EJXYEYCW3KQADFCITW3U6454","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":"83ff6ba5732f7fb0afb206ec0e81141212d2cb84626b00b0a1faaa9236b91ebd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-03-30T18:34:55Z","title_canon_sha256":"bdc01b115d59b7397da0105c127a6ff9c01cda992c1f31f0fc834ced04391d9b"},"schema_version":"1.0","source":{"id":"1503.08781","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.08781","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"1503.08781v4","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.08781","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"N6EJXYEYCW3K","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"N6EJXYEYCW3KQADF","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"N6EJXYEY","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:6978fde1ef17cd57d9fea9abf5908e4442b2d4c6450cd9be49906d5e9d197f8a","target":"graph","created_at":"2026-05-18T00:46:32Z","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 study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove:\n  $\\mathbf{Theorem}$\n  If $K$ is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough $\\lambda$:\n  * The union of an increasing chain of $\\lambda$-saturated models is $\\lambda$-saturated.\n  * There exists a type-full good $\\lambda$-frame with underlying class the saturated models of size $\\lambda$.\n  * There exists a unique limit model of siz","authors_text":"Sebastien Vasey, Will Boney","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-03-30T18:34:55Z","title":"Chains of saturated models in AECs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08781","kind":"arxiv","version":4},"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":"sha256:654229e17fc41a304a0971690c3f296e6ea00616416c8170c7f7865ca9021abe","target":"record","created_at":"2026-05-18T00:46:32Z","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":"83ff6ba5732f7fb0afb206ec0e81141212d2cb84626b00b0a1faaa9236b91ebd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-03-30T18:34:55Z","title_canon_sha256":"bdc01b115d59b7397da0105c127a6ff9c01cda992c1f31f0fc834ced04391d9b"},"schema_version":"1.0","source":{"id":"1503.08781","kind":"arxiv","version":4}},"canonical_sha256":"6f889be09815b6a8006512276dd3dcef3ee0db62ee4daf23838193bdc78531b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6f889be09815b6a8006512276dd3dcef3ee0db62ee4daf23838193bdc78531b2","first_computed_at":"2026-05-18T00:46:32.217876Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:46:32.217876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PWVeyNUa23Os/Jpy/HHUSpwImboLsyglKH6JRKQLfO4xG/O7zhJgXNoYDrsfHpMuhY5UlTyHteZCs8c2stbVCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:46:32.218591Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.08781","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:654229e17fc41a304a0971690c3f296e6ea00616416c8170c7f7865ca9021abe","sha256:6978fde1ef17cd57d9fea9abf5908e4442b2d4c6450cd9be49906d5e9d197f8a"],"state_sha256":"1201d51e9d1ab9676cf10fb298723df992366e55c985db8e24da38a7333a2a52"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uEkQRkkZx1z6knLY2Z8MCLP/9141nRHg3sWgGDBoeU2/gK+q1FWUKqKziT24LoAtna6VygetNia8kmBkAOW5DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T17:44:16.878708Z","bundle_sha256":"aad60c7f01295aaa1d966fc4e1ab37b54f19a34aeed69a2601ecaa7b1b456dcd"}}