{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:NLCAFEYXSWS2M6I65ST453WYRG","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":"5c8bccdc798eb22a53e4d7ffa328f625122569024da540b2fcd6128eda565903","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-11-03T16:57:25Z","title_canon_sha256":"6c3e90b3b8044f03d761622b6ce1ca415236c12c86e4d578b82b2a78e86c5f1a"},"schema_version":"1.0","source":{"id":"1511.00982","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.00982","created_at":"2026-05-18T00:37:54Z"},{"alias_kind":"arxiv_version","alias_value":"1511.00982v4","created_at":"2026-05-18T00:37:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.00982","created_at":"2026-05-18T00:37:54Z"},{"alias_kind":"pith_short_12","alias_value":"NLCAFEYXSWS2","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"NLCAFEYXSWS2M6I6","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"NLCAFEYX","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:86b912b723caee88c9e7a3c10e3cf69934b7825dd227c04c8959dd2db579e9c1","target":"graph","created_at":"2026-05-18T00:37:54Z","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 consider an ultraproduct that is designed to omit a fixed set of unary types $\\Gamma$, called the $\\Gamma$-ultraproduct. The $\\Gamma$-ultraproduct is not always well-behaved, but we discuss several general conditions under which it is and several examples.","authors_text":"Will Boney","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-11-03T16:57:25Z","title":"The \\Gamma-Ultraproduct and Averageable Classes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.00982","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":"sha256:1c665c94a6f8401419ddf355f0944857f5faf4a4796bc9ba6421dea359994aeb","target":"record","created_at":"2026-05-18T00:37:54Z","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":"5c8bccdc798eb22a53e4d7ffa328f625122569024da540b2fcd6128eda565903","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-11-03T16:57:25Z","title_canon_sha256":"6c3e90b3b8044f03d761622b6ce1ca415236c12c86e4d578b82b2a78e86c5f1a"},"schema_version":"1.0","source":{"id":"1511.00982","kind":"arxiv","version":4}},"canonical_sha256":"6ac402931795a5a6791eeca7ceeed889a83e3043de4986c45e4fe1e403173571","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6ac402931795a5a6791eeca7ceeed889a83e3043de4986c45e4fe1e403173571","first_computed_at":"2026-05-18T00:37:54.970443Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:37:54.970443Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dmz504JT7xfbS8lNiM5LNSCYE+5WwDkTX8SEK/UmVPqHt+owUzKR6GmtrCSpI7cDWp1S0p1mEQE5ud78ZhQ3CA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:37:54.970979Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.00982","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1c665c94a6f8401419ddf355f0944857f5faf4a4796bc9ba6421dea359994aeb","sha256:86b912b723caee88c9e7a3c10e3cf69934b7825dd227c04c8959dd2db579e9c1"],"state_sha256":"8a97269a279f200b71574d0e8ffcd83649d244296cc68a38fd4d323a5ee5fa1d"}