{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:NLCAFEYXSWS2M6I65ST453WYRG","short_pith_number":"pith:NLCAFEYX","canonical_record":{"source":{"id":"1511.00982","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-11-03T16:57:25Z","cross_cats_sorted":[],"title_canon_sha256":"6c3e90b3b8044f03d761622b6ce1ca415236c12c86e4d578b82b2a78e86c5f1a","abstract_canon_sha256":"5c8bccdc798eb22a53e4d7ffa328f625122569024da540b2fcd6128eda565903"},"schema_version":"1.0"},"canonical_sha256":"6ac402931795a5a6791eeca7ceeed889a83e3043de4986c45e4fe1e403173571","source":{"kind":"arxiv","id":"1511.00982","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"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:NLCAFEYXSWS2M6I65ST453WYRG","target":"record","payload":{"canonical_record":{"source":{"id":"1511.00982","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-11-03T16:57:25Z","cross_cats_sorted":[],"title_canon_sha256":"6c3e90b3b8044f03d761622b6ce1ca415236c12c86e4d578b82b2a78e86c5f1a","abstract_canon_sha256":"5c8bccdc798eb22a53e4d7ffa328f625122569024da540b2fcd6128eda565903"},"schema_version":"1.0"},"canonical_sha256":"6ac402931795a5a6791eeca7ceeed889a83e3043de4986c45e4fe1e403173571","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:37:54.970979Z","signature_b64":"dmz504JT7xfbS8lNiM5LNSCYE+5WwDkTX8SEK/UmVPqHt+owUzKR6GmtrCSpI7cDWp1S0p1mEQE5ud78ZhQ3CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ac402931795a5a6791eeca7ceeed889a83e3043de4986c45e4fe1e403173571","last_reissued_at":"2026-05-18T00:37:54.970443Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:37:54.970443Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.00982","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":"2026-05-18T00:37:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g0BDgSVWYQx5Cf+I+Ec/kU/lT9UV9eAvHcDd+0sub++tsIW1IE2W4xtNdyBApG+ZY7Qc0il6D2IxRwcFBvmfCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T13:59:40.139339Z"},"content_sha256":"1c665c94a6f8401419ddf355f0944857f5faf4a4796bc9ba6421dea359994aeb","schema_version":"1.0","event_id":"sha256:1c665c94a6f8401419ddf355f0944857f5faf4a4796bc9ba6421dea359994aeb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:NLCAFEYXSWS2M6I65ST453WYRG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The \\Gamma-Ultraproduct and Averageable Classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Will Boney","submitted_at":"2015-11-03T16:57:25Z","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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.00982","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":"2026-05-18T00:37:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BMjpyE7EfJyklMEyQQT06n0kFeXgrS/r2oOpQFZVAKNCm59kPa3gKJgdkn+xGd7qK4RT//au1WuqQSkNTvopBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T13:59:40.140088Z"},"content_sha256":"86b912b723caee88c9e7a3c10e3cf69934b7825dd227c04c8959dd2db579e9c1","schema_version":"1.0","event_id":"sha256:86b912b723caee88c9e7a3c10e3cf69934b7825dd227c04c8959dd2db579e9c1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NLCAFEYXSWS2M6I65ST453WYRG/bundle.json","state_url":"https://pith.science/pith/NLCAFEYXSWS2M6I65ST453WYRG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NLCAFEYXSWS2M6I65ST453WYRG/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":"2026-05-25T13:59:40Z","links":{"resolver":"https://pith.science/pith/NLCAFEYXSWS2M6I65ST453WYRG","bundle":"https://pith.science/pith/NLCAFEYXSWS2M6I65ST453WYRG/bundle.json","state":"https://pith.science/pith/NLCAFEYXSWS2M6I65ST453WYRG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NLCAFEYXSWS2M6I65ST453WYRG/bundle.json"},"state":{"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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DS5v7hlMCv7cDOoHiod7gkvaGc4TWJwwKYcmu+vGx+J8i1D2pO+mQgkV8zve5eMvtDrSdD2PoSHp/sLuEBfhCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T13:59:40.144185Z","bundle_sha256":"7d0839d2e6c1439c3e5310be105a7aec05650ac3c9d6372ed2022ca03d3bbffa"}}