{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:IJJKH7B3QFT4CXMAMYPGZ3OMLO","short_pith_number":"pith:IJJKH7B3","canonical_record":{"source":{"id":"1510.05679","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-19T20:45:09Z","cross_cats_sorted":[],"title_canon_sha256":"3616bf3d8e37a4e7f40ec90ed447500146c7e410eb593237b82547504d48f1f6","abstract_canon_sha256":"eed59b971973a543576e4fafefd5b80f2307d9b256217cb6c5e5a7b721c7d7c8"},"schema_version":"1.0"},"canonical_sha256":"4252a3fc3b8167c15d80661e6cedcc5b9768cd117d2261fad136092a8bc944f3","source":{"kind":"arxiv","id":"1510.05679","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.05679","created_at":"2026-05-18T00:58:55Z"},{"alias_kind":"arxiv_version","alias_value":"1510.05679v3","created_at":"2026-05-18T00:58:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.05679","created_at":"2026-05-18T00:58:55Z"},{"alias_kind":"pith_short_12","alias_value":"IJJKH7B3QFT4","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"IJJKH7B3QFT4CXMA","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"IJJKH7B3","created_at":"2026-05-18T12:29:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:IJJKH7B3QFT4CXMAMYPGZ3OMLO","target":"record","payload":{"canonical_record":{"source":{"id":"1510.05679","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-19T20:45:09Z","cross_cats_sorted":[],"title_canon_sha256":"3616bf3d8e37a4e7f40ec90ed447500146c7e410eb593237b82547504d48f1f6","abstract_canon_sha256":"eed59b971973a543576e4fafefd5b80f2307d9b256217cb6c5e5a7b721c7d7c8"},"schema_version":"1.0"},"canonical_sha256":"4252a3fc3b8167c15d80661e6cedcc5b9768cd117d2261fad136092a8bc944f3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:55.463432Z","signature_b64":"uSGMMwXsbwc9m2zZoQ7IF00d9JvfMgCHHj2EzrCEbqieL67GmcMDfOCYPZKqRaxbJUV/+mNlmDUoa7UFM5SbAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4252a3fc3b8167c15d80661e6cedcc5b9768cd117d2261fad136092a8bc944f3","last_reissued_at":"2026-05-18T00:58:55.462879Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:55.462879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.05679","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:58:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EJ27BwdAyc+KB/em9lflBsL9+lUfGxH4wugCjLR2OsNzJHKUTE3Tdh7rnRn3rnDYGqqIEDzT0nBH+y+384dODA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T08:51:37.674592Z"},"content_sha256":"8c43387b046d978f16500738920a2027ff7f4889a229d80bc693d82c4af9245e","schema_version":"1.0","event_id":"sha256:8c43387b046d978f16500738920a2027ff7f4889a229d80bc693d82c4af9245e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:IJJKH7B3QFT4CXMAMYPGZ3OMLO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Borel Complexity and Potential Canonical Scott Sentences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Douglas Ulrich, Michael C. Laskowski, Richard Rast","submitted_at":"2015-10-19T20:45:09Z","abstract_excerpt":"We define and investigate HC-forcing invariant formulas of set theory, whose interpretations in the hereditarily countable sets are well behaved under forcing extensions. This leads naturally to a notion of cardinality ||Phi|| for sentences Phi of $L_{\\omega_1,\\omega}$, which counts the number of sentences of $L_{\\infty,\\omega}$ that, in some forcing extension, become a canonical Scott sentence of a model of Phi. We show this cardinal bounds the complexity of (Mod(Phi), iso), the class of models of Phi with universe omega, by proving that (Mod(Phi),iso) is not Borel reducible to (Mod(Psi),iso)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05679","kind":"arxiv","version":3},"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":"2026-05-18T00:58:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OyRFxT1IZ9aq+lgK9FP+yJyXchPTtp44Adr9GfztaUdAkbpnKe38TFQZ3JMdHwbTtngHo049NhmAkbeEOTccAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T08:51:37.674947Z"},"content_sha256":"6b93f9804f4855aee98d3155b8bea454ff40b4164b5ed740db226694add78df2","schema_version":"1.0","event_id":"sha256:6b93f9804f4855aee98d3155b8bea454ff40b4164b5ed740db226694add78df2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IJJKH7B3QFT4CXMAMYPGZ3OMLO/bundle.json","state_url":"https://pith.science/pith/IJJKH7B3QFT4CXMAMYPGZ3OMLO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IJJKH7B3QFT4CXMAMYPGZ3OMLO/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":"2026-05-30T08:51:37Z","links":{"resolver":"https://pith.science/pith/IJJKH7B3QFT4CXMAMYPGZ3OMLO","bundle":"https://pith.science/pith/IJJKH7B3QFT4CXMAMYPGZ3OMLO/bundle.json","state":"https://pith.science/pith/IJJKH7B3QFT4CXMAMYPGZ3OMLO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IJJKH7B3QFT4CXMAMYPGZ3OMLO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:IJJKH7B3QFT4CXMAMYPGZ3OMLO","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":"eed59b971973a543576e4fafefd5b80f2307d9b256217cb6c5e5a7b721c7d7c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-19T20:45:09Z","title_canon_sha256":"3616bf3d8e37a4e7f40ec90ed447500146c7e410eb593237b82547504d48f1f6"},"schema_version":"1.0","source":{"id":"1510.05679","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.05679","created_at":"2026-05-18T00:58:55Z"},{"alias_kind":"arxiv_version","alias_value":"1510.05679v3","created_at":"2026-05-18T00:58:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.05679","created_at":"2026-05-18T00:58:55Z"},{"alias_kind":"pith_short_12","alias_value":"IJJKH7B3QFT4","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"IJJKH7B3QFT4CXMA","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"IJJKH7B3","created_at":"2026-05-18T12:29:25Z"}],"graph_snapshots":[{"event_id":"sha256:6b93f9804f4855aee98d3155b8bea454ff40b4164b5ed740db226694add78df2","target":"graph","created_at":"2026-05-18T00:58:55Z","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 define and investigate HC-forcing invariant formulas of set theory, whose interpretations in the hereditarily countable sets are well behaved under forcing extensions. This leads naturally to a notion of cardinality ||Phi|| for sentences Phi of $L_{\\omega_1,\\omega}$, which counts the number of sentences of $L_{\\infty,\\omega}$ that, in some forcing extension, become a canonical Scott sentence of a model of Phi. We show this cardinal bounds the complexity of (Mod(Phi), iso), the class of models of Phi with universe omega, by proving that (Mod(Phi),iso) is not Borel reducible to (Mod(Psi),iso)","authors_text":"Douglas Ulrich, Michael C. Laskowski, Richard Rast","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-19T20:45:09Z","title":"Borel Complexity and Potential Canonical Scott Sentences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05679","kind":"arxiv","version":3},"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":"sha256:8c43387b046d978f16500738920a2027ff7f4889a229d80bc693d82c4af9245e","target":"record","created_at":"2026-05-18T00:58:55Z","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":"eed59b971973a543576e4fafefd5b80f2307d9b256217cb6c5e5a7b721c7d7c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-10-19T20:45:09Z","title_canon_sha256":"3616bf3d8e37a4e7f40ec90ed447500146c7e410eb593237b82547504d48f1f6"},"schema_version":"1.0","source":{"id":"1510.05679","kind":"arxiv","version":3}},"canonical_sha256":"4252a3fc3b8167c15d80661e6cedcc5b9768cd117d2261fad136092a8bc944f3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4252a3fc3b8167c15d80661e6cedcc5b9768cd117d2261fad136092a8bc944f3","first_computed_at":"2026-05-18T00:58:55.462879Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:58:55.462879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uSGMMwXsbwc9m2zZoQ7IF00d9JvfMgCHHj2EzrCEbqieL67GmcMDfOCYPZKqRaxbJUV/+mNlmDUoa7UFM5SbAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:58:55.463432Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.05679","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8c43387b046d978f16500738920a2027ff7f4889a229d80bc693d82c4af9245e","sha256:6b93f9804f4855aee98d3155b8bea454ff40b4164b5ed740db226694add78df2"],"state_sha256":"e6aff177b495d483789b20d14345a4c2633f4c118509df1dfee42f3e52cde119"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S9kf+l0TKOHO10yCVyHhTkLFy3a6s8W/bLPbi7zzcKNy4Sw0YCr4xkxldkJ+oarJgI2O+SlDaOCQt+adFL66Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T08:51:37.676953Z","bundle_sha256":"672133ff73d16a14dcac71e4306c793e9da713c8d4862229586ff27a136ed35e"}}