{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:QQUFIPIFA5SQ525D7S6XCRZBT7","short_pith_number":"pith:QQUFIPIF","canonical_record":{"source":{"id":"1906.04351","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-06-11T02:10:05Z","cross_cats_sorted":[],"title_canon_sha256":"052a496f5e5ed068e7f932fbb430dc71e6e4b83156a6f27a7ed0b605b7f059db","abstract_canon_sha256":"f3e283effb867d05097a60ddb208cc79717c60f0b9ece24921c4ec63041f632f"},"schema_version":"1.0"},"canonical_sha256":"8428543d0507650eeba3fcbd7147219ff1d20fc2540a2ef3689b1edc50ed47ce","source":{"kind":"arxiv","id":"1906.04351","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.04351","created_at":"2026-05-17T23:43:39Z"},{"alias_kind":"arxiv_version","alias_value":"1906.04351v1","created_at":"2026-05-17T23:43:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.04351","created_at":"2026-05-17T23:43:39Z"},{"alias_kind":"pith_short_12","alias_value":"QQUFIPIFA5SQ","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"QQUFIPIFA5SQ525D","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"QQUFIPIF","created_at":"2026-05-18T12:33:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:QQUFIPIFA5SQ525D7S6XCRZBT7","target":"record","payload":{"canonical_record":{"source":{"id":"1906.04351","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-06-11T02:10:05Z","cross_cats_sorted":[],"title_canon_sha256":"052a496f5e5ed068e7f932fbb430dc71e6e4b83156a6f27a7ed0b605b7f059db","abstract_canon_sha256":"f3e283effb867d05097a60ddb208cc79717c60f0b9ece24921c4ec63041f632f"},"schema_version":"1.0"},"canonical_sha256":"8428543d0507650eeba3fcbd7147219ff1d20fc2540a2ef3689b1edc50ed47ce","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:39.586387Z","signature_b64":"OgvXJqNosWp3zI/Mg87P3TBh86t2+rZUlcZBACRSKtLScdzKqBbnQJVN+ZR2U8RlUgQiWN2am08gvbv3huGXCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8428543d0507650eeba3fcbd7147219ff1d20fc2540a2ef3689b1edc50ed47ce","last_reissued_at":"2026-05-17T23:43:39.585845Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:39.585845Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.04351","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":"2026-05-17T23:43:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dRMNUewPxSHcJAuHZk7x6hOmoDibkuGdwla1QKQkity67ptOWlcOg7vTBkLm/PEjlNfLc/BTAhmXvfW0aNpPBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T23:22:26.588411Z"},"content_sha256":"455a694fbfb6cfa22e2a5835d3528d7757c3a4de956c34f87974a0de0b56096e","schema_version":"1.0","event_id":"sha256:455a694fbfb6cfa22e2a5835d3528d7757c3a4de956c34f87974a0de0b56096e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:QQUFIPIFA5SQ525D7S6XCRZBT7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bounds on Scott Ranks of Some Polish Metric Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"William Chan","submitted_at":"2019-06-11T02:10:05Z","abstract_excerpt":"If $\\mathcal{N}$ is a proper Polish metric space and $\\mathcal{M}$ is any countable dense submetric space of $\\mathcal{N}$, then the Scott rank of $\\mathcal{N}$ in the natural first order language of metric spaces is countable and in fact at most $\\omega_1^{\\mathcal{M}} + 1$, where $\\omega_1^{\\mathcal{M}}$ is the Church-Kleene ordinal of $\\mathcal{M}$ (construed as a subset of $\\omega$) which is the least ordinal with no presentation on $\\omega$ computable from $\\mathcal{M}$.\n  If $\\mathcal{N}$ is a rigid Polish metric space and $\\mathcal{M}$ is any countable dense submetric space, then the Sc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.04351","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":"2026-05-17T23:43:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eIzlEXgW/Zp5T0E77XYS5SnSBV4fqElGeZ3W7JOM6axcvcsBAQrZbxihE4tSBPcm36HlrggNhDRtcIQ1GUxTBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T23:22:26.589093Z"},"content_sha256":"a83eef6be5430065e9c9af46233bc4dfeff2ff42743b5c14bf50dd0f8f1f5145","schema_version":"1.0","event_id":"sha256:a83eef6be5430065e9c9af46233bc4dfeff2ff42743b5c14bf50dd0f8f1f5145"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QQUFIPIFA5SQ525D7S6XCRZBT7/bundle.json","state_url":"https://pith.science/pith/QQUFIPIFA5SQ525D7S6XCRZBT7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QQUFIPIFA5SQ525D7S6XCRZBT7/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":"2026-06-07T23:22:26Z","links":{"resolver":"https://pith.science/pith/QQUFIPIFA5SQ525D7S6XCRZBT7","bundle":"https://pith.science/pith/QQUFIPIFA5SQ525D7S6XCRZBT7/bundle.json","state":"https://pith.science/pith/QQUFIPIFA5SQ525D7S6XCRZBT7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QQUFIPIFA5SQ525D7S6XCRZBT7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:QQUFIPIFA5SQ525D7S6XCRZBT7","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":"f3e283effb867d05097a60ddb208cc79717c60f0b9ece24921c4ec63041f632f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-06-11T02:10:05Z","title_canon_sha256":"052a496f5e5ed068e7f932fbb430dc71e6e4b83156a6f27a7ed0b605b7f059db"},"schema_version":"1.0","source":{"id":"1906.04351","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.04351","created_at":"2026-05-17T23:43:39Z"},{"alias_kind":"arxiv_version","alias_value":"1906.04351v1","created_at":"2026-05-17T23:43:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.04351","created_at":"2026-05-17T23:43:39Z"},{"alias_kind":"pith_short_12","alias_value":"QQUFIPIFA5SQ","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"QQUFIPIFA5SQ525D","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"QQUFIPIF","created_at":"2026-05-18T12:33:27Z"}],"graph_snapshots":[{"event_id":"sha256:a83eef6be5430065e9c9af46233bc4dfeff2ff42743b5c14bf50dd0f8f1f5145","target":"graph","created_at":"2026-05-17T23:43:39Z","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":"If $\\mathcal{N}$ is a proper Polish metric space and $\\mathcal{M}$ is any countable dense submetric space of $\\mathcal{N}$, then the Scott rank of $\\mathcal{N}$ in the natural first order language of metric spaces is countable and in fact at most $\\omega_1^{\\mathcal{M}} + 1$, where $\\omega_1^{\\mathcal{M}}$ is the Church-Kleene ordinal of $\\mathcal{M}$ (construed as a subset of $\\omega$) which is the least ordinal with no presentation on $\\omega$ computable from $\\mathcal{M}$.\n  If $\\mathcal{N}$ is a rigid Polish metric space and $\\mathcal{M}$ is any countable dense submetric space, then the Sc","authors_text":"William Chan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-06-11T02:10:05Z","title":"Bounds on Scott Ranks of Some Polish Metric Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.04351","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":"sha256:455a694fbfb6cfa22e2a5835d3528d7757c3a4de956c34f87974a0de0b56096e","target":"record","created_at":"2026-05-17T23:43:39Z","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":"f3e283effb867d05097a60ddb208cc79717c60f0b9ece24921c4ec63041f632f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-06-11T02:10:05Z","title_canon_sha256":"052a496f5e5ed068e7f932fbb430dc71e6e4b83156a6f27a7ed0b605b7f059db"},"schema_version":"1.0","source":{"id":"1906.04351","kind":"arxiv","version":1}},"canonical_sha256":"8428543d0507650eeba3fcbd7147219ff1d20fc2540a2ef3689b1edc50ed47ce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8428543d0507650eeba3fcbd7147219ff1d20fc2540a2ef3689b1edc50ed47ce","first_computed_at":"2026-05-17T23:43:39.585845Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:43:39.585845Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OgvXJqNosWp3zI/Mg87P3TBh86t2+rZUlcZBACRSKtLScdzKqBbnQJVN+ZR2U8RlUgQiWN2am08gvbv3huGXCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:43:39.586387Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.04351","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:455a694fbfb6cfa22e2a5835d3528d7757c3a4de956c34f87974a0de0b56096e","sha256:a83eef6be5430065e9c9af46233bc4dfeff2ff42743b5c14bf50dd0f8f1f5145"],"state_sha256":"3449e060ae0329c74fc0a9e299d3947ea9c44ffd91a558a5c6fcf5be4252b091"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ip+97QVNvZwbNPoKTVPa3zhDgLkYNWSvZ46KTIpCdr+SKPWqE1YWIyqrtgbsM9jLBoKkZdr4c+i1b8dBtDBFCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T23:22:26.592872Z","bundle_sha256":"0dbc437a8e5204403d0aad61a69d1ec765201faf52c58432f1f5d0b64e9936dd"}}