{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:PJNUSBG7FTHMH4QPR6ROEOJWDE","short_pith_number":"pith:PJNUSBG7","canonical_record":{"source":{"id":"1803.03881","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-03-11T02:03:21Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"79ee36faa1dd066801a6e7cba481ef846c8bf9727af8bc300f5799e6a4247172","abstract_canon_sha256":"093b44e0d6c8a406d20231c28b634f3694574c46ec74e13e963c05ef5f62f92b"},"schema_version":"1.0"},"canonical_sha256":"7a5b4904df2ccec3f20f8fa2e2393619363dccd4459ff96472c78b1f838081c9","source":{"kind":"arxiv","id":"1803.03881","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.03881","created_at":"2026-05-18T00:14:59Z"},{"alias_kind":"arxiv_version","alias_value":"1803.03881v2","created_at":"2026-05-18T00:14:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.03881","created_at":"2026-05-18T00:14:59Z"},{"alias_kind":"pith_short_12","alias_value":"PJNUSBG7FTHM","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"PJNUSBG7FTHMH4QP","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"PJNUSBG7","created_at":"2026-05-18T12:32:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:PJNUSBG7FTHMH4QPR6ROEOJWDE","target":"record","payload":{"canonical_record":{"source":{"id":"1803.03881","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-03-11T02:03:21Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"79ee36faa1dd066801a6e7cba481ef846c8bf9727af8bc300f5799e6a4247172","abstract_canon_sha256":"093b44e0d6c8a406d20231c28b634f3694574c46ec74e13e963c05ef5f62f92b"},"schema_version":"1.0"},"canonical_sha256":"7a5b4904df2ccec3f20f8fa2e2393619363dccd4459ff96472c78b1f838081c9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:59.415273Z","signature_b64":"U8PLhhHlyhDpQupn79wUEVw2q8wYEnTycrMAKY3YMdCJYb1WNV6bxHy1uNtOBvzp+9eraa1nbKrNVUWV1icJCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7a5b4904df2ccec3f20f8fa2e2393619363dccd4459ff96472c78b1f838081c9","last_reissued_at":"2026-05-18T00:14:59.414661Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:59.414661Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.03881","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:14:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cNCMznHBQb9eRKum0VlqI/QvUHbUNQkNXGeqB3GlzWPagxRs1DWgJ1WPSyOQQeEzqSLOs3m5VehSLS3ML7KODQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T23:27:15.011194Z"},"content_sha256":"19a8c1e5d185bf1091537a5f5fb67e1e4ac48c09d0eb6170d8e750808cefb6fb","schema_version":"1.0","event_id":"sha256:19a8c1e5d185bf1091537a5f5fb67e1e4ac48c09d0eb6170d8e750808cefb6fb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:PJNUSBG7FTHMH4QPR6ROEOJWDE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The linear stability of the Schwarzschild spacetime in the harmonic gauge: odd part","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Pei-Ken Hung","submitted_at":"2018-03-11T02:03:21Z","abstract_excerpt":"In this paper, we study the odd solution of the linearlized Einstein equation on the Schwarzschild background and in the harmonic gauge. With the aid of Regge-Wheeler quantities, we are able to estimate the odd part of Lichnerowicz d'Alembertian equation. In particular, we prove the solution decays at rate $\\tau^{-1+\\delta}$ to a linearlized Kerr solution."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.03881","kind":"arxiv","version":2},"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":"2026-05-18T00:14:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/1EGiYAadiHK8O1OP+U1TwuaErhV20IdvXHQRftHPyjP4f8MI4EA7oaUy4cMEyOEZp9EbPuOswKsPbVLu9FADg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T23:27:15.011547Z"},"content_sha256":"9061d054a47613015f771cd3ca86be592c6fd2740bafb4120b02996f719c677e","schema_version":"1.0","event_id":"sha256:9061d054a47613015f771cd3ca86be592c6fd2740bafb4120b02996f719c677e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PJNUSBG7FTHMH4QPR6ROEOJWDE/bundle.json","state_url":"https://pith.science/pith/PJNUSBG7FTHMH4QPR6ROEOJWDE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PJNUSBG7FTHMH4QPR6ROEOJWDE/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":"2026-06-29T23:27:15Z","links":{"resolver":"https://pith.science/pith/PJNUSBG7FTHMH4QPR6ROEOJWDE","bundle":"https://pith.science/pith/PJNUSBG7FTHMH4QPR6ROEOJWDE/bundle.json","state":"https://pith.science/pith/PJNUSBG7FTHMH4QPR6ROEOJWDE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PJNUSBG7FTHMH4QPR6ROEOJWDE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:PJNUSBG7FTHMH4QPR6ROEOJWDE","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":"093b44e0d6c8a406d20231c28b634f3694574c46ec74e13e963c05ef5f62f92b","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-03-11T02:03:21Z","title_canon_sha256":"79ee36faa1dd066801a6e7cba481ef846c8bf9727af8bc300f5799e6a4247172"},"schema_version":"1.0","source":{"id":"1803.03881","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.03881","created_at":"2026-05-18T00:14:59Z"},{"alias_kind":"arxiv_version","alias_value":"1803.03881v2","created_at":"2026-05-18T00:14:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.03881","created_at":"2026-05-18T00:14:59Z"},{"alias_kind":"pith_short_12","alias_value":"PJNUSBG7FTHM","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"PJNUSBG7FTHMH4QP","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"PJNUSBG7","created_at":"2026-05-18T12:32:43Z"}],"graph_snapshots":[{"event_id":"sha256:9061d054a47613015f771cd3ca86be592c6fd2740bafb4120b02996f719c677e","target":"graph","created_at":"2026-05-18T00:14:59Z","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":"In this paper, we study the odd solution of the linearlized Einstein equation on the Schwarzschild background and in the harmonic gauge. With the aid of Regge-Wheeler quantities, we are able to estimate the odd part of Lichnerowicz d'Alembertian equation. In particular, we prove the solution decays at rate $\\tau^{-1+\\delta}$ to a linearlized Kerr solution.","authors_text":"Pei-Ken Hung","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-03-11T02:03:21Z","title":"The linear stability of the Schwarzschild spacetime in the harmonic gauge: odd part"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.03881","kind":"arxiv","version":2},"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":"sha256:19a8c1e5d185bf1091537a5f5fb67e1e4ac48c09d0eb6170d8e750808cefb6fb","target":"record","created_at":"2026-05-18T00:14:59Z","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":"093b44e0d6c8a406d20231c28b634f3694574c46ec74e13e963c05ef5f62f92b","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-03-11T02:03:21Z","title_canon_sha256":"79ee36faa1dd066801a6e7cba481ef846c8bf9727af8bc300f5799e6a4247172"},"schema_version":"1.0","source":{"id":"1803.03881","kind":"arxiv","version":2}},"canonical_sha256":"7a5b4904df2ccec3f20f8fa2e2393619363dccd4459ff96472c78b1f838081c9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7a5b4904df2ccec3f20f8fa2e2393619363dccd4459ff96472c78b1f838081c9","first_computed_at":"2026-05-18T00:14:59.414661Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:59.414661Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U8PLhhHlyhDpQupn79wUEVw2q8wYEnTycrMAKY3YMdCJYb1WNV6bxHy1uNtOBvzp+9eraa1nbKrNVUWV1icJCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:59.415273Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.03881","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:19a8c1e5d185bf1091537a5f5fb67e1e4ac48c09d0eb6170d8e750808cefb6fb","sha256:9061d054a47613015f771cd3ca86be592c6fd2740bafb4120b02996f719c677e"],"state_sha256":"9fcffc26e3c1039a659a6b67368fe6f4b497c436cf1e1abcb5625701f12d4c5a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P5q0zvEIbuGFl5nJPw2rNveozftxHw9dZ7vsGTzZvcweoOa4AkrEyQgUvA5RBlufDu8wUW6ESumyvBp3hJbdCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T23:27:15.013418Z","bundle_sha256":"ce7666d2e0e29071d5b59762e191d4de889e9a09ca47b0feb96af4de4a5fb36b"}}