{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:MMBLLVEB2LK3BFQXD7WDAHBKYO","short_pith_number":"pith:MMBLLVEB","canonical_record":{"source":{"id":"2211.04239","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"gr-qc","submitted_at":"2022-11-08T13:34:37Z","cross_cats_sorted":[],"title_canon_sha256":"815a3df2ae0cd57f394194f1f10776c42ffd4f143dac17b7082176657521b575","abstract_canon_sha256":"c170f5c5e1bc1ddb64aa0cc9b8f46ed425f751f10f860c89f4d4554e09c457e5"},"schema_version":"1.0"},"canonical_sha256":"6302b5d481d2d5b096171fec301c2ac3ac96ed3f37173d28c76a6f3fc82b1690","source":{"kind":"arxiv","id":"2211.04239","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2211.04239","created_at":"2026-07-05T06:00:33Z"},{"alias_kind":"arxiv_version","alias_value":"2211.04239v2","created_at":"2026-07-05T06:00:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.04239","created_at":"2026-07-05T06:00:33Z"},{"alias_kind":"pith_short_12","alias_value":"MMBLLVEB2LK3","created_at":"2026-07-05T06:00:33Z"},{"alias_kind":"pith_short_16","alias_value":"MMBLLVEB2LK3BFQX","created_at":"2026-07-05T06:00:33Z"},{"alias_kind":"pith_short_8","alias_value":"MMBLLVEB","created_at":"2026-07-05T06:00:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:MMBLLVEB2LK3BFQXD7WDAHBKYO","target":"record","payload":{"canonical_record":{"source":{"id":"2211.04239","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"gr-qc","submitted_at":"2022-11-08T13:34:37Z","cross_cats_sorted":[],"title_canon_sha256":"815a3df2ae0cd57f394194f1f10776c42ffd4f143dac17b7082176657521b575","abstract_canon_sha256":"c170f5c5e1bc1ddb64aa0cc9b8f46ed425f751f10f860c89f4d4554e09c457e5"},"schema_version":"1.0"},"canonical_sha256":"6302b5d481d2d5b096171fec301c2ac3ac96ed3f37173d28c76a6f3fc82b1690","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:00:33.259860Z","signature_b64":"zOzjDdXq/YgxiGwCp0NRpxmg/v68sNyqyKCrTFGEF2dBTZAJiEvJc47yAsS68/uUV4Dyy1xMxXL16RuwxMtyDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6302b5d481d2d5b096171fec301c2ac3ac96ed3f37173d28c76a6f3fc82b1690","last_reissued_at":"2026-07-05T06:00:33.259375Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:00:33.259375Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2211.04239","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":"2026-07-05T06:00:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TgquVxyr566XAgvvB8+J9eJLbcgtkX0wBvUJ2VM9PwNGq9dskd8ZC+pyOj8/sF8dXCLIwGFKzTMFVxh7YcX5BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T10:12:02.743330Z"},"content_sha256":"4a904a34aaaff963c12167682f586445187593132135040fc406c28b1ca32a54","schema_version":"1.0","event_id":"sha256:4a904a34aaaff963c12167682f586445187593132135040fc406c28b1ca32a54"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:MMBLLVEB2LK3BFQXD7WDAHBKYO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gravitational lensing by using the 0th order of affine perturbation series of the deflection angle of a ray near a photon sphere","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"gr-qc","authors_text":"Naoki Tsukamoto","submitted_at":"2022-11-08T13:34:37Z","abstract_excerpt":"The 0th order of affine perturbation series of the deflection angle of a ray near a photon sphere is more accurate than a deflection angle in a strong deflection limit, which is used often, because the later has hidden error terms. We investigate gravitational lensing by using 0th order affine perturbation series of the deflection angle in a general asymptotically-flat, static, and spherical symmetric spacetime with the photon sphere. We apply our formula to Schwarzschild black hole, Reissner-Nordstr\\\"om black hole, and Ellis-Bronnikov wormhole spacetimes as examples. By comparing observables "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.04239","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2211.04239/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T06:00:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zW0ZnDXPIf4wI4r9paZlZ1hsHuobn1j/9k2jiPZrTKMMe4dvtowSCHpjsKS3DDt1CaznpdioZvoQuTY4k3r0Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T10:12:02.743824Z"},"content_sha256":"8ad898be2b37d5a48b53b7e933179d1a6f074b0c742c4cb0afa9c5d08cdbbf3d","schema_version":"1.0","event_id":"sha256:8ad898be2b37d5a48b53b7e933179d1a6f074b0c742c4cb0afa9c5d08cdbbf3d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MMBLLVEB2LK3BFQXD7WDAHBKYO/bundle.json","state_url":"https://pith.science/pith/MMBLLVEB2LK3BFQXD7WDAHBKYO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MMBLLVEB2LK3BFQXD7WDAHBKYO/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":"2026-08-13T10:12:02Z","links":{"resolver":"https://pith.science/pith/MMBLLVEB2LK3BFQXD7WDAHBKYO","bundle":"https://pith.science/pith/MMBLLVEB2LK3BFQXD7WDAHBKYO/bundle.json","state":"https://pith.science/pith/MMBLLVEB2LK3BFQXD7WDAHBKYO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MMBLLVEB2LK3BFQXD7WDAHBKYO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:MMBLLVEB2LK3BFQXD7WDAHBKYO","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":"c170f5c5e1bc1ddb64aa0cc9b8f46ed425f751f10f860c89f4d4554e09c457e5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"gr-qc","submitted_at":"2022-11-08T13:34:37Z","title_canon_sha256":"815a3df2ae0cd57f394194f1f10776c42ffd4f143dac17b7082176657521b575"},"schema_version":"1.0","source":{"id":"2211.04239","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2211.04239","created_at":"2026-07-05T06:00:33Z"},{"alias_kind":"arxiv_version","alias_value":"2211.04239v2","created_at":"2026-07-05T06:00:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.04239","created_at":"2026-07-05T06:00:33Z"},{"alias_kind":"pith_short_12","alias_value":"MMBLLVEB2LK3","created_at":"2026-07-05T06:00:33Z"},{"alias_kind":"pith_short_16","alias_value":"MMBLLVEB2LK3BFQX","created_at":"2026-07-05T06:00:33Z"},{"alias_kind":"pith_short_8","alias_value":"MMBLLVEB","created_at":"2026-07-05T06:00:33Z"}],"graph_snapshots":[{"event_id":"sha256:8ad898be2b37d5a48b53b7e933179d1a6f074b0c742c4cb0afa9c5d08cdbbf3d","target":"graph","created_at":"2026-07-05T06:00:33Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2211.04239/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The 0th order of affine perturbation series of the deflection angle of a ray near a photon sphere is more accurate than a deflection angle in a strong deflection limit, which is used often, because the later has hidden error terms. We investigate gravitational lensing by using 0th order affine perturbation series of the deflection angle in a general asymptotically-flat, static, and spherical symmetric spacetime with the photon sphere. We apply our formula to Schwarzschild black hole, Reissner-Nordstr\\\"om black hole, and Ellis-Bronnikov wormhole spacetimes as examples. By comparing observables ","authors_text":"Naoki Tsukamoto","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"gr-qc","submitted_at":"2022-11-08T13:34:37Z","title":"Gravitational lensing by using the 0th order of affine perturbation series of the deflection angle of a ray near a photon sphere"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.04239","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":"sha256:4a904a34aaaff963c12167682f586445187593132135040fc406c28b1ca32a54","target":"record","created_at":"2026-07-05T06:00:33Z","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":"c170f5c5e1bc1ddb64aa0cc9b8f46ed425f751f10f860c89f4d4554e09c457e5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"gr-qc","submitted_at":"2022-11-08T13:34:37Z","title_canon_sha256":"815a3df2ae0cd57f394194f1f10776c42ffd4f143dac17b7082176657521b575"},"schema_version":"1.0","source":{"id":"2211.04239","kind":"arxiv","version":2}},"canonical_sha256":"6302b5d481d2d5b096171fec301c2ac3ac96ed3f37173d28c76a6f3fc82b1690","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6302b5d481d2d5b096171fec301c2ac3ac96ed3f37173d28c76a6f3fc82b1690","first_computed_at":"2026-07-05T06:00:33.259375Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:00:33.259375Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zOzjDdXq/YgxiGwCp0NRpxmg/v68sNyqyKCrTFGEF2dBTZAJiEvJc47yAsS68/uUV4Dyy1xMxXL16RuwxMtyDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T06:00:33.259860Z","signed_message":"canonical_sha256_bytes"},"source_id":"2211.04239","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a904a34aaaff963c12167682f586445187593132135040fc406c28b1ca32a54","sha256:8ad898be2b37d5a48b53b7e933179d1a6f074b0c742c4cb0afa9c5d08cdbbf3d"],"state_sha256":"85d4c793cc0edd25e504a01308f13cb1c86a8f9e406b0fedcee8d0f1f7423b32"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RUlccCS/CDNDI09AfJKerT5pp45n95gUvez6XQn4QJ39ReG72QGLWth2gJBjZaA5idlnmXqoGDwwsoEJABtsBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T10:12:02.749077Z","bundle_sha256":"915043d516e10c53c9094282f94d0a62744984497a2107950b0c4a12511054d1"}}