{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:WKQ435EVVUE5PE767SR4KIBGGG","short_pith_number":"pith:WKQ435EV","canonical_record":{"source":{"id":"2204.01879","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2022-04-04T22:54:42Z","cross_cats_sorted":["astro-ph.CO","astro-ph.GA","astro-ph.HE","hep-th"],"title_canon_sha256":"2cea4de35177d0a242c862ecb1835cb88390664cb16dc7403b361ebb99dea81d","abstract_canon_sha256":"e5bb21975cdc7650d3732b9cf510e42c550ebfd83ee7078767ed889f1cd9ede2"},"schema_version":"1.0"},"canonical_sha256":"b2a1cdf495ad09d793fefca3c5202631884a221f8eac3fa1792146b94640e661","source":{"kind":"arxiv","id":"2204.01879","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2204.01879","created_at":"2026-07-05T05:00:00Z"},{"alias_kind":"arxiv_version","alias_value":"2204.01879v2","created_at":"2026-07-05T05:00:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2204.01879","created_at":"2026-07-05T05:00:00Z"},{"alias_kind":"pith_short_12","alias_value":"WKQ435EVVUE5","created_at":"2026-07-05T05:00:00Z"},{"alias_kind":"pith_short_16","alias_value":"WKQ435EVVUE5PE76","created_at":"2026-07-05T05:00:00Z"},{"alias_kind":"pith_short_8","alias_value":"WKQ435EV","created_at":"2026-07-05T05:00:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:WKQ435EVVUE5PE767SR4KIBGGG","target":"record","payload":{"canonical_record":{"source":{"id":"2204.01879","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2022-04-04T22:54:42Z","cross_cats_sorted":["astro-ph.CO","astro-ph.GA","astro-ph.HE","hep-th"],"title_canon_sha256":"2cea4de35177d0a242c862ecb1835cb88390664cb16dc7403b361ebb99dea81d","abstract_canon_sha256":"e5bb21975cdc7650d3732b9cf510e42c550ebfd83ee7078767ed889f1cd9ede2"},"schema_version":"1.0"},"canonical_sha256":"b2a1cdf495ad09d793fefca3c5202631884a221f8eac3fa1792146b94640e661","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:00:00.224235Z","signature_b64":"bBWg3Fecs8p++D2zpuF5ZB3yXObPJ7+6YBpNo/STapNnOy5VH2jAM21Bw7Z/Tz+lKMknZ2wu7lnCPWXc/t50Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2a1cdf495ad09d793fefca3c5202631884a221f8eac3fa1792146b94640e661","last_reissued_at":"2026-07-05T05:00:00.223835Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:00:00.223835Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2204.01879","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":"2026-07-05T05:00:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"66ve1wR3f5+IPzgGL0T5JnbeKuz4bZ28tXIxjBGQmjxRD7p/Hi/6wOEIFGiMs2RJ1j664mWzkb4wz2Ncpz9jBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T12:15:24.721894Z"},"content_sha256":"03db14b30ee2d54d2c643584e82bea51a7986e06e665a945249abe037c625401","schema_version":"1.0","event_id":"sha256:03db14b30ee2d54d2c643584e82bea51a7986e06e665a945249abe037c625401"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:WKQ435EVVUE5PE767SR4KIBGGG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Distortions of images of Schwarzschild lensing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["astro-ph.CO","astro-ph.GA","astro-ph.HE","hep-th"],"primary_cat":"gr-qc","authors_text":"K. S. Virbhadra","submitted_at":"2022-04-04T22:54:42Z","abstract_excerpt":"We model the supermassive dark object $M87^*$ as a Schwarzschild lens and study the variations in tangential, radial, and total (the product of tangential and radial) magnifications of images (primary, secondary, and relativistic) against the changes in angular source position and the ratio of lens-source to the observer-source distance. Further, we study the behavior of partial derivatives (with respect to the angular source position) of total magnifications of images against the angular source position. Finally, we model supermassive dark objects at centers of 40 galaxies as Schwarzschild le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.01879","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/2204.01879/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":"2026-07-05T05:00:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"afotQ7RNwdU+swAA5rB3AUi/nVFlAy4kEdtMDdM6tk7RUYtbLVP37zSx5R/RSg3m4uaPcr6oe5V0NPMJ3L4FBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T12:15:24.722429Z"},"content_sha256":"e288a382a3c1d63d6f65172d32ae277adf1af83ef56f4ff041791383681b6f80","schema_version":"1.0","event_id":"sha256:e288a382a3c1d63d6f65172d32ae277adf1af83ef56f4ff041791383681b6f80"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WKQ435EVVUE5PE767SR4KIBGGG/bundle.json","state_url":"https://pith.science/pith/WKQ435EVVUE5PE767SR4KIBGGG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WKQ435EVVUE5PE767SR4KIBGGG/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":"2026-08-07T12:15:24Z","links":{"resolver":"https://pith.science/pith/WKQ435EVVUE5PE767SR4KIBGGG","bundle":"https://pith.science/pith/WKQ435EVVUE5PE767SR4KIBGGG/bundle.json","state":"https://pith.science/pith/WKQ435EVVUE5PE767SR4KIBGGG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WKQ435EVVUE5PE767SR4KIBGGG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:WKQ435EVVUE5PE767SR4KIBGGG","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":"e5bb21975cdc7650d3732b9cf510e42c550ebfd83ee7078767ed889f1cd9ede2","cross_cats_sorted":["astro-ph.CO","astro-ph.GA","astro-ph.HE","hep-th"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2022-04-04T22:54:42Z","title_canon_sha256":"2cea4de35177d0a242c862ecb1835cb88390664cb16dc7403b361ebb99dea81d"},"schema_version":"1.0","source":{"id":"2204.01879","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2204.01879","created_at":"2026-07-05T05:00:00Z"},{"alias_kind":"arxiv_version","alias_value":"2204.01879v2","created_at":"2026-07-05T05:00:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2204.01879","created_at":"2026-07-05T05:00:00Z"},{"alias_kind":"pith_short_12","alias_value":"WKQ435EVVUE5","created_at":"2026-07-05T05:00:00Z"},{"alias_kind":"pith_short_16","alias_value":"WKQ435EVVUE5PE76","created_at":"2026-07-05T05:00:00Z"},{"alias_kind":"pith_short_8","alias_value":"WKQ435EV","created_at":"2026-07-05T05:00:00Z"}],"graph_snapshots":[{"event_id":"sha256:e288a382a3c1d63d6f65172d32ae277adf1af83ef56f4ff041791383681b6f80","target":"graph","created_at":"2026-07-05T05:00:00Z","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/2204.01879/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We model the supermassive dark object $M87^*$ as a Schwarzschild lens and study the variations in tangential, radial, and total (the product of tangential and radial) magnifications of images (primary, secondary, and relativistic) against the changes in angular source position and the ratio of lens-source to the observer-source distance. Further, we study the behavior of partial derivatives (with respect to the angular source position) of total magnifications of images against the angular source position. Finally, we model supermassive dark objects at centers of 40 galaxies as Schwarzschild le","authors_text":"K. S. Virbhadra","cross_cats":["astro-ph.CO","astro-ph.GA","astro-ph.HE","hep-th"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2022-04-04T22:54:42Z","title":"Distortions of images of Schwarzschild lensing"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.01879","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":"sha256:03db14b30ee2d54d2c643584e82bea51a7986e06e665a945249abe037c625401","target":"record","created_at":"2026-07-05T05:00:00Z","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":"e5bb21975cdc7650d3732b9cf510e42c550ebfd83ee7078767ed889f1cd9ede2","cross_cats_sorted":["astro-ph.CO","astro-ph.GA","astro-ph.HE","hep-th"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2022-04-04T22:54:42Z","title_canon_sha256":"2cea4de35177d0a242c862ecb1835cb88390664cb16dc7403b361ebb99dea81d"},"schema_version":"1.0","source":{"id":"2204.01879","kind":"arxiv","version":2}},"canonical_sha256":"b2a1cdf495ad09d793fefca3c5202631884a221f8eac3fa1792146b94640e661","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b2a1cdf495ad09d793fefca3c5202631884a221f8eac3fa1792146b94640e661","first_computed_at":"2026-07-05T05:00:00.223835Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:00:00.223835Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bBWg3Fecs8p++D2zpuF5ZB3yXObPJ7+6YBpNo/STapNnOy5VH2jAM21Bw7Z/Tz+lKMknZ2wu7lnCPWXc/t50Cw==","signature_status":"signed_v1","signed_at":"2026-07-05T05:00:00.224235Z","signed_message":"canonical_sha256_bytes"},"source_id":"2204.01879","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:03db14b30ee2d54d2c643584e82bea51a7986e06e665a945249abe037c625401","sha256:e288a382a3c1d63d6f65172d32ae277adf1af83ef56f4ff041791383681b6f80"],"state_sha256":"1f2ef1bafb6a1365d69116134e5810a8e9b2a7043f5a89fa72308b893f30dd48"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NbI6NsonWGQo/2RE1ULIOTchACnSWO0U4rx7xeRNQDt9s91ZcSPWY14Bpm4MjKFSmRavPKTAkjv8k0KClCCKBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T12:15:24.727987Z","bundle_sha256":"38a623449e7e52e8f8a2242e3cff16b29eb9f2b5d95626de7c721af15dfafab3"}}