{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:I3KRVDZ4YVPPAALQ3K6PTLTT4H","short_pith_number":"pith:I3KRVDZ4","canonical_record":{"source":{"id":"2404.18815","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2024-04-29T15:52:05Z","cross_cats_sorted":["math.CA","math.DG","math.FA"],"title_canon_sha256":"efdcf1407d8858ceec05cb966bc3c64fbf797bb7149dcea849ccd3649370af10","abstract_canon_sha256":"559113ff8369194921f7947674895d35c4e48bac959996e1586e5ddf2e5beb09"},"schema_version":"1.0"},"canonical_sha256":"46d51a8f3cc55ef00170dabcf9ae73e1e3857a3c184d5ccb2b141caa16c36f2b","source":{"kind":"arxiv","id":"2404.18815","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.18815","created_at":"2026-06-19T16:12:03Z"},{"alias_kind":"arxiv_version","alias_value":"2404.18815v4","created_at":"2026-06-19T16:12:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.18815","created_at":"2026-06-19T16:12:03Z"},{"alias_kind":"pith_short_12","alias_value":"I3KRVDZ4YVPP","created_at":"2026-06-19T16:12:03Z"},{"alias_kind":"pith_short_16","alias_value":"I3KRVDZ4YVPPAALQ","created_at":"2026-06-19T16:12:03Z"},{"alias_kind":"pith_short_8","alias_value":"I3KRVDZ4","created_at":"2026-06-19T16:12:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:I3KRVDZ4YVPPAALQ3K6PTLTT4H","target":"record","payload":{"canonical_record":{"source":{"id":"2404.18815","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2024-04-29T15:52:05Z","cross_cats_sorted":["math.CA","math.DG","math.FA"],"title_canon_sha256":"efdcf1407d8858ceec05cb966bc3c64fbf797bb7149dcea849ccd3649370af10","abstract_canon_sha256":"559113ff8369194921f7947674895d35c4e48bac959996e1586e5ddf2e5beb09"},"schema_version":"1.0"},"canonical_sha256":"46d51a8f3cc55ef00170dabcf9ae73e1e3857a3c184d5ccb2b141caa16c36f2b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:12:03.040277Z","signature_b64":"o/AyC8XeGcdtdmN3lDtAlrRAnXdjsb7d1VzOakArxsCMpGF5WsopuiZHEfh5pt9CBuI+QYZMD2FSD+Wr48myCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46d51a8f3cc55ef00170dabcf9ae73e1e3857a3c184d5ccb2b141caa16c36f2b","last_reissued_at":"2026-06-19T16:12:03.039884Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:12:03.039884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2404.18815","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-19T16:12:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YukmCplRVlBF9/ubOCCySFIvd775fZ1FFPJLFXyyHd3oF+otJTCYzLRO+eUTQP5PjsHkymqF/7KT6JkAm2aYDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T14:08:37.413120Z"},"content_sha256":"ac57be2912ab6fcf1457fa1bbef8de7b05927325cbf238b4054ea521c45c85d0","schema_version":"1.0","event_id":"sha256:ac57be2912ab6fcf1457fa1bbef8de7b05927325cbf238b4054ea521c45c85d0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:I3KRVDZ4YVPPAALQ3K6PTLTT4H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bifurcations for Lagrangian systems and geodesics II","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.DG","math.FA"],"primary_cat":"math.DS","authors_text":"Guangcun Lu","submitted_at":"2024-04-29T15:52:05Z","abstract_excerpt":"This is the second part of a two--part series investigating bifurcation phenomena in autonomous Lagrangian systems and geodesic flows on Finsler and Riemannian manifolds. Building upon the abstract bifurcation theorems established in earlier work and the results of Part I, this study makes contributions in two main directions. In Part A, we focus on bifurcations of generalized periodic solutions in autonomous Lagrangian systems. By employing Morse index and nullity techniques within the normal space to the $\\mathbb{R}$-orbits of solutions, we derive necessary and sufficient conditions for bifu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.18815","kind":"arxiv","version":4},"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/2404.18815/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":"2026-06-19T16:12:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o2NApY22TXxnqtqcldm3FPzd6AVBi8Nx/c2Fuc0vZ+D7tFMAKnSaBIQmWBlbYxjtXY/JMJ/AkL8Tvhhm1EgwBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T14:08:37.415363Z"},"content_sha256":"f9d2c3c79cb1eeabd416742895b033c679772870d8fd48c4aa7ad36d4da9dbb5","schema_version":"1.0","event_id":"sha256:f9d2c3c79cb1eeabd416742895b033c679772870d8fd48c4aa7ad36d4da9dbb5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I3KRVDZ4YVPPAALQ3K6PTLTT4H/bundle.json","state_url":"https://pith.science/pith/I3KRVDZ4YVPPAALQ3K6PTLTT4H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I3KRVDZ4YVPPAALQ3K6PTLTT4H/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":"2026-06-22T14:08:37Z","links":{"resolver":"https://pith.science/pith/I3KRVDZ4YVPPAALQ3K6PTLTT4H","bundle":"https://pith.science/pith/I3KRVDZ4YVPPAALQ3K6PTLTT4H/bundle.json","state":"https://pith.science/pith/I3KRVDZ4YVPPAALQ3K6PTLTT4H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I3KRVDZ4YVPPAALQ3K6PTLTT4H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:I3KRVDZ4YVPPAALQ3K6PTLTT4H","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":"559113ff8369194921f7947674895d35c4e48bac959996e1586e5ddf2e5beb09","cross_cats_sorted":["math.CA","math.DG","math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2024-04-29T15:52:05Z","title_canon_sha256":"efdcf1407d8858ceec05cb966bc3c64fbf797bb7149dcea849ccd3649370af10"},"schema_version":"1.0","source":{"id":"2404.18815","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.18815","created_at":"2026-06-19T16:12:03Z"},{"alias_kind":"arxiv_version","alias_value":"2404.18815v4","created_at":"2026-06-19T16:12:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.18815","created_at":"2026-06-19T16:12:03Z"},{"alias_kind":"pith_short_12","alias_value":"I3KRVDZ4YVPP","created_at":"2026-06-19T16:12:03Z"},{"alias_kind":"pith_short_16","alias_value":"I3KRVDZ4YVPPAALQ","created_at":"2026-06-19T16:12:03Z"},{"alias_kind":"pith_short_8","alias_value":"I3KRVDZ4","created_at":"2026-06-19T16:12:03Z"}],"graph_snapshots":[{"event_id":"sha256:f9d2c3c79cb1eeabd416742895b033c679772870d8fd48c4aa7ad36d4da9dbb5","target":"graph","created_at":"2026-06-19T16:12:03Z","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/2404.18815/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This is the second part of a two--part series investigating bifurcation phenomena in autonomous Lagrangian systems and geodesic flows on Finsler and Riemannian manifolds. Building upon the abstract bifurcation theorems established in earlier work and the results of Part I, this study makes contributions in two main directions. In Part A, we focus on bifurcations of generalized periodic solutions in autonomous Lagrangian systems. By employing Morse index and nullity techniques within the normal space to the $\\mathbb{R}$-orbits of solutions, we derive necessary and sufficient conditions for bifu","authors_text":"Guangcun Lu","cross_cats":["math.CA","math.DG","math.FA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2024-04-29T15:52:05Z","title":"Bifurcations for Lagrangian systems and geodesics II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.18815","kind":"arxiv","version":4},"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:ac57be2912ab6fcf1457fa1bbef8de7b05927325cbf238b4054ea521c45c85d0","target":"record","created_at":"2026-06-19T16:12:03Z","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":"559113ff8369194921f7947674895d35c4e48bac959996e1586e5ddf2e5beb09","cross_cats_sorted":["math.CA","math.DG","math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2024-04-29T15:52:05Z","title_canon_sha256":"efdcf1407d8858ceec05cb966bc3c64fbf797bb7149dcea849ccd3649370af10"},"schema_version":"1.0","source":{"id":"2404.18815","kind":"arxiv","version":4}},"canonical_sha256":"46d51a8f3cc55ef00170dabcf9ae73e1e3857a3c184d5ccb2b141caa16c36f2b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"46d51a8f3cc55ef00170dabcf9ae73e1e3857a3c184d5ccb2b141caa16c36f2b","first_computed_at":"2026-06-19T16:12:03.039884Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:12:03.039884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o/AyC8XeGcdtdmN3lDtAlrRAnXdjsb7d1VzOakArxsCMpGF5WsopuiZHEfh5pt9CBuI+QYZMD2FSD+Wr48myCA==","signature_status":"signed_v1","signed_at":"2026-06-19T16:12:03.040277Z","signed_message":"canonical_sha256_bytes"},"source_id":"2404.18815","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ac57be2912ab6fcf1457fa1bbef8de7b05927325cbf238b4054ea521c45c85d0","sha256:f9d2c3c79cb1eeabd416742895b033c679772870d8fd48c4aa7ad36d4da9dbb5"],"state_sha256":"d7db9a78a505bbc3eb2945c2a7013bba044992b92b7e0f25d55b6ee13268b4f6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EYlB5cajR/qMkMOL+g1D8aPxIA7AQOdKa2Jz4pUK1Cxm4y3rBewXHK8d6MhUPWbVZuWBcgUZDXlL9R+330LgBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T14:08:37.426470Z","bundle_sha256":"d693135af4c2a63437dfa9a2d4c558af9fef9bff3b75c2d63aed9cd153972cca"}}