{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:QHEE6RO73OTK5EGPTQEUVF5CG7","short_pith_number":"pith:QHEE6RO7","canonical_record":{"source":{"id":"1112.3231","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-12-14T14:32:30Z","cross_cats_sorted":["math.DG","nlin.CD"],"title_canon_sha256":"e640d721f955ae82ae3d36dc3be42f2af88d19d4567ae1b4bac24452ab5d04d0","abstract_canon_sha256":"1a19210ce7b3da4ebe9c789019f973de45076efceabd9527a0b8e64e37190866"},"schema_version":"1.0"},"canonical_sha256":"81c84f45dfdba6ae90cf9c094a97a237c9b2b9cffaee5ec1bd91330da5345713","source":{"kind":"arxiv","id":"1112.3231","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.3231","created_at":"2026-05-18T04:06:19Z"},{"alias_kind":"arxiv_version","alias_value":"1112.3231v1","created_at":"2026-05-18T04:06:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.3231","created_at":"2026-05-18T04:06:19Z"},{"alias_kind":"pith_short_12","alias_value":"QHEE6RO73OTK","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QHEE6RO73OTK5EGP","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QHEE6RO7","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:QHEE6RO73OTK5EGPTQEUVF5CG7","target":"record","payload":{"canonical_record":{"source":{"id":"1112.3231","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-12-14T14:32:30Z","cross_cats_sorted":["math.DG","nlin.CD"],"title_canon_sha256":"e640d721f955ae82ae3d36dc3be42f2af88d19d4567ae1b4bac24452ab5d04d0","abstract_canon_sha256":"1a19210ce7b3da4ebe9c789019f973de45076efceabd9527a0b8e64e37190866"},"schema_version":"1.0"},"canonical_sha256":"81c84f45dfdba6ae90cf9c094a97a237c9b2b9cffaee5ec1bd91330da5345713","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:19.741753Z","signature_b64":"AP3BvW9LFGtMX6pEN7TSC/GKVAhVWL1FagPTurP9jPRVX/rMUjZEdc+VQx5TJyCj31VmvE+xv4c2jrD75bZpBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81c84f45dfdba6ae90cf9c094a97a237c9b2b9cffaee5ec1bd91330da5345713","last_reissued_at":"2026-05-18T04:06:19.740924Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:19.740924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.3231","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":"2026-05-18T04:06:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LMDyEnn0RImHcg/U7JGfcIbfiORdFrD6RkfU0yXmbxEc99BuRaems4mqCEgO0D4AmgI3gpwu0e1eyHNyhLlBAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T05:08:44.516563Z"},"content_sha256":"f05e557da2542ca67c2a0534b1e1ee150566fe127b05115e406cd4f62c2b08cf","schema_version":"1.0","event_id":"sha256:f05e557da2542ca67c2a0534b1e1ee150566fe127b05115e406cd4f62c2b08cf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:QHEE6RO73OTK5EGPTQEUVF5CG7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Regular and irregular geodesics on spherical harmonic surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","nlin.CD"],"primary_cat":"math.DS","authors_text":"Thomas J. Waters","submitted_at":"2011-12-14T14:32:30Z","abstract_excerpt":"The behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex. In this paper we use the Hamiltonian formulation of the geodesic equations to analyze their integrability properties. In particular, we examine the behavior of geodesics on surfaces defined by the spherical harmonics. Using the Morales-Ramis theorem and Kovacic algorithm we are able to prove that the geodesic equations on all surfaces defined by the sectoral harmonics are not integrable, and we use Poincar\\'{e} sections to demonstrate the breakdown of regular motion."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3231","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":"2026-05-18T04:06:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rIoVXOBY+PvEB8YAsFNgsQ2odRLuDdkV78pFpPmWxwcxvHc74VKxPDD6HpuhcrtXWYf2CAz+VX4sSLWIPNfVBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T05:08:44.516915Z"},"content_sha256":"8bdd6dba74635f02b511ea741afecd543ac34432474f42a157a22ec8a219c597","schema_version":"1.0","event_id":"sha256:8bdd6dba74635f02b511ea741afecd543ac34432474f42a157a22ec8a219c597"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QHEE6RO73OTK5EGPTQEUVF5CG7/bundle.json","state_url":"https://pith.science/pith/QHEE6RO73OTK5EGPTQEUVF5CG7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QHEE6RO73OTK5EGPTQEUVF5CG7/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":"2026-05-30T05:08:44Z","links":{"resolver":"https://pith.science/pith/QHEE6RO73OTK5EGPTQEUVF5CG7","bundle":"https://pith.science/pith/QHEE6RO73OTK5EGPTQEUVF5CG7/bundle.json","state":"https://pith.science/pith/QHEE6RO73OTK5EGPTQEUVF5CG7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QHEE6RO73OTK5EGPTQEUVF5CG7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:QHEE6RO73OTK5EGPTQEUVF5CG7","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":"1a19210ce7b3da4ebe9c789019f973de45076efceabd9527a0b8e64e37190866","cross_cats_sorted":["math.DG","nlin.CD"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-12-14T14:32:30Z","title_canon_sha256":"e640d721f955ae82ae3d36dc3be42f2af88d19d4567ae1b4bac24452ab5d04d0"},"schema_version":"1.0","source":{"id":"1112.3231","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.3231","created_at":"2026-05-18T04:06:19Z"},{"alias_kind":"arxiv_version","alias_value":"1112.3231v1","created_at":"2026-05-18T04:06:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.3231","created_at":"2026-05-18T04:06:19Z"},{"alias_kind":"pith_short_12","alias_value":"QHEE6RO73OTK","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QHEE6RO73OTK5EGP","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QHEE6RO7","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:8bdd6dba74635f02b511ea741afecd543ac34432474f42a157a22ec8a219c597","target":"graph","created_at":"2026-05-18T04:06:19Z","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":"The behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex. In this paper we use the Hamiltonian formulation of the geodesic equations to analyze their integrability properties. In particular, we examine the behavior of geodesics on surfaces defined by the spherical harmonics. Using the Morales-Ramis theorem and Kovacic algorithm we are able to prove that the geodesic equations on all surfaces defined by the sectoral harmonics are not integrable, and we use Poincar\\'{e} sections to demonstrate the breakdown of regular motion.","authors_text":"Thomas J. Waters","cross_cats":["math.DG","nlin.CD"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-12-14T14:32:30Z","title":"Regular and irregular geodesics on spherical harmonic surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3231","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:f05e557da2542ca67c2a0534b1e1ee150566fe127b05115e406cd4f62c2b08cf","target":"record","created_at":"2026-05-18T04:06:19Z","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":"1a19210ce7b3da4ebe9c789019f973de45076efceabd9527a0b8e64e37190866","cross_cats_sorted":["math.DG","nlin.CD"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-12-14T14:32:30Z","title_canon_sha256":"e640d721f955ae82ae3d36dc3be42f2af88d19d4567ae1b4bac24452ab5d04d0"},"schema_version":"1.0","source":{"id":"1112.3231","kind":"arxiv","version":1}},"canonical_sha256":"81c84f45dfdba6ae90cf9c094a97a237c9b2b9cffaee5ec1bd91330da5345713","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81c84f45dfdba6ae90cf9c094a97a237c9b2b9cffaee5ec1bd91330da5345713","first_computed_at":"2026-05-18T04:06:19.740924Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:06:19.740924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AP3BvW9LFGtMX6pEN7TSC/GKVAhVWL1FagPTurP9jPRVX/rMUjZEdc+VQx5TJyCj31VmvE+xv4c2jrD75bZpBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:06:19.741753Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.3231","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f05e557da2542ca67c2a0534b1e1ee150566fe127b05115e406cd4f62c2b08cf","sha256:8bdd6dba74635f02b511ea741afecd543ac34432474f42a157a22ec8a219c597"],"state_sha256":"9c78729fa3b14ca3024902965166cf8623e08d7582912037c590d2ffd7b20905"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"llYhXHDGqPxJQah7wW8XgHdzTkdvsLIMeo15v323ZFYcHckwHyHPE5K9Cb20EKBXchJuyboRKQ3qLYjd9EYkCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T05:08:44.518884Z","bundle_sha256":"0a5d82dfe6839abbb03df74054ca76d43691bf0018fd6a20b727193bf02f96a6"}}