{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:AUMZLWBS6XSL4GCXUPH3NSH4X7","short_pith_number":"pith:AUMZLWBS","canonical_record":{"source":{"id":"1908.04496","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-13T05:39:57Z","cross_cats_sorted":["math-ph","math.MP","nlin.SI","physics.class-ph"],"title_canon_sha256":"22ded73023a7398ecdfa85a8b85563e20ad0a21c3cfcdc3e62a6e9c3a18a18e4","abstract_canon_sha256":"0a57c316f156b2502dceadccd638c3493cc93e8863b89f61dd582b408058d027"},"schema_version":"1.0"},"canonical_sha256":"051995d832f5e4be1857a3cfb6c8fcbfd1ff0a9675c9fa991a8cd69a3236cf21","source":{"kind":"arxiv","id":"1908.04496","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.04496","created_at":"2026-07-05T01:32:56Z"},{"alias_kind":"arxiv_version","alias_value":"1908.04496v1","created_at":"2026-07-05T01:32:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.04496","created_at":"2026-07-05T01:32:56Z"},{"alias_kind":"pith_short_12","alias_value":"AUMZLWBS6XSL","created_at":"2026-07-05T01:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"AUMZLWBS6XSL4GCX","created_at":"2026-07-05T01:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"AUMZLWBS","created_at":"2026-07-05T01:32:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:AUMZLWBS6XSL4GCXUPH3NSH4X7","target":"record","payload":{"canonical_record":{"source":{"id":"1908.04496","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-13T05:39:57Z","cross_cats_sorted":["math-ph","math.MP","nlin.SI","physics.class-ph"],"title_canon_sha256":"22ded73023a7398ecdfa85a8b85563e20ad0a21c3cfcdc3e62a6e9c3a18a18e4","abstract_canon_sha256":"0a57c316f156b2502dceadccd638c3493cc93e8863b89f61dd582b408058d027"},"schema_version":"1.0"},"canonical_sha256":"051995d832f5e4be1857a3cfb6c8fcbfd1ff0a9675c9fa991a8cd69a3236cf21","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:32:56.440504Z","signature_b64":"Kt2/Q5o5kOfqasC9fqzyK5PSGe9WZiz1YSC1xx3O12Y81QA426OHDk3Xse8EYTfCRPQStWqsT8DoYT29jIpyBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"051995d832f5e4be1857a3cfb6c8fcbfd1ff0a9675c9fa991a8cd69a3236cf21","last_reissued_at":"2026-07-05T01:32:56.440126Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:32:56.440126Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.04496","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":"2026-07-05T01:32:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RK0iA16aWHjzf/JGaVVTIJnKvGfZhWOO7qIMfMYhJWhoG0SsmTmUiXexSrqFWFl6U9g3LRzVp0dT6LTn79JtAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T05:07:05.159435Z"},"content_sha256":"bd8335e07d97df153104c977f5a6f03624978f7c13da63ba333640d0f3f998ae","schema_version":"1.0","event_id":"sha256:bd8335e07d97df153104c977f5a6f03624978f7c13da63ba333640d0f3f998ae"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:AUMZLWBS6XSL4GCXUPH3NSH4X7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Symmetry reduction of the 3-body problem in $\\mathbb{R}^4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","nlin.SI","physics.class-ph"],"primary_cat":"math.DS","authors_text":"Holger R. Dullin, J\\\"urgen Scheurle","submitted_at":"2019-08-13T05:39:57Z","abstract_excerpt":"The 3-body problem in $\\mathbb{R}^4$ has 24 dimensions and is invariant under translations and rotations. We do the full symplectic symmetry reduction and obtain a reduced Hamiltonian in local symplectic coordinates on a reduced phase space with 8 dimensions. The Hamiltonian depends on two parameters $\\mu_1 > \\mu_2 \\ge 0$, related to the conserved angular momentum. The limit $\\mu_2 \\to 0$ corresponds to the 3-dimensional limit. We show that the reduced Hamiltonian has relative equilibria that are local minima and hence Lyapunov stable when $\\mu_2$ is sufficiently small. This proves the existen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.04496","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.04496/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":"2026-07-05T01:32:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jao0pTsecfA+beqMemCNXSv3zqLvTPd87V8kLU32ulN2+ewKRAit3jxP1HQzIx33btqkCXHX6zBt9i/XbaRCBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T05:07:05.161906Z"},"content_sha256":"52bfaf0ab037cf4cecf93937242f50ba405fab894f289f4486b1f650d2704df9","schema_version":"1.0","event_id":"sha256:52bfaf0ab037cf4cecf93937242f50ba405fab894f289f4486b1f650d2704df9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AUMZLWBS6XSL4GCXUPH3NSH4X7/bundle.json","state_url":"https://pith.science/pith/AUMZLWBS6XSL4GCXUPH3NSH4X7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AUMZLWBS6XSL4GCXUPH3NSH4X7/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":"2026-08-16T05:07:05Z","links":{"resolver":"https://pith.science/pith/AUMZLWBS6XSL4GCXUPH3NSH4X7","bundle":"https://pith.science/pith/AUMZLWBS6XSL4GCXUPH3NSH4X7/bundle.json","state":"https://pith.science/pith/AUMZLWBS6XSL4GCXUPH3NSH4X7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AUMZLWBS6XSL4GCXUPH3NSH4X7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:AUMZLWBS6XSL4GCXUPH3NSH4X7","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":"0a57c316f156b2502dceadccd638c3493cc93e8863b89f61dd582b408058d027","cross_cats_sorted":["math-ph","math.MP","nlin.SI","physics.class-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-13T05:39:57Z","title_canon_sha256":"22ded73023a7398ecdfa85a8b85563e20ad0a21c3cfcdc3e62a6e9c3a18a18e4"},"schema_version":"1.0","source":{"id":"1908.04496","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.04496","created_at":"2026-07-05T01:32:56Z"},{"alias_kind":"arxiv_version","alias_value":"1908.04496v1","created_at":"2026-07-05T01:32:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.04496","created_at":"2026-07-05T01:32:56Z"},{"alias_kind":"pith_short_12","alias_value":"AUMZLWBS6XSL","created_at":"2026-07-05T01:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"AUMZLWBS6XSL4GCX","created_at":"2026-07-05T01:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"AUMZLWBS","created_at":"2026-07-05T01:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:52bfaf0ab037cf4cecf93937242f50ba405fab894f289f4486b1f650d2704df9","target":"graph","created_at":"2026-07-05T01:32:56Z","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/1908.04496/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The 3-body problem in $\\mathbb{R}^4$ has 24 dimensions and is invariant under translations and rotations. We do the full symplectic symmetry reduction and obtain a reduced Hamiltonian in local symplectic coordinates on a reduced phase space with 8 dimensions. The Hamiltonian depends on two parameters $\\mu_1 > \\mu_2 \\ge 0$, related to the conserved angular momentum. The limit $\\mu_2 \\to 0$ corresponds to the 3-dimensional limit. We show that the reduced Hamiltonian has relative equilibria that are local minima and hence Lyapunov stable when $\\mu_2$ is sufficiently small. This proves the existen","authors_text":"Holger R. Dullin, J\\\"urgen Scheurle","cross_cats":["math-ph","math.MP","nlin.SI","physics.class-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-13T05:39:57Z","title":"Symmetry reduction of the 3-body problem in $\\mathbb{R}^4$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.04496","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:bd8335e07d97df153104c977f5a6f03624978f7c13da63ba333640d0f3f998ae","target":"record","created_at":"2026-07-05T01:32:56Z","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":"0a57c316f156b2502dceadccd638c3493cc93e8863b89f61dd582b408058d027","cross_cats_sorted":["math-ph","math.MP","nlin.SI","physics.class-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-13T05:39:57Z","title_canon_sha256":"22ded73023a7398ecdfa85a8b85563e20ad0a21c3cfcdc3e62a6e9c3a18a18e4"},"schema_version":"1.0","source":{"id":"1908.04496","kind":"arxiv","version":1}},"canonical_sha256":"051995d832f5e4be1857a3cfb6c8fcbfd1ff0a9675c9fa991a8cd69a3236cf21","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"051995d832f5e4be1857a3cfb6c8fcbfd1ff0a9675c9fa991a8cd69a3236cf21","first_computed_at":"2026-07-05T01:32:56.440126Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:32:56.440126Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Kt2/Q5o5kOfqasC9fqzyK5PSGe9WZiz1YSC1xx3O12Y81QA426OHDk3Xse8EYTfCRPQStWqsT8DoYT29jIpyBg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:32:56.440504Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.04496","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bd8335e07d97df153104c977f5a6f03624978f7c13da63ba333640d0f3f998ae","sha256:52bfaf0ab037cf4cecf93937242f50ba405fab894f289f4486b1f650d2704df9"],"state_sha256":"af5a690ffb948d03fc3e7cde2aa7682c0c9b9d419aa28d8227a750a280cdad70"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"361lyOCoLTorQB2n+qco3uVOfJ3oDeAC8XNYESCmOeRRVRpakCYMqHTceqySqx38ZIDQsVYUuQhUAYy8bpd4BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T05:07:05.171625Z","bundle_sha256":"0a6c6ff7137ddda7d2dd512931ec051997b9a3ba544ea59e831a81244cecda6b"}}