{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:7NZ4FFISGCUX2LQVBLQEO62CYT","short_pith_number":"pith:7NZ4FFIS","canonical_record":{"source":{"id":"2311.08920","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2023-11-15T12:46:46Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"b2d0b24be12c21823153d09886df102c839176af6b8aa02944a92435b85dfbf9","abstract_canon_sha256":"f6918e0339c0c1f67bf74b750616cc8187fb1bffb056adacba4bb5379ba1236b"},"schema_version":"1.0"},"canonical_sha256":"fb73c2951230a97d2e150ae0477b42c4e57ce80381ef809d3bfaa3063435ec53","source":{"kind":"arxiv","id":"2311.08920","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2311.08920","created_at":"2026-07-05T07:13:06Z"},{"alias_kind":"arxiv_version","alias_value":"2311.08920v1","created_at":"2026-07-05T07:13:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2311.08920","created_at":"2026-07-05T07:13:06Z"},{"alias_kind":"pith_short_12","alias_value":"7NZ4FFISGCUX","created_at":"2026-07-05T07:13:06Z"},{"alias_kind":"pith_short_16","alias_value":"7NZ4FFISGCUX2LQV","created_at":"2026-07-05T07:13:06Z"},{"alias_kind":"pith_short_8","alias_value":"7NZ4FFIS","created_at":"2026-07-05T07:13:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:7NZ4FFISGCUX2LQVBLQEO62CYT","target":"record","payload":{"canonical_record":{"source":{"id":"2311.08920","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2023-11-15T12:46:46Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"b2d0b24be12c21823153d09886df102c839176af6b8aa02944a92435b85dfbf9","abstract_canon_sha256":"f6918e0339c0c1f67bf74b750616cc8187fb1bffb056adacba4bb5379ba1236b"},"schema_version":"1.0"},"canonical_sha256":"fb73c2951230a97d2e150ae0477b42c4e57ce80381ef809d3bfaa3063435ec53","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:13:06.221749Z","signature_b64":"DqucAaJkN8oWdsNPjrz1CyVae9kLBbwP9DrKUpQPQLTyIL2YONQvlo2XLlDzGkercqcAdLC4eWiqIBAq4SCLCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb73c2951230a97d2e150ae0477b42c4e57ce80381ef809d3bfaa3063435ec53","last_reissued_at":"2026-07-05T07:13:06.221280Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:13:06.221280Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2311.08920","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":"2026-07-05T07:13:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lqG8DPT+/tPFHcWelzeYMEndWG/Zt0rCJ9+yrQqjzY4KW//hFIxHfUU6Ktz1Zdm7+AelO7zh8Su9Igr+iA5ICw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T17:53:51.035405Z"},"content_sha256":"e074e7f046795d2699ba6e88b1d2b299b2475458e5b845d46e5d747664b45759","schema_version":"1.0","event_id":"sha256:e074e7f046795d2699ba6e88b1d2b299b2475458e5b845d46e5d747664b45759"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:7NZ4FFISGCUX2LQVBLQEO62CYT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kustaanheimo-Stiefel Transformation, Birkhoff-Waldvogel Transformation and Integrable Mechanical Billiards","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.DS","authors_text":"Airi Takeuchi, Lei Zhao","submitted_at":"2023-11-15T12:46:46Z","abstract_excerpt":"The three-dimensional Kepler problem is related to the four-dimensional isotropic harmonic oscillators by the Kustaanheimo-Stiefel Transformations. In the first part of this paper, we study how certain integrable mechanical billiards are related by this transformation. This in part illustrates the rotation-invariance of integrable reflection walls in the three-dimensional Kepler billiards found till so far. The second part of this paper deals with Birkhoff-Waldvogel's Transformation of the three-dimensional problem with two Kepler centers. In particular, we establish an analogous theory of Lev"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.08920","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/2311.08920/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":"2026-07-05T07:13:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QIESoqGM19DChcaq3NQ/KbHXPW6A69rgHk6TicMVOagJKrwLdc8lSLmyoR6C2szaX+6yipPkowpAtlheDJrTCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T17:53:51.036331Z"},"content_sha256":"79f9569ca0d67203dc4ccad238b7bd486ad57652b0aa16b2ac2eea660d4b41da","schema_version":"1.0","event_id":"sha256:79f9569ca0d67203dc4ccad238b7bd486ad57652b0aa16b2ac2eea660d4b41da"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7NZ4FFISGCUX2LQVBLQEO62CYT/bundle.json","state_url":"https://pith.science/pith/7NZ4FFISGCUX2LQVBLQEO62CYT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7NZ4FFISGCUX2LQVBLQEO62CYT/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":"2026-08-12T17:53:51Z","links":{"resolver":"https://pith.science/pith/7NZ4FFISGCUX2LQVBLQEO62CYT","bundle":"https://pith.science/pith/7NZ4FFISGCUX2LQVBLQEO62CYT/bundle.json","state":"https://pith.science/pith/7NZ4FFISGCUX2LQVBLQEO62CYT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7NZ4FFISGCUX2LQVBLQEO62CYT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:7NZ4FFISGCUX2LQVBLQEO62CYT","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":"f6918e0339c0c1f67bf74b750616cc8187fb1bffb056adacba4bb5379ba1236b","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2023-11-15T12:46:46Z","title_canon_sha256":"b2d0b24be12c21823153d09886df102c839176af6b8aa02944a92435b85dfbf9"},"schema_version":"1.0","source":{"id":"2311.08920","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2311.08920","created_at":"2026-07-05T07:13:06Z"},{"alias_kind":"arxiv_version","alias_value":"2311.08920v1","created_at":"2026-07-05T07:13:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2311.08920","created_at":"2026-07-05T07:13:06Z"},{"alias_kind":"pith_short_12","alias_value":"7NZ4FFISGCUX","created_at":"2026-07-05T07:13:06Z"},{"alias_kind":"pith_short_16","alias_value":"7NZ4FFISGCUX2LQV","created_at":"2026-07-05T07:13:06Z"},{"alias_kind":"pith_short_8","alias_value":"7NZ4FFIS","created_at":"2026-07-05T07:13:06Z"}],"graph_snapshots":[{"event_id":"sha256:79f9569ca0d67203dc4ccad238b7bd486ad57652b0aa16b2ac2eea660d4b41da","target":"graph","created_at":"2026-07-05T07:13:06Z","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/2311.08920/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The three-dimensional Kepler problem is related to the four-dimensional isotropic harmonic oscillators by the Kustaanheimo-Stiefel Transformations. In the first part of this paper, we study how certain integrable mechanical billiards are related by this transformation. This in part illustrates the rotation-invariance of integrable reflection walls in the three-dimensional Kepler billiards found till so far. The second part of this paper deals with Birkhoff-Waldvogel's Transformation of the three-dimensional problem with two Kepler centers. In particular, we establish an analogous theory of Lev","authors_text":"Airi Takeuchi, Lei Zhao","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2023-11-15T12:46:46Z","title":"Kustaanheimo-Stiefel Transformation, Birkhoff-Waldvogel Transformation and Integrable Mechanical Billiards"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.08920","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:e074e7f046795d2699ba6e88b1d2b299b2475458e5b845d46e5d747664b45759","target":"record","created_at":"2026-07-05T07:13:06Z","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":"f6918e0339c0c1f67bf74b750616cc8187fb1bffb056adacba4bb5379ba1236b","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2023-11-15T12:46:46Z","title_canon_sha256":"b2d0b24be12c21823153d09886df102c839176af6b8aa02944a92435b85dfbf9"},"schema_version":"1.0","source":{"id":"2311.08920","kind":"arxiv","version":1}},"canonical_sha256":"fb73c2951230a97d2e150ae0477b42c4e57ce80381ef809d3bfaa3063435ec53","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb73c2951230a97d2e150ae0477b42c4e57ce80381ef809d3bfaa3063435ec53","first_computed_at":"2026-07-05T07:13:06.221280Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:13:06.221280Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DqucAaJkN8oWdsNPjrz1CyVae9kLBbwP9DrKUpQPQLTyIL2YONQvlo2XLlDzGkercqcAdLC4eWiqIBAq4SCLCw==","signature_status":"signed_v1","signed_at":"2026-07-05T07:13:06.221749Z","signed_message":"canonical_sha256_bytes"},"source_id":"2311.08920","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e074e7f046795d2699ba6e88b1d2b299b2475458e5b845d46e5d747664b45759","sha256:79f9569ca0d67203dc4ccad238b7bd486ad57652b0aa16b2ac2eea660d4b41da"],"state_sha256":"f0b42ada06db1ec2b251aa74efa3f8a6c74c84e6423df09a87217137327de578"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SP/aKDfSnAcMqRXz2eqouncbQd0LSkfemYUCfEWUM1oZwc28xTGzRoMGNgaXUVPm3xewmK65aLlbry3aalL3Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T17:53:51.041883Z","bundle_sha256":"948c385ba7048bf8aac0da1b8ae10cee5e2e68d57a658b1d8e82b8392de56ab0"}}