{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:GJRQBFLK4PLZLPKI7DSEAUF3PV","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":"e4eb09f808d35641fcfaa7fa28b960c6010584c61ef4c7032c4e303ca72595f4","cross_cats_sorted":["nlin.SI"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-03-28T08:05:56Z","title_canon_sha256":"b67e7d89b62cfdc330d6fabed768900909ec7f4c21de10f992fdc039b293c2f0"},"schema_version":"1.0","source":{"id":"1203.6188","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.6188","created_at":"2026-05-18T02:17:22Z"},{"alias_kind":"arxiv_version","alias_value":"1203.6188v1","created_at":"2026-05-18T02:17:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.6188","created_at":"2026-05-18T02:17:22Z"},{"alias_kind":"pith_short_12","alias_value":"GJRQBFLK4PLZ","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GJRQBFLK4PLZLPKI","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GJRQBFLK","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:399586e8f973235a595b7263835611a4685500600f1d9160526f4916115cafd0","target":"graph","created_at":"2026-05-18T02:17:22Z","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":"We classify nonsingular symmetric periodic trajectories (SPTs) of billiards inside ellipsoids of R^{n+1} without any symmetry of revolution. SPTs are defined as periodic trajectories passing through some symmetry set. We prove that there are exactly 2^{2n}(2^{n+1}-1) classes of such trajectories. We have implemented an algorithm to find minimal SPTs of each of the 12 classes in the 2D case (R^2) and each of the 112 classes in the 3D case (R^3). They have periods 3, 4 or 6 in the 2D case; and 4, 5, 6, 8 or 10 in the 3D case. We display a selection of 3D minimal SPTs. Some of them have propertie","authors_text":"Pablo S. Casas, Rafael Ram\\'irez-Ros","cross_cats":["nlin.SI"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-03-28T08:05:56Z","title":"Classification of symmetric periodic trajectories in ellipsoidal billiards"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6188","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":"sha256:41067d58a21b54fb72e85da72e4ac450950d4f0be23854edc57562d5ce34a983","target":"record","created_at":"2026-05-18T02:17:22Z","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":"e4eb09f808d35641fcfaa7fa28b960c6010584c61ef4c7032c4e303ca72595f4","cross_cats_sorted":["nlin.SI"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-03-28T08:05:56Z","title_canon_sha256":"b67e7d89b62cfdc330d6fabed768900909ec7f4c21de10f992fdc039b293c2f0"},"schema_version":"1.0","source":{"id":"1203.6188","kind":"arxiv","version":1}},"canonical_sha256":"326300956ae3d795bd48f8e44050bb7d7e44dc93961c3f89ef37654e9d021cab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"326300956ae3d795bd48f8e44050bb7d7e44dc93961c3f89ef37654e9d021cab","first_computed_at":"2026-05-18T02:17:22.177762Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:22.177762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jmfqTSFqHFVMobldKc24aAEMqleg/iIW1w8Kr+cAiGPis7FOVGI0+atYBBYA5gVCiknh4I9w6A4KYFC7FNdbAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:22.178450Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.6188","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:41067d58a21b54fb72e85da72e4ac450950d4f0be23854edc57562d5ce34a983","sha256:399586e8f973235a595b7263835611a4685500600f1d9160526f4916115cafd0"],"state_sha256":"590a1feb5a2fbdc24cf8e8b1605069a5a16b8956773dcb22195c97a1f865cb84"}