{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:EDV6CM5SLTOKRFXTXW7UVQMV7K","short_pith_number":"pith:EDV6CM5S","canonical_record":{"source":{"id":"1012.2490","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-12-11T21:41:58Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"56c4f4aa7d65474080ad636cbe62037c86f2eeeb5394688cbe7e5f789b1e10fb","abstract_canon_sha256":"f3007301a692c0a11d8691d7804b6b269d178c1a3a998c39aee39a3e079c5f78"},"schema_version":"1.0"},"canonical_sha256":"20ebe133b25cdca896f3bdbf4ac195fa91d8ea2082cc9ec5c0c2b38efc702488","source":{"kind":"arxiv","id":"1012.2490","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.2490","created_at":"2026-05-18T04:02:00Z"},{"alias_kind":"arxiv_version","alias_value":"1012.2490v2","created_at":"2026-05-18T04:02:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.2490","created_at":"2026-05-18T04:02:00Z"},{"alias_kind":"pith_short_12","alias_value":"EDV6CM5SLTOK","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"EDV6CM5SLTOKRFXT","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"EDV6CM5S","created_at":"2026-05-18T12:26:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:EDV6CM5SLTOKRFXTXW7UVQMV7K","target":"record","payload":{"canonical_record":{"source":{"id":"1012.2490","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-12-11T21:41:58Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"56c4f4aa7d65474080ad636cbe62037c86f2eeeb5394688cbe7e5f789b1e10fb","abstract_canon_sha256":"f3007301a692c0a11d8691d7804b6b269d178c1a3a998c39aee39a3e079c5f78"},"schema_version":"1.0"},"canonical_sha256":"20ebe133b25cdca896f3bdbf4ac195fa91d8ea2082cc9ec5c0c2b38efc702488","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:02:00.946165Z","signature_b64":"3CcSd5ehYkLRt39t7Wh2mabeaSoU1v4L/RCa9SrYaFxbz7iZKMltM1VZKQa9Aihn5giSCYZ/g3EN/bGMV2b5CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"20ebe133b25cdca896f3bdbf4ac195fa91d8ea2082cc9ec5c0c2b38efc702488","last_reissued_at":"2026-05-18T04:02:00.945451Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:02:00.945451Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.2490","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:02:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZLPSLioFg2ufmjYUBR9l7yXnU5pFbiC5Kk9+SS8hzPDEnX4tT74JDBbzLw7bBgH3TnKbe4o1EVEunBnq+MIRCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T02:40:56.406894Z"},"content_sha256":"9b4d48f0970d9b63447bc32d7b414a26613520fe47a4e83b8d8440d49371a3b0","schema_version":"1.0","event_id":"sha256:9b4d48f0970d9b63447bc32d7b414a26613520fe47a4e83b8d8440d49371a3b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:EDV6CM5SLTOKRFXTXW7UVQMV7K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Polygonal Homographic Orbits of the Curved n-Body Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.DS","authors_text":"Florin Diacu","submitted_at":"2010-12-11T21:41:58Z","abstract_excerpt":"In the $2$-dimensional $n$-body problem, $n\\ge 3$, in spaces of constant curvature, $\\kappa\\ne 0$, we study polygonal homographic solutions. We first provide necessary and sufficient conditions for the existence of these orbits and then consider the case of regular polygons. We further use this criterion to show that, for any $n\\ge 3$, the regular $n$-gon is a polygonal homographic orbit if and only if all masses are equal. Then we prove the existence of relative equilibria of non-equal masses on the sphere of curvature $\\kappa>0$ for $n=3$ in the case of scalene triangles. Such triangular rel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2490","kind":"arxiv","version":2},"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":"2026-05-18T04:02:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5BYgcTmmfFEohpWVLQgS5uBYgqU933nBGvFxYBMhOlI/xOsHQRUIHCN7g92iiwdCdfuelxJwd3K/ivpHcpmSAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T02:40:56.407550Z"},"content_sha256":"1b770c201833d1df2b192e9fae414131d988a761bb4d25522e50ea1070c24ecd","schema_version":"1.0","event_id":"sha256:1b770c201833d1df2b192e9fae414131d988a761bb4d25522e50ea1070c24ecd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EDV6CM5SLTOKRFXTXW7UVQMV7K/bundle.json","state_url":"https://pith.science/pith/EDV6CM5SLTOKRFXTXW7UVQMV7K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EDV6CM5SLTOKRFXTXW7UVQMV7K/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":"2026-06-11T02:40:56Z","links":{"resolver":"https://pith.science/pith/EDV6CM5SLTOKRFXTXW7UVQMV7K","bundle":"https://pith.science/pith/EDV6CM5SLTOKRFXTXW7UVQMV7K/bundle.json","state":"https://pith.science/pith/EDV6CM5SLTOKRFXTXW7UVQMV7K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EDV6CM5SLTOKRFXTXW7UVQMV7K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:EDV6CM5SLTOKRFXTXW7UVQMV7K","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":"f3007301a692c0a11d8691d7804b6b269d178c1a3a998c39aee39a3e079c5f78","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-12-11T21:41:58Z","title_canon_sha256":"56c4f4aa7d65474080ad636cbe62037c86f2eeeb5394688cbe7e5f789b1e10fb"},"schema_version":"1.0","source":{"id":"1012.2490","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.2490","created_at":"2026-05-18T04:02:00Z"},{"alias_kind":"arxiv_version","alias_value":"1012.2490v2","created_at":"2026-05-18T04:02:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.2490","created_at":"2026-05-18T04:02:00Z"},{"alias_kind":"pith_short_12","alias_value":"EDV6CM5SLTOK","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"EDV6CM5SLTOKRFXT","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"EDV6CM5S","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:1b770c201833d1df2b192e9fae414131d988a761bb4d25522e50ea1070c24ecd","target":"graph","created_at":"2026-05-18T04:02:00Z","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":"In the $2$-dimensional $n$-body problem, $n\\ge 3$, in spaces of constant curvature, $\\kappa\\ne 0$, we study polygonal homographic solutions. We first provide necessary and sufficient conditions for the existence of these orbits and then consider the case of regular polygons. We further use this criterion to show that, for any $n\\ge 3$, the regular $n$-gon is a polygonal homographic orbit if and only if all masses are equal. Then we prove the existence of relative equilibria of non-equal masses on the sphere of curvature $\\kappa>0$ for $n=3$ in the case of scalene triangles. Such triangular rel","authors_text":"Florin Diacu","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-12-11T21:41:58Z","title":"Polygonal Homographic Orbits of the Curved n-Body Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2490","kind":"arxiv","version":2},"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":"sha256:9b4d48f0970d9b63447bc32d7b414a26613520fe47a4e83b8d8440d49371a3b0","target":"record","created_at":"2026-05-18T04:02:00Z","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":"f3007301a692c0a11d8691d7804b6b269d178c1a3a998c39aee39a3e079c5f78","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-12-11T21:41:58Z","title_canon_sha256":"56c4f4aa7d65474080ad636cbe62037c86f2eeeb5394688cbe7e5f789b1e10fb"},"schema_version":"1.0","source":{"id":"1012.2490","kind":"arxiv","version":2}},"canonical_sha256":"20ebe133b25cdca896f3bdbf4ac195fa91d8ea2082cc9ec5c0c2b38efc702488","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"20ebe133b25cdca896f3bdbf4ac195fa91d8ea2082cc9ec5c0c2b38efc702488","first_computed_at":"2026-05-18T04:02:00.945451Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:02:00.945451Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3CcSd5ehYkLRt39t7Wh2mabeaSoU1v4L/RCa9SrYaFxbz7iZKMltM1VZKQa9Aihn5giSCYZ/g3EN/bGMV2b5CA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:02:00.946165Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.2490","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9b4d48f0970d9b63447bc32d7b414a26613520fe47a4e83b8d8440d49371a3b0","sha256:1b770c201833d1df2b192e9fae414131d988a761bb4d25522e50ea1070c24ecd"],"state_sha256":"14cd0fcd75edd0566b44fd6a0a916532060572baa733d7d25eafa146bc7134cd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HzygUMiamTKnJXS4ITSw2s848POPYUxGDomFeeRos6msT/cNI+LyUS6wWjwi/JCpzkI8kpCKYuzvVciA2ujaDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T02:40:56.410743Z","bundle_sha256":"96aaf91a1ff0041bf8b7b616d4caa3e2bf42465f4fb5a76c399d5ae7c0292dae"}}