{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:KBG4PYQBCYMF4DXEBNYX5SAFMO","short_pith_number":"pith:KBG4PYQB","canonical_record":{"source":{"id":"1810.13011","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-10-30T21:17:46Z","cross_cats_sorted":[],"title_canon_sha256":"9be6ab69a4c4e62ed7bb42f8032156d41d893734d6d112cddbf47074c2454bb8","abstract_canon_sha256":"7987e66debe389b00a2de8afa72242aef5690e50b7b6db5d86fad354518ad73e"},"schema_version":"1.0"},"canonical_sha256":"504dc7e20116185e0ee40b717ec805638ea29ca81df01dbcd2b4b2e2586f4eb6","source":{"kind":"arxiv","id":"1810.13011","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.13011","created_at":"2026-05-17T23:45:57Z"},{"alias_kind":"arxiv_version","alias_value":"1810.13011v2","created_at":"2026-05-17T23:45:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.13011","created_at":"2026-05-17T23:45:57Z"},{"alias_kind":"pith_short_12","alias_value":"KBG4PYQBCYMF","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_16","alias_value":"KBG4PYQBCYMF4DXE","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_8","alias_value":"KBG4PYQB","created_at":"2026-05-18T12:32:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:KBG4PYQBCYMF4DXEBNYX5SAFMO","target":"record","payload":{"canonical_record":{"source":{"id":"1810.13011","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-10-30T21:17:46Z","cross_cats_sorted":[],"title_canon_sha256":"9be6ab69a4c4e62ed7bb42f8032156d41d893734d6d112cddbf47074c2454bb8","abstract_canon_sha256":"7987e66debe389b00a2de8afa72242aef5690e50b7b6db5d86fad354518ad73e"},"schema_version":"1.0"},"canonical_sha256":"504dc7e20116185e0ee40b717ec805638ea29ca81df01dbcd2b4b2e2586f4eb6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:57.938519Z","signature_b64":"qrwCevZTsVn2/JdB0CF98nbzHYtgkIq/fWj9ZYjYPQLEZkMVkU8+F92xLRB8+/APbw6H2yAM06VRS0l0w2zhDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"504dc7e20116185e0ee40b717ec805638ea29ca81df01dbcd2b4b2e2586f4eb6","last_reissued_at":"2026-05-17T23:45:57.937971Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:57.937971Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1810.13011","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":"2026-05-17T23:45:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"plTp2jeTebiBYsvkUNiFFwN9oZBSb+0O/q1+WW4R+IeJ/w0Edd2gNi71S1rAjZfDKbP8q9qGFMOITcxXwhfVDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T10:19:48.856754Z"},"content_sha256":"7ed24039b5695ed34e1a387e1e88feffc2950a1bc98a9e7e819ffae46689655c","schema_version":"1.0","event_id":"sha256:7ed24039b5695ed34e1a387e1e88feffc2950a1bc98a9e7e819ffae46689655c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:KBG4PYQBCYMF4DXEBNYX5SAFMO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Planar $N$-body central configurations with a homogeneous potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Marshall Hampton","submitted_at":"2018-10-30T21:17:46Z","abstract_excerpt":"Central configurations give rise to self-similar solutions to the Newtonian $N$-body problem, and play important roles in understanding its complicated dynamics. Even the simple question of whether or not there are finitely many planar central configurations for $N$ positive masses remains unsolved in most cases. Considering central configurations as critical points of a function $f$, we explicity compute the eigenvalues of the Hessian of $f$ for all $N$ for the point vortex potential for the regular polygon with equal masses. For homogeneous potentials including the Newtonian case we compute "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.13011","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":"2026-05-17T23:45:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eEVfJiBGn2hzQqd7nBgnkd7oZkYjT33ls93F1vMJ4/vzutlzwaolWEPtmHJsvjWNOSYjJH0AOq97fdWq7rJdCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T10:19:48.857102Z"},"content_sha256":"1bd75d15896dab870c112e5cd2bcbae28898bd6dee14e3f8f80fe99fbc64fc79","schema_version":"1.0","event_id":"sha256:1bd75d15896dab870c112e5cd2bcbae28898bd6dee14e3f8f80fe99fbc64fc79"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KBG4PYQBCYMF4DXEBNYX5SAFMO/bundle.json","state_url":"https://pith.science/pith/KBG4PYQBCYMF4DXEBNYX5SAFMO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KBG4PYQBCYMF4DXEBNYX5SAFMO/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":"2026-06-11T10:19:48Z","links":{"resolver":"https://pith.science/pith/KBG4PYQBCYMF4DXEBNYX5SAFMO","bundle":"https://pith.science/pith/KBG4PYQBCYMF4DXEBNYX5SAFMO/bundle.json","state":"https://pith.science/pith/KBG4PYQBCYMF4DXEBNYX5SAFMO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KBG4PYQBCYMF4DXEBNYX5SAFMO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:KBG4PYQBCYMF4DXEBNYX5SAFMO","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":"7987e66debe389b00a2de8afa72242aef5690e50b7b6db5d86fad354518ad73e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-10-30T21:17:46Z","title_canon_sha256":"9be6ab69a4c4e62ed7bb42f8032156d41d893734d6d112cddbf47074c2454bb8"},"schema_version":"1.0","source":{"id":"1810.13011","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.13011","created_at":"2026-05-17T23:45:57Z"},{"alias_kind":"arxiv_version","alias_value":"1810.13011v2","created_at":"2026-05-17T23:45:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.13011","created_at":"2026-05-17T23:45:57Z"},{"alias_kind":"pith_short_12","alias_value":"KBG4PYQBCYMF","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_16","alias_value":"KBG4PYQBCYMF4DXE","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_8","alias_value":"KBG4PYQB","created_at":"2026-05-18T12:32:33Z"}],"graph_snapshots":[{"event_id":"sha256:1bd75d15896dab870c112e5cd2bcbae28898bd6dee14e3f8f80fe99fbc64fc79","target":"graph","created_at":"2026-05-17T23:45:57Z","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":"Central configurations give rise to self-similar solutions to the Newtonian $N$-body problem, and play important roles in understanding its complicated dynamics. Even the simple question of whether or not there are finitely many planar central configurations for $N$ positive masses remains unsolved in most cases. Considering central configurations as critical points of a function $f$, we explicity compute the eigenvalues of the Hessian of $f$ for all $N$ for the point vortex potential for the regular polygon with equal masses. For homogeneous potentials including the Newtonian case we compute ","authors_text":"Marshall Hampton","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-10-30T21:17:46Z","title":"Planar $N$-body central configurations with a homogeneous potential"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.13011","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":"sha256:7ed24039b5695ed34e1a387e1e88feffc2950a1bc98a9e7e819ffae46689655c","target":"record","created_at":"2026-05-17T23:45:57Z","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":"7987e66debe389b00a2de8afa72242aef5690e50b7b6db5d86fad354518ad73e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-10-30T21:17:46Z","title_canon_sha256":"9be6ab69a4c4e62ed7bb42f8032156d41d893734d6d112cddbf47074c2454bb8"},"schema_version":"1.0","source":{"id":"1810.13011","kind":"arxiv","version":2}},"canonical_sha256":"504dc7e20116185e0ee40b717ec805638ea29ca81df01dbcd2b4b2e2586f4eb6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"504dc7e20116185e0ee40b717ec805638ea29ca81df01dbcd2b4b2e2586f4eb6","first_computed_at":"2026-05-17T23:45:57.937971Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:45:57.937971Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qrwCevZTsVn2/JdB0CF98nbzHYtgkIq/fWj9ZYjYPQLEZkMVkU8+F92xLRB8+/APbw6H2yAM06VRS0l0w2zhDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:45:57.938519Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.13011","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7ed24039b5695ed34e1a387e1e88feffc2950a1bc98a9e7e819ffae46689655c","sha256:1bd75d15896dab870c112e5cd2bcbae28898bd6dee14e3f8f80fe99fbc64fc79"],"state_sha256":"57602768c4cb72caf102adca84fbc01ebd3ad5f809f9397c36294165870f3eeb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8CE1PfVbG8fnYAqBtpx+30U0Rh5lK1phuBbw9vr6rSHrULhGNWuXCpgz6VXCR3hqGFoWGpxcosPAWt8rQAfWAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T10:19:48.859043Z","bundle_sha256":"6209b4bb72511df5a9db1386d11f58a6cfc96723440f609e5ef6a34055f85a2b"}}