{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:5DMWNC63XULVQPIXOOUGFFZO65","short_pith_number":"pith:5DMWNC63","canonical_record":{"source":{"id":"1410.8456","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-10-30T17:29:09Z","cross_cats_sorted":[],"title_canon_sha256":"68dbfe379b89f6e2015e369d463baaef932f2008618493c24e9f424cc1c66fae","abstract_canon_sha256":"8729537da50f3d35d6ddfc08cce0e4c144dc95bfdc9c42cfec3766265e9238fd"},"schema_version":"1.0"},"canonical_sha256":"e8d9668bdbbd17583d1773a862972ef7427763277a79b40462d89c0e57251429","source":{"kind":"arxiv","id":"1410.8456","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.8456","created_at":"2026-05-18T02:38:59Z"},{"alias_kind":"arxiv_version","alias_value":"1410.8456v1","created_at":"2026-05-18T02:38:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.8456","created_at":"2026-05-18T02:38:59Z"},{"alias_kind":"pith_short_12","alias_value":"5DMWNC63XULV","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5DMWNC63XULVQPIX","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5DMWNC63","created_at":"2026-05-18T12:28:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:5DMWNC63XULVQPIXOOUGFFZO65","target":"record","payload":{"canonical_record":{"source":{"id":"1410.8456","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-10-30T17:29:09Z","cross_cats_sorted":[],"title_canon_sha256":"68dbfe379b89f6e2015e369d463baaef932f2008618493c24e9f424cc1c66fae","abstract_canon_sha256":"8729537da50f3d35d6ddfc08cce0e4c144dc95bfdc9c42cfec3766265e9238fd"},"schema_version":"1.0"},"canonical_sha256":"e8d9668bdbbd17583d1773a862972ef7427763277a79b40462d89c0e57251429","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:59.871513Z","signature_b64":"niui9r8HV7zSwXuOj80cFmjkbn+YnVQBm11wKU1OTwkyJy5puvpXVyCtL63DjBS8gBkiIXrtyp+V4y5KBtTBBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e8d9668bdbbd17583d1773a862972ef7427763277a79b40462d89c0e57251429","last_reissued_at":"2026-05-18T02:38:59.871153Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:59.871153Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.8456","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":"2026-05-18T02:38:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5CNV4nNtIp6v5DphTpXQjPRx+2fTUIaleoJDg5mENrsIDkJU9aHjbCQ0aLT3fuFF9M23Z6BT0zaCGQXrpo0NAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T14:09:44.107337Z"},"content_sha256":"f71eb35b4b8c8be5f68d7515e956a75e4d2e78c10387cbee0e748dd0a47c0e23","schema_version":"1.0","event_id":"sha256:f71eb35b4b8c8be5f68d7515e956a75e4d2e78c10387cbee0e748dd0a47c0e23"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:5DMWNC63XULVQPIXOOUGFFZO65","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lengths of three simple periodic geodesics on a Riemannian $2$-sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Alexander Nabutovsky, Regina Rotman, Yevgeny Liokumovich","submitted_at":"2014-10-30T17:29:09Z","abstract_excerpt":"Let $M$ be a Riemannian $2$-sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on $M$. In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed $20d$, where $d$ is the diameter of $M$.\n  We also present an upper bound that depends only on the area and diameter for the lengths of the three simple periodic geodesics with positive indices that appear as minimax critical values in the classical proofs of the Lyusternik-Shnirelman theorem.\n  Finally, we present better b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8456","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":""},"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":"2026-05-18T02:38:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dDP6MAtdkx9W2lS02XnyFaATq55Fm1JWNux7SR4eDPlIEjfFb8S3ida7noBAmOZM2Sl+7sy1tpoeDfyxx5ePCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T14:09:44.108039Z"},"content_sha256":"e325fde3bc64080d86b7f507618e6db275ea54e043160fe474a2700f1f400afd","schema_version":"1.0","event_id":"sha256:e325fde3bc64080d86b7f507618e6db275ea54e043160fe474a2700f1f400afd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5DMWNC63XULVQPIXOOUGFFZO65/bundle.json","state_url":"https://pith.science/pith/5DMWNC63XULVQPIXOOUGFFZO65/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5DMWNC63XULVQPIXOOUGFFZO65/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":"2026-06-06T14:09:44Z","links":{"resolver":"https://pith.science/pith/5DMWNC63XULVQPIXOOUGFFZO65","bundle":"https://pith.science/pith/5DMWNC63XULVQPIXOOUGFFZO65/bundle.json","state":"https://pith.science/pith/5DMWNC63XULVQPIXOOUGFFZO65/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5DMWNC63XULVQPIXOOUGFFZO65/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:5DMWNC63XULVQPIXOOUGFFZO65","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":"8729537da50f3d35d6ddfc08cce0e4c144dc95bfdc9c42cfec3766265e9238fd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-10-30T17:29:09Z","title_canon_sha256":"68dbfe379b89f6e2015e369d463baaef932f2008618493c24e9f424cc1c66fae"},"schema_version":"1.0","source":{"id":"1410.8456","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.8456","created_at":"2026-05-18T02:38:59Z"},{"alias_kind":"arxiv_version","alias_value":"1410.8456v1","created_at":"2026-05-18T02:38:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.8456","created_at":"2026-05-18T02:38:59Z"},{"alias_kind":"pith_short_12","alias_value":"5DMWNC63XULV","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5DMWNC63XULVQPIX","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5DMWNC63","created_at":"2026-05-18T12:28:14Z"}],"graph_snapshots":[{"event_id":"sha256:e325fde3bc64080d86b7f507618e6db275ea54e043160fe474a2700f1f400afd","target":"graph","created_at":"2026-05-18T02:38:59Z","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":"Let $M$ be a Riemannian $2$-sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on $M$. In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed $20d$, where $d$ is the diameter of $M$.\n  We also present an upper bound that depends only on the area and diameter for the lengths of the three simple periodic geodesics with positive indices that appear as minimax critical values in the classical proofs of the Lyusternik-Shnirelman theorem.\n  Finally, we present better b","authors_text":"Alexander Nabutovsky, Regina Rotman, Yevgeny Liokumovich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-10-30T17:29:09Z","title":"Lengths of three simple periodic geodesics on a Riemannian $2$-sphere"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8456","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:f71eb35b4b8c8be5f68d7515e956a75e4d2e78c10387cbee0e748dd0a47c0e23","target":"record","created_at":"2026-05-18T02:38:59Z","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":"8729537da50f3d35d6ddfc08cce0e4c144dc95bfdc9c42cfec3766265e9238fd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-10-30T17:29:09Z","title_canon_sha256":"68dbfe379b89f6e2015e369d463baaef932f2008618493c24e9f424cc1c66fae"},"schema_version":"1.0","source":{"id":"1410.8456","kind":"arxiv","version":1}},"canonical_sha256":"e8d9668bdbbd17583d1773a862972ef7427763277a79b40462d89c0e57251429","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e8d9668bdbbd17583d1773a862972ef7427763277a79b40462d89c0e57251429","first_computed_at":"2026-05-18T02:38:59.871153Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:59.871153Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"niui9r8HV7zSwXuOj80cFmjkbn+YnVQBm11wKU1OTwkyJy5puvpXVyCtL63DjBS8gBkiIXrtyp+V4y5KBtTBBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:59.871513Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.8456","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f71eb35b4b8c8be5f68d7515e956a75e4d2e78c10387cbee0e748dd0a47c0e23","sha256:e325fde3bc64080d86b7f507618e6db275ea54e043160fe474a2700f1f400afd"],"state_sha256":"50e667b3ef3250d642185175c11fe9b4a34db31ed436fe8e14f095505163a37f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y/yjX9UcK6fX97Gq8Vd2/mkX9M/vTg5CuhIL5L3hTy46iyAqHYo5TFUsAdVNYP3RvYdZO2fJaA1J1Pur2wt5CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T14:09:44.112232Z","bundle_sha256":"d5b3940f1892c494157836bd5207e1e6d082ce20e0e633b4feb32012c6d03eaa"}}