{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:AGHC3N3ORU6M6E7S27AZBUAKQZ","short_pith_number":"pith:AGHC3N3O","canonical_record":{"source":{"id":"1311.6568","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-26T07:13:03Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"3527d2a8403750c90329c25079ba9dad2b0fa15ffe709ae7188ad720beae291c","abstract_canon_sha256":"bb9fbcce344a92a18abacf2822297e8cce203ff01c6fcc249bffa4ba71a7944f"},"schema_version":"1.0"},"canonical_sha256":"018e2db76e8d3ccf13f2d7c190d00a866b31d50d4c012618781fdef6ee73978a","source":{"kind":"arxiv","id":"1311.6568","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.6568","created_at":"2026-05-18T02:54:46Z"},{"alias_kind":"arxiv_version","alias_value":"1311.6568v3","created_at":"2026-05-18T02:54:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.6568","created_at":"2026-05-18T02:54:46Z"},{"alias_kind":"pith_short_12","alias_value":"AGHC3N3ORU6M","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"AGHC3N3ORU6M6E7S","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"AGHC3N3O","created_at":"2026-05-18T12:27:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:AGHC3N3ORU6M6E7S27AZBUAKQZ","target":"record","payload":{"canonical_record":{"source":{"id":"1311.6568","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-26T07:13:03Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"3527d2a8403750c90329c25079ba9dad2b0fa15ffe709ae7188ad720beae291c","abstract_canon_sha256":"bb9fbcce344a92a18abacf2822297e8cce203ff01c6fcc249bffa4ba71a7944f"},"schema_version":"1.0"},"canonical_sha256":"018e2db76e8d3ccf13f2d7c190d00a866b31d50d4c012618781fdef6ee73978a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:46.716625Z","signature_b64":"gvLLhplpLU2yxe1Qi0v9yqgb+KFf+FLAU6f2c0LENoUlKLg0WnUJB/P/RuHMi82UpQJ/ZlHMxP0jyTNw53RbDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"018e2db76e8d3ccf13f2d7c190d00a866b31d50d4c012618781fdef6ee73978a","last_reissued_at":"2026-05-18T02:54:46.716227Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:46.716227Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.6568","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:54:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TMrAHCmuboyaNGL0bH6BOgLx/LPHR1Jbxs0iRkm/TVWIjX0Vo1XUEsJXHqiINGXmZISrjlJQ0LwiolDrhoM8AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T15:55:53.008845Z"},"content_sha256":"6628e2e7a5a0a4b6af5504e93baef2c06f0cb6c5ab847cc2e75a58cac14ade7b","schema_version":"1.0","event_id":"sha256:6628e2e7a5a0a4b6af5504e93baef2c06f0cb6c5ab847cc2e75a58cac14ade7b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:AGHC3N3ORU6M6E7S27AZBUAKQZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Depth $0$ Nonsingular Morse Smale flows on $S^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DS","authors_text":"Bin Yu","submitted_at":"2013-11-26T07:13:03Z","abstract_excerpt":"In this paper, we first develope the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tools to describe nonsingular Morse-Smale flows without heteroclinic trajectories connecting saddle orbits (abbreviated as depth $0$ NMS flows). It mainly contains the following several directions: \\begin{enumerate}\n  \\item we use WLG to list depth $0$ NMS flows on $S^3$;\n  \\item wit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6568","kind":"arxiv","version":3},"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":"2026-05-18T02:54:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nS65DciQyX9pmHMNsEaYbMoX/Ujj6IF6kOOIZj9/yCDEd5PlJa+IegcCjKBE79cfmlGWHWUXim5XOeYQ6ioQCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T15:55:53.009191Z"},"content_sha256":"83bc4131cb936fd8782f62f77aa40a121c0dbc2192d49df7040175b937cdaf14","schema_version":"1.0","event_id":"sha256:83bc4131cb936fd8782f62f77aa40a121c0dbc2192d49df7040175b937cdaf14"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AGHC3N3ORU6M6E7S27AZBUAKQZ/bundle.json","state_url":"https://pith.science/pith/AGHC3N3ORU6M6E7S27AZBUAKQZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AGHC3N3ORU6M6E7S27AZBUAKQZ/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":"2026-06-27T15:55:53Z","links":{"resolver":"https://pith.science/pith/AGHC3N3ORU6M6E7S27AZBUAKQZ","bundle":"https://pith.science/pith/AGHC3N3ORU6M6E7S27AZBUAKQZ/bundle.json","state":"https://pith.science/pith/AGHC3N3ORU6M6E7S27AZBUAKQZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AGHC3N3ORU6M6E7S27AZBUAKQZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:AGHC3N3ORU6M6E7S27AZBUAKQZ","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":"bb9fbcce344a92a18abacf2822297e8cce203ff01c6fcc249bffa4ba71a7944f","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-26T07:13:03Z","title_canon_sha256":"3527d2a8403750c90329c25079ba9dad2b0fa15ffe709ae7188ad720beae291c"},"schema_version":"1.0","source":{"id":"1311.6568","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.6568","created_at":"2026-05-18T02:54:46Z"},{"alias_kind":"arxiv_version","alias_value":"1311.6568v3","created_at":"2026-05-18T02:54:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.6568","created_at":"2026-05-18T02:54:46Z"},{"alias_kind":"pith_short_12","alias_value":"AGHC3N3ORU6M","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"AGHC3N3ORU6M6E7S","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"AGHC3N3O","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:83bc4131cb936fd8782f62f77aa40a121c0dbc2192d49df7040175b937cdaf14","target":"graph","created_at":"2026-05-18T02:54:46Z","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 this paper, we first develope the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tools to describe nonsingular Morse-Smale flows without heteroclinic trajectories connecting saddle orbits (abbreviated as depth $0$ NMS flows). It mainly contains the following several directions: \\begin{enumerate}\n  \\item we use WLG to list depth $0$ NMS flows on $S^3$;\n  \\item wit","authors_text":"Bin Yu","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-26T07:13:03Z","title":"Depth $0$ Nonsingular Morse Smale flows on $S^3$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6568","kind":"arxiv","version":3},"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":"sha256:6628e2e7a5a0a4b6af5504e93baef2c06f0cb6c5ab847cc2e75a58cac14ade7b","target":"record","created_at":"2026-05-18T02:54:46Z","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":"bb9fbcce344a92a18abacf2822297e8cce203ff01c6fcc249bffa4ba71a7944f","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-26T07:13:03Z","title_canon_sha256":"3527d2a8403750c90329c25079ba9dad2b0fa15ffe709ae7188ad720beae291c"},"schema_version":"1.0","source":{"id":"1311.6568","kind":"arxiv","version":3}},"canonical_sha256":"018e2db76e8d3ccf13f2d7c190d00a866b31d50d4c012618781fdef6ee73978a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"018e2db76e8d3ccf13f2d7c190d00a866b31d50d4c012618781fdef6ee73978a","first_computed_at":"2026-05-18T02:54:46.716227Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:54:46.716227Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gvLLhplpLU2yxe1Qi0v9yqgb+KFf+FLAU6f2c0LENoUlKLg0WnUJB/P/RuHMi82UpQJ/ZlHMxP0jyTNw53RbDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:54:46.716625Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.6568","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6628e2e7a5a0a4b6af5504e93baef2c06f0cb6c5ab847cc2e75a58cac14ade7b","sha256:83bc4131cb936fd8782f62f77aa40a121c0dbc2192d49df7040175b937cdaf14"],"state_sha256":"6add7ef1a52f5de6ebd418e493b642821003015e02414de64958ae3a63fc7d68"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TYEJaoN/qjPwSHh9bSiDpKzOYkL0c7dEDkbQDwS+px5pGO/ujGQTXjwKcnLBIRVpI+BRlI+y/VkRjrOYv0lbCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T15:55:53.011042Z","bundle_sha256":"79129748c53934c0951b4a5834fb22c5bf102049087a5ec8bce27665eea39eb9"}}