{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:OYR5R7Z2LXJ3Y3SOVNN4XBZ3JN","short_pith_number":"pith:OYR5R7Z2","canonical_record":{"source":{"id":"1506.04789","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-06-15T22:13:38Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"b45a393846929419dba3e2c34f2334f2dd67819ffccfc36d7e3a8a08c2668a82","abstract_canon_sha256":"6c57cbacf9b96fd694fb968b54404d6dd0a953e8811993872cdb9160fb4f40a0"},"schema_version":"1.0"},"canonical_sha256":"7623d8ff3a5dd3bc6e4eab5bcb873b4b7f84cb387d6b54e61050bc8ba5f710db","source":{"kind":"arxiv","id":"1506.04789","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.04789","created_at":"2026-05-18T01:01:58Z"},{"alias_kind":"arxiv_version","alias_value":"1506.04789v1","created_at":"2026-05-18T01:01:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.04789","created_at":"2026-05-18T01:01:58Z"},{"alias_kind":"pith_short_12","alias_value":"OYR5R7Z2LXJ3","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"OYR5R7Z2LXJ3Y3SO","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"OYR5R7Z2","created_at":"2026-05-18T12:29:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:OYR5R7Z2LXJ3Y3SOVNN4XBZ3JN","target":"record","payload":{"canonical_record":{"source":{"id":"1506.04789","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-06-15T22:13:38Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"b45a393846929419dba3e2c34f2334f2dd67819ffccfc36d7e3a8a08c2668a82","abstract_canon_sha256":"6c57cbacf9b96fd694fb968b54404d6dd0a953e8811993872cdb9160fb4f40a0"},"schema_version":"1.0"},"canonical_sha256":"7623d8ff3a5dd3bc6e4eab5bcb873b4b7f84cb387d6b54e61050bc8ba5f710db","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:58.209852Z","signature_b64":"PdrCugIarWErfnmjyvLyP4L39rZN+ThM7fByzO8cebfLC8bU24aBGz/ySi02qfuS3rxTv25nAJwbq1ZIM2XMAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7623d8ff3a5dd3bc6e4eab5bcb873b4b7f84cb387d6b54e61050bc8ba5f710db","last_reissued_at":"2026-05-18T01:01:58.209074Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:58.209074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.04789","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":"2026-05-18T01:01:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QcDn+UnT3p6S/KzPXv3Qx4YCaZI9QLZ+KiBQaseGckuBHa/KOuefgvCQTPNw4mPrTLu6THTUVE8CqaPHvfnFDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T04:02:41.159897Z"},"content_sha256":"2b823ac17c60bdc0d7e9458b967fd5f4dc276511f6b673387d751c9fdd77dfd5","schema_version":"1.0","event_id":"sha256:2b823ac17c60bdc0d7e9458b967fd5f4dc276511f6b673387d751c9fdd77dfd5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:OYR5R7Z2LXJ3Y3SOVNN4XBZ3JN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Using simplicial volume to count maximally broken Morse trajectories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.GT","authors_text":"Hannah Alpert","submitted_at":"2015-06-15T22:13:38Z","abstract_excerpt":"Given a closed Riemannian manifold of dimension $n$ and a Morse-Smale function, there are finitely many $n$-part broken trajectories of the negative gradient flow. We show that if the manifold admits a hyperbolic metric, then the number of $n$-part broken trajectories is always at least the hyperbolic volume. The proof combines known theorems in Morse theory with lemmas of Gromov about simplicial volumes of stratified spaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.04789","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":"2026-05-18T01:01:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o1VmRtS9gsq6JQhMEROiLoAxUuTrrvoAihf2qUUmuQZ+QtLkbISatkj21Py+RTC8eTkMCWz5hG16VGXfyeHtDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T04:02:41.160232Z"},"content_sha256":"c761ebfddbebe2517cc9a28c7b01509e863bbde019fb3a70617c73a1555466f4","schema_version":"1.0","event_id":"sha256:c761ebfddbebe2517cc9a28c7b01509e863bbde019fb3a70617c73a1555466f4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OYR5R7Z2LXJ3Y3SOVNN4XBZ3JN/bundle.json","state_url":"https://pith.science/pith/OYR5R7Z2LXJ3Y3SOVNN4XBZ3JN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OYR5R7Z2LXJ3Y3SOVNN4XBZ3JN/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":"2026-07-03T04:02:41Z","links":{"resolver":"https://pith.science/pith/OYR5R7Z2LXJ3Y3SOVNN4XBZ3JN","bundle":"https://pith.science/pith/OYR5R7Z2LXJ3Y3SOVNN4XBZ3JN/bundle.json","state":"https://pith.science/pith/OYR5R7Z2LXJ3Y3SOVNN4XBZ3JN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OYR5R7Z2LXJ3Y3SOVNN4XBZ3JN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:OYR5R7Z2LXJ3Y3SOVNN4XBZ3JN","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":"6c57cbacf9b96fd694fb968b54404d6dd0a953e8811993872cdb9160fb4f40a0","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-06-15T22:13:38Z","title_canon_sha256":"b45a393846929419dba3e2c34f2334f2dd67819ffccfc36d7e3a8a08c2668a82"},"schema_version":"1.0","source":{"id":"1506.04789","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.04789","created_at":"2026-05-18T01:01:58Z"},{"alias_kind":"arxiv_version","alias_value":"1506.04789v1","created_at":"2026-05-18T01:01:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.04789","created_at":"2026-05-18T01:01:58Z"},{"alias_kind":"pith_short_12","alias_value":"OYR5R7Z2LXJ3","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"OYR5R7Z2LXJ3Y3SO","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"OYR5R7Z2","created_at":"2026-05-18T12:29:34Z"}],"graph_snapshots":[{"event_id":"sha256:c761ebfddbebe2517cc9a28c7b01509e863bbde019fb3a70617c73a1555466f4","target":"graph","created_at":"2026-05-18T01:01:58Z","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":"Given a closed Riemannian manifold of dimension $n$ and a Morse-Smale function, there are finitely many $n$-part broken trajectories of the negative gradient flow. We show that if the manifold admits a hyperbolic metric, then the number of $n$-part broken trajectories is always at least the hyperbolic volume. The proof combines known theorems in Morse theory with lemmas of Gromov about simplicial volumes of stratified spaces.","authors_text":"Hannah Alpert","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-06-15T22:13:38Z","title":"Using simplicial volume to count maximally broken Morse trajectories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.04789","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":"sha256:2b823ac17c60bdc0d7e9458b967fd5f4dc276511f6b673387d751c9fdd77dfd5","target":"record","created_at":"2026-05-18T01:01:58Z","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":"6c57cbacf9b96fd694fb968b54404d6dd0a953e8811993872cdb9160fb4f40a0","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-06-15T22:13:38Z","title_canon_sha256":"b45a393846929419dba3e2c34f2334f2dd67819ffccfc36d7e3a8a08c2668a82"},"schema_version":"1.0","source":{"id":"1506.04789","kind":"arxiv","version":1}},"canonical_sha256":"7623d8ff3a5dd3bc6e4eab5bcb873b4b7f84cb387d6b54e61050bc8ba5f710db","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7623d8ff3a5dd3bc6e4eab5bcb873b4b7f84cb387d6b54e61050bc8ba5f710db","first_computed_at":"2026-05-18T01:01:58.209074Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:01:58.209074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PdrCugIarWErfnmjyvLyP4L39rZN+ThM7fByzO8cebfLC8bU24aBGz/ySi02qfuS3rxTv25nAJwbq1ZIM2XMAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:01:58.209852Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.04789","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2b823ac17c60bdc0d7e9458b967fd5f4dc276511f6b673387d751c9fdd77dfd5","sha256:c761ebfddbebe2517cc9a28c7b01509e863bbde019fb3a70617c73a1555466f4"],"state_sha256":"aece7ba6f669d5f6eb8b7e3ff5e2b0e6a297c904d85082a0cbaddb241ba9611e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GT3M41miMqED+xk1qgogG7u8gu5iLQcl7B5lbMWjSzJMle2k4RFZKpUL5OltAFkXjHgJBVUFLIbvA9P/IRJQCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T04:02:41.162116Z","bundle_sha256":"53050dc7f57d9d1fabd49902f3b0e1120d22e5cf3565525dbe6f694ac220d82f"}}