{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1996:YKL2GF5GV722HR264QYHHJEBWQ","short_pith_number":"pith:YKL2GF5G","canonical_record":{"source":{"id":"dg-ga/9612004","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"dg-ga","submitted_at":"1996-12-03T17:58:11Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"151cf67d9a4a4090664001cc7356d43be491f6a8f87a7fcf4921fa8693242809","abstract_canon_sha256":"f4a41d2fb9368de995ca64a672aeac2432bac3e683f573322c779b029a756578"},"schema_version":"1.0"},"canonical_sha256":"c297a317a6aff5a3c75ee43073a481b40d1ede6e222432f958db1e0a783a8e80","source":{"kind":"arxiv","id":"dg-ga/9612004","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"dg-ga/9612004","created_at":"2026-05-18T01:06:48Z"},{"alias_kind":"arxiv_version","alias_value":"dg-ga/9612004v1","created_at":"2026-05-18T01:06:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.dg-ga/9612004","created_at":"2026-05-18T01:06:48Z"},{"alias_kind":"pith_short_12","alias_value":"YKL2GF5GV722","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_16","alias_value":"YKL2GF5GV722HR26","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_8","alias_value":"YKL2GF5G","created_at":"2026-05-18T12:25:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1996:YKL2GF5GV722HR264QYHHJEBWQ","target":"record","payload":{"canonical_record":{"source":{"id":"dg-ga/9612004","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"dg-ga","submitted_at":"1996-12-03T17:58:11Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"151cf67d9a4a4090664001cc7356d43be491f6a8f87a7fcf4921fa8693242809","abstract_canon_sha256":"f4a41d2fb9368de995ca64a672aeac2432bac3e683f573322c779b029a756578"},"schema_version":"1.0"},"canonical_sha256":"c297a317a6aff5a3c75ee43073a481b40d1ede6e222432f958db1e0a783a8e80","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:48.548921Z","signature_b64":"q9hu7vgsGJWJmSfD4WoIH0TRdy/7ZSt0ztnLsAHNmGKa1D7TfmWvzTBoOOlw5bl4oxMKzAHS3Gca+NWHume7Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c297a317a6aff5a3c75ee43073a481b40d1ede6e222432f958db1e0a783a8e80","last_reissued_at":"2026-05-18T01:06:48.548256Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:48.548256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"dg-ga/9612004","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":"2026-05-18T01:06:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4vJKHMPocVmVQfDwgDxv5gzNy/MZmPO12R0XZDOQqGqn6k93i5sGBP1JQXgTeLxjbXrKqzaIXKiiBGQHLLi3CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T17:09:21.109240Z"},"content_sha256":"ce7dde0e583dcd673daeedf8b23f544f315f6e60c877d00e094ad737f4cea0da","schema_version":"1.0","event_id":"sha256:ce7dde0e583dcd673daeedf8b23f544f315f6e60c877d00e094ad737f4cea0da"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1996:YKL2GF5GV722HR264QYHHJEBWQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds","license":"","headline":"","cross_cats":["math.DG"],"primary_cat":"dg-ga","authors_text":"Michael Hutchings, Yi-Jen Lee","submitted_at":"1996-12-03T17:58:11Z","abstract_excerpt":"Let X be a compact oriented Riemannian manifold and let $\\phi:X\\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $\\phi$, we prove a formula relating:\n  (a) the number of closed orbits of the gradient flow of $\\phi$ of any given degree;\n  (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of $\\phi$; and\n  (c) a kind of Reidemeister torsion of X determined by the homotopy class of $\\phi$.\n  When $\\dim(X)=3$ and $b_1(X)>0$, we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"dg-ga/9612004","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":"2026-05-18T01:06:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r6AJawTNwfksbrIEde3G/93divOwh1+PxeCa1rGaSHhg4R6iA6+qaCDB3x7Q5pTQd5TESVqqW2TlpxwFzs83Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T17:09:21.109721Z"},"content_sha256":"9950de450d180df6ee71438d2e09e14d6c5514b3802fdc5e0d6cef266c8ef1f8","schema_version":"1.0","event_id":"sha256:9950de450d180df6ee71438d2e09e14d6c5514b3802fdc5e0d6cef266c8ef1f8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ/bundle.json","state_url":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YKL2GF5GV722HR264QYHHJEBWQ/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":"2026-08-10T17:09:21Z","links":{"resolver":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ","bundle":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ/bundle.json","state":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YKL2GF5GV722HR264QYHHJEBWQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1996:YKL2GF5GV722HR264QYHHJEBWQ","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":"f4a41d2fb9368de995ca64a672aeac2432bac3e683f573322c779b029a756578","cross_cats_sorted":["math.DG"],"license":"","primary_cat":"dg-ga","submitted_at":"1996-12-03T17:58:11Z","title_canon_sha256":"151cf67d9a4a4090664001cc7356d43be491f6a8f87a7fcf4921fa8693242809"},"schema_version":"1.0","source":{"id":"dg-ga/9612004","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"dg-ga/9612004","created_at":"2026-05-18T01:06:48Z"},{"alias_kind":"arxiv_version","alias_value":"dg-ga/9612004v1","created_at":"2026-05-18T01:06:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.dg-ga/9612004","created_at":"2026-05-18T01:06:48Z"},{"alias_kind":"pith_short_12","alias_value":"YKL2GF5GV722","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_16","alias_value":"YKL2GF5GV722HR26","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_8","alias_value":"YKL2GF5G","created_at":"2026-05-18T12:25:48Z"}],"graph_snapshots":[{"event_id":"sha256:9950de450d180df6ee71438d2e09e14d6c5514b3802fdc5e0d6cef266c8ef1f8","target":"graph","created_at":"2026-05-18T01:06:48Z","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 X be a compact oriented Riemannian manifold and let $\\phi:X\\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $\\phi$, we prove a formula relating:\n  (a) the number of closed orbits of the gradient flow of $\\phi$ of any given degree;\n  (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of $\\phi$; and\n  (c) a kind of Reidemeister torsion of X determined by the homotopy class of $\\phi$.\n  When $\\dim(X)=3$ and $b_1(X)>0$, we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed m","authors_text":"Michael Hutchings, Yi-Jen Lee","cross_cats":["math.DG"],"headline":"","license":"","primary_cat":"dg-ga","submitted_at":"1996-12-03T17:58:11Z","title":"Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"dg-ga/9612004","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:ce7dde0e583dcd673daeedf8b23f544f315f6e60c877d00e094ad737f4cea0da","target":"record","created_at":"2026-05-18T01:06:48Z","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":"f4a41d2fb9368de995ca64a672aeac2432bac3e683f573322c779b029a756578","cross_cats_sorted":["math.DG"],"license":"","primary_cat":"dg-ga","submitted_at":"1996-12-03T17:58:11Z","title_canon_sha256":"151cf67d9a4a4090664001cc7356d43be491f6a8f87a7fcf4921fa8693242809"},"schema_version":"1.0","source":{"id":"dg-ga/9612004","kind":"arxiv","version":1}},"canonical_sha256":"c297a317a6aff5a3c75ee43073a481b40d1ede6e222432f958db1e0a783a8e80","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c297a317a6aff5a3c75ee43073a481b40d1ede6e222432f958db1e0a783a8e80","first_computed_at":"2026-05-18T01:06:48.548256Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:06:48.548256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q9hu7vgsGJWJmSfD4WoIH0TRdy/7ZSt0ztnLsAHNmGKa1D7TfmWvzTBoOOlw5bl4oxMKzAHS3Gca+NWHume7Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:06:48.548921Z","signed_message":"canonical_sha256_bytes"},"source_id":"dg-ga/9612004","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce7dde0e583dcd673daeedf8b23f544f315f6e60c877d00e094ad737f4cea0da","sha256:9950de450d180df6ee71438d2e09e14d6c5514b3802fdc5e0d6cef266c8ef1f8"],"state_sha256":"fbe8afbcd91ea0965e46afabb9010a94cfd3abb2e095fb9d8ceffd89467c5a45"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1AIZ0FRFDnl1lablPd6U99K2YGm6zouMsoTTTvBjVdR+RO9r502e96FBbwxdx4M5xbpBXo03bVJNe+RyOOxMAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T17:09:21.114895Z","bundle_sha256":"bbd3f1daabea07907288df1ee318876cfdf3278aec79dd6074c12e535330cb87"}}