{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:FD5XILGQK57UBBVCGLMW6I5CKF","short_pith_number":"pith:FD5XILGQ","canonical_record":{"source":{"id":"1406.6004","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2014-06-23T17:47:07Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"38ad2fc2413ce52cb32ef253cbdd43d2e2c9ac5fd31201905038529ed1a121be","abstract_canon_sha256":"d16a5a5630daeec3cc9658949e372b8fa3fc7c00eed495436d7da6fb66f0f5f4"},"schema_version":"1.0"},"canonical_sha256":"28fb742cd0577f4086a232d96f23a251725a28d6ba19a4cfa9fe35e70af68b7d","source":{"kind":"arxiv","id":"1406.6004","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.6004","created_at":"2026-05-18T02:27:44Z"},{"alias_kind":"arxiv_version","alias_value":"1406.6004v2","created_at":"2026-05-18T02:27:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.6004","created_at":"2026-05-18T02:27:44Z"},{"alias_kind":"pith_short_12","alias_value":"FD5XILGQK57U","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"FD5XILGQK57UBBVC","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"FD5XILGQ","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:FD5XILGQK57UBBVCGLMW6I5CKF","target":"record","payload":{"canonical_record":{"source":{"id":"1406.6004","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2014-06-23T17:47:07Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"38ad2fc2413ce52cb32ef253cbdd43d2e2c9ac5fd31201905038529ed1a121be","abstract_canon_sha256":"d16a5a5630daeec3cc9658949e372b8fa3fc7c00eed495436d7da6fb66f0f5f4"},"schema_version":"1.0"},"canonical_sha256":"28fb742cd0577f4086a232d96f23a251725a28d6ba19a4cfa9fe35e70af68b7d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:44.521429Z","signature_b64":"DBAoONA1DFFWDji2HBe95UdHzzxWHPYsiDPItiiL0LMQMgNdKGyYD3oba6/LtUodElKCu7/exXLGvfeLUfdABw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28fb742cd0577f4086a232d96f23a251725a28d6ba19a4cfa9fe35e70af68b7d","last_reissued_at":"2026-05-18T02:27:44.520934Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:44.520934Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1406.6004","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":"2026-05-18T02:27:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3yQvqk1cD5VdAqhtDJC1q9adC+C/ok9wvR1A6Qtp/HjR996TAjOhsvgiyaB6yW9FVg+McFZ/nvgzrOvz/h+RAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T15:56:36.222289Z"},"content_sha256":"70d39857259ccee690e731dd235a5ff3da5a2b253046cddb8daf06c5aa8847fd","schema_version":"1.0","event_id":"sha256:70d39857259ccee690e731dd235a5ff3da5a2b253046cddb8daf06c5aa8847fd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:FD5XILGQK57UBBVCGLMW6I5CKF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Lagrangian Cubic Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.SG","authors_text":"Cedric Membrez, Paul Biran","submitted_at":"2014-06-23T17:47:07Z","abstract_excerpt":"Let $M$ be a closed symplectic manifold and $L \\subset M$ a Lagrangian submanifold. Denote by $[L]$ the homology class induced by $L$ viewed as a class in the quantum homology of $M$. The present paper is concerned with properties and identities involving the class $[L]$ in the quantum homology ring. We also study the relations between these identities and invariants of $L$ coming from Lagrangian Floer theory. We pay special attention to the case when $L$ is a Lagrangian sphere."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.6004","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":"2026-05-18T02:27:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ylPYJnsXwOLk+ehyNuh4BleYznotNx10LFgpxNlGYArzblkKUbUJZBq7wZZ5SOO2O4VZrTmvmYOhP/nn8iFuDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T15:56:36.222650Z"},"content_sha256":"9e9d036a8638e7b230aa03d66480bce201dce100115918845f95dc5cf763b020","schema_version":"1.0","event_id":"sha256:9e9d036a8638e7b230aa03d66480bce201dce100115918845f95dc5cf763b020"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FD5XILGQK57UBBVCGLMW6I5CKF/bundle.json","state_url":"https://pith.science/pith/FD5XILGQK57UBBVCGLMW6I5CKF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FD5XILGQK57UBBVCGLMW6I5CKF/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":"2026-06-04T15:56:36Z","links":{"resolver":"https://pith.science/pith/FD5XILGQK57UBBVCGLMW6I5CKF","bundle":"https://pith.science/pith/FD5XILGQK57UBBVCGLMW6I5CKF/bundle.json","state":"https://pith.science/pith/FD5XILGQK57UBBVCGLMW6I5CKF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FD5XILGQK57UBBVCGLMW6I5CKF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:FD5XILGQK57UBBVCGLMW6I5CKF","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":"d16a5a5630daeec3cc9658949e372b8fa3fc7c00eed495436d7da6fb66f0f5f4","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2014-06-23T17:47:07Z","title_canon_sha256":"38ad2fc2413ce52cb32ef253cbdd43d2e2c9ac5fd31201905038529ed1a121be"},"schema_version":"1.0","source":{"id":"1406.6004","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.6004","created_at":"2026-05-18T02:27:44Z"},{"alias_kind":"arxiv_version","alias_value":"1406.6004v2","created_at":"2026-05-18T02:27:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.6004","created_at":"2026-05-18T02:27:44Z"},{"alias_kind":"pith_short_12","alias_value":"FD5XILGQK57U","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"FD5XILGQK57UBBVC","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"FD5XILGQ","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:9e9d036a8638e7b230aa03d66480bce201dce100115918845f95dc5cf763b020","target":"graph","created_at":"2026-05-18T02:27:44Z","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 closed symplectic manifold and $L \\subset M$ a Lagrangian submanifold. Denote by $[L]$ the homology class induced by $L$ viewed as a class in the quantum homology of $M$. The present paper is concerned with properties and identities involving the class $[L]$ in the quantum homology ring. We also study the relations between these identities and invariants of $L$ coming from Lagrangian Floer theory. We pay special attention to the case when $L$ is a Lagrangian sphere.","authors_text":"Cedric Membrez, Paul Biran","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2014-06-23T17:47:07Z","title":"The Lagrangian Cubic Equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.6004","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":"sha256:70d39857259ccee690e731dd235a5ff3da5a2b253046cddb8daf06c5aa8847fd","target":"record","created_at":"2026-05-18T02:27:44Z","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":"d16a5a5630daeec3cc9658949e372b8fa3fc7c00eed495436d7da6fb66f0f5f4","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2014-06-23T17:47:07Z","title_canon_sha256":"38ad2fc2413ce52cb32ef253cbdd43d2e2c9ac5fd31201905038529ed1a121be"},"schema_version":"1.0","source":{"id":"1406.6004","kind":"arxiv","version":2}},"canonical_sha256":"28fb742cd0577f4086a232d96f23a251725a28d6ba19a4cfa9fe35e70af68b7d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"28fb742cd0577f4086a232d96f23a251725a28d6ba19a4cfa9fe35e70af68b7d","first_computed_at":"2026-05-18T02:27:44.520934Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:27:44.520934Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DBAoONA1DFFWDji2HBe95UdHzzxWHPYsiDPItiiL0LMQMgNdKGyYD3oba6/LtUodElKCu7/exXLGvfeLUfdABw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:27:44.521429Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.6004","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:70d39857259ccee690e731dd235a5ff3da5a2b253046cddb8daf06c5aa8847fd","sha256:9e9d036a8638e7b230aa03d66480bce201dce100115918845f95dc5cf763b020"],"state_sha256":"f3b6b9ee6cb3582d798ab0a8ecf129d261250ddff66b39b83f1ef590150b6a8e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CTbhfWk7uMlJDreNubk9sl7qsjUNwUquA7ixzgMBPTdw7xd/mvRZUYYhFYQ79kBUkYMpdSkMA2+bW4FFhqQJDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T15:56:36.224627Z","bundle_sha256":"a400d788d668f11994eccc3f26c7bdbdd2d023f2d23b590a2121f432cc9fa66c"}}