{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:OEUEV6TNM47AX7KXULCD34CI37","short_pith_number":"pith:OEUEV6TN","canonical_record":{"source":{"id":"1001.2534","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-01-14T19:20:29Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"dca9a489832f446a17469ca754b7fd3c0073159ae905a827f4512e91c70b7be2","abstract_canon_sha256":"1402f5af8b6ad18bd7be3753edb65c886b973883b9c16fc6bdb6a1195087253f"},"schema_version":"1.0"},"canonical_sha256":"71284afa6d673e0bfd57a2c43df048dfe8b87c923f63b52f744a5f3241cd4fb0","source":{"kind":"arxiv","id":"1001.2534","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.2534","created_at":"2026-05-18T02:58:02Z"},{"alias_kind":"arxiv_version","alias_value":"1001.2534v1","created_at":"2026-05-18T02:58:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.2534","created_at":"2026-05-18T02:58:02Z"},{"alias_kind":"pith_short_12","alias_value":"OEUEV6TNM47A","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"OEUEV6TNM47AX7KX","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"OEUEV6TN","created_at":"2026-05-18T12:26:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:OEUEV6TNM47AX7KXULCD34CI37","target":"record","payload":{"canonical_record":{"source":{"id":"1001.2534","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-01-14T19:20:29Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"dca9a489832f446a17469ca754b7fd3c0073159ae905a827f4512e91c70b7be2","abstract_canon_sha256":"1402f5af8b6ad18bd7be3753edb65c886b973883b9c16fc6bdb6a1195087253f"},"schema_version":"1.0"},"canonical_sha256":"71284afa6d673e0bfd57a2c43df048dfe8b87c923f63b52f744a5f3241cd4fb0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:02.476085Z","signature_b64":"M8i2Lk3sm3/mbfHEhCV9Wu/ou+FMPu/0Ipv3p6MwOf8freqet656DvxfPgr23BBwSZgRAhL3IPo+xPEFsbqoDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71284afa6d673e0bfd57a2c43df048dfe8b87c923f63b52f744a5f3241cd4fb0","last_reissued_at":"2026-05-18T02:58:02.475356Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:02.475356Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1001.2534","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":"2026-05-18T02:58:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j0EkqypecDgFD/dwMo8KOd54XGkYKKrhKvHsPORyFe7EiN4THg8L99RK3oQfS7lFWWh46JlmNoXDJeAa1nsWDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T00:55:40.582227Z"},"content_sha256":"8a6f14b0ae8cb33f731aeb345963cacec69584a6bc9f24141224d00c1943ba84","schema_version":"1.0","event_id":"sha256:8a6f14b0ae8cb33f731aeb345963cacec69584a6bc9f24141224d00c1943ba84"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:OEUEV6TNM47AX7KXULCD34CI37","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multi-Parameter Div-Curl Lemmas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.CA","authors_text":"Brett D. Wick, Jill C. Pipher, Michael T. Lacey, Stefanie Petermichl","submitted_at":"2010-01-14T19:20:29Z","abstract_excerpt":"We study the possible analogous of the Div-Curl Lemma in classical harmonic analysis and partial differential equations, but from the point of view of the multi-parameter setting. In this context we see two possible Div-Curl lemmas that arise. Extensions to differential forms are also given."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.2534","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":"2026-05-18T02:58:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JCkh0e1SxRChx4zrTOR15XbgXzij74HhwhMYiwFn5AE8wnRS/yv/MUkHLWleJf50FU+OenYnLdLXqOA7LrpoBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T00:55:40.582769Z"},"content_sha256":"2445efc86d4b6ffecf7549c1f4752f518baf62b3dc7b6f330c234d181285070a","schema_version":"1.0","event_id":"sha256:2445efc86d4b6ffecf7549c1f4752f518baf62b3dc7b6f330c234d181285070a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OEUEV6TNM47AX7KXULCD34CI37/bundle.json","state_url":"https://pith.science/pith/OEUEV6TNM47AX7KXULCD34CI37/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OEUEV6TNM47AX7KXULCD34CI37/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":"2026-06-01T00:55:40Z","links":{"resolver":"https://pith.science/pith/OEUEV6TNM47AX7KXULCD34CI37","bundle":"https://pith.science/pith/OEUEV6TNM47AX7KXULCD34CI37/bundle.json","state":"https://pith.science/pith/OEUEV6TNM47AX7KXULCD34CI37/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OEUEV6TNM47AX7KXULCD34CI37/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:OEUEV6TNM47AX7KXULCD34CI37","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":"1402f5af8b6ad18bd7be3753edb65c886b973883b9c16fc6bdb6a1195087253f","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-01-14T19:20:29Z","title_canon_sha256":"dca9a489832f446a17469ca754b7fd3c0073159ae905a827f4512e91c70b7be2"},"schema_version":"1.0","source":{"id":"1001.2534","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.2534","created_at":"2026-05-18T02:58:02Z"},{"alias_kind":"arxiv_version","alias_value":"1001.2534v1","created_at":"2026-05-18T02:58:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.2534","created_at":"2026-05-18T02:58:02Z"},{"alias_kind":"pith_short_12","alias_value":"OEUEV6TNM47A","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"OEUEV6TNM47AX7KX","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"OEUEV6TN","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:2445efc86d4b6ffecf7549c1f4752f518baf62b3dc7b6f330c234d181285070a","target":"graph","created_at":"2026-05-18T02:58:02Z","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":"We study the possible analogous of the Div-Curl Lemma in classical harmonic analysis and partial differential equations, but from the point of view of the multi-parameter setting. In this context we see two possible Div-Curl lemmas that arise. Extensions to differential forms are also given.","authors_text":"Brett D. Wick, Jill C. Pipher, Michael T. Lacey, Stefanie Petermichl","cross_cats":["math.CV"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-01-14T19:20:29Z","title":"Multi-Parameter Div-Curl Lemmas"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.2534","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":"sha256:8a6f14b0ae8cb33f731aeb345963cacec69584a6bc9f24141224d00c1943ba84","target":"record","created_at":"2026-05-18T02:58:02Z","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":"1402f5af8b6ad18bd7be3753edb65c886b973883b9c16fc6bdb6a1195087253f","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-01-14T19:20:29Z","title_canon_sha256":"dca9a489832f446a17469ca754b7fd3c0073159ae905a827f4512e91c70b7be2"},"schema_version":"1.0","source":{"id":"1001.2534","kind":"arxiv","version":1}},"canonical_sha256":"71284afa6d673e0bfd57a2c43df048dfe8b87c923f63b52f744a5f3241cd4fb0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"71284afa6d673e0bfd57a2c43df048dfe8b87c923f63b52f744a5f3241cd4fb0","first_computed_at":"2026-05-18T02:58:02.475356Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:02.475356Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"M8i2Lk3sm3/mbfHEhCV9Wu/ou+FMPu/0Ipv3p6MwOf8freqet656DvxfPgr23BBwSZgRAhL3IPo+xPEFsbqoDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:02.476085Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.2534","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8a6f14b0ae8cb33f731aeb345963cacec69584a6bc9f24141224d00c1943ba84","sha256:2445efc86d4b6ffecf7549c1f4752f518baf62b3dc7b6f330c234d181285070a"],"state_sha256":"e38c3f1afecf0eca309d3146a0456c7402e4539935d6762fb82610ea3bcbfe30"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nr6cu0OhO3CcuCLfREutep5izHEstzF/Dj1LSa52JtIRpzJvmqjSM9ZilDvHWto98ipwKdOVdbcWBGQ/NhmLDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T00:55:40.586264Z","bundle_sha256":"c722d12150fffec1b2c39ee55df1b8a53011184ab7245b55d3d6ff76de5dd761"}}