{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:GGBCE4ZNEGYOR7PJU5L7UZ2GCI","short_pith_number":"pith:GGBCE4ZN","canonical_record":{"source":{"id":"1203.5821","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-03-26T21:18:06Z","cross_cats_sorted":[],"title_canon_sha256":"ebf8e2a0f86b1150eafad08014ee0c4229879c751cf0554c93ff7cdbe00908a2","abstract_canon_sha256":"6aae995ea15e81094fd8c2922d0107e0da07493287917cece3262815e33d58c5"},"schema_version":"1.0"},"canonical_sha256":"318222732d21b0e8fde9a757fa6746123fc101cbcd2381e6e0e96dee0f0b14f6","source":{"kind":"arxiv","id":"1203.5821","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.5821","created_at":"2026-05-18T03:59:13Z"},{"alias_kind":"arxiv_version","alias_value":"1203.5821v1","created_at":"2026-05-18T03:59:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.5821","created_at":"2026-05-18T03:59:13Z"},{"alias_kind":"pith_short_12","alias_value":"GGBCE4ZNEGYO","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GGBCE4ZNEGYOR7PJ","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GGBCE4ZN","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:GGBCE4ZNEGYOR7PJU5L7UZ2GCI","target":"record","payload":{"canonical_record":{"source":{"id":"1203.5821","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-03-26T21:18:06Z","cross_cats_sorted":[],"title_canon_sha256":"ebf8e2a0f86b1150eafad08014ee0c4229879c751cf0554c93ff7cdbe00908a2","abstract_canon_sha256":"6aae995ea15e81094fd8c2922d0107e0da07493287917cece3262815e33d58c5"},"schema_version":"1.0"},"canonical_sha256":"318222732d21b0e8fde9a757fa6746123fc101cbcd2381e6e0e96dee0f0b14f6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:13.355696Z","signature_b64":"Aiq9L/o1x3hqjgTeOkIxiGWgAjXbzDOQUKVEF+h/CVS5U2M8HwaMDqVPUROVoTgliyBDfB6XKDYy3m9gnpjnCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"318222732d21b0e8fde9a757fa6746123fc101cbcd2381e6e0e96dee0f0b14f6","last_reissued_at":"2026-05-18T03:59:13.355029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:13.355029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1203.5821","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":"2026-05-18T03:59:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H6l6lo3Qn8Xth5DOpjREHj9fNwDNWgEMY/iDaqZ6FVtDrTAOBOP05tCpSDODOMfVp2ioF8Ufll8tMRpzcOq1AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T09:25:12.435347Z"},"content_sha256":"fb8b9b192d0f2749cb78ee440e09c7e69012c42099c13274eb9a70936c7b32f5","schema_version":"1.0","event_id":"sha256:fb8b9b192d0f2749cb78ee440e09c7e69012c42099c13274eb9a70936c7b32f5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:GGBCE4ZNEGYOR7PJU5L7UZ2GCI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A note on the rank of positive closed currents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Romain Dujardin","submitted_at":"2012-03-26T21:18:06Z","abstract_excerpt":"In this short note we prove an estimate on the rank a.e. of the tangent (p,p) vector to a a positive closed current of bidimension (p,p) in CP^k, in terms of the dimension of its trace measure."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5821","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":"2026-05-18T03:59:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U9yNRXGhjP75GOzxQjd0Kr4XXi7dyCd3DPNTTFsL4aZ3NGFCA6t2is2RYlFNq0tsPLizbx9H5pmsDc9NSQJuCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T09:25:12.435710Z"},"content_sha256":"9649f5004291029703e0d7a8788cb1b7ca3bf90838c84d96c3fcd6770ca09147","schema_version":"1.0","event_id":"sha256:9649f5004291029703e0d7a8788cb1b7ca3bf90838c84d96c3fcd6770ca09147"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GGBCE4ZNEGYOR7PJU5L7UZ2GCI/bundle.json","state_url":"https://pith.science/pith/GGBCE4ZNEGYOR7PJU5L7UZ2GCI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GGBCE4ZNEGYOR7PJU5L7UZ2GCI/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":"2026-06-29T09:25:12Z","links":{"resolver":"https://pith.science/pith/GGBCE4ZNEGYOR7PJU5L7UZ2GCI","bundle":"https://pith.science/pith/GGBCE4ZNEGYOR7PJU5L7UZ2GCI/bundle.json","state":"https://pith.science/pith/GGBCE4ZNEGYOR7PJU5L7UZ2GCI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GGBCE4ZNEGYOR7PJU5L7UZ2GCI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:GGBCE4ZNEGYOR7PJU5L7UZ2GCI","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":"6aae995ea15e81094fd8c2922d0107e0da07493287917cece3262815e33d58c5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-03-26T21:18:06Z","title_canon_sha256":"ebf8e2a0f86b1150eafad08014ee0c4229879c751cf0554c93ff7cdbe00908a2"},"schema_version":"1.0","source":{"id":"1203.5821","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.5821","created_at":"2026-05-18T03:59:13Z"},{"alias_kind":"arxiv_version","alias_value":"1203.5821v1","created_at":"2026-05-18T03:59:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.5821","created_at":"2026-05-18T03:59:13Z"},{"alias_kind":"pith_short_12","alias_value":"GGBCE4ZNEGYO","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GGBCE4ZNEGYOR7PJ","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GGBCE4ZN","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:9649f5004291029703e0d7a8788cb1b7ca3bf90838c84d96c3fcd6770ca09147","target":"graph","created_at":"2026-05-18T03:59:13Z","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 short note we prove an estimate on the rank a.e. of the tangent (p,p) vector to a a positive closed current of bidimension (p,p) in CP^k, in terms of the dimension of its trace measure.","authors_text":"Romain Dujardin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-03-26T21:18:06Z","title":"A note on the rank of positive closed currents"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5821","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:fb8b9b192d0f2749cb78ee440e09c7e69012c42099c13274eb9a70936c7b32f5","target":"record","created_at":"2026-05-18T03:59:13Z","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":"6aae995ea15e81094fd8c2922d0107e0da07493287917cece3262815e33d58c5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-03-26T21:18:06Z","title_canon_sha256":"ebf8e2a0f86b1150eafad08014ee0c4229879c751cf0554c93ff7cdbe00908a2"},"schema_version":"1.0","source":{"id":"1203.5821","kind":"arxiv","version":1}},"canonical_sha256":"318222732d21b0e8fde9a757fa6746123fc101cbcd2381e6e0e96dee0f0b14f6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"318222732d21b0e8fde9a757fa6746123fc101cbcd2381e6e0e96dee0f0b14f6","first_computed_at":"2026-05-18T03:59:13.355029Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:59:13.355029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Aiq9L/o1x3hqjgTeOkIxiGWgAjXbzDOQUKVEF+h/CVS5U2M8HwaMDqVPUROVoTgliyBDfB6XKDYy3m9gnpjnCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:59:13.355696Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.5821","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb8b9b192d0f2749cb78ee440e09c7e69012c42099c13274eb9a70936c7b32f5","sha256:9649f5004291029703e0d7a8788cb1b7ca3bf90838c84d96c3fcd6770ca09147"],"state_sha256":"6383ece22380812a5e1a6329f191db30b034c135574e64f1ace8b6ca516d4056"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QR+OM70rdumNFr2vaIQUyYzLNuyYpQL61MneiBKBkjTwlWr17+csHPaWBrmE1BCY2kLV+e5y21ZRWtSV/QEPDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T09:25:12.437696Z","bundle_sha256":"6543b13e1290f6095699c68fced6891d70e0600472e05f99f10c883c7ab02fea"}}