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We also prove that $F^\\nabla=0$ is a necessary (but not sufficient) condition for $d^\\nabla$ to be a chain complex. Instead we prove that $F^\\nabla=0$ and $d^\\nabla\\circ d^\\nabla\\circ\\nabla=0$ are necessary and sufficient conditions for $d^\\nabla$ to be a {\\em chain $2$-complex}, i.e., $d^\\nabla\\circ d^\\nabl"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1409.3479","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-09-11T15:39:04Z","cross_cats_sorted":[],"title_canon_sha256":"9c7e16b738c2bfa3078460f59f1fc81e05e27120c85084dd5e86cbc460a1774f","abstract_canon_sha256":"2bcfe1f7004d1c5d6533e754c7e226681737525223b39e24362809cba4847988"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:01.220123Z","signature_b64":"daaciVHvFqFHEzAaMyML36Oo6AG66dtHsZfd0diys9MRUgsFH4YtmjfdM9PtXSheafTWJFf4LpuJyaxcxnaWAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b68db921b84418dbc8d5f4b06dc12bb9b6e66581cc5ce87b409a89f755b1369","last_reissued_at":"2026-05-18T02:43:01.219634Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:01.219634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A curvature form for pseudoconnections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"C.A. Morales, M. Vilches","submitted_at":"2014-09-11T15:39:04Z","abstract_excerpt":"We obtain the curvature form $F^\\nabla=P\\circ d^\\nabla\\circ\\nabla-d^\\nabla\\circ P\\circ\\nabla+d^\\nabla\\circ\\nabla\\circ P$ for a vector bundle pseudoconnection $\\nabla$, where $d^\\nabla$ is the exterior derivative associated to $\\nabla$. We use $F^\\nabla$ to obtain the curvature of $\\nabla$. We also prove that $F^\\nabla=0$ is a necessary (but not sufficient) condition for $d^\\nabla$ to be a chain complex. Instead we prove that $F^\\nabla=0$ and $d^\\nabla\\circ d^\\nabla\\circ\\nabla=0$ are necessary and sufficient conditions for $d^\\nabla$ to be a {\\em chain $2$-complex}, i.e., $d^\\nabla\\circ d^\\nabl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.3479","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1409.3479","created_at":"2026-05-18T02:43:01.219712+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.3479v1","created_at":"2026-05-18T02:43:01.219712+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.3479","created_at":"2026-05-18T02:43:01.219712+00:00"},{"alias_kind":"pith_short_12","alias_value":"NNUNXEQ3QRAY","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_16","alias_value":"NNUNXEQ3QRAY3PEN","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_8","alias_value":"NNUNXEQ3","created_at":"2026-05-18T12:28:41.024544+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NNUNXEQ3QRAY3PENL5FQNXASXO","json":"https://pith.science/pith/NNUNXEQ3QRAY3PENL5FQNXASXO.json","graph_json":"https://pith.science/api/pith-number/NNUNXEQ3QRAY3PENL5FQNXASXO/graph.json","events_json":"https://pith.science/api/pith-number/NNUNXEQ3QRAY3PENL5FQNXASXO/events.json","paper":"https://pith.science/paper/NNUNXEQ3"},"agent_actions":{"view_html":"https://pith.science/pith/NNUNXEQ3QRAY3PENL5FQNXASXO","download_json":"https://pith.science/pith/NNUNXEQ3QRAY3PENL5FQNXASXO.json","view_paper":"https://pith.science/paper/NNUNXEQ3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.3479&json=true","fetch_graph":"https://pith.science/api/pith-number/NNUNXEQ3QRAY3PENL5FQNXASXO/graph.json","fetch_events":"https://pith.science/api/pith-number/NNUNXEQ3QRAY3PENL5FQNXASXO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NNUNXEQ3QRAY3PENL5FQNXASXO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NNUNXEQ3QRAY3PENL5FQNXASXO/action/storage_attestation","attest_author":"https://pith.science/pith/NNUNXEQ3QRAY3PENL5FQNXASXO/action/author_attestation","sign_citation":"https://pith.science/pith/NNUNXEQ3QRAY3PENL5FQNXASXO/action/citation_signature","submit_replication":"https://pith.science/pith/NNUNXEQ3QRAY3PENL5FQNXASXO/action/replication_record"}},"created_at":"2026-05-18T02:43:01.219712+00:00","updated_at":"2026-05-18T02:43:01.219712+00:00"}