{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:WPWHOSFQ7EJCK5JWNV7YFXWLA3","short_pith_number":"pith:WPWHOSFQ","canonical_record":{"source":{"id":"2607.07243","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.QA","submitted_at":"2026-07-08T10:22:31Z","cross_cats_sorted":["math-ph","math.MP","math.RA"],"title_canon_sha256":"b46ee6694b82b6ea0b412f72dc577857950af60a8302820c19f0150ba1fa5e35","abstract_canon_sha256":"081baa7b74c968f90669b36d32948e6925362c508ab5c80435e883fbebdab566"},"schema_version":"1.0"},"canonical_sha256":"b3ec7748b0f9122575366d7f82decb06c77b4d6cd7a2fbb2c98b3eac14cf5732","source":{"kind":"arxiv","id":"2607.07243","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.07243","created_at":"2026-07-09T01:20:18Z"},{"alias_kind":"arxiv_version","alias_value":"2607.07243v1","created_at":"2026-07-09T01:20:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.07243","created_at":"2026-07-09T01:20:18Z"},{"alias_kind":"pith_short_12","alias_value":"WPWHOSFQ7EJC","created_at":"2026-07-09T01:20:18Z"},{"alias_kind":"pith_short_16","alias_value":"WPWHOSFQ7EJCK5JW","created_at":"2026-07-09T01:20:18Z"},{"alias_kind":"pith_short_8","alias_value":"WPWHOSFQ","created_at":"2026-07-09T01:20:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:WPWHOSFQ7EJCK5JWNV7YFXWLA3","target":"record","payload":{"canonical_record":{"source":{"id":"2607.07243","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.QA","submitted_at":"2026-07-08T10:22:31Z","cross_cats_sorted":["math-ph","math.MP","math.RA"],"title_canon_sha256":"b46ee6694b82b6ea0b412f72dc577857950af60a8302820c19f0150ba1fa5e35","abstract_canon_sha256":"081baa7b74c968f90669b36d32948e6925362c508ab5c80435e883fbebdab566"},"schema_version":"1.0"},"canonical_sha256":"b3ec7748b0f9122575366d7f82decb06c77b4d6cd7a2fbb2c98b3eac14cf5732","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-09T01:20:18.667444Z","signature_b64":"VGLdz8TIkdh9cp+5RZa1AYTJVKUk7+twoLKLQTtEItSFr8EYXxgYjxI55OK7AWn2BuEh4plSwEJqOAc2swlxDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b3ec7748b0f9122575366d7f82decb06c77b4d6cd7a2fbb2c98b3eac14cf5732","last_reissued_at":"2026-07-09T01:20:18.666960Z","signature_status":"signed_v1","first_computed_at":"2026-07-09T01:20:18.666960Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.07243","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":"2026-07-09T01:20:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tqSp0N/l6d6rM5U8hfj8wBTwkrA9EjwoWdDbHc2SaM+WAVA/XZCaQbw/xw0fJObiXJUCfasVS1FdXk5bmgNoDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T23:21:51.637794Z"},"content_sha256":"3faba765408202c68b679a2178cb8f1bf7823fb4d226dc5271569b312c7fb52c","schema_version":"1.0","event_id":"sha256:3faba765408202c68b679a2178cb8f1bf7823fb4d226dc5271569b312c7fb52c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:WPWHOSFQ7EJCK5JWNV7YFXWLA3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Noncommutative vector field calculi","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math-ph","math.MP","math.RA"],"primary_cat":"math.QA","authors_text":"Emanuele Latini, Rita Fioresi, Thomas Weber","submitted_at":"2026-07-08T10:22:31Z","abstract_excerpt":"We discuss noncommutative differential geometry from a vector field centric point of view. This is based on the notion of first order vector field calculus (FOVC), which has been previously introduced by Borowiec under the name Cartan pair. We define the universal FOVC and construct an adjunction between the categories of FOVC and that of first order differential calculi, showing that the vector field approach is dual, though not equivalent, to the differential form one. This correspondence is then extended to covariant vector field and differential calculi. On Hopf algebras, (bi)covariant FOV"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07243","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.07243/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-09T01:20:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LxMzhUUtmyhODdISClqlaMPmszgGodP6WyAEaNojnD4DveHWQVByWSVo58Fsz8MmlybM9IfznVDBN880JeocAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T23:21:51.638109Z"},"content_sha256":"c716824c5d493709b3aff8ccacf50ac7bd6a8f8daba843ee4d703a9b8b10a279","schema_version":"1.0","event_id":"sha256:c716824c5d493709b3aff8ccacf50ac7bd6a8f8daba843ee4d703a9b8b10a279"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WPWHOSFQ7EJCK5JWNV7YFXWLA3/bundle.json","state_url":"https://pith.science/pith/WPWHOSFQ7EJCK5JWNV7YFXWLA3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WPWHOSFQ7EJCK5JWNV7YFXWLA3/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":"2026-08-13T23:21:51Z","links":{"resolver":"https://pith.science/pith/WPWHOSFQ7EJCK5JWNV7YFXWLA3","bundle":"https://pith.science/pith/WPWHOSFQ7EJCK5JWNV7YFXWLA3/bundle.json","state":"https://pith.science/pith/WPWHOSFQ7EJCK5JWNV7YFXWLA3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WPWHOSFQ7EJCK5JWNV7YFXWLA3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:WPWHOSFQ7EJCK5JWNV7YFXWLA3","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":"081baa7b74c968f90669b36d32948e6925362c508ab5c80435e883fbebdab566","cross_cats_sorted":["math-ph","math.MP","math.RA"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.QA","submitted_at":"2026-07-08T10:22:31Z","title_canon_sha256":"b46ee6694b82b6ea0b412f72dc577857950af60a8302820c19f0150ba1fa5e35"},"schema_version":"1.0","source":{"id":"2607.07243","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.07243","created_at":"2026-07-09T01:20:18Z"},{"alias_kind":"arxiv_version","alias_value":"2607.07243v1","created_at":"2026-07-09T01:20:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.07243","created_at":"2026-07-09T01:20:18Z"},{"alias_kind":"pith_short_12","alias_value":"WPWHOSFQ7EJC","created_at":"2026-07-09T01:20:18Z"},{"alias_kind":"pith_short_16","alias_value":"WPWHOSFQ7EJCK5JW","created_at":"2026-07-09T01:20:18Z"},{"alias_kind":"pith_short_8","alias_value":"WPWHOSFQ","created_at":"2026-07-09T01:20:18Z"}],"graph_snapshots":[{"event_id":"sha256:c716824c5d493709b3aff8ccacf50ac7bd6a8f8daba843ee4d703a9b8b10a279","target":"graph","created_at":"2026-07-09T01:20:18Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.07243/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We discuss noncommutative differential geometry from a vector field centric point of view. This is based on the notion of first order vector field calculus (FOVC), which has been previously introduced by Borowiec under the name Cartan pair. We define the universal FOVC and construct an adjunction between the categories of FOVC and that of first order differential calculi, showing that the vector field approach is dual, though not equivalent, to the differential form one. This correspondence is then extended to covariant vector field and differential calculi. On Hopf algebras, (bi)covariant FOV","authors_text":"Emanuele Latini, Rita Fioresi, Thomas Weber","cross_cats":["math-ph","math.MP","math.RA"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.QA","submitted_at":"2026-07-08T10:22:31Z","title":"Noncommutative vector field calculi"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07243","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":"sha256:3faba765408202c68b679a2178cb8f1bf7823fb4d226dc5271569b312c7fb52c","target":"record","created_at":"2026-07-09T01:20:18Z","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":"081baa7b74c968f90669b36d32948e6925362c508ab5c80435e883fbebdab566","cross_cats_sorted":["math-ph","math.MP","math.RA"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.QA","submitted_at":"2026-07-08T10:22:31Z","title_canon_sha256":"b46ee6694b82b6ea0b412f72dc577857950af60a8302820c19f0150ba1fa5e35"},"schema_version":"1.0","source":{"id":"2607.07243","kind":"arxiv","version":1}},"canonical_sha256":"b3ec7748b0f9122575366d7f82decb06c77b4d6cd7a2fbb2c98b3eac14cf5732","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b3ec7748b0f9122575366d7f82decb06c77b4d6cd7a2fbb2c98b3eac14cf5732","first_computed_at":"2026-07-09T01:20:18.666960Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-09T01:20:18.666960Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VGLdz8TIkdh9cp+5RZa1AYTJVKUk7+twoLKLQTtEItSFr8EYXxgYjxI55OK7AWn2BuEh4plSwEJqOAc2swlxDw==","signature_status":"signed_v1","signed_at":"2026-07-09T01:20:18.667444Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.07243","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3faba765408202c68b679a2178cb8f1bf7823fb4d226dc5271569b312c7fb52c","sha256:c716824c5d493709b3aff8ccacf50ac7bd6a8f8daba843ee4d703a9b8b10a279"],"state_sha256":"c2f7949f2c2bfd94994ca035f17f8cf545009322b5a659d567b05ca9124ab798"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O2BOUGUMkuoJhY9tePcRK9NsrFURw4Jix6u9fd33DPKR+qZRJTqaqd9KN8jj5VHjn6b+wo3L4PxcTL8ycNLmCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T23:21:51.642928Z","bundle_sha256":"c19cc4ee9690de2177eb34927f03bf13c1dd8959aa54161d425588bd11e7d5e8"}}