{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:GOUUNTLNVAV2ZNPOKPLSKVSF2B","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":"b474963e50c998b2358fb5b60694ce03ff1b52d39db16e21e08a3a30ef162467","cross_cats_sorted":["math.AG","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-17T13:20:19Z","title_canon_sha256":"979ef8542a1361a0342e5075bb5962995fa82ca76e77df0f103acb982336dc56"},"schema_version":"1.0","source":{"id":"1607.04868","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.04868","created_at":"2026-05-18T00:10:13Z"},{"alias_kind":"arxiv_version","alias_value":"1607.04868v4","created_at":"2026-05-18T00:10:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.04868","created_at":"2026-05-18T00:10:13Z"},{"alias_kind":"pith_short_12","alias_value":"GOUUNTLNVAV2","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"GOUUNTLNVAV2ZNPO","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"GOUUNTLN","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:f8753f4bdc546c9b53ceec25b95d42556c51ba0604ee03bc008906e2c666cf7b","target":"graph","created_at":"2026-05-18T00:10:13Z","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":"We enrich Baker and Bowler's theory of matroids over tracts with notions of vectors and covectors. In the case of oriented matroids, these $F$-vectors and $F$-covectors coincide with the usual signed vectors and signed covectors. In the case of matroids over a field $F$, the $F$-covector set resp. $F$-vector set of an $F$-matroid is a linear subspace of $F^E$ resp. its orthogonal complement.","authors_text":"Laura Anderson","cross_cats":["math.AG","math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-17T13:20:19Z","title":"Vectors of matroids over tracts"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04868","kind":"arxiv","version":4},"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":"sha256:3fc6ec2299887a680a97d7f2deacee92c8c949ce254ba3bf7a6b876e44fcfdc8","target":"record","created_at":"2026-05-18T00:10:13Z","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":"b474963e50c998b2358fb5b60694ce03ff1b52d39db16e21e08a3a30ef162467","cross_cats_sorted":["math.AG","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-17T13:20:19Z","title_canon_sha256":"979ef8542a1361a0342e5075bb5962995fa82ca76e77df0f103acb982336dc56"},"schema_version":"1.0","source":{"id":"1607.04868","kind":"arxiv","version":4}},"canonical_sha256":"33a946cd6da82bacb5ee53d7255645d0463c9395b1a5d66046d70f4d4a24bb31","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"33a946cd6da82bacb5ee53d7255645d0463c9395b1a5d66046d70f4d4a24bb31","first_computed_at":"2026-05-18T00:10:13.993774Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:13.993774Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1vytRbjGlfK5CO015Y++5QyAuZA0FVARakZNLokcF7mxYiRAaFS6MGNs+sKzYYoUwc3yE/afPPxmJ3IS6hAfBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:13.994460Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.04868","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3fc6ec2299887a680a97d7f2deacee92c8c949ce254ba3bf7a6b876e44fcfdc8","sha256:f8753f4bdc546c9b53ceec25b95d42556c51ba0604ee03bc008906e2c666cf7b"],"state_sha256":"393dc02af9c474a3a107095952caa4b0589a95a69ad4cb63eb61601eb6be4c6d"}