{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:76OYCTJB3N32ZHOWGLHSHQU4LR","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":"ab5ec70f4498a6d46cc28bee16e4bf142b58d90f0feb3ea521fe235f494c0c7c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-11T16:48:03Z","title_canon_sha256":"0c893ba1fb9d1caa5f61a8a0a7bf845eb5dc45e55b3f52debd2de37e56def482"},"schema_version":"1.0","source":{"id":"1511.03568","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.03568","created_at":"2026-05-17T23:52:34Z"},{"alias_kind":"arxiv_version","alias_value":"1511.03568v4","created_at":"2026-05-17T23:52:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.03568","created_at":"2026-05-17T23:52:34Z"},{"alias_kind":"pith_short_12","alias_value":"76OYCTJB3N32","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"76OYCTJB3N32ZHOW","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"76OYCTJB","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:550ea368be2bc6f8acb3bdc1a1f07e8825b55187948d531869e4ed4de452bbb1","target":"graph","created_at":"2026-05-17T23:52:34Z","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":"Baker and Norine proved a Riemann--Roch theorem for divisors on undirected graphs. The notions of graph divisor theory are in duality with the notions of the chip-firing game of Bj\\\"orner, Lov\\'asz and Shor. We use this connection to prove Riemann--Roch-type results on directed graphs. We give a simple proof for a Riemann--Roch inequality on Eulerian directed graphs, improving a result of Amini and Manjunath. We also study possibilities and impossibilities of Riemann--Roch-type equalities in strongly connected digraphs and give examples. We intend to make the connections of this theory to grap","authors_text":"B\\'alint Hujter, Lilla T\\'othm\\'er\\'esz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-11T16:48:03Z","title":"Chip-firing based methods in the Riemann--Roch theory of directed graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03568","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":"sha256:422101943f4b304e06cb1455eb7d43c650d82d9f49a435fbbfd8df081f0f6af4","target":"record","created_at":"2026-05-17T23:52:34Z","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":"ab5ec70f4498a6d46cc28bee16e4bf142b58d90f0feb3ea521fe235f494c0c7c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-11T16:48:03Z","title_canon_sha256":"0c893ba1fb9d1caa5f61a8a0a7bf845eb5dc45e55b3f52debd2de37e56def482"},"schema_version":"1.0","source":{"id":"1511.03568","kind":"arxiv","version":4}},"canonical_sha256":"ff9d814d21db77ac9dd632cf23c29c5c74a4071bc84d266fc20f4c6c615ab50f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ff9d814d21db77ac9dd632cf23c29c5c74a4071bc84d266fc20f4c6c615ab50f","first_computed_at":"2026-05-17T23:52:34.513052Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:52:34.513052Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qYfwkHt1Pgs4UkCM3GDJew1iIDjWl5SSRaf03+6vJe5R6Z0QphWU/QgbR6tvAMuY1Kj7jlrNy6gNaX/vF3bHBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:52:34.513450Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.03568","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:422101943f4b304e06cb1455eb7d43c650d82d9f49a435fbbfd8df081f0f6af4","sha256:550ea368be2bc6f8acb3bdc1a1f07e8825b55187948d531869e4ed4de452bbb1"],"state_sha256":"cb4797ea7054ab87cf0a67cf76db1add7177164bae466ba26bef3ee1c63d914c"}