{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:N7WISSJPFRQHBXNZMY2HAKRNIG","short_pith_number":"pith:N7WISSJP","canonical_record":{"source":{"id":"1401.5730","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-01-22T16:50:40Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"172c1675d5c7d0c30f5c8440edc1361100d65451d309773d72f946c6ac6913f2","abstract_canon_sha256":"9de46788881e6879a26ecec39cf2a5693e4f12fc6a797445dd8fa72371655ab1"},"schema_version":"1.0"},"canonical_sha256":"6fec89492f2c6070ddb96634702a2d41a8f9f3a30190a0c6ac4874fc22291264","source":{"kind":"arxiv","id":"1401.5730","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.5730","created_at":"2026-05-18T02:32:54Z"},{"alias_kind":"arxiv_version","alias_value":"1401.5730v2","created_at":"2026-05-18T02:32:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.5730","created_at":"2026-05-18T02:32:54Z"},{"alias_kind":"pith_short_12","alias_value":"N7WISSJPFRQH","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"N7WISSJPFRQHBXNZ","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"N7WISSJP","created_at":"2026-05-18T12:28:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:N7WISSJPFRQHBXNZMY2HAKRNIG","target":"record","payload":{"canonical_record":{"source":{"id":"1401.5730","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-01-22T16:50:40Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"172c1675d5c7d0c30f5c8440edc1361100d65451d309773d72f946c6ac6913f2","abstract_canon_sha256":"9de46788881e6879a26ecec39cf2a5693e4f12fc6a797445dd8fa72371655ab1"},"schema_version":"1.0"},"canonical_sha256":"6fec89492f2c6070ddb96634702a2d41a8f9f3a30190a0c6ac4874fc22291264","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:54.212728Z","signature_b64":"yHhlL63JXSjosQjlL5OCXapOGEWRiVAK0e837VH1fBjVQxElpAFCDx4iNWQkVMIFTJ9t8CFsv4/laHOBBrx8BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6fec89492f2c6070ddb96634702a2d41a8f9f3a30190a0c6ac4874fc22291264","last_reissued_at":"2026-05-18T02:32:54.212200Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:54.212200Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1401.5730","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:32:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sJScBGPzOF9SwmWqHRXDeAXwFscYHQlwIaJN2jg9Hi9Dac//LbzxJW2FohkKN7Pd+SX9xPessf93BXBHj/JzBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T21:39:14.036831Z"},"content_sha256":"fd5736db9aca504f8b3a5e2af1623c682e246812c2cbc0250e9791c3cb3c5551","schema_version":"1.0","event_id":"sha256:fd5736db9aca504f8b3a5e2af1623c682e246812c2cbc0250e9791c3cb3c5551"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:N7WISSJPFRQHBXNZMY2HAKRNIG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algebraic and combinatorial rank of divisors on finite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Lucia Caporaso, Margarida Melo, Yoav Len","submitted_at":"2014-01-22T16:50:40Z","abstract_excerpt":"We study the algebraic rank of a divisor on a graph, an invariant defined using divisors on algebraic curves dual to the graph. We prove it satisfies the Riemann-Roch formula, a specialization property, and the Clifford inequality. We prove that it is at most equal to the (usual) combinatorial rank, and that equality holds in many cases, though not in general."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.5730","kind":"arxiv","version":2},"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":"2026-05-18T02:32:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VFAcaOneeojtVnoUKtNP8NKkITCUvDukzYw2UMRFu4Qs8vvmr1/eyd+iYrgbW+aXLa8/bqCl1ZH6b2J01VX5CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T21:39:14.037159Z"},"content_sha256":"b611aed1fb21154134fe39cf9ade24befd43b4453a68536e306512d317ac5700","schema_version":"1.0","event_id":"sha256:b611aed1fb21154134fe39cf9ade24befd43b4453a68536e306512d317ac5700"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N7WISSJPFRQHBXNZMY2HAKRNIG/bundle.json","state_url":"https://pith.science/pith/N7WISSJPFRQHBXNZMY2HAKRNIG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N7WISSJPFRQHBXNZMY2HAKRNIG/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":"2026-06-28T21:39:14Z","links":{"resolver":"https://pith.science/pith/N7WISSJPFRQHBXNZMY2HAKRNIG","bundle":"https://pith.science/pith/N7WISSJPFRQHBXNZMY2HAKRNIG/bundle.json","state":"https://pith.science/pith/N7WISSJPFRQHBXNZMY2HAKRNIG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N7WISSJPFRQHBXNZMY2HAKRNIG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:N7WISSJPFRQHBXNZMY2HAKRNIG","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":"9de46788881e6879a26ecec39cf2a5693e4f12fc6a797445dd8fa72371655ab1","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-01-22T16:50:40Z","title_canon_sha256":"172c1675d5c7d0c30f5c8440edc1361100d65451d309773d72f946c6ac6913f2"},"schema_version":"1.0","source":{"id":"1401.5730","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.5730","created_at":"2026-05-18T02:32:54Z"},{"alias_kind":"arxiv_version","alias_value":"1401.5730v2","created_at":"2026-05-18T02:32:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.5730","created_at":"2026-05-18T02:32:54Z"},{"alias_kind":"pith_short_12","alias_value":"N7WISSJPFRQH","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"N7WISSJPFRQHBXNZ","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"N7WISSJP","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:b611aed1fb21154134fe39cf9ade24befd43b4453a68536e306512d317ac5700","target":"graph","created_at":"2026-05-18T02:32:54Z","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 study the algebraic rank of a divisor on a graph, an invariant defined using divisors on algebraic curves dual to the graph. We prove it satisfies the Riemann-Roch formula, a specialization property, and the Clifford inequality. We prove that it is at most equal to the (usual) combinatorial rank, and that equality holds in many cases, though not in general.","authors_text":"Lucia Caporaso, Margarida Melo, Yoav Len","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-01-22T16:50:40Z","title":"Algebraic and combinatorial rank of divisors on finite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.5730","kind":"arxiv","version":2},"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:fd5736db9aca504f8b3a5e2af1623c682e246812c2cbc0250e9791c3cb3c5551","target":"record","created_at":"2026-05-18T02:32:54Z","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":"9de46788881e6879a26ecec39cf2a5693e4f12fc6a797445dd8fa72371655ab1","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-01-22T16:50:40Z","title_canon_sha256":"172c1675d5c7d0c30f5c8440edc1361100d65451d309773d72f946c6ac6913f2"},"schema_version":"1.0","source":{"id":"1401.5730","kind":"arxiv","version":2}},"canonical_sha256":"6fec89492f2c6070ddb96634702a2d41a8f9f3a30190a0c6ac4874fc22291264","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6fec89492f2c6070ddb96634702a2d41a8f9f3a30190a0c6ac4874fc22291264","first_computed_at":"2026-05-18T02:32:54.212200Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:54.212200Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yHhlL63JXSjosQjlL5OCXapOGEWRiVAK0e837VH1fBjVQxElpAFCDx4iNWQkVMIFTJ9t8CFsv4/laHOBBrx8BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:54.212728Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.5730","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fd5736db9aca504f8b3a5e2af1623c682e246812c2cbc0250e9791c3cb3c5551","sha256:b611aed1fb21154134fe39cf9ade24befd43b4453a68536e306512d317ac5700"],"state_sha256":"2f9ea32eba70ddc9381d6cc38f60682b24149cdb172e166458c13369afd6ed4d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NhUigelqgsq0XtH3VXrbH84dvCsjQmCURk3HiKRjizZyCIzx2xQw8nbP3AYDuaSy6Q/foODETNZm3LTHOcwbCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T21:39:14.039029Z","bundle_sha256":"42c00c7c51d7020e5207f50395ad6897aa77737d511f26a61d24e92a44aa20b1"}}