{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:NFMP5U6BPOCMRQ3EFP6EKCIBCA","short_pith_number":"pith:NFMP5U6B","schema_version":"1.0","canonical_sha256":"6958fed3c17b84c8c3642bfc450901100be4b77815d8661ed260d8f6eb9f98c7","source":{"kind":"arxiv","id":"1509.03380","version":3},"attestation_state":"computed","paper":{"title":"A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Alexander Lazar","submitted_at":"2015-09-11T03:14:11Z","abstract_excerpt":"In his preprint https://arxiv.org/abs/1308.3813, Cartwright introduced the notion of a weak tropical complex in order to generalize the concepts of divisors and the Picard group on graphs from Baker and Norine's paper Riemann-Roch and Abel-Jacobi Theory on a Finite Graph. A tropical complex $\\Gamma$ is a $\\Delta$-complex equipped with certain algebraic data. Divisors in a tropical complex are formal linear combinations of ridges, and piecewise-linear functions on a tropical complex give rise in a natural way to divisors. Divisors that arise from PL-functions are called principal, and divisors "},"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":"1509.03380","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-11T03:14:11Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"7baaa4be6c28ccfb3ba6f5cb8df7f15deda52b026130dec2b60151603d3ffbb5","abstract_canon_sha256":"8c320069b6659a51dec5fb0a622c1768a4b46bd3b4462ab79943520ab3d70589"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:12.080492Z","signature_b64":"WD8384PSbJmgZ6J7+ovjIGMtgx2HVc2fWEOPR0UrYX0Xs42Cxqc4yGNh8Zu1ONYWmyTBSTW2/Z94U3CVRkd5BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6958fed3c17b84c8c3642bfc450901100be4b77815d8661ed260d8f6eb9f98c7","last_reissued_at":"2026-05-18T00:25:12.079871Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:12.079871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Alexander Lazar","submitted_at":"2015-09-11T03:14:11Z","abstract_excerpt":"In his preprint https://arxiv.org/abs/1308.3813, Cartwright introduced the notion of a weak tropical complex in order to generalize the concepts of divisors and the Picard group on graphs from Baker and Norine's paper Riemann-Roch and Abel-Jacobi Theory on a Finite Graph. A tropical complex $\\Gamma$ is a $\\Delta$-complex equipped with certain algebraic data. Divisors in a tropical complex are formal linear combinations of ridges, and piecewise-linear functions on a tropical complex give rise in a natural way to divisors. Divisors that arise from PL-functions are called principal, and divisors "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03380","kind":"arxiv","version":3},"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":"1509.03380","created_at":"2026-05-18T00:25:12.079970+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.03380v3","created_at":"2026-05-18T00:25:12.079970+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.03380","created_at":"2026-05-18T00:25:12.079970+00:00"},{"alias_kind":"pith_short_12","alias_value":"NFMP5U6BPOCM","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_16","alias_value":"NFMP5U6BPOCMRQ3E","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_8","alias_value":"NFMP5U6B","created_at":"2026-05-18T12:29:32.376354+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/NFMP5U6BPOCMRQ3EFP6EKCIBCA","json":"https://pith.science/pith/NFMP5U6BPOCMRQ3EFP6EKCIBCA.json","graph_json":"https://pith.science/api/pith-number/NFMP5U6BPOCMRQ3EFP6EKCIBCA/graph.json","events_json":"https://pith.science/api/pith-number/NFMP5U6BPOCMRQ3EFP6EKCIBCA/events.json","paper":"https://pith.science/paper/NFMP5U6B"},"agent_actions":{"view_html":"https://pith.science/pith/NFMP5U6BPOCMRQ3EFP6EKCIBCA","download_json":"https://pith.science/pith/NFMP5U6BPOCMRQ3EFP6EKCIBCA.json","view_paper":"https://pith.science/paper/NFMP5U6B","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.03380&json=true","fetch_graph":"https://pith.science/api/pith-number/NFMP5U6BPOCMRQ3EFP6EKCIBCA/graph.json","fetch_events":"https://pith.science/api/pith-number/NFMP5U6BPOCMRQ3EFP6EKCIBCA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NFMP5U6BPOCMRQ3EFP6EKCIBCA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NFMP5U6BPOCMRQ3EFP6EKCIBCA/action/storage_attestation","attest_author":"https://pith.science/pith/NFMP5U6BPOCMRQ3EFP6EKCIBCA/action/author_attestation","sign_citation":"https://pith.science/pith/NFMP5U6BPOCMRQ3EFP6EKCIBCA/action/citation_signature","submit_replication":"https://pith.science/pith/NFMP5U6BPOCMRQ3EFP6EKCIBCA/action/replication_record"}},"created_at":"2026-05-18T00:25:12.079970+00:00","updated_at":"2026-05-18T00:25:12.079970+00:00"}