{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2003:45A24P3VTBKKWZP4DOXW53YDL6","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":"84d84deddafe5cb39974eb940245909d384a8af998911e724145027698ac09c0","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2003-04-21T12:16:01Z","title_canon_sha256":"de3a06faf088980e199a75def6f0ea7129be5b1fd81e784cea995ee36b734bc6"},"schema_version":"1.0","source":{"id":"math/0304289","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0304289","created_at":"2026-05-18T04:35:57Z"},{"alias_kind":"arxiv_version","alias_value":"math/0304289v1","created_at":"2026-05-18T04:35:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0304289","created_at":"2026-05-18T04:35:57Z"},{"alias_kind":"pith_short_12","alias_value":"45A24P3VTBKK","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_16","alias_value":"45A24P3VTBKKWZP4","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_8","alias_value":"45A24P3V","created_at":"2026-05-18T12:25:51Z"}],"graph_snapshots":[{"event_id":"sha256:db0db30735a4204fc414b411e21398e213a13281e61646b4fd36c02d089b20a2","target":"graph","created_at":"2026-05-18T04:35:57Z","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":"Let $G=(V(G),E(G))$ be a planar digraph embedded in the plane in which all inner faces are equilateral triangles (with three edges in each), and let the union $\\Rscr$ of these faces forms a convex polygon. The question is: given a function $\\sigma$ on the boundary edges of $G$, does there exist a concave function $f$ on $\\Rscr$ which is affinely linear within each bounded face and satisfies $f(v)-f(u)=\\sigma(e)$ for each boundary edge $e=(u,v)$?\n  The functions $\\sigma$ admitting such an $f$ form a polyhedral cone $C$, and when the region $\\Rscr$ is a triangle, $C$ turns out to be exactly the ","authors_text":"Alexander V. Karzanov","cross_cats":[],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2003-04-21T12:16:01Z","title":"Concave cocirculations in a triangular grid"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0304289","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":"sha256:25ec1312fa3ecef7deea28c2d5b61e375439241a235033885547a3b6c0a0dcff","target":"record","created_at":"2026-05-18T04:35:57Z","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":"84d84deddafe5cb39974eb940245909d384a8af998911e724145027698ac09c0","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2003-04-21T12:16:01Z","title_canon_sha256":"de3a06faf088980e199a75def6f0ea7129be5b1fd81e784cea995ee36b734bc6"},"schema_version":"1.0","source":{"id":"math/0304289","kind":"arxiv","version":1}},"canonical_sha256":"e741ae3f759854ab65fc1baf6eef035f8bef95808112bccbda4a8aebac38a016","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e741ae3f759854ab65fc1baf6eef035f8bef95808112bccbda4a8aebac38a016","first_computed_at":"2026-05-18T04:35:57.888844Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:35:57.888844Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"w0o++9I8NUssuo7d6ixt22INqECcrwVsAhiUw1Lit0JG7sAD7ZBab7l/S8Los6/4DUUhwGo1oNYhhuBL3Z6fAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:35:57.889622Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0304289","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:25ec1312fa3ecef7deea28c2d5b61e375439241a235033885547a3b6c0a0dcff","sha256:db0db30735a4204fc414b411e21398e213a13281e61646b4fd36c02d089b20a2"],"state_sha256":"9934dc7712ac91afd5b0894de1909df219f7011f6fb7cce9e90026f327c9cbb1"}