{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:4PAYFDPFX56TBSK7NHJWP4CMZH","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":"66be4ad75f4d9fc7073dae773f273bf76fe6c9eb3e2befdcb4ee8f1e49f56d0a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2021-10-20T15:20:26Z","title_canon_sha256":"cc0ce9f2026a76b2e2d34fc9b04aa6fb5adb142bb069b289feea56679eb22727"},"schema_version":"1.0","source":{"id":"2110.10618","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2110.10618","created_at":"2026-07-05T03:24:19Z"},{"alias_kind":"arxiv_version","alias_value":"2110.10618v1","created_at":"2026-07-05T03:24:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2110.10618","created_at":"2026-07-05T03:24:19Z"},{"alias_kind":"pith_short_12","alias_value":"4PAYFDPFX56T","created_at":"2026-07-05T03:24:19Z"},{"alias_kind":"pith_short_16","alias_value":"4PAYFDPFX56TBSK7","created_at":"2026-07-05T03:24:19Z"},{"alias_kind":"pith_short_8","alias_value":"4PAYFDPF","created_at":"2026-07-05T03:24:19Z"}],"graph_snapshots":[{"event_id":"sha256:3bdfdf3c95ec4fdbf15a9e483b307ea9a191216c072fe782be1ebc3499a90f78","target":"graph","created_at":"2026-07-05T03:24:19Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2110.10618/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Length density is a recently introduced factorization invariant, assigned to each element $n$ of a cancellative commutative atomic semigroup $S$, that measures how far the set of factorization lengths of $n$ is from being a full interval. We examine length density of elements of numerical semigroups (that is, additive subsemigroups of the non-negative integers).","authors_text":"Christopher O'Neill, Cole Brower, Joseph McDonough, Scott Chapman, Travis Kulhanek, Vadim Ponomarenko, Vody Pavlyuk","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2021-10-20T15:20:26Z","title":"Length density and numerical semigroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.10618","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":"sha256:5194d6e28efc12a7f92a9ac34613ffcad19ca66af9ae70464ad0e689c71bf3d0","target":"record","created_at":"2026-07-05T03:24:19Z","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":"66be4ad75f4d9fc7073dae773f273bf76fe6c9eb3e2befdcb4ee8f1e49f56d0a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2021-10-20T15:20:26Z","title_canon_sha256":"cc0ce9f2026a76b2e2d34fc9b04aa6fb5adb142bb069b289feea56679eb22727"},"schema_version":"1.0","source":{"id":"2110.10618","kind":"arxiv","version":1}},"canonical_sha256":"e3c1828de5bf7d30c95f69d367f04cc9dcdb6c5f90a64e7c76c27fc45ee4c8c0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e3c1828de5bf7d30c95f69d367f04cc9dcdb6c5f90a64e7c76c27fc45ee4c8c0","first_computed_at":"2026-07-05T03:24:19.388280Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:24:19.388280Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1dkeJP6rtQseuVQrR96wRXBbse9rMeJqDme8VFc+m02P0Tq6r7sAitflqeMOzVQqlhQzDS0wpphI5FFVdHoMAg==","signature_status":"signed_v1","signed_at":"2026-07-05T03:24:19.388661Z","signed_message":"canonical_sha256_bytes"},"source_id":"2110.10618","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5194d6e28efc12a7f92a9ac34613ffcad19ca66af9ae70464ad0e689c71bf3d0","sha256:3bdfdf3c95ec4fdbf15a9e483b307ea9a191216c072fe782be1ebc3499a90f78"],"state_sha256":"63e982a6178f6f271cbbb794707ef1243bc703a29cf24691ecc98fea41c1963a"}