{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:E33NNX2JGNOG65ECP4MHAEBGFZ","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":"67f59d966335759e48b52d67f1c1230cf1d4a5448f5c445d2564109062a72c78","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-11T16:04:45Z","title_canon_sha256":"754a4d3d567d882f1cd3ee60e77f9ef4ee0cb86624dc83f356d715160ba60777"},"schema_version":"1.0","source":{"id":"1705.04256","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.04256","created_at":"2026-05-18T00:22:21Z"},{"alias_kind":"arxiv_version","alias_value":"1705.04256v2","created_at":"2026-05-18T00:22:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.04256","created_at":"2026-05-18T00:22:21Z"},{"alias_kind":"pith_short_12","alias_value":"E33NNX2JGNOG","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_16","alias_value":"E33NNX2JGNOG65EC","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_8","alias_value":"E33NNX2J","created_at":"2026-05-18T12:31:12Z"}],"graph_snapshots":[{"event_id":"sha256:b16ff40412387f6f4f5814541cd5f99f4f0b5bf2a3328551ba25a7d870a5adab","target":"graph","created_at":"2026-05-18T00:22:21Z","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":"In this paper, we generalize the work of Tuenter to give an identity which completely characterizes the complement of a numerical semigroup in terms of its Ap\\'ery sets. Using this result, we compute the $m$th power Sylvester and alternating Sylvester sums for free numerical semigroups. Explicit formulas are given for small $m$.","authors_text":"Caleb McKinley Shor, T. Alden Gassert","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-11T16:04:45Z","title":"Characterizations of numerical semigroup complements via Ap\\'ery sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04256","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":"sha256:35fcfe7becd655b0c79f4a2bfb7ecb878b588cd81bb764c5b7b54dbc403b368f","target":"record","created_at":"2026-05-18T00:22:21Z","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":"67f59d966335759e48b52d67f1c1230cf1d4a5448f5c445d2564109062a72c78","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-11T16:04:45Z","title_canon_sha256":"754a4d3d567d882f1cd3ee60e77f9ef4ee0cb86624dc83f356d715160ba60777"},"schema_version":"1.0","source":{"id":"1705.04256","kind":"arxiv","version":2}},"canonical_sha256":"26f6d6df49335c6f74827f187010262e4cbb644d048a32d0bc6fca6bd6e43a92","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"26f6d6df49335c6f74827f187010262e4cbb644d048a32d0bc6fca6bd6e43a92","first_computed_at":"2026-05-18T00:22:21.002992Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:21.002992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"19KAxTj30T3JBy9u2Pv35uAIigfr9Uh6J78r0D42SwAG2dFB6Hl3yP9daLqibRXXZPrmOyJtV+P35+/Hd88QBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:21.003553Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.04256","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:35fcfe7becd655b0c79f4a2bfb7ecb878b588cd81bb764c5b7b54dbc403b368f","sha256:b16ff40412387f6f4f5814541cd5f99f4f0b5bf2a3328551ba25a7d870a5adab"],"state_sha256":"8e5abc65f22dd7d629a1a1d129743a6fab090405ae33b16ec7970f2b5ef7f252"}