{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:F4LJUEHZB43NRZ6XOAKYFUVAU2","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":"ce6d2f4e2535e3a81dec6a03d63e6f1614fc9796bd79705c206032afbf1c7e8e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-04-15T23:26:48Z","title_canon_sha256":"38cc2aea7480aab6ba8a30efafc72adf3227d3fa6d7338d53444e88e69547797"},"schema_version":"1.0","source":{"id":"2604.14478","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.14478","created_at":"2026-06-19T16:09:58Z"},{"alias_kind":"arxiv_version","alias_value":"2604.14478v2","created_at":"2026-06-19T16:09:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.14478","created_at":"2026-06-19T16:09:58Z"},{"alias_kind":"pith_short_12","alias_value":"F4LJUEHZB43N","created_at":"2026-06-19T16:09:58Z"},{"alias_kind":"pith_short_16","alias_value":"F4LJUEHZB43NRZ6X","created_at":"2026-06-19T16:09:58Z"},{"alias_kind":"pith_short_8","alias_value":"F4LJUEHZ","created_at":"2026-06-19T16:09:58Z"}],"graph_snapshots":[{"event_id":"sha256:50a86bf2359bf8bdf56108ebdc66f8768640c5e9013e98a15feb56f7e2bbfa31","target":"graph","created_at":"2026-06-19T16:09:58Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"We develop a semigroup-theoretic analogue of liaison for relative ideals of a numerical semigroup. Two parallel linkage notions are proposed: a theory based on translates of the semigroup and a theory based on translates of the canonical ideal."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That the classical notions of liaison and linkage admit a faithful translation to the setting of relative ideals in numerical semigroups while preserving essential algebraic properties such as symmetry or linkage invariants."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"A new linkage theory for relative ideals in numerical semigroups is introduced via two notions: principal links using semigroup translates and canonical links using canonical ideal translates."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Relative ideals in numerical semigroups admit two parallel linkage theories, one via semigroup translates and one via canonical ideal translates."}],"snapshot_sha256":"937573a8d0b7b64b9c63b061cd8bea315eec6f7a0c7aeb09905cd2d847a083e8"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2604.14478/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We develop a semigroup-theoretic analogue of liaison for relative ideals of a numerical semigroup. Two parallel linkage notions are proposed: a theory based on translates of the semigroup and a theory based on translates of the canonical ideal.","authors_text":"Ignacio Ojeda","cross_cats":[],"headline":"Relative ideals in numerical semigroups admit two parallel linkage theories, one via semigroup translates and one via canonical ideal translates.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-04-15T23:26:48Z","title":"A semigroup-theoretic linkage theory for relative ideals: principal and canonical links"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.14478","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-10T11:16:08.419582Z","id":"017b60f4-c848-41ec-9daa-39e0339b68c0","model_set":{"reader":"grok-4.3"},"one_line_summary":"A new linkage theory for relative ideals in numerical semigroups is introduced via two notions: principal links using semigroup translates and canonical links using canonical ideal translates.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Relative ideals in numerical semigroups admit two parallel linkage theories, one via semigroup translates and one via canonical ideal translates.","strongest_claim":"We develop a semigroup-theoretic analogue of liaison for relative ideals of a numerical semigroup. Two parallel linkage notions are proposed: a theory based on translates of the semigroup and a theory based on translates of the canonical ideal.","weakest_assumption":"That the classical notions of liaison and linkage admit a faithful translation to the setting of relative ideals in numerical semigroups while preserving essential algebraic properties such as symmetry or linkage invariants."}},"verdict_id":"017b60f4-c848-41ec-9daa-39e0339b68c0"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ea392601e94324c3517936a0c6e02211cd932aaffd18a219f4aa2b32157330bb","target":"record","created_at":"2026-06-19T16:09:58Z","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":"ce6d2f4e2535e3a81dec6a03d63e6f1614fc9796bd79705c206032afbf1c7e8e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-04-15T23:26:48Z","title_canon_sha256":"38cc2aea7480aab6ba8a30efafc72adf3227d3fa6d7338d53444e88e69547797"},"schema_version":"1.0","source":{"id":"2604.14478","kind":"arxiv","version":2}},"canonical_sha256":"2f169a10f90f36d8e7d7701582d2a0a6ad7fc393306addcd6454c35fcd9d0b4e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f169a10f90f36d8e7d7701582d2a0a6ad7fc393306addcd6454c35fcd9d0b4e","first_computed_at":"2026-06-19T16:09:58.193454Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:09:58.193454Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"G/7Y8KEhuJxfsrVn4At3zgmr8nHMgPdlUQaOFzQGgFrL+6tDjYTKdvtBZnbewHx7D1MIq6lB1TVYeSnLlPvLAA==","signature_status":"signed_v1","signed_at":"2026-06-19T16:09:58.193827Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.14478","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea392601e94324c3517936a0c6e02211cd932aaffd18a219f4aa2b32157330bb","sha256:50a86bf2359bf8bdf56108ebdc66f8768640c5e9013e98a15feb56f7e2bbfa31"],"state_sha256":"c438bbf0a742403cd2f5d9974b3b9d2a0a44066546bf70f4df71533113310d11"}