{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:WK7BELVEP57MR5S54FDHZT4SYJ","short_pith_number":"pith:WK7BELVE","canonical_record":{"source":{"id":"1309.7476","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-28T16:38:18Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e24905c12290f06db26e6d55cefdb700cd98e4d1aeae63f9ea69151dad9c424f","abstract_canon_sha256":"65773ee6fe95b91d6aeb601f692f502fbf6cd04fff72a56f081fb2c9d4eccff9"},"schema_version":"1.0"},"canonical_sha256":"b2be122ea47f7ec8f65de1467ccf92c259c0c7aaccea7547ea6d053992685cf2","source":{"kind":"arxiv","id":"1309.7476","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.7476","created_at":"2026-05-18T00:08:16Z"},{"alias_kind":"arxiv_version","alias_value":"1309.7476v3","created_at":"2026-05-18T00:08:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.7476","created_at":"2026-05-18T00:08:16Z"},{"alias_kind":"pith_short_12","alias_value":"WK7BELVEP57M","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"WK7BELVEP57MR5S5","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"WK7BELVE","created_at":"2026-05-18T12:28:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:WK7BELVEP57MR5S54FDHZT4SYJ","target":"record","payload":{"canonical_record":{"source":{"id":"1309.7476","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-28T16:38:18Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e24905c12290f06db26e6d55cefdb700cd98e4d1aeae63f9ea69151dad9c424f","abstract_canon_sha256":"65773ee6fe95b91d6aeb601f692f502fbf6cd04fff72a56f081fb2c9d4eccff9"},"schema_version":"1.0"},"canonical_sha256":"b2be122ea47f7ec8f65de1467ccf92c259c0c7aaccea7547ea6d053992685cf2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:16.542333Z","signature_b64":"z55BVstE7bFbh7/BD5141KxW96SqaIb3/vLtzBQST+voS6hicGuYa1BVyBFm4OVQfZyD+6NZDcdgNXBUYwM3Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2be122ea47f7ec8f65de1467ccf92c259c0c7aaccea7547ea6d053992685cf2","last_reissued_at":"2026-05-18T00:08:16.541852Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:16.541852Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.7476","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:08:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J0ZeW7S3LSW2qKaMifDlIDEszS4cKoQ2OKY7Ss8vBOFg3jLI+0rnHfW2wRNNaZqGll8EgJz0vFAiQaaypOErDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T02:46:11.510536Z"},"content_sha256":"3434fb9e0dea9008db8ea6983aa8a720fb59480de039e27af56be2a0497471b4","schema_version":"1.0","event_id":"sha256:3434fb9e0dea9008db8ea6983aa8a720fb59480de039e27af56be2a0497471b4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:WK7BELVEP57MR5S54FDHZT4SYJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Linearity of \\omega-primality in Numerical Monoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Christopher O'Neill, Roberto Pelayo","submitted_at":"2013-09-28T16:38:18Z","abstract_excerpt":"In an atomic, cancellative, commutative monoid, the \\omega-value measures how far an element is from being prime. In numerical monoids, we show that this invariant exhibits eventual quasilinearity (i.e., periodic linearity). We apply this result to describe the asymptotic behavior of the \\omega-function for a general numerical monoid and give an explicit formula when the monoid has embedding dimension 2."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7476","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:08:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hAxKEyboext6jkMy5NGhEoPiWpO/F1Wfz3mBpPgvuE47VajDatcX4M/i4PtWRpsGYh8SzEXnQ0cAucjI05QrBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T02:46:11.510880Z"},"content_sha256":"3027c2d5415ea7f84d099d84deefacc377405c8ba3a3721c878b9ccbd7e298c7","schema_version":"1.0","event_id":"sha256:3027c2d5415ea7f84d099d84deefacc377405c8ba3a3721c878b9ccbd7e298c7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WK7BELVEP57MR5S54FDHZT4SYJ/bundle.json","state_url":"https://pith.science/pith/WK7BELVEP57MR5S54FDHZT4SYJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WK7BELVEP57MR5S54FDHZT4SYJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-04T02:46:11Z","links":{"resolver":"https://pith.science/pith/WK7BELVEP57MR5S54FDHZT4SYJ","bundle":"https://pith.science/pith/WK7BELVEP57MR5S54FDHZT4SYJ/bundle.json","state":"https://pith.science/pith/WK7BELVEP57MR5S54FDHZT4SYJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WK7BELVEP57MR5S54FDHZT4SYJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:WK7BELVEP57MR5S54FDHZT4SYJ","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":"65773ee6fe95b91d6aeb601f692f502fbf6cd04fff72a56f081fb2c9d4eccff9","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-28T16:38:18Z","title_canon_sha256":"e24905c12290f06db26e6d55cefdb700cd98e4d1aeae63f9ea69151dad9c424f"},"schema_version":"1.0","source":{"id":"1309.7476","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.7476","created_at":"2026-05-18T00:08:16Z"},{"alias_kind":"arxiv_version","alias_value":"1309.7476v3","created_at":"2026-05-18T00:08:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.7476","created_at":"2026-05-18T00:08:16Z"},{"alias_kind":"pith_short_12","alias_value":"WK7BELVEP57M","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"WK7BELVEP57MR5S5","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"WK7BELVE","created_at":"2026-05-18T12:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:3027c2d5415ea7f84d099d84deefacc377405c8ba3a3721c878b9ccbd7e298c7","target":"graph","created_at":"2026-05-18T00:08:16Z","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 an atomic, cancellative, commutative monoid, the \\omega-value measures how far an element is from being prime. In numerical monoids, we show that this invariant exhibits eventual quasilinearity (i.e., periodic linearity). We apply this result to describe the asymptotic behavior of the \\omega-function for a general numerical monoid and give an explicit formula when the monoid has embedding dimension 2.","authors_text":"Christopher O'Neill, Roberto Pelayo","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-28T16:38:18Z","title":"On the Linearity of \\omega-primality in Numerical Monoids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7476","kind":"arxiv","version":3},"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":"sha256:3434fb9e0dea9008db8ea6983aa8a720fb59480de039e27af56be2a0497471b4","target":"record","created_at":"2026-05-18T00:08:16Z","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":"65773ee6fe95b91d6aeb601f692f502fbf6cd04fff72a56f081fb2c9d4eccff9","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-28T16:38:18Z","title_canon_sha256":"e24905c12290f06db26e6d55cefdb700cd98e4d1aeae63f9ea69151dad9c424f"},"schema_version":"1.0","source":{"id":"1309.7476","kind":"arxiv","version":3}},"canonical_sha256":"b2be122ea47f7ec8f65de1467ccf92c259c0c7aaccea7547ea6d053992685cf2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b2be122ea47f7ec8f65de1467ccf92c259c0c7aaccea7547ea6d053992685cf2","first_computed_at":"2026-05-18T00:08:16.541852Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:08:16.541852Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z55BVstE7bFbh7/BD5141KxW96SqaIb3/vLtzBQST+voS6hicGuYa1BVyBFm4OVQfZyD+6NZDcdgNXBUYwM3Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:08:16.542333Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.7476","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3434fb9e0dea9008db8ea6983aa8a720fb59480de039e27af56be2a0497471b4","sha256:3027c2d5415ea7f84d099d84deefacc377405c8ba3a3721c878b9ccbd7e298c7"],"state_sha256":"756bebb5a955121ca225a793f1840ebad4f1f8bb7504e098485ee2703fda5617"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G1ocgXRqObM9beiNp5lgHnvSdfhme4lmyhL7UE2cCWQMzJstSLMayPX/SmqixfU8ydZnch0eXJbgUHSuefyKBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T02:46:11.512879Z","bundle_sha256":"3b281fffc39b1807da30e96e31b13c42088f9a197dcc7decc6cdd6e62de80c5e"}}