{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:6I5Y3L65S4IOHOS3U6UTNWJA5N","short_pith_number":"pith:6I5Y3L65","canonical_record":{"source":{"id":"1503.08351","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-03-28T21:08:23Z","cross_cats_sorted":[],"title_canon_sha256":"bf496f06e757ba0b3508cc7a9ae0e3ffca3de88b8ab92ee3261415aaf4e81b53","abstract_canon_sha256":"f82acdca96961e7c9fac4b90a2ca8c97a3ae64952163b4cc221a0b95fad0d7b3"},"schema_version":"1.0"},"canonical_sha256":"f23b8dafdd9710e3ba5ba7a936d920eb5124b1a0425f25e03b697961da245c84","source":{"kind":"arxiv","id":"1503.08351","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.08351","created_at":"2026-05-18T00:08:16Z"},{"alias_kind":"arxiv_version","alias_value":"1503.08351v4","created_at":"2026-05-18T00:08:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.08351","created_at":"2026-05-18T00:08:16Z"},{"alias_kind":"pith_short_12","alias_value":"6I5Y3L65S4IO","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6I5Y3L65S4IOHOS3","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6I5Y3L65","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:6I5Y3L65S4IOHOS3U6UTNWJA5N","target":"record","payload":{"canonical_record":{"source":{"id":"1503.08351","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-03-28T21:08:23Z","cross_cats_sorted":[],"title_canon_sha256":"bf496f06e757ba0b3508cc7a9ae0e3ffca3de88b8ab92ee3261415aaf4e81b53","abstract_canon_sha256":"f82acdca96961e7c9fac4b90a2ca8c97a3ae64952163b4cc221a0b95fad0d7b3"},"schema_version":"1.0"},"canonical_sha256":"f23b8dafdd9710e3ba5ba7a936d920eb5124b1a0425f25e03b697961da245c84","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:16.492988Z","signature_b64":"sVhJ/uSbbj3DIRuZlOgV969AfAj45gCMc6enXsFyy5bZmKg62HCtkeKjpVUrLeddJ+XsjzW01jSA20JS7WLhAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f23b8dafdd9710e3ba5ba7a936d920eb5124b1a0425f25e03b697961da245c84","last_reissued_at":"2026-05-18T00:08:16.492404Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:16.492404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.08351","source_version":4,"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":"vl0w5+6KfjXkkA0Of0Z/nEUtBrlaZLBPQ07btEuIPjNFic33mh4Y4zrXx2NnKPg4LUQOtqlPwC/v9Qi3BAGnAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T03:53:11.852348Z"},"content_sha256":"509dc45a937e538157e6789ce4bd6570d93aee35947546ae42489a45f1ba9734","schema_version":"1.0","event_id":"sha256:509dc45a937e538157e6789ce4bd6570d93aee35947546ae42489a45f1ba9734"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:6I5Y3L65S4IOHOS3U6UTNWJA5N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On factorization invariants and Hilbert functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Christopher O'Neill","submitted_at":"2015-03-28T21:08:23Z","abstract_excerpt":"Nonunique factorization in cancellative commutative semigroups is often studied using combinatorial factorization invariants, which assign to each semigroup element a quantity determined by the factorization structure. For numerical semigroups (additive subsemigroups of the natural numbers), several factorization invariants are known to admit predictable behavior for sufficiently large semigroup elements. In particular, the catenary degree and delta set invariants are both eventually periodic, and the omega-primality invariant is eventually quasilinear. In this paper, we demonstrate how each o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08351","kind":"arxiv","version":4},"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":"j1wtgajvMlWIloHOpCeUCJuI7lBWRM2MJ7SpqJueuNBMyk1wU/qMsi+YW+oR84utrtVbiF8/H8l3WvaEVFhsDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T03:53:11.852702Z"},"content_sha256":"0aef61964b176b15a62f77ec5e7072ce39ea08e0e3f36b91f58109751a1d3078","schema_version":"1.0","event_id":"sha256:0aef61964b176b15a62f77ec5e7072ce39ea08e0e3f36b91f58109751a1d3078"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6I5Y3L65S4IOHOS3U6UTNWJA5N/bundle.json","state_url":"https://pith.science/pith/6I5Y3L65S4IOHOS3U6UTNWJA5N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6I5Y3L65S4IOHOS3U6UTNWJA5N/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":"2026-06-04T03:53:11Z","links":{"resolver":"https://pith.science/pith/6I5Y3L65S4IOHOS3U6UTNWJA5N","bundle":"https://pith.science/pith/6I5Y3L65S4IOHOS3U6UTNWJA5N/bundle.json","state":"https://pith.science/pith/6I5Y3L65S4IOHOS3U6UTNWJA5N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6I5Y3L65S4IOHOS3U6UTNWJA5N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6I5Y3L65S4IOHOS3U6UTNWJA5N","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":"f82acdca96961e7c9fac4b90a2ca8c97a3ae64952163b4cc221a0b95fad0d7b3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-03-28T21:08:23Z","title_canon_sha256":"bf496f06e757ba0b3508cc7a9ae0e3ffca3de88b8ab92ee3261415aaf4e81b53"},"schema_version":"1.0","source":{"id":"1503.08351","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.08351","created_at":"2026-05-18T00:08:16Z"},{"alias_kind":"arxiv_version","alias_value":"1503.08351v4","created_at":"2026-05-18T00:08:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.08351","created_at":"2026-05-18T00:08:16Z"},{"alias_kind":"pith_short_12","alias_value":"6I5Y3L65S4IO","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6I5Y3L65S4IOHOS3","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6I5Y3L65","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:0aef61964b176b15a62f77ec5e7072ce39ea08e0e3f36b91f58109751a1d3078","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":"Nonunique factorization in cancellative commutative semigroups is often studied using combinatorial factorization invariants, which assign to each semigroup element a quantity determined by the factorization structure. For numerical semigroups (additive subsemigroups of the natural numbers), several factorization invariants are known to admit predictable behavior for sufficiently large semigroup elements. In particular, the catenary degree and delta set invariants are both eventually periodic, and the omega-primality invariant is eventually quasilinear. In this paper, we demonstrate how each o","authors_text":"Christopher O'Neill","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-03-28T21:08:23Z","title":"On factorization invariants and Hilbert functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08351","kind":"arxiv","version":4},"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":"sha256:509dc45a937e538157e6789ce4bd6570d93aee35947546ae42489a45f1ba9734","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":"f82acdca96961e7c9fac4b90a2ca8c97a3ae64952163b4cc221a0b95fad0d7b3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-03-28T21:08:23Z","title_canon_sha256":"bf496f06e757ba0b3508cc7a9ae0e3ffca3de88b8ab92ee3261415aaf4e81b53"},"schema_version":"1.0","source":{"id":"1503.08351","kind":"arxiv","version":4}},"canonical_sha256":"f23b8dafdd9710e3ba5ba7a936d920eb5124b1a0425f25e03b697961da245c84","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f23b8dafdd9710e3ba5ba7a936d920eb5124b1a0425f25e03b697961da245c84","first_computed_at":"2026-05-18T00:08:16.492404Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:08:16.492404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sVhJ/uSbbj3DIRuZlOgV969AfAj45gCMc6enXsFyy5bZmKg62HCtkeKjpVUrLeddJ+XsjzW01jSA20JS7WLhAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:08:16.492988Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.08351","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:509dc45a937e538157e6789ce4bd6570d93aee35947546ae42489a45f1ba9734","sha256:0aef61964b176b15a62f77ec5e7072ce39ea08e0e3f36b91f58109751a1d3078"],"state_sha256":"8edc68a25a7e93e627f1188ecd4ecb0985bc0fccf56a7f94806fc6bd0b4b4c84"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aHMEirFIoJx406esiGqtp4pV+V/HFGkV7zVn7zat5Sa5Ru1DwRCQmQtkezqtuv7fEB9eqzO2qJHy7XHEvyZgDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T03:53:11.854648Z","bundle_sha256":"17368c2b26d756970132a6da475c86feead4d9f8d59e07fbd0b0128ab6f4a854"}}