{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:EAPDUSCQGZPREMYBWYT7AZDYBP","short_pith_number":"pith:EAPDUSCQ","canonical_record":{"source":{"id":"2406.02503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2024-06-04T17:21:43Z","cross_cats_sorted":[],"title_canon_sha256":"96be76579e128a18f7313d1e4a348f492ab8c938e9638929e2eca2d77155d4e5","abstract_canon_sha256":"93455338523074be2a16cd98adb36c1a47dff084e59a7aa4851d8b29734c5528"},"schema_version":"1.0"},"canonical_sha256":"201e3a4850365f123301b627f064780bf2e01fc2bf9324ffeecac036dad52b3d","source":{"kind":"arxiv","id":"2406.02503","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2406.02503","created_at":"2026-07-05T08:27:20Z"},{"alias_kind":"arxiv_version","alias_value":"2406.02503v1","created_at":"2026-07-05T08:27:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.02503","created_at":"2026-07-05T08:27:20Z"},{"alias_kind":"pith_short_12","alias_value":"EAPDUSCQGZPR","created_at":"2026-07-05T08:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"EAPDUSCQGZPREMYB","created_at":"2026-07-05T08:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"EAPDUSCQ","created_at":"2026-07-05T08:27:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:EAPDUSCQGZPREMYBWYT7AZDYBP","target":"record","payload":{"canonical_record":{"source":{"id":"2406.02503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2024-06-04T17:21:43Z","cross_cats_sorted":[],"title_canon_sha256":"96be76579e128a18f7313d1e4a348f492ab8c938e9638929e2eca2d77155d4e5","abstract_canon_sha256":"93455338523074be2a16cd98adb36c1a47dff084e59a7aa4851d8b29734c5528"},"schema_version":"1.0"},"canonical_sha256":"201e3a4850365f123301b627f064780bf2e01fc2bf9324ffeecac036dad52b3d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:27:20.692730Z","signature_b64":"kivYCtchbJOjbDEgbds8qkoIvGnZ3TIew7WPEjidPCgk1fpISKUEzpOuKSCj+3UO3570OyXj8pIYQ5h/OwUCCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"201e3a4850365f123301b627f064780bf2e01fc2bf9324ffeecac036dad52b3d","last_reissued_at":"2026-07-05T08:27:20.692332Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:27:20.692332Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2406.02503","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:27:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XuyEdIvcT+V+sk62EyeaU1mwRSg03vf556PiyhKnBkjupq11MsuSJjUOUzbjVNBp48PMqGt2M8MFSgBPhsk3AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T05:53:12.491437Z"},"content_sha256":"b01f856e6f124debf6e1cf2919cf326889616bebb1567eef6228e43ad93c4515","schema_version":"1.0","event_id":"sha256:b01f856e6f124debf6e1cf2919cf326889616bebb1567eef6228e43ad93c4515"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:EAPDUSCQGZPREMYBWYT7AZDYBP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Atomicity in integral domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Felix Gotti, Jim Coykendall","submitted_at":"2024-06-04T17:21:43Z","abstract_excerpt":"In algebra, atomicity is the study of divisibility by and factorizations into atoms (also called irreducibles). In one side of the spectrum of atomicity we find the antimatter algebraic structures, inside which there are no atoms and, therefore, divisibility by and factorizations into atoms are not possible. In the other (more interesting) side of the spectrum, we find the atomic algebraic structures, where essentially every element factors into atoms (the study of such objects is known as factorization theory). In this paper, we survey some of the most fundamental results on the atomicity of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.02503","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2406.02503/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T08:27:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ptnfRFUZdP9i7gvRvlfExeICiVkFkL27TTPPqeeEMrx+HOTAW6tmPjjHuqXlAaVogKSlxVomdCoJwretMy4IDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T05:53:12.492078Z"},"content_sha256":"bf33e26fd19fca568c78e5ad212eb24d931b24e9876ff922b874da04ef36cc9d","schema_version":"1.0","event_id":"sha256:bf33e26fd19fca568c78e5ad212eb24d931b24e9876ff922b874da04ef36cc9d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EAPDUSCQGZPREMYBWYT7AZDYBP/bundle.json","state_url":"https://pith.science/pith/EAPDUSCQGZPREMYBWYT7AZDYBP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EAPDUSCQGZPREMYBWYT7AZDYBP/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":"2026-08-13T05:53:12Z","links":{"resolver":"https://pith.science/pith/EAPDUSCQGZPREMYBWYT7AZDYBP","bundle":"https://pith.science/pith/EAPDUSCQGZPREMYBWYT7AZDYBP/bundle.json","state":"https://pith.science/pith/EAPDUSCQGZPREMYBWYT7AZDYBP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EAPDUSCQGZPREMYBWYT7AZDYBP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:EAPDUSCQGZPREMYBWYT7AZDYBP","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":"93455338523074be2a16cd98adb36c1a47dff084e59a7aa4851d8b29734c5528","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2024-06-04T17:21:43Z","title_canon_sha256":"96be76579e128a18f7313d1e4a348f492ab8c938e9638929e2eca2d77155d4e5"},"schema_version":"1.0","source":{"id":"2406.02503","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2406.02503","created_at":"2026-07-05T08:27:20Z"},{"alias_kind":"arxiv_version","alias_value":"2406.02503v1","created_at":"2026-07-05T08:27:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.02503","created_at":"2026-07-05T08:27:20Z"},{"alias_kind":"pith_short_12","alias_value":"EAPDUSCQGZPR","created_at":"2026-07-05T08:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"EAPDUSCQGZPREMYB","created_at":"2026-07-05T08:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"EAPDUSCQ","created_at":"2026-07-05T08:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:bf33e26fd19fca568c78e5ad212eb24d931b24e9876ff922b874da04ef36cc9d","target":"graph","created_at":"2026-07-05T08:27:20Z","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/2406.02503/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In algebra, atomicity is the study of divisibility by and factorizations into atoms (also called irreducibles). In one side of the spectrum of atomicity we find the antimatter algebraic structures, inside which there are no atoms and, therefore, divisibility by and factorizations into atoms are not possible. In the other (more interesting) side of the spectrum, we find the atomic algebraic structures, where essentially every element factors into atoms (the study of such objects is known as factorization theory). In this paper, we survey some of the most fundamental results on the atomicity of ","authors_text":"Felix Gotti, Jim Coykendall","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2024-06-04T17:21:43Z","title":"Atomicity in integral domains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.02503","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:b01f856e6f124debf6e1cf2919cf326889616bebb1567eef6228e43ad93c4515","target":"record","created_at":"2026-07-05T08:27:20Z","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":"93455338523074be2a16cd98adb36c1a47dff084e59a7aa4851d8b29734c5528","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2024-06-04T17:21:43Z","title_canon_sha256":"96be76579e128a18f7313d1e4a348f492ab8c938e9638929e2eca2d77155d4e5"},"schema_version":"1.0","source":{"id":"2406.02503","kind":"arxiv","version":1}},"canonical_sha256":"201e3a4850365f123301b627f064780bf2e01fc2bf9324ffeecac036dad52b3d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"201e3a4850365f123301b627f064780bf2e01fc2bf9324ffeecac036dad52b3d","first_computed_at":"2026-07-05T08:27:20.692332Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:27:20.692332Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kivYCtchbJOjbDEgbds8qkoIvGnZ3TIew7WPEjidPCgk1fpISKUEzpOuKSCj+3UO3570OyXj8pIYQ5h/OwUCCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T08:27:20.692730Z","signed_message":"canonical_sha256_bytes"},"source_id":"2406.02503","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b01f856e6f124debf6e1cf2919cf326889616bebb1567eef6228e43ad93c4515","sha256:bf33e26fd19fca568c78e5ad212eb24d931b24e9876ff922b874da04ef36cc9d"],"state_sha256":"727d091c860af3b886f37d49f9c8b2403448c8158162dd24ce0e6e5a41eef9f0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NblX4uHInn6Jm3u6rcdvCB1NyQdwJN0xPqSSXow91jgYQi+2uW6xUX2RW03UKpFspdFptkusVXNJYFz6yzWNCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T05:53:12.497745Z","bundle_sha256":"c9dc0a0e82ff69bcd5d5f33f2ed9439cc7d7d5c10f6fa02ac93ed5674ba88af0"}}