{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:PL42RP4DN7GBXV3XMPICCK5UBI","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":"5021d6c45c976acfb1077d649a511d07033675ac04d5926110bb9f60bd8746f0","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2024-12-27T17:22:04Z","title_canon_sha256":"409022247886b5391ba3b541d0f6b14c6429ef72fa1a9190ee76882152af2f0a"},"schema_version":"1.0","source":{"id":"2412.19753","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.19753","created_at":"2026-07-05T11:13:17Z"},{"alias_kind":"arxiv_version","alias_value":"2412.19753v2","created_at":"2026-07-05T11:13:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.19753","created_at":"2026-07-05T11:13:17Z"},{"alias_kind":"pith_short_12","alias_value":"PL42RP4DN7GB","created_at":"2026-07-05T11:13:17Z"},{"alias_kind":"pith_short_16","alias_value":"PL42RP4DN7GBXV3X","created_at":"2026-07-05T11:13:17Z"},{"alias_kind":"pith_short_8","alias_value":"PL42RP4D","created_at":"2026-07-05T11:13:17Z"}],"graph_snapshots":[{"event_id":"sha256:24da85f3c502779c5bf4646743d13768ef8c7bf2d47d782f3d53bdaae2b7cf15","target":"graph","created_at":"2026-07-05T11:13:17Z","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/2412.19753/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A divisibility relation on ultrafilters on the set $\\mathbb{N}$ of natural numbers is defined as follows: ${\\cal F}\\hspace{1mm}\\widetilde{\\mid}\\hspace{1mm}{\\cal G}$ if and only if every set in $\\cal F$ upward closed for divisibility also belongs to $\\cal G$. Previously we isolated basic classes: powers of prime ultrafilters, and described the pattern of an ultrafilter, measuring the quantity of members of each basic class dividing a given ultrafilter. In this paper we define a topology on the set of basic classes which will allow us to calculate the pattern of the limit of a $\\widetilde{\\mid}$","authors_text":"Boris \\v{S}obot","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2024-12-27T17:22:04Z","title":"Divisibility classes of ultrafilters and their patterns"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.19753","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":"sha256:12f01a4289d2c5aee0b320aa32908140838b68401611fdd55ea8760174ed6ffb","target":"record","created_at":"2026-07-05T11:13:17Z","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":"5021d6c45c976acfb1077d649a511d07033675ac04d5926110bb9f60bd8746f0","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2024-12-27T17:22:04Z","title_canon_sha256":"409022247886b5391ba3b541d0f6b14c6429ef72fa1a9190ee76882152af2f0a"},"schema_version":"1.0","source":{"id":"2412.19753","kind":"arxiv","version":2}},"canonical_sha256":"7af9a8bf836fcc1bd77763d0212bb40a1dff11a1a27e66d23b52f2b973b34c78","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7af9a8bf836fcc1bd77763d0212bb40a1dff11a1a27e66d23b52f2b973b34c78","first_computed_at":"2026-07-05T11:13:17.770081Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:13:17.770081Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nuwoy2vbamaZgmmfgCyLVBRp1qeKmwwcnCST/Ks0h+dRTdZ9l4ah1sr8uBWGsWAhpeTWXA7vcPsi9dHa4anaAA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:13:17.770591Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.19753","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:12f01a4289d2c5aee0b320aa32908140838b68401611fdd55ea8760174ed6ffb","sha256:24da85f3c502779c5bf4646743d13768ef8c7bf2d47d782f3d53bdaae2b7cf15"],"state_sha256":"646a32f2fd6996cd2163653419c833f2eee184b6a667c216fb54c9e8e16c7d41"}