{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:F7XETGJK5QSJCNQIZWPMBPRGNO","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":"74fb1fc422685703abf688cfcd7ad5cd1b5cdc6dcd9c30202ead8e36536e2b11","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.OA","submitted_at":"2026-05-20T19:10:13Z","title_canon_sha256":"7f254884a74c557c8f78b08418fedd323c4c1d8f7c0c646f5e53aa3fd84d9ca2"},"schema_version":"1.0","source":{"id":"2605.21655","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.21655","created_at":"2026-05-22T01:03:27Z"},{"alias_kind":"arxiv_version","alias_value":"2605.21655v1","created_at":"2026-05-22T01:03:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.21655","created_at":"2026-05-22T01:03:27Z"},{"alias_kind":"pith_short_12","alias_value":"F7XETGJK5QSJ","created_at":"2026-05-22T01:03:27Z"},{"alias_kind":"pith_short_16","alias_value":"F7XETGJK5QSJCNQI","created_at":"2026-05-22T01:03:27Z"},{"alias_kind":"pith_short_8","alias_value":"F7XETGJK","created_at":"2026-05-22T01:03:27Z"}],"graph_snapshots":[{"event_id":"sha256:1a77310eb62427ce5ad30b757735a57212dd56b45ba604f0c6b0c075193bc04e","target":"graph","created_at":"2026-05-22T01:03:27Z","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/2605.21655/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $A$ be a simple separable exact $C^*$-algebra that has traces. We show the following existed regularity properties are equivalent:\n  \\quad(1) $l^\\infty(A)/J_A$ has real rank zero, where $J_A$ is the trace kernel ideal.\n  \\quad(2) $A$ is tracially almost divisible.\n  \\quad(3) $A$ is tracially $m$-almost divisible for some $m\\in\\N\\cup\\{0\\}.$\n  \\quad(4) $A$ has tracial approximate oscillation zero.\n  \\quad(5) $A$ has Property (TM).\n  We also show that for an algebraically simple separable stable rank one \\CA\\ $B$ with non-empty compact ${\\rm T}(B)$ and locally finite nuclear dimension, its un","authors_text":"Xuanlong Fu","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.OA","submitted_at":"2026-05-20T19:10:13Z","title":"Divisibility and Real Rank Zero"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21655","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:d0a67fd0e6ea2d51d230151e008522394a25d9e7a2fe2c143e6866a4ff2c6f92","target":"record","created_at":"2026-05-22T01:03:27Z","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":"74fb1fc422685703abf688cfcd7ad5cd1b5cdc6dcd9c30202ead8e36536e2b11","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.OA","submitted_at":"2026-05-20T19:10:13Z","title_canon_sha256":"7f254884a74c557c8f78b08418fedd323c4c1d8f7c0c646f5e53aa3fd84d9ca2"},"schema_version":"1.0","source":{"id":"2605.21655","kind":"arxiv","version":1}},"canonical_sha256":"2fee49992aec24913608cd9ec0be266b8cd4031ffcce85fb6deeb4a6d19aead9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2fee49992aec24913608cd9ec0be266b8cd4031ffcce85fb6deeb4a6d19aead9","first_computed_at":"2026-05-22T01:03:27.009962Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:03:27.009962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Afm7+6Bzvq43M8f6TsCltXdRZ1ydoVWF5Jk6M7RevY2rjJS7aGrFqsN5fOsQgFXdAd1V1Cc2POLySLL+3zxUAw==","signature_status":"signed_v1","signed_at":"2026-05-22T01:03:27.010457Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.21655","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d0a67fd0e6ea2d51d230151e008522394a25d9e7a2fe2c143e6866a4ff2c6f92","sha256:1a77310eb62427ce5ad30b757735a57212dd56b45ba604f0c6b0c075193bc04e"],"state_sha256":"79ba3696aec2dadbe0551e2d1e81fe33e06150eb0623e4e03bb3d6cc2880f938"}