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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.21655","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.OA","submitted_at":"2026-05-20T19:10:13Z","cross_cats_sorted":[],"title_canon_sha256":"7f254884a74c557c8f78b08418fedd323c4c1d8f7c0c646f5e53aa3fd84d9ca2","abstract_canon_sha256":"74fb1fc422685703abf688cfcd7ad5cd1b5cdc6dcd9c30202ead8e36536e2b11"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:03:27.010457Z","signature_b64":"Afm7+6Bzvq43M8f6TsCltXdRZ1ydoVWF5Jk6M7RevY2rjJS7aGrFqsN5fOsQgFXdAd1V1Cc2POLySLL+3zxUAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2fee49992aec24913608cd9ec0be266b8cd4031ffcce85fb6deeb4a6d19aead9","last_reissued_at":"2026-05-22T01:03:27.009962Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:03:27.009962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Divisibility and Real Rank Zero","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Xuanlong Fu","submitted_at":"2026-05-20T19:10:13Z","abstract_excerpt":"Let $A$ be a simple separable exact $C^*$-algebra that has traces. 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