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We give a sharp answer: \\[D_C\\text{ is an RB-domain}\\quad\\Longleftrightarrow\\quad C\\text{ is simplicial}. \\] Thus every non-simplicial proper cone gives an FS-domain which is not an RB-domain. The proof converts the RB approximation property into finite-valued $C$-isoton"},"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":"2607.02251","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.GN","submitted_at":"2026-07-02T14:42:58Z","cross_cats_sorted":[],"title_canon_sha256":"ebb4f8036adf988060d9a2da0dfe5362f678d6cc7bda8d523cb890e2b1580f44","abstract_canon_sha256":"8396ac185d7d44057365fc0f53796acf83b53bb5a75fbd13ff01878f620eb3d3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:46.444367Z","signature_b64":"h5dnrAQ6hGJKupPmAvlDRc28iLS86kK/tZk4i/jLt3xa0miQz8/kRLkUf1nIj1F0I6ZvqGuYYWJ9bI9df+PrCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94d5f04f622f80a0bde6d8416667a90aef04191f51b3ea9f212e077993b40e23","last_reissued_at":"2026-07-03T01:17:46.443965Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:46.443965Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cone domains separate FS-domains from RB-domains","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Yuxu Chen","submitted_at":"2026-07-02T14:42:58Z","abstract_excerpt":"Let $C$ be a closed, convex, pointed and generating cone in a finite-dimensional real vector space $V$, and let \\( D_C=(-C)\\cup\\{\\bot\\}\\) be the negative cone with a new least element, ordered by the cone order. Keimel proved that these cone domains are FS-domains and asked whether they are always retracts of bifinite domains. We give a sharp answer: \\[D_C\\text{ is an RB-domain}\\quad\\Longleftrightarrow\\quad C\\text{ is simplicial}. \\] Thus every non-simplicial proper cone gives an FS-domain which is not an RB-domain. 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