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A standard $\\varepsilon$-isometry $f:X\\rightarrow Y$ is said to be $(\\alpha,\\gamma)$-stable (with respect to $T:L(f)\\equiv\\overline{{\\rm span}}f(X)\\rightarrow X$ for some $\\alpha, \\gamma>0$) if $T$ is a linear operator with $\\|T\\|\\leq\\alpha$ so that $Tf-Id$ is uniformly bounded by $\\gamma\\varepsilon$ on $X$. The pair $(X,Y)$ is said to be stable if every standard $\\varepsilon$-isometry $f:X\\rightarrow Y$ is $(\\alpha,\\gamma)$-stable for some $\\alpha,\\gamma>0$. $X (Y)$ is said to be universally left (right)-stable, if $(X,Y)$ is always "},"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":"1301.3374","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-01-15T14:58:32Z","cross_cats_sorted":[],"title_canon_sha256":"84005cff1637591638650098954cd9ccef3f14591f7e2a85a1637b9a67521bae","abstract_canon_sha256":"f9075bfce218f8523e15fc6d81d12ba51628f6acf9bdc646c7d8b3b13e2bcdc9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:35.208836Z","signature_b64":"vWNE/Xk8IhRDEO4DBYnObqWZ8NbL4NG399IAzGFWUJTYySMQsfUkWkVJjD5spAQwUvCuS1Qh+SkMB2whosbwDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6e5953c8482afc2e76f693a71988caa2585b34ada88fdfe26176e9507a22ab32","last_reissued_at":"2026-05-18T03:06:35.208265Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:35.208265Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Universal stability of Banach spaces for $\\varepsilon$-isometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Duanxu Dai, Lixin Cheng, Yunbai Dong, Yu Zhou","submitted_at":"2013-01-15T14:58:32Z","abstract_excerpt":"Let $X$, $Y$ be two real Banach spaces and $\\varepsilon>0$. 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The pair $(X,Y)$ is said to be stable if every standard $\\varepsilon$-isometry $f:X\\rightarrow Y$ is $(\\alpha,\\gamma)$-stable for some $\\alpha,\\gamma>0$. $X (Y)$ is said to be universally left (right)-stable, if $(X,Y)$ is always "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.3374","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1301.3374","created_at":"2026-05-18T03:06:35.208349+00:00"},{"alias_kind":"arxiv_version","alias_value":"1301.3374v4","created_at":"2026-05-18T03:06:35.208349+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.3374","created_at":"2026-05-18T03:06:35.208349+00:00"},{"alias_kind":"pith_short_12","alias_value":"NZMVHSCIFL6C","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_16","alias_value":"NZMVHSCIFL6C45XW","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_8","alias_value":"NZMVHSCI","created_at":"2026-05-18T12:27:54.935989+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NZMVHSCIFL6C45XWSOTRTCGKUJ","json":"https://pith.science/pith/NZMVHSCIFL6C45XWSOTRTCGKUJ.json","graph_json":"https://pith.science/api/pith-number/NZMVHSCIFL6C45XWSOTRTCGKUJ/graph.json","events_json":"https://pith.science/api/pith-number/NZMVHSCIFL6C45XWSOTRTCGKUJ/events.json","paper":"https://pith.science/paper/NZMVHSCI"},"agent_actions":{"view_html":"https://pith.science/pith/NZMVHSCIFL6C45XWSOTRTCGKUJ","download_json":"https://pith.science/pith/NZMVHSCIFL6C45XWSOTRTCGKUJ.json","view_paper":"https://pith.science/paper/NZMVHSCI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1301.3374&json=true","fetch_graph":"https://pith.science/api/pith-number/NZMVHSCIFL6C45XWSOTRTCGKUJ/graph.json","fetch_events":"https://pith.science/api/pith-number/NZMVHSCIFL6C45XWSOTRTCGKUJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NZMVHSCIFL6C45XWSOTRTCGKUJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NZMVHSCIFL6C45XWSOTRTCGKUJ/action/storage_attestation","attest_author":"https://pith.science/pith/NZMVHSCIFL6C45XWSOTRTCGKUJ/action/author_attestation","sign_citation":"https://pith.science/pith/NZMVHSCIFL6C45XWSOTRTCGKUJ/action/citation_signature","submit_replication":"https://pith.science/pith/NZMVHSCIFL6C45XWSOTRTCGKUJ/action/replication_record"}},"created_at":"2026-05-18T03:06:35.208349+00:00","updated_at":"2026-05-18T03:06:35.208349+00:00"}