{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:VJSHALFWDU7BMPPHSTM2WRIQBU","short_pith_number":"pith:VJSHALFW","schema_version":"1.0","canonical_sha256":"aa64702cb61d3e163de794d9ab45100d10c90994095530d3e0ebf11168854a9f","source":{"kind":"arxiv","id":"2607.17672","version":1},"attestation_state":"computed","paper":{"title":"Kirszbraun extensions preserving uniform distance in Hilbert spaces","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Krzysztof J. Ciosmak","submitted_at":"2026-07-20T08:24:35Z","abstract_excerpt":"Let $X$ be a subset of a real Hilbert space and let $v\\colon X\\to Y$, where $Y$ is a real Hilbert space. We prove that the following conditions are equivalent: whenever $A\\subset X$, $\\rho\\geq0$, and $u\\colon A\\to Y$ is $1$-Lipschitz with $\\left\\lVert u(x)-v(x)\\right\\lVert\\leq\\rho$ for $x\\in A$, there is a $1$-Lipschitz extension $\\widetilde u\\colon X\\to Y$ with $\\left\\lVert \\widetilde u(x)-v(x)\\right \\lVert\\leq\\rho$ for $x\\in X$; and for every $1\\leq k\\leq\\dim Y$, $$\n  \\left\\lVert v(x_0)-\\sum_{i=1}^k t_i v(x_i)\\right\\lVert\n  \\leq\n  \\left\\lVert x_0-\\sum_{i=1}^k t_i x_i \\right \\lVert$$ whenever"},"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":"2607.17672","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.FA","submitted_at":"2026-07-20T08:24:35Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"9b854a49e2c94714c9597e9afabd43196a689184e4c559988965c61d2968c311","abstract_canon_sha256":"0a6455962784a938a08e59ec39ddf0211296e1ee65d2fce5a6d16a3b77db1b45"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-21T02:21:53.880238Z","signature_b64":"GKEAG2mVBkk26k5LrasHVa5wvubBEKgIPmmcI89WVX/yAIyx1LrLxPnpeEghtMl99l0dU9cT32USXXeHzqFKCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa64702cb61d3e163de794d9ab45100d10c90994095530d3e0ebf11168854a9f","last_reissued_at":"2026-07-21T02:21:53.879386Z","signature_status":"signed_v1","first_computed_at":"2026-07-21T02:21:53.879386Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Kirszbraun extensions preserving uniform distance in Hilbert spaces","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Krzysztof J. Ciosmak","submitted_at":"2026-07-20T08:24:35Z","abstract_excerpt":"Let $X$ be a subset of a real Hilbert space and let $v\\colon X\\to Y$, where $Y$ is a real Hilbert space. We prove that the following conditions are equivalent: whenever $A\\subset X$, $\\rho\\geq0$, and $u\\colon A\\to Y$ is $1$-Lipschitz with $\\left\\lVert u(x)-v(x)\\right\\lVert\\leq\\rho$ for $x\\in A$, there is a $1$-Lipschitz extension $\\widetilde u\\colon X\\to Y$ with $\\left\\lVert \\widetilde u(x)-v(x)\\right \\lVert\\leq\\rho$ for $x\\in X$; and for every $1\\leq k\\leq\\dim Y$, $$\n  \\left\\lVert v(x_0)-\\sum_{i=1}^k t_i v(x_i)\\right\\lVert\n  \\leq\n  \\left\\lVert x_0-\\sum_{i=1}^k t_i x_i \\right \\lVert$$ whenever"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.17672","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.17672/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2607.17672","created_at":"2026-07-21T02:21:53.879806+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.17672v1","created_at":"2026-07-21T02:21:53.879806+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.17672","created_at":"2026-07-21T02:21:53.879806+00:00"},{"alias_kind":"pith_short_12","alias_value":"VJSHALFWDU7B","created_at":"2026-07-21T02:21:53.879806+00:00"},{"alias_kind":"pith_short_16","alias_value":"VJSHALFWDU7BMPPH","created_at":"2026-07-21T02:21:53.879806+00:00"},{"alias_kind":"pith_short_8","alias_value":"VJSHALFW","created_at":"2026-07-21T02:21:53.879806+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/VJSHALFWDU7BMPPHSTM2WRIQBU","json":"https://pith.science/pith/VJSHALFWDU7BMPPHSTM2WRIQBU.json","graph_json":"https://pith.science/api/pith-number/VJSHALFWDU7BMPPHSTM2WRIQBU/graph.json","events_json":"https://pith.science/api/pith-number/VJSHALFWDU7BMPPHSTM2WRIQBU/events.json","paper":"https://pith.science/paper/VJSHALFW"},"agent_actions":{"view_html":"https://pith.science/pith/VJSHALFWDU7BMPPHSTM2WRIQBU","download_json":"https://pith.science/pith/VJSHALFWDU7BMPPHSTM2WRIQBU.json","view_paper":"https://pith.science/paper/VJSHALFW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.17672&json=true","fetch_graph":"https://pith.science/api/pith-number/VJSHALFWDU7BMPPHSTM2WRIQBU/graph.json","fetch_events":"https://pith.science/api/pith-number/VJSHALFWDU7BMPPHSTM2WRIQBU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VJSHALFWDU7BMPPHSTM2WRIQBU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VJSHALFWDU7BMPPHSTM2WRIQBU/action/storage_attestation","attest_author":"https://pith.science/pith/VJSHALFWDU7BMPPHSTM2WRIQBU/action/author_attestation","sign_citation":"https://pith.science/pith/VJSHALFWDU7BMPPHSTM2WRIQBU/action/citation_signature","submit_replication":"https://pith.science/pith/VJSHALFWDU7BMPPHSTM2WRIQBU/action/replication_record"}},"created_at":"2026-07-21T02:21:53.879806+00:00","updated_at":"2026-07-21T02:21:53.879806+00:00"}