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We consider the completions of $\\cal F_0$ of the form $\\cal F=(\\cal F_0,\\cal G)$ obtained by appending a sequence $\\cal G=\\{g_i\\}_{i\\in\\mathbb{I}_k}$ of vectors in $\\mathbb C^d$ such that $\\|g_i\\|^2=a_i$ for $i\\in\\mathbb{I}_k$, and endow the set of completions with the metric $d(\\cal F,\\tilde {\\mathcal F}) =\\max\\{ \\,\\|g_i-\\tilde g_i\\|: \\ i\\in\\mathbb{I}_k\\}$ where $\\tilde {\\cal F}=(\\cal F_0,\\,\\tilde {\\cal G})$."},"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":"1610.02378","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-10-07T19:10:22Z","cross_cats_sorted":[],"title_canon_sha256":"2ae90c9959642f9f0583d12a8eb0741ddf4e7e078b96559277e41bdd230a51e3","abstract_canon_sha256":"8d13d8882db9a8c5552fb5c47850f05988c239811f81a9f16bd595df7530bcd2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:57.557757Z","signature_b64":"98E5OZJvM1NTDclZi3mUGWyG5C0AEJiIro9msCtxQEUHIvZcn5Lpm8oJaqJAGM2nrBKiVG7XtsNUBlg87SctDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8a28c77b7c6a0daff20c3f82da3307f8ba486bd97dffbe797a088f8f60038972","last_reissued_at":"2026-05-18T01:02:57.557221Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:57.557221Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Frame completions with prescribed norms: local minimizers and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Demetrio Stojanoff, Noelia B. 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We consider the completions of $\\cal F_0$ of the form $\\cal F=(\\cal F_0,\\cal G)$ obtained by appending a sequence $\\cal G=\\{g_i\\}_{i\\in\\mathbb{I}_k}$ of vectors in $\\mathbb C^d$ such that $\\|g_i\\|^2=a_i$ for $i\\in\\mathbb{I}_k$, and endow the set of completions with the metric $d(\\cal F,\\tilde {\\mathcal F}) =\\max\\{ \\,\\|g_i-\\tilde g_i\\|: \\ i\\in\\mathbb{I}_k\\}$ where $\\tilde {\\cal F}=(\\cal F_0,\\,\\tilde {\\cal G})$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.02378","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":""},"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":"1610.02378","created_at":"2026-05-18T01:02:57.557298+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.02378v1","created_at":"2026-05-18T01:02:57.557298+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.02378","created_at":"2026-05-18T01:02:57.557298+00:00"},{"alias_kind":"pith_short_12","alias_value":"RIUMO634NIG2","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_16","alias_value":"RIUMO634NIG274QM","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_8","alias_value":"RIUMO634","created_at":"2026-05-18T12:30:41.710351+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/RIUMO634NIG274QMH6BNUMYH7C","json":"https://pith.science/pith/RIUMO634NIG274QMH6BNUMYH7C.json","graph_json":"https://pith.science/api/pith-number/RIUMO634NIG274QMH6BNUMYH7C/graph.json","events_json":"https://pith.science/api/pith-number/RIUMO634NIG274QMH6BNUMYH7C/events.json","paper":"https://pith.science/paper/RIUMO634"},"agent_actions":{"view_html":"https://pith.science/pith/RIUMO634NIG274QMH6BNUMYH7C","download_json":"https://pith.science/pith/RIUMO634NIG274QMH6BNUMYH7C.json","view_paper":"https://pith.science/paper/RIUMO634","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.02378&json=true","fetch_graph":"https://pith.science/api/pith-number/RIUMO634NIG274QMH6BNUMYH7C/graph.json","fetch_events":"https://pith.science/api/pith-number/RIUMO634NIG274QMH6BNUMYH7C/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RIUMO634NIG274QMH6BNUMYH7C/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RIUMO634NIG274QMH6BNUMYH7C/action/storage_attestation","attest_author":"https://pith.science/pith/RIUMO634NIG274QMH6BNUMYH7C/action/author_attestation","sign_citation":"https://pith.science/pith/RIUMO634NIG274QMH6BNUMYH7C/action/citation_signature","submit_replication":"https://pith.science/pith/RIUMO634NIG274QMH6BNUMYH7C/action/replication_record"}},"created_at":"2026-05-18T01:02:57.557298+00:00","updated_at":"2026-05-18T01:02:57.557298+00:00"}