{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:C2CZEYLS4JYPL74ETKNOPM4MVU","short_pith_number":"pith:C2CZEYLS","schema_version":"1.0","canonical_sha256":"1685926172e270f5ff849a9ae7b38cad20dbe5b251d54e7dbc39c1b82b3b34be","source":{"kind":"arxiv","id":"2608.09866","version":1},"attestation_state":"computed","paper":{"title":"Sharp spectral constants for scaled $q$-numerical ranges","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mohamed Amine Aouichaoui, Ryan O'Loughlin","submitted_at":"2026-08-10T17:23:51Z","abstract_excerpt":"For $ n \\geq 2$, $A\\in M_n(\\mathbb C)$ and $0<|q|\\leq 1$, let $\\Omega_q(A)=q^{-1}W_q(A)$ be the scaled $q$-numerical range. We prove that for every $\\gamma \\geq 1$, \\[\n  \\Omega_{\\eta(\\gamma)}(A)\n  =\\bigcup_{\\kappa(S)\\leq\\gamma}W(S^{-1}AS),\n  \\qquad\n  \\eta(\\gamma)=\\frac{2}{\\gamma+\\gamma^{-1}}, \\] where $\\kappa(S)=\\|S\\|\\,\\|S^{-1}\\|$. As a consequence, we prove the sharp inequality \\[\n  \\|p(A)\\|\\leq\n  \\max\\!\\left\\{1,\\frac{2|q|}{1+\\sqrt{1-|q|^2}}\\right\\}\n  \\max_{z\\in\\Omega_q(A)}|p(z)|, \\] for all polynomials $p$."},"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":"2608.09866","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.FA","submitted_at":"2026-08-10T17:23:51Z","cross_cats_sorted":[],"title_canon_sha256":"175baec8f1d31316d7f9ce0248bfbb7bc886d19470c8e81d7548d3d7bb2a7bae","abstract_canon_sha256":"712bc6b553172635265b1e95635058fa500f80201a5d14b697e12fcb2e4ce6c5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-11T02:25:17.861278Z","signature_b64":"6yFGqGhGm3iGqAb0zKDn1YtVKTc6balZHf8/VrzHc2NjLdlxryFADSSkcI3vrefEc67KWp+RV0sdF+jFxiNGAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1685926172e270f5ff849a9ae7b38cad20dbe5b251d54e7dbc39c1b82b3b34be","last_reissued_at":"2026-08-11T02:25:17.859790Z","signature_status":"signed_v1","first_computed_at":"2026-08-11T02:25:17.859790Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp spectral constants for scaled $q$-numerical ranges","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mohamed Amine Aouichaoui, Ryan O'Loughlin","submitted_at":"2026-08-10T17:23:51Z","abstract_excerpt":"For $ n \\geq 2$, $A\\in M_n(\\mathbb C)$ and $0<|q|\\leq 1$, let $\\Omega_q(A)=q^{-1}W_q(A)$ be the scaled $q$-numerical range. We prove that for every $\\gamma \\geq 1$, \\[\n  \\Omega_{\\eta(\\gamma)}(A)\n  =\\bigcup_{\\kappa(S)\\leq\\gamma}W(S^{-1}AS),\n  \\qquad\n  \\eta(\\gamma)=\\frac{2}{\\gamma+\\gamma^{-1}}, \\] where $\\kappa(S)=\\|S\\|\\,\\|S^{-1}\\|$. As a consequence, we prove the sharp inequality \\[\n  \\|p(A)\\|\\leq\n  \\max\\!\\left\\{1,\\frac{2|q|}{1+\\sqrt{1-|q|^2}}\\right\\}\n  \\max_{z\\in\\Omega_q(A)}|p(z)|, \\] for all polynomials $p$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.09866","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/2608.09866/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":"2608.09866","created_at":"2026-08-11T02:25:17.860380+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.09866v1","created_at":"2026-08-11T02:25:17.860380+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.09866","created_at":"2026-08-11T02:25:17.860380+00:00"},{"alias_kind":"pith_short_12","alias_value":"C2CZEYLS4JYP","created_at":"2026-08-11T02:25:17.860380+00:00"},{"alias_kind":"pith_short_16","alias_value":"C2CZEYLS4JYPL74E","created_at":"2026-08-11T02:25:17.860380+00:00"},{"alias_kind":"pith_short_8","alias_value":"C2CZEYLS","created_at":"2026-08-11T02:25:17.860380+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/C2CZEYLS4JYPL74ETKNOPM4MVU","json":"https://pith.science/pith/C2CZEYLS4JYPL74ETKNOPM4MVU.json","graph_json":"https://pith.science/api/pith-number/C2CZEYLS4JYPL74ETKNOPM4MVU/graph.json","events_json":"https://pith.science/api/pith-number/C2CZEYLS4JYPL74ETKNOPM4MVU/events.json","paper":"https://pith.science/paper/C2CZEYLS"},"agent_actions":{"view_html":"https://pith.science/pith/C2CZEYLS4JYPL74ETKNOPM4MVU","download_json":"https://pith.science/pith/C2CZEYLS4JYPL74ETKNOPM4MVU.json","view_paper":"https://pith.science/paper/C2CZEYLS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.09866&json=true","fetch_graph":"https://pith.science/api/pith-number/C2CZEYLS4JYPL74ETKNOPM4MVU/graph.json","fetch_events":"https://pith.science/api/pith-number/C2CZEYLS4JYPL74ETKNOPM4MVU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C2CZEYLS4JYPL74ETKNOPM4MVU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C2CZEYLS4JYPL74ETKNOPM4MVU/action/storage_attestation","attest_author":"https://pith.science/pith/C2CZEYLS4JYPL74ETKNOPM4MVU/action/author_attestation","sign_citation":"https://pith.science/pith/C2CZEYLS4JYPL74ETKNOPM4MVU/action/citation_signature","submit_replication":"https://pith.science/pith/C2CZEYLS4JYPL74ETKNOPM4MVU/action/replication_record"}},"created_at":"2026-08-11T02:25:17.860380+00:00","updated_at":"2026-08-11T02:25:17.860380+00:00"}