{"paper":{"title":"The joint numerical range of three hermitian $4\\times 4$ matrices","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The joint numerical range of three Hermitian 4x4 matrices has non-generic boundaries classifiable into fifteen types based on non-elliptic faces.","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ilya Spitkovsky, Karol \\.Zyczkowski, Konrad Szyma\\'nski, Piotr Pikul, Stephan Weis","submitted_at":"2025-10-31T17:34:54Z","abstract_excerpt":"We analyze the joint numerical range $W$ of three hermitian matrices of order four. In the generic case, this three-dimensional convex set has a smooth boundary. We analyze non-generic structures. Fifteen possible classes regarding the numbers of non-elliptic faces in the boundary of $W$ are identified and an explicit example is presented for each class. Secondly, it is shown that a nonempty intersection of three mutually distinct one-dimensional faces is a corner point. Thirdly, introducing a tensor product structure into $\\mathbb C^4=\\mathbb C^2\\otimes\\mathbb C^2$, one defines the separable "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Fifteen possible classes regarding the numbers of non-elliptic faces in the boundary of W are identified and an explicit example is presented for each class. A nonempty intersection of three mutually distinct one-dimensional faces is a corner point.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis assumes that varying only the number of non-elliptic faces fully captures all possible non-generic boundary structures for three Hermitian 4x4 matrices without additional geometric features or degeneracies arising, as implied by the identification of exactly fifteen classes in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Classifies non-generic joint numerical ranges of three Hermitian 4x4 matrices into 15 classes with examples, proves corner points from face intersections, and compares the separable numerical range boundary for entanglement analysis.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The joint numerical range of three Hermitian 4x4 matrices has non-generic boundaries classifiable into fifteen types based on non-elliptic faces.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9de2e237d544232c9fba57f29caffcb53bf426897ca2549e01f66ebd8d7a365e"},"source":{"id":"2510.27670","kind":"arxiv","version":2},"verdict":{"id":"4576030c-bc1e-4ed7-8175-de083007f681","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T02:42:59.509916Z","strongest_claim":"Fifteen possible classes regarding the numbers of non-elliptic faces in the boundary of W are identified and an explicit example is presented for each class. A nonempty intersection of three mutually distinct one-dimensional faces is a corner point.","one_line_summary":"Classifies non-generic joint numerical ranges of three Hermitian 4x4 matrices into 15 classes with examples, proves corner points from face intersections, and compares the separable numerical range boundary for entanglement analysis.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis assumes that varying only the number of non-elliptic faces fully captures all possible non-generic boundary structures for three Hermitian 4x4 matrices without additional geometric features or degeneracies arising, as implied by the identification of exactly fifteen classes in the abstract.","pith_extraction_headline":"The joint numerical range of three Hermitian 4x4 matrices has non-generic boundaries classifiable into fifteen types based on non-elliptic faces."},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"90fd9df962dcbf30fced2238a231408677f4f50e5c5ae3e474df80a58e4a96d0"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}